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\section{Introduction}
There are two interesting ways to formulate the field equations of an open
NS superstring. The first comes from Berkovits' nonpolynomial string field
theory\cite{Berkovits}, and involves a ghost and picture number $0$
string field $\Phi$ in the large Hilbert space subject to the equations of
motion
\begin{equation}\eta_0(e^{-\Phi}Q e^\Phi)=0.\label{eq:berkovits}\end{equation}
The second comes from cubic superstring field theory\cite{PTY,AZ}, and
involves a ghost number $1$, picture number $0$ string field $\Psi$ in the
small Hilbert space subject to the equations of motion
\begin{equation}Q\Psi+\Psi^2 = 0. \label{eq:pty}
\end{equation}
The cubic equations of motion are simpler, in that they are polynomial and
directly analogous to the field equations for the open bosonic
string\cite{Witten}, but suffer
from the disadvantage that they are difficult to derive from a completely
reliable action\footnote{To evaluate the energy
in this paper, we will use the action originally proposed by Preitschopf,
Thorne, and Yost\cite{PTY}:
\begin{equation}S=\frac{1}{2}\langle\!\langle \Psi Q\Psi\rangle\!\rangle+\frac{1}{3}\langle\!\langle
\Psi^3\rangle\!\rangle.\end{equation}
The bracket $\langle\!\langle\cdot\rangle\!\rangle$ is defined using the Witten vertex
with a midpoint insertion
\begin{equation}Y_{-2}=Y(i)\tilde{Y}(i),\ \ \ Y(z)=-\partial\xi e^{-2\phi}c(z).
\label{eq:Ym2}\end{equation} See appendix \ref{app:conventions}.
The problems with this
action are well-known, including difficulties with the convergence of level
truncation\cite{Russians_level,Ohmori,Ohmori_rev,Raeymaekers}, complications
with gauge fixing and perturbation theory\cite{PTY}, problems with the
Ramond sector\cite{equivalence_Ramond,Kroyter_philosophy},
and the existence of a singular kernel for the bracket\cite{Berkovits_critic}.
Recently there has been some interest in finding a more suitable
action\cite{Kroyter_philosophy,democratic,BS}, though the success of these
proposals remains unclear.}. Nevertheless
the Berkovits and cubic equations are known to be perturbatively
equivalent\cite{equivalence}, and nonperturbatively any Berkovits solution
generates a cubic solution via the equation
\begin{equation}\Psi=e^{-\Phi}Qe^{\Phi}.\label{eq:Berkcub}\end{equation}
However, the reverse is not true. The existence of a cubic solution $\Psi$
does not guarantee the existence of a Berkovits solution $e^{\Phi}$ satisfying
\eq{Berkcub}. For example, cubic superstring field theory has a ``tachyon
vacuum'' on a BPS D-brane\cite{Erler,Kroyter_myvac}. There is no evidence for
such a solution in Berkovits' string field theory, either
analytically\cite{Erler,ES} or numerically\cite{BSZ}.
In this paper we present a new example of this phenomenon. We show that
the cubic equations of motion on a non-BPS D-brane possess an unexpected
class of universal solutions which appear not to exist in Berkovits string
field theory. The existence of these solutions is highly nontrivial, but
their physical interpretation is unknown. They possess a number of
surprising properties which may be important for our understanding of string
field theory:
\begin{itemize}
\item The solutions are not real. In fact, every solution appears to be
related to its conjugate by a topologically nontrivial gauge transformation.
\item The solutions appear not to exist in Berkovits' string field theory.
\item If we ignore the reality condition and compute observables,
the solutions turn out to carry half the tension of a non-BPS D-brane.
\end{itemize}
We will call them {\it half-brane solutions}, in accordance with their
tension. The solutions are significant in that they appear to be the
first examples of topological solutions in open string field theory.
We hope that they can provide a deeper understanding of the
topology of the string field algebra, with the ultimate goal of providing a
``microscopic'' description of D-brane charges in the context of string
field theory.
This paper is organized as follows. In section \ref{sec:solution} we construct
half-brane solutions by extending the wedge algebra to include
generators of worldsheet supersymmetry. We attempt an analogous
construction in Berkovits string field theory, and show that it fails.
In section \ref{sec:reality} we prove that
half-brane solutions do not satisfy the string field reality condition. We
also show, within a controlled subalgebra of states, that every half-brane
solution is related to its conjugate by a topologically nontrivial gauge
transformation. In section \ref{sec:phantom} we discuss the regularization
and phantom
piece for the half-brane solution. The phantom term offers an interesting
perspective on the nature of convergence in the wedge algebra, and suggests
a more general technique for constructing states in the wedge
algebra---including, possibly, projector states distinct from
the sliver and identity string field. In section
\ref{sec:observables} we calculate the action and closed string tadpole.
We find highly nontrivial agreement between these observables, indicating
that the solutions represent a state with half the tension of a non-BPS
D-brane. We end with some discussion.
\section{Solution}
\label{sec:solution}
\subsection{Algebra}
\label{subsec:algebra}
To begin we need to recall some facts about the algebra of string
fields\footnote{In this paper we use the left handed convention for the star
product\cite{simple}. Other standard sources for the
superstring\cite{Ohmori,Raeymaekers,BSZ} use the right handed
convention\cite{Okawa}, and there are some important
sign differences in the GSO($-$) sector. See appendix
\ref{app:conventions}.}
on a non-BPS D-brane. The algebra has two $\mathbb{Z}_2$ gradings: Grassmann
parity $\epsilon$, which corresponds to the Grassmann parity of the vertex
operator creating the string field; and worldsheet spinor
number $F$, which tells us whether the field is in the GSO($+$) or GSO($-$)
sector. Fields in the algebra are assigned internal Chan-Paton factors
according to the table:
\begin{center}
\begin{tabular}{|c|c|c|c|}\hline
$\ \ \ \epsilon\ \ \ $ & $\ \ \ F\ \ \ $ & CP factor
\\ \hline
$0$ & $0$ & $\mathbb{I}$ \\ \hline
$1$ & $0$ & $\sigma_3$ \\ \hline
$0$ & $1$ & $\sigma_2$ \\ \hline
$1$ & $1$ & $\sigma_1$ \\ \hline
\end{tabular}.
\end{center}
The BRST charge $Q$ and the midpoint insertion $Y_{-2}$ both implicitly
carry an internal CP factor of $\sigma_3$, and the 1-string vertex
$\langle\!\langle \cdot\rangle\!\rangle$ automatically contains a factor of $1/2$ times
the trace over internal CP matrices. To keep track of signs when commuting
vertex operators and CP factors past each other, it is helpful to define what
we will call {\it effective Grassmann parity}
\begin{equation}E=\epsilon+F\ \ \ \ (\mathrm{mod}\ 2).\end{equation}
In particular, the star algebra has a natural graded
commutator\footnote{This ``double bracket'' commutator should be
distinguished from the graded commutator
$[\Psi,\Phi]=\Psi\Phi-(-1)^{E(\Psi)E(\Phi)}\Phi\Psi$ which emerges naturally
from the action, both in the infinitesimal gauge transformation and the
kinetic operator around a nontrivial solution. The single bracket $[,]$ is
only graded according to effective Grassmann parity.}
\begin{equation}\llbracket\Psi,\Phi\rrbracket = \Psi\Phi -
(-1)^{E(\Psi)E(\Phi)+F(\Psi)F(\Phi)}\Phi\Psi,\end{equation}
where $\Psi,\Phi$ implicitly carry the appropriate CP factor. This
suggests that the star product on a non-BPS brane has a structure
analogous to a product of matrices whose entries contain two mutually
commuting types of Grassmann number, the first has a Grassmannality measured
by $E$ and the second by $F$. However, only effective Grassmann parity
enters into the string field theory
axioms:
\begin{eqnarray}Q(\Psi\Phi)\!\!\!\!\!\!\!\!&& = (Q\Psi)\Phi+(-1)^{E(\Psi)}\Psi(Q\Phi),
\nonumber\\
\langle\!\langle \Psi\Phi\rangle\!\rangle \!\!\!\!\!\!\!\!&&= (-1)^{E(\Psi)E(\Phi)}
\langle\!\langle \Phi\Psi \rangle\!\rangle.\end{eqnarray}
In particular, the physical string field $\Psi$ on a non-BPS D-brane
must be {\it effective} Grassmann odd. For example, the
tachyon vertex operator $\gamma(0)$ is Grassmann even in the traditional
sense, but since it carries worldsheet spinor number, it counts
as ``effectively'' Grassmann odd.
\begin{table}[t]
\begin{center}
\begin{tabular}{|c|c|c|c|c|c|c|}\hline
& ${\mathrm{ghost} \atop \mathrm{number}}$
& ${\mathrm{effective}\atop \mathrm{Grassmann\ parity}}$
& ${\mathrm{worldsheet}\atop \mathrm{spinor\ number}}$
& ${\mathrm{scaling}\atop\mathrm{dimension}}$
& reality
& twist \\ \hline
$K$ & $0$ & $0$ & $0$ & $1$ & real & $1$ \\ \hline
$B$ & $-1$ & $1$ & $0$ & $1$ & real & $1$ \\ \hline
$c$ & $1$ & $1$ & $0$ & $-1$ & real & $-1$ \\ \hline
$G$ & $0$ & $0$ & $1$ & $\frac{1}{2}$ & real & $-i$ \\ \hline
$\gamma$ & $1$ & $1$ & $1$ & $-\frac{1}{2}$ & real & $-i$ \\ \hline
\end{tabular}
\end{center}
\caption{\label{tab:alg} Some important quantum numbers for the
atomic fields. Scaling dimension refers to the eigenvalue of the
field under the action of the operator
$\frac{1}{2}\mathcal{L}^-=\frac{1}{2}(\mathcal{L}_0-\mathcal{L}_0^\star)$.
Reality and twist refer to the eigenvalues of the fields under
reality and twist conjugation, defined in appendix
\ref{app:conventions}. By ``real'' we mean that the fields have eigenvalue
$1$ under reality conjugation.}
\end{table}
With these preparations we are ready to give the algebraic setup for our
solution. The solution is constructed by taking star products of
four ``atomic'' string fields:
\begin{equation}K,\ \ \ \ B,\ \ \ \ c,\ \ \ \ G.\ \end{equation}
The ghost number, effective Grassmann parity, and some other important
quantum numbers of these fields are summarized in table \ref{tab:alg}.
We can construct $K,B,c,G$ by acting certain operators on the
identity string field $|I\rangle$:
\begin{eqnarray}K \!\!\!\!\!\!\!\!&& = \mathbb{I}\otimes \mathcal{L}^+_L|I\rangle,
\ \ \ \ \ \ \ \ \ \ \ B = \sigma_3\otimes \mathcal{B}^+_L|I\rangle,\nonumber\\
c \!\!\!\!\!\!\!\!&& = \sigma_3\otimes\frac{1}{\pi}c(1)|I\rangle,\ \ \ \ \ \
G=\sigma_1\otimes \mathcal{G}_L|I\rangle.
\end{eqnarray}
The subscript $L$ above denotes taking the left half of the charges:
\begin{eqnarray}
\mathcal{L}^+ \!\!\!\!\!\!\!\!&& = \mathcal{L}_0+\mathcal{L}_0^\star,
\ \ \ \ \ \ \ \ \ \ \ \
\mathcal{L}_0 = f_\mathcal{S}^{-1}\circ L_0,\nonumber\\
\mathcal{B}^+ \!\!\!\!\!\!\!\!&& = \mathcal{B}_0+\mathcal{B}_0^\star,
\ \ \ \ \ \ \ \ \ \ \ \
\mathcal{B}_0 = f_\mathcal{S}^{-1}\circ b_0,\nonumber\\
\mathcal{G}\ \, \!\!\!\!\!\!\!\!&& = f_\mathcal{S}^{-1}\circ G_{-1/2},
\label{eq:SS_op}\end{eqnarray}
where $f_\mathcal{S}^{-1}(z) = \tan\frac{\pi}{2}z$ is the inverse of the
sliver conformal map\cite{Schnabl,RZO} and the star $^\star$ denotes BPZ
conjugation. Another definition of these fields is given by mapping them
to operator insertions inside correlation functions on the cylinder:
\begin{eqnarray}
K\!\!\!\!\!\!\!\!&& \rightarrow \mathbb{I}
\int_{-i\infty}^{i\infty}\frac{dz}{2\pi i} T(z),
\ \ \ \ \ \ \,
B\rightarrow \sigma_3\int_{-i\infty}^{i\infty}\frac{dz}{2\pi i}b(z),
\nonumber\\
c\!\!\!\!\!\!\!\!&& \rightarrow \sigma_3 c(z),\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
G\rightarrow \sigma_1\int_{-i\infty}^{i\infty}\frac{dz}{2\pi i}G(z).
\end{eqnarray}
See \cite{Okawa,SSF1} and the appendix of \cite{simple} for an explanation
of how this mapping works. The essentially new ingredient in our algebraic
setup is the string field $G$. It lives in the GSO($-$) sector, and corresponds
to a line integral insertion of the worldsheet supercurrent $G(z)$.
Note that to define $G$ we need to ``split'' the operator $\mathcal{G}$ into
left and right halves. Such splittings are potentially anomalous\cite{RZ},
but in this case the splitting appears to be regular (see appendix
\ref{app:G} for more details).
The fields $K,B,c,G$ freely generate a subalgebra of the open
string star algebra subject to the relations
\begin{eqnarray}\!\!\!\!\!\!\!\!&& G^2 = K, \ \ \ \ \ Bc+cB = 1,\ \ \ \ \ B^2=c^2=0,
\nonumber\\
\!\!\!\!\!\!\!\!&& K,B,G\ \mathrm{mutually\ commute}.\label{eq:alg}\end{eqnarray}
$K$ generates the algebra of wedge states\cite{Okawa,RZ_wedge} in the sense
that any star-algebra power of the $SL(2,\mathbb{R})$ vacuum
$\Omega=|0\rangle$ can be written $\Omega^\alpha = e^{-\alpha K}$. It is
useful to define the operators:
\begin{equation}\partial = [K,\cdot],\ \ \ \ \ \
\delta = \llbracket G,\cdot\rrbracket. \end{equation}
In the cylinder coordinate frame, $\partial$ generates an infinitesimal
worldsheet translation and $\delta$ generates a worldsheet
supersymmetry variation. They are derivations of the star product:
\begin{eqnarray}
\partial(\Psi\Phi)\!\!\!\!\!\!\!\!&& = (\partial\Psi)\Phi + \Psi(\partial\Phi),\nonumber\\
\delta(\Psi\Phi)\!\!\!\!\!\!\!\!&& = (\delta\Psi)\Phi + (-1)^{F(\Psi)}\Psi(\delta\Phi).
\label{eq:dderiv}\end{eqnarray}
Since the supersymmetry variation of $c$ produces the $\gamma$
ghost, it is helpful to introduce the corresponding string field:
\begin{equation}\gamma =
\sigma_2 \otimes \frac{1}{\sqrt{\pi}}\gamma(1)|I\rangle \ \rightarrow\
\sigma_2\gamma(z)= \sigma_2\eta e^\phi(z).\end{equation}
Then we have
\begin{eqnarray}\delta c\ = \!\!\!\!\!\!\!\!&& 2i\gamma,
\ \ \ \ \ \ \ \delta\gamma \ = -\frac{i}{2}\partial c,\nonumber\\
\delta G = \!\!\!\!\!\!\!\!&& 2K,\ \ \ \ \ \ \ \delta K=0,\ \ \ \ \ \ \ \ \ \ \ \ \ \
\delta B=0.
\label{eq:dother}\end{eqnarray}
Note that $\delta$ satisfies the supersymmetry algebra $\delta^2=\partial$.
Since $K$ is the worldsheet superpartner of $G$, together these fields
generate a supersymmetric extension of the wedge algebra, which we will call
the {\it wedge superalgebra}.
The algebra generated by $K,B,c,G$ is closed under the action of the
BRST operator
\begin{equation}QK = 0,\ \ \ \ \ QB=K,\ \ \ \ \
Qc = cKc-\gamma^2,\ \ \ \ \ \ QG=0.\label{eq:BRST}\end{equation}
Therefore it makes sense to look for solutions to the cubic equations of
motion
\begin{equation}Q\Psi+\Psi^2=0\end{equation}
within this subalgebra.
\subsection{Half-Brane Solutions}
\label{subsec:solutions}
In this paper we study solutions of the form
\begin{equation}\Psi[f] = \left(c\frac{KB}{1-f}c + B\gamma^2\right)f,
\label{eq:sol1}\end{equation}
where $f$ is a string field in the wedge superalgebra. Multiplying and
dividing by $\sqrt{f}$ gives a gauge equivalent solution
\begin{equation}\hat{\Psi}[f] =
\sqrt{f}\left(c\frac{KB}{1-f}c+B\gamma^2\right)\sqrt{f},\label{eq:sol_real}
\end{equation}
which is twist symmetric\footnote{By twist symmetric, we mean that the GSO($+$)
and GSO($-$) components of the solution separately have definite eigenvalue
under twist conjugation. In particular, the GSO($+$) component is made of
states whose $L_0+1$ eigenvalues are even integers, and the GSO($-$)
component is made of states whose $L_0+\frac{1}{2}$ eigenvalues are odd
integers.}. These are exactly the
same formal expressions which give the pure gauge and tachyon vacuum
solutions of \cite{Erler}. The only
new ingredient here is $f$, which can depend on $G$. Explicitly,
\begin{equation}f=f_++Gf_-,\end{equation} where $f_\pm=f_\pm(K)$ are functions
of $K$ only. In terms of $f_\pm$ the solution takes the form
\begin{equation}\Psi[f] = \left(cKB\frac{1-f_++Gf_-}{(1-f_+)^2-Kf_-^2}c
+B\gamma^2\right)(f_++Gf_-).\label{eq:sol2}\end{equation}
With a little extra work we can also compute $\sqrt{f_++Gf_-}$ to find the
twist symmetric solution.
The physical interpretation of these solutions depends on the choice of
$f_\pm$. To see how, it is helpful
to employ a formal analysis in the $\mathcal{L}^-$ level expansion, which is
an easy and apparently reliable method for identifying gauge orbits in
solutions of this type\cite{simple}. Recall that
$\mathcal{L}^-=\mathcal{L}_0-\mathcal{L}_0^\star$
is a reparameterization generator and a derivation. This means that
the star product of two $\mathcal{L}^-$ eigenstates is itself an eigenstate,
and the eigenvalues add. Since $K,B,c,G$ are eigenstates of $\mathcal{L}^-$
(see table \ref{tab:alg}), we can find the $\mathcal{L}^-$ level expansion
of $\Psi[f]$ by expanding in powers of $K$ and ordering the terms in
sequence of increasing scaling dimension. The expansion can take one of three
different forms, depending on the behavior of $f_\pm$ at $K=0$:
\begin{eqnarray}\mathrm{Pure\ Gauge}: \!\!\!\!\!\!\!\!&&\ \ \ \ f_+(0)\neq 1,
\label{eq:bc_gauge}\\
\mathrm{Half\ Brane}: \!\!\!\!\!\!\!\!&&\ \ \ \ f_+(0)=1,\ \ \ f_-(0) \neq 0,
\label{eq:bc_exotic}\\
\mathrm{Tachyon\ Vacuum}: \!\!\!\!\!\!\!\!&&\ \ \ \ f_+(0)= 1,\ \ \ f_-(0)=0,\ \ \
f_+'(0)\neq 0, \label{eq:bc_vacuum}
\end{eqnarray}
corresponding to the expansions
\begin{eqnarray}\mathrm{Pure\ Gauge}:\ \ \ \!\!\!\!\!\!\!\!&&
\Psi = \frac{f_+(0)}{1-f_+(0)}\,Q(Bc)- \frac{f_+(0)^2}{1-f_+(0)}\,B\gamma^2
\ + \ ...,\nonumber\\
\mathrm{Half\ Brane}:\ \ \ \!\!\!\!\!\!\!\!&& \Psi = -\frac{1}{f_-(0)}\,cGBc\ +
\ ...,\nonumber\\
\mathrm{Tachyon\ Vacuum}:\ \ \ \!\!\!\!\!\!\!\!&& \Psi = -\frac{1}{f_+'(0)}\,c\ +
\ ...,\end{eqnarray}
where $...$ denotes higher level terms. Each expansion formally
corresponds to a physically distinct gauge orbit\footnote{Following
\cite{simple}, one can construct a formal gauge transformation
relating solutions with different choices of $f$:
$\Psi[f'] = g^{-1}(Q+\Psi[f])g$. However, the gauge transformation breaks
down if $f$ and $f'$ do not share the same boundary conditions at $K=0$,
\eq{bc_gauge}-\eq{bc_vacuum}, since either $g$ or $g^{-1}$ would formally
require inverse powers of $K$ in its $\mathcal{L}^-$ level expansion. Inverse
powers of $K$ are not constructible states within the wedge algebra.}
within our general ansatz (see figure \ref{fig:exotic_3sol}). The pure gauge
and tachyon vacuum solutions are known\cite{Erler}, but the so-called
{\it half-brane solutions} are new. These are the main subject of this
paper.
\begin{figure}
\begin{center}
\resizebox{3.5in}{1.9in}{\includegraphics{exotic_3sol.eps}}
\end{center}
\caption{\label{fig:exotic_3sol} Three dimensional ``phase space'' of
solutions, parameterized by $f_+(0),f_-(0)$ and $f_+'(0)$. Tachyon vacuum
solutions sit on a line embedded in a plane of half-brane solutions, which
themselves are embedded in an ambient space of pure gauge solutions.
Note that the point $f_+(0)=1,f_-(0)=0,f_+'(0)=0$ appears to represent
singular solutions.}
\end{figure}
Often one can get insight into the physics of a solution by inspecting its
leading term in the $\mathcal{L}^-$ level expansion. For the
tachyon vacuum the leading term is proportional to the $c$ ghost, which
is responsible for the absence of cohomology at the vacuum\cite{Erler}. For
pure gauge solutions, the leading term is BRST exact to linear order,
corresponding to the fact that pure gauge solutions represent a deformation
of the perturbative vacuum by a trivial element of the BRST cohomology.
For half brane solutions, the full meaning of the leading term $cGBc$ is
not clear to us. However, it is worth noting that $cGBc$ has twist
eigenvalue $+i$:
\begin{equation}(cGBc)^\S = +i\, cGBc.\end{equation}
Therefore half-brane solutions result from condensation of
states in the GSO($-$) sector carrying odd integer eigenvalues of
$L_0+\frac{1}{2}$. This is peculiar since all of these states carry positive
mass squared. The more familiar states responsible for tachyon
condensation carry even integer $L_0+\frac{1}{2}$, and in fact these states
have vanishing expectation value in the twist even solution \eq{sol_real}.
The fact that half-brane solutions result from ``condensation'' of massive
modes of the open string is one way to anticipate that they cannot satisfy the
reality condition.
Let us give two explicit examples of half-brane solutions. The first comes
by setting
\begin{equation}f= f_++Gf_- = \frac{1}{1-iG},\end{equation}
and takes the form
\begin{equation}{\Psi_\mathrm{simp}} = \Big[i cGBc +Q(Bc)\Big]\frac{1+iG}{1+K}.
\label{eq:simple}\end{equation}
We will explain the factor of $i$ shortly. We will call this the
{\it simple} half-brane solution,
since it is in many ways analogous to the ``simple'' tachyon vacuum
introduced in \cite{simple}. In
particular, \eq{simple} requires no phantom term, and gives
the most straightforward calculation of the action and gauge invariant
overlap. Another solution, which is likely to be better behaved in the
level expansion (see appendix \ref{app:level} and \cite{simple}), comes from
setting
\begin{equation}f= f_++Gf_- = (1+ia G)\Omega,\label{eq:Sch_f}\end{equation}
where $a\neq 0$ is a parameter. It takes the form,
\begin{equation}
{\Psi_\mathrm{Sch}} = \left[c\frac{KB(1-\Omega+ia G\Omega)}{(1-\Omega)^2+a^2K\Omega^2}c
+B\gamma^2\right](1+ia G)\Omega.\label{eq:Schnabl}\end{equation}
Unlike \eq{simple}, this solution is composed of wedge states whose angles
have strictly positive lower bound. We will call it the
{\it Schnabl-like} solution. To compute the action
or gauge invariant overlap, we should express
the solution as a regularized sum subtracted against a phantom term. We will
explain how to do this in section \ref{sec:phantom}.
\subsection{Half-Brane Solutions in Berkovits' String Field Theory}
\label{subsec:berkovits}
We would now like to know whether half-brane solutions exist in Berkovits'
string field theory. The task is to find a pair of string fields
$(g,g^{-1})$ at ghost and picture number zero, and in the large Hilbert space,
satisfying
\begin{eqnarray}Qg \!\!\!\!\!\!\!\!&& = g\Psi, \label{eq:Berk1}\\ g^{-1}g \!\!\!\!\!\!\!\!&& =
gg^{-1}= 1.\label{eq:Berk2}\end{eqnarray}
where $\Psi$ is a cubic half-brane solution. Within a certain subalgebra of
states, we will show that these equations have no solutions for $g$ and
$g^{-1}$. A similar approach can be used to argue that Berkovits' string
field theory does not have a tachyon vacuum solution on a BPS D-brane\cite{ES}.
To solve equations \eq{Berk1} and \eq{Berk2}, we must extend our subalgebra
to include fields in the large Hilbert space. The minimal and most natural
extension is to include the string field
\begin{equation}A = -\sigma_3\otimes \xi\partial\xi e^{-2\phi}c(1)|I\rangle
\end{equation}
which satisfies
\begin{equation}QA=1,\ \ \ \ A\gamma^2=-c,\ \ \ \ Ac=cA=0,\ \ \ \
\llbracket \gamma,A\rrbracket=0,\ \ \ \ \llbracket \partial c,A\rrbracket=0.
\end{equation}
$A$ describes an insertion of an inverse picture changing operator
multiplied by the $\xi$ zero mode. It has ghost number $-1$, is
effective Grassmann odd, carries even worldsheet spinor number, and has
scaling dimension $0$. Naively, the field $A$ is enough to generate any
Berkovits solution given any cubic solution. To see how, note that
\begin{equation}g=1+A\Psi.\label{eq:formal_berk}\end{equation}
solves \eq{Berk1}\cite{equivalence,super_photon}. Then, we can {\it almost}
solve \eq{Berk2} by expressing $g^{-1}$ as an infinite geometric series in
powers of $-A\Psi$. However, this series is not guaranteed to converge.
This is why the cubic and Berkovits equations of motion are not
{\it a priori} equivalent.
We search for a Berkovits half-brane by making the most general possible ansatz
in the subalgebra generated by $K,B,c,G$ and $A$. Expand $\Psi$ and
$(g,g^{-1})$ into $\mathcal{L}^-$ eigenstates as follows:
\begin{eqnarray}\Psi\ \!\!\!\!\!\!\!\!&& = \,
\Psi_{-1/2}+\Psi_0+\Psi_{1/2}+...,\nonumber\\
g\ \!\!\!\!\!\!\!\!&& = \, g_{-1/2}\, +\,g_0\,+\,g_{1/2}\,+...,\nonumber\\
g^{-1}\!\!\!\!\!\!\!\!&& = \, \bar{g}_{-1/2}\, +\,\bar{g}_0\,+\,\bar{g}_{1/2}\,+...,
\end{eqnarray}
where the subscripts refers to the $\frac{1}{2}\mathcal{L}^-$ eigenvalue of the
fields. If $\Psi$ is a cubic half-brane solution, its expansion
takes the general form
\begin{eqnarray}\Psi_{-1/2} \!\!\!\!\!\!\!\!&&=\, -\frac{1}{\alpha_1+\alpha_2}cGBc,
\nonumber\\
\Psi_0\ \, \!\!\!\!\!\!\!\!&&= \,
-\frac{\alpha_1}{\alpha_1+\alpha_2}GcGBc-\frac{\alpha_2}{\alpha_1+\alpha_2}
cGBcG+\frac{\beta_1+\beta_2}{(\alpha_1+\alpha_2)^2}cKBc+B\gamma^2,
\nonumber\\
\Psi_{1/2}\ \!\!\!\!\!\!\!\!&& =\, ...,
\label{eq:lowest_psi}\end{eqnarray}
where $\alpha_1,\alpha_2,\beta_1,\beta_2$ are constants parameterizing the
gauge orbit up to this level. The most general ansatz for $g$ is
\begin{eqnarray}g_{-1/2}\,\!\!\!\!\!\!\!\!&& = x\, A\gamma,\nonumber\\
g_0\ \!\!\!\!\!\!\!\!&& = y_1+y_2\, Bc+y_3\,A \partial c+y_4\,GA\gamma+y_5\,A\gamma G,
\nonumber\\
g_{1/2}\,\!\!\!\!\!\!\!\!&& = ...,
\label{eq:lowest_g}\end{eqnarray}
where $x$ and $y_1,...,y_5$ are
coefficients to be determined by solving the equations of motion. We make a
similar ansatz for $g^{-1}$. Now plug
these formulas into \eq{Berk1} and solve level by level:
\begin{eqnarray}0\!\!\!\!\!\!\!\!&& = g_{-1/2}\Psi_{-1/2},\nonumber\\
Qg_{-1/2}\!\!\!\!\!\!\!\!&& = g_{-1/2}\Psi_0+g_0\Psi_{-1/2},\nonumber\\
\!\!\!\!\!\!\!\!&&\,\vdots\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ .
\label{eq:Berk1_levels}\end{eqnarray}
The lowest level equation is trivially satisfied. Plugging \eq{lowest_psi}
and \eq{lowest_g} into the next equation gives
\begin{equation}xQ(A\gamma) =
\frac{2i \alpha_1 x-y_1+2iy_5}{\alpha_1+\alpha_2} cGBc+x\, \gamma cB.
\end{equation}
Note that the right hand side is in the small Hilbert space. Acting with
$\eta_0$ therefore gives
\begin{equation}x\,Q(\eta_0(A\gamma))=0\label{eq:xcoeff}\end{equation}
The field $\eta_0(A\gamma)$ is the zero momentum tachyon in the $-1$
picture. Since the zero momentum tachyon is off-shell, \eq{xcoeff} implies
that the coefficient $x$ vanishes, i.e. $g_{-1/2}=0$.
A similar argument also shows that $\bar{g}_{-1/2}=0$.
Equation \eq{Berk1_levels} then implies that $g_0$ has a right kernel:
\begin{equation}g_0\Psi_{-1/2}=0.\end{equation}
To construct $g^{-1}$, we must solve \eq{Berk2} level by level:
\begin{eqnarray}\bar{g}_{-1/2}g_{-1/2} \!\!\!\!\!\!\!\!&& = 0,\nonumber\\
\bar{g}_{-1/2}g_0 + \bar{g}_0g_{-1/2} \!\!\!\!\!\!\!\!&& =0,\nonumber\\
\bar{g}_{-1/2}g_{1/2} +\bar{g}_0g_0+\bar{g}_{1/2}g_{-1/2}\!\!\!\!\!\!\!\!&& = 1,
\nonumber\\
\!\!\!\!\!\!\!\!&&\ \vdots \ \ \, .\end{eqnarray}
Since $g_{-1/2}=\bar{g}_{-1/2}=0$ this implies
\begin{equation}\bar{g}_0g_0=1,\end{equation}
but this contradicts the fact that $g_0$ has a right kernel. Therefore,
the Berkovits half-brane solution does not exist in the $K,B,c,G,A$
subalgebra. While it is possible that a more general ansatz is necessary,
we believe that this subalgebra is rich enough to capture a half-brane
solution, if one were to exist.\footnote{Fuchs and Kroyter
suggest\cite{equivalence} a general mapping between cubic and Berkovits
solutions $g=1+\tilde{A}\Psi$, where $\tilde{A}$ is a midpoint insertion of
$\xi \partial\xi e^{-2\phi}c$. However, this solution appears to be too singular
to allow for a computation of the Berkovits action.}
\section{Reality Condition}
\label{sec:reality}
Physical solutions in cubic superstring field theory are expected to satisfy
the reality condition\footnote{The dagger ($^\ddag$) refers to a composition
of Hermitian and BPZ conjugation. See appendix
\ref{app:conventions}. Note that this form of the reality condition is correct
only for the left-handed star product convention.}\cite{Ohmori,Zwiebach}
\begin{equation}\Psi^\ddag = \Psi,\label{eq:strong_real}\end{equation}
It is important to ask whether half-brane solutions meet this criterion.
Surprisingly, the answer is no, according to the following theorem:
\begin{theorem}
Under assumptions {\bf 1)}-{\bf 4)} stated below, there are no half-brane
solutions in the $K,B,c,G$ subalgebra satisfying the reality condition.
\end{theorem}
\begin{proof} Every solution $\Psi$ in the $K,B,c,G$ subalgebra is
associated with a pair of states in wedge algebra:
\begin{equation}f_+(K),\ \ \ \ \ f_-(K).\end{equation}
We can reconstruct $f_+$ and $f_-$ from the solution by
solving the equations\footnote{The most general solution in the
$K,B,c,G$ subalgebra can be found by making the most general
(formal) pure gauge ansatz, following Okawa\cite{Okawa}. In equation \eq{fpm}
$\beta$ represents a line integral insertion of the $\beta$ ghost in the
sliver coordinate frame.}
\begin{equation}B\Psi B = B\frac{K(f_++Gf_-)}{1-(f_++Gf_-)}. \ \ \ \ \ \ \ \
\llbracket\beta,\llbracket\beta,\Psi\rrbracket\rrbracket
= B(f_++Gf_-),\label{eq:fpm}
\end{equation}
To prove the theorem, we show that the reality condition is inconsistent with
certain regularity conditions which must be imposed on $f_+(K)$ and $f_-(K)$.
The regularity conditions are:
\begin{description}
\item {\bf 1)} $f_+(0)=1$ and $f_-(0)\neq 0$ and in particular $f_+'(0)$ is
finite.
\item {\bf 2)} $\lim_{K\to\infty}f_+(K) =0 $ and
$\lim_{K\to\infty}\sqrt{K}f_-(K)=0$.
\item {\bf 3)} $f_+$ and $f_-$ are continuous functions of $K$ for all
$K\geq 0$.
\item {\bf 4)} The field $\frac{K}{(1-f_+)^2-Kf_-^2}$, is a continuous
function of $K$ for all $K\geq 0$.
\end{description}
Condition {\bf 1)} is essentially the definition of the half-brane solution.
Condition {\bf 2)} ensures that the solution is not too ``identity-like,'' so
that it can have well-defined action and gauge invariant overlap. Conditions
{\bf 3)} and {\bf 4)} are motivated by a conjecture due to
Rastelli\cite{Rastelli} suggesting that the algebra of wedge states should be
identified with the $C^*$-algebra of bounded, continuous functions on the
positive real line\footnote{The definition of the algebra of wedge states is
not known, but discontinuous functions of $K$ appear to be problematic in
the level expansion. For related discussions,
see \cite{Schnabl_lightning}.}. In particular, {\bf 3)} and {\bf 4)}
assume that $f_+,f_-$ and $\frac{K}{(1-f_+)^2-Kf_-^2}$ must
separately be well-defined states in order for the solution itself
to be well-defined. Since these fields can be extracted directly from the
solution via equation \eq{fpm}, this assumption appears necessary.
The reality condition implies that $f_+$ and $f_-$ are real
functions of $K$. To see why this contradicts regularity, consider the
denominator of the expression appearing in {\bf 4)}, which we will call $D(K)$:
\begin{equation}D(K) = (1-f_+)^2 - Kf_-^2.\end{equation}
By assumption {\bf 1)} we have
\begin{equation}D(0)=0\end{equation}
and
\begin{equation}D'(0) = -f_-(0)^2. \end{equation}
Since the slope is negative, we have
\begin{equation}D(K)<0\ \ \mathrm{for\ some\ positive}\ K.
\label{eq:pf1}\end{equation}
Now by assumption {\bf 2)}
\begin{equation}\lim_{K\to\infty}D(K) = 1.\label{eq:pf2}\end{equation}
Since $D$ is continuous by assumption {\bf 3)}, this means
\begin{equation}D(K)=0\ \ \mathrm{for\ some\ strictly\ positive}\ K.
\end{equation}
See figure \ref{fig:exotic_real}. Since $D(K)$ has a zero, the ratio
$K/D$ cannot be continuous for all $K\geq 0$ which violates assumption
{\bf 4)}.
\end{proof}
\begin{figure}
\begin{center}
\resizebox{3in}{1.55in}{\includegraphics{exotic_real.eps}}
\end{center}
\caption{\label{fig:exotic_real}
If $f_\pm(K)$ are real, boundary conditions for the half-brane solution
at $K=0$ and $\infty$ require that the denominator of the solution
\eq{sol2} has a zero for positive $K$.}
\end{figure}
It is helpful to see why real $f_+$ and $f_-$ are problematic in specific
examples. Suppose we defined the simple solution in \eq{simple}
without the factor of $i$:
\begin{equation}f=f_++Gf_-=\frac{1}{1-G},\end{equation}
In this case condition {\bf 4)} is satisfied since
\begin{equation}\frac{K}{(1-f_+)^2-Kf_-^2}=K-1\end{equation}
is a continuous function of $K$. But condition {\bf 3)} is not satisfied:
both $f_+$ and $f_-$ are equal to $\frac{1}{1-K}$, which has a pole at $K=1$. One could try to define
$\frac{1}{1-K}$ using the Schwinger parameterization\cite{simple}
\begin{equation}\frac{1}{1-K} = -\int_0^\infty dt\, e^t \Omega^t,\end{equation}
but since the wedge state $\Omega^t$ approaches a constant
(the sliver) for large $t$, this integral diverges exponentially. A
second example
is the Schnabl-like solution with $a=-i$, so that the factor of $i$ cancels
in \eq{Sch_f}. In this case
\begin{equation}f_+=f_-=\Omega\end{equation}
are both real and satisfy {\bf 3)}, but
\begin{equation}\frac{K}{(1-f_+)^2-Kf_-^2} =
\frac{K}{(1-\Omega)^2 - K\Omega^2}\label{eq:ex2}\end{equation}
has a pole at $K\approx 0.931$ and violates {\bf 4)}. One can try to define
this state by a geometric series
\begin{equation}\frac{K(1-\Omega)}{(1-\Omega)^2 - K\Omega^2} =
K(1-\Omega)\left[\sum_{n=0}^\infty (2\Omega-(1-K)
\Omega^2)^n \right],\label{eq:geom_ex}\end{equation}
and evaluate the contribution of each term in the series
to a typical state in the Fock space, for example $L_{-2}|0\rangle$.
We have found numerically that the contributions to this coefficient
eventually grow exponentially with $n$. By contrast,
contributions from the analogous sum at $a=1$ decay quite rapidly
(as $1/n^4$) to give the coefficient $2.86\,L_{-2}|0\rangle$.
\subsection{Topology of Half-Brane Solutions}
\label{subsec:Topology}
Though half-brane solutions are not real, every half-brane solution is
related to its conjugate by a complex gauge transformation:
\begin{equation}\Psi^\ddag = U^{-1}(Q+\Psi)U.\label{eq:weak_real}
\end{equation}
The required $U$ is straightforward to compute (see appendix B of
\cite{simple}) and is regular, in as far as the solutions themselves are
regular. This raises an interesting issue. From the perspective of gauge
invariant observables, a solution satisfying \eq{weak_real} is naively
equivalent to a real solution. In fact, such solutions have been useful
for studying marginal deformations with singular OPEs\cite{FK,KO}, solutions
in Berkovits' string field
theory\cite{super_photon,super_marg,Okawa_supermarg1,Okawa_supermarg2,KOsuper},
and the tachyon vacuum \cite{simple}.
A second thought, however, reveals that \eq{weak_real} is not quite
enough to guarantee the reality of observables. We must also require that $U$
can be implemented as a sequence of infinitesimal gauge transformations,
that is, $U$ can be continuously deformed to the identity. Remarkably, for
half-brane solutions, this appears not to be possible. That is, $U$ is a
topologically nontrivial gauge transformation. This
is the first explicit example of a topologically nontrivial
gauge transformation in string field theory, and is especially interesting
since the topology is not related to any spacetime geometry in the
$\alpha'\to0$ limit, but appears to be intrinsic to the internal structure
of the string.
To start we must define what it means to ``continuously deform'' the gauge
transformation $U$. For simplicity, we will restrict ourselves to the $K,B,c,G$
subalgebra, though we presume that our results are more general.
We assume that a continuous deformation of the gauge transformation $U$
will effect a continuous deformation of half brane solutions,
in the following sense:
\begin{definition}
(Continuity.) Let $\Psi(t),t\in[0,1]$ be a 1-parameter family of half-brane
solutions in the $K,B,c,G$ subalgebra. We say that this family is
{\bf continuous} only if $f_+(K,t)$ and $f_-(K,t)$, defined via \eq{fpm},
satisfy the following properties
\begin{description}
\item{\bf A1)-A4)} $f_+(K,t)$ and $f_-(K,t)$ satisfy conditions
{\bf 1)}-{\bf 4)} for every fixed $t\in[0,1]$.
\item{\bf B)} $f_+(K,t)$ and $f_-(K,t)$ are continuous functions of
$K$ and $t$ for $K\geq 0$ and $t\in[0,1]$.
\end{description}
\end{definition}
\noindent Conditions {\bf A1)-A4)} ensure that $\Psi(t)$ is a regular
half-brane solution for all $t$. Condition {\bf B)} ensures that there are
no ``jumps'' as we change $t$, that is, $f_+$ and $f_-$ should change
continuously with $t$ if $\Psi(t)$ does. We now come to our central claim:
\begin{theorem}
There is no continuous 1-parameter family of half-brane solutions
$\Psi(t),t\in[0,1]$ in the $K,B,c,G$ subalgebra such that
$\Psi(0)=\Psi(1)^\ddag$.
\end{theorem}
\noindent This means, in particular, that the gauge transformation $U$ relating
a half-brane solution to its conjugate cannot be continuously deformed to
the identity.
\begin{proof} Since $\Psi(0)=\Psi(1)^\ddag$, the family of states
$f_+(K,t)$ and $f_-(K,t)$ associated with $\Psi(t)$ must satisfy the
boundary condition,
\begin{equation}f_+(K,0)=f_+(K,1)^*,\ \ \ \ f_-(K,0)=f_-(K,1)^*.
\label{eq:fpmconj}\end{equation}
Analogous to the proof of Theorem 1, we will show that this boundary
condition is incompatible with the continuity conditions stated above. In
particular, we will show that the boundary condition, together with
conditions {\bf A1)-A3)} and {\bf B)} imply that the field
\begin{equation}D=(1-f_+)^2-Kf_-^2\end{equation}
has a zero at some point $(K,t)$. Therefore condition {\bf A4)} is
violated, and the sought after continuous family of solutions does not exist.
It is useful to think of $K$ and $t$ as coordinates on a semi-infinite strip
\begin{equation}\Sigma = \mathbb{R}_+\otimes[0,1],\end{equation}
Consider the function
\begin{equation}\left.\Theta\right|_{\delta\Sigma}
=\left.\frac{D}{|D|}\right|_{\delta\Sigma},
\end{equation}
which maps the boundary of $\Sigma$ into complex numbers of unit modulus.
The boundary includes the point at $K=\infty$, so that $\delta\Sigma$ has
the topology of a circle. We make the following claims:
\begin{claim}$\Theta|_{\delta\Sigma}$ is a continuous map from $\delta\Sigma$
into complex numbers of unit modulus.
\end{claim}
\begin{proof} By assumption we take the half-brane solution and its conjugate
at $t=0$ and $t=1$ to be well-defined solutions. Therefore, $D$ cannot have
any zeros for positive $K$ at $t=0$ and $t=1$. Conditions
{\bf A1)}, {\bf A2)}, {\bf B)} then imply continuity on all of $\delta\Sigma$.
\end{proof}
\begin{claim} If $\Theta|_{\delta\Sigma}$ has nonzero winding number,
then $D$ has a zero inside $\Sigma$.
\end{claim}
\begin{proof} Suppose $D$ has no zeros in $\Sigma$. Since {\bf B)} implies
that $D$ is continuous, we can extend
$\Theta|_{\delta\Sigma}$ to a continuous function on the entire semi-infinite
strip by simply taking $\Theta=\frac{D}{|D|}$. Shrinking $\delta\Sigma$
to a point, this function gives a continuous homotopy from
$\Theta|_{\delta\Sigma}$ to the identity map. Since the identity map has zero
winding number, the result follows.
\end{proof}
\begin{claim} The winding number of $\Theta|_{\delta\Sigma}$ is odd.\end{claim}
\begin{proof} The proof of this claim is the most technical part of the
argument. As a first step, it is helpful to introduce a notion of ``winding
number'' for maps from a closed interval into complex numbers of
unit modulus. Let $g$ be a continuous map from a
closed oriented interval $I$ into complex numbers of unit modulus.
We can lift $g$ to a continuous map $\phi:I\to\mathbb{R}$ such that
$g=e^{i\phi}$. Parameterizing $I$ by
$\lambda\in[0,1]$, we define the {\it winding number} of $g$ to be the
unique integer $n$ such that
\begin{equation}\phi(1)-\phi(0)=2\pi n +R,\ \ \ \ \ 0\leq R<2\pi.
\label{eq:open_winding}\end{equation}
We will write $n=w[g]$.
Now consider two closed oriented intervals $I_1$ and $I_2$ which
intersect at their endpoints to form a circle $S^1$. Assume that the
orientation of $I_1$ is the same as that of the circle, and the
orientation of $I_2$ is opposite. If $g$ is a continuous map from $S^1$
into complex numbers of unit modulus, then
\begin{equation}w[g] = w\left[g|_{I_1}\right]-w\left[g|_{I_2}\right],
\label{eq:seg_add}\end{equation}
where $g|_{I_1}$ and $g|_{I_2}$ is the restriction of $g$ to the
intervals $I_1$ and $I_2$, respectively. The proof is
straightforward.
\begin{figure}
\begin{center}
\resizebox{5in}{1.2in}{\includegraphics{exotic_Sigma.eps}}
\end{center}
\caption{\label{fig:Sigma} The boundary segments $I_1$ and $I_2$ of $\Sigma$.
}
\end{figure}
We compute the winding number of $\Theta|_{\delta\Sigma}$ by
splitting $\delta\Sigma$ into two segments, computing the winding numbers on
each segment separately, and taking the difference following \eq{seg_add}.
The segments will be:
\begin{eqnarray}
\!\!\!\!\!\!\!\!&& I_1:\ \mathrm{the}\ K=0\ \mathrm{boundary\ of}\ \Sigma,\nonumber\\
\!\!\!\!\!\!\!\!&& I_2:\ \mathrm{the}\ t=0\ \mathrm{and}\ t=1\ \mathrm{boundaries\ of}\
\Sigma,\ \mathrm{connected\ through}\ K=\infty.\nonumber
\end{eqnarray}
See figure \ref{fig:Sigma}. First we compute the winding number of
$\Theta|_{I_1}$ in terms of the winding number of the function
\begin{equation}\left.\frac{f_-}{|f_-|}\right|_{I_1}=e^{i\theta},
\label{eq:fmodfm}\end{equation}
Conditions {\bf A1)} and {\bf B)} implies that $\theta$ is a continuous map
from $I_1$ into $\mathbb{R}$. The winding number of \eq{fmodfm} is the
integer $n$ satisfying
\begin{equation}\theta(1)-\theta(0)=2\pi n +R\ \ \ \ 0\leq R<2\pi.
\label{eq:FmW}\end{equation}
From {\bf A1)} we also have
\begin{equation}\Theta|_{I_1}=e^{i(2\theta+\pi)}.
\label{eq:DmodD_I1}\end{equation}
The winding number of $\Theta|_{I_1}$ follows:
\begin{equation}(2\theta(1)+\pi)-(2\theta(0)+\pi)=4\pi n+2R.\end{equation}
If $0\leq R<\pi$, the winding number is $2n$, and if $\pi\leq R<2\pi$
the winding number is $2n+1$. Thus
\begin{eqnarray}w[\Theta|_{I_1}]\!\!\!\!\!\!\!\!&&
\in 2\mathbb{Z}\ \ \ \ \ \ \ \ \ \ \ \mathrm{if}\ \ 0\leq R<\pi,\nonumber\\
w[\Theta|_{I_1}]\!\!\!\!\!\!\!\!&& \in 2\mathbb{Z}+1
\ \ \ \ \ \ \mathrm{if}\ \ \pi\leq R<2\pi.
\end{eqnarray}
Now compute the winding number on $I_2$. Parameterize $I_2$
by $\lambda\in[0,1]$ such that: 1) $\lambda=0$ corresponds to the
corner of the strip where $K$ and $t$ both vanish; 2)
$\lambda=\frac{1}{2}$ corresponds the the point at $K=\infty$, and
3) $\lambda=1$ corresponds to the corner where $K$ vanishes and $t=1$.
Also write
\begin{equation}\Theta|_{I_2}=e^{i\psi},
\end{equation}
where $\psi$ is a continuous map from $I_2$ into $\mathbb{R}$.
Because of \eq{fpmconj}, $\Theta|_{I_2}$ evaluated on the $t=1$ boundary
is the complex conjugate of its value on the $t=0$ boundary, which implies
\begin{equation}\psi(1)-\psi(0) = 2(\psi(1)-\psi(\fraction{1}{2})).\end{equation}
Continuity requires that $\Theta=1$ at $K=\infty$, and therefore
$\psi(\frac{1}{2})=2\pi m$ for some integer $m$:
\begin{equation}\psi(1)-\psi(0) = 2\psi(1)-4\pi m.\end{equation}
Comparing with equation \eq{DmodD_I1} we are free to assume
\begin{equation}\psi(1)=2\theta(1)+\pi\end{equation}
Also \eq{fpmconj} implies
\begin{equation}\theta(1)=-\theta(0)+2\pi k,\end{equation}
for some integer $k$. Therefore
\begin{eqnarray}\psi(1)-\psi(0) \!\!\!\!\!\!\!\!&& = 4\theta(1)+2\pi(-2m+1)\nonumber\\
\!\!\!\!\!\!\!\!&& = 2(\theta(1)-\theta(0))+2\pi(2k-2m+1).
\end{eqnarray}
Now we substitute equation \eq{FmW} we find
\begin{equation}\psi(1)-\psi(0) = 2\pi(2n+2k-2m+1)+2R.\end{equation}
Therefore
\begin{eqnarray}w[\Theta|_{I_2}]\!\!\!\!\!\!\!\!&&
\in 2\mathbb{Z}+1\ \ \ \ \ \ \ \mathrm{if}\ \ 0\leq R<\pi,\nonumber\\
w[\Theta|_{I_2}]\!\!\!\!\!\!\!\!&& \in 2\mathbb{Z}
\ \ \ \ \ \ \ \ \ \ \ \ \mathrm{if}\ \ \pi\leq R<2\pi.
\end{eqnarray}
Now we invoke \eq{seg_add} to find the final result:
\begin{equation}w[\Theta|_{\delta \Sigma}]
=w[\Theta|_{I_1}]-w[\Theta|_{I_2}]\in 2\mathbb{Z}+1.\end{equation}
\end{proof}
\noindent Since the winding number of $\Theta|_{\delta \Sigma}$ is odd, it
cannot be zero. Therefore $D$ must vanish at some point inside the
semi-infinite strip. This completes the proof of Theorem 2.
\end{proof}
Theorem 2 implies that half-brane solutions come in at least two
topologically distinct sectors in the $K,B,c,G$ subalgebra. With some
extra work, one can show that there are precisely two. It would be nice
to find a way to characterize these sectors in terms of an invariant which is
computable in terms of $f_+$ and $f_-$. A related question is that, though
we have claimed that every half-brane solution is physically distinguishable
from its conjugate on account of a topologically nontrivial gauge
transformation, we have not found an observable which would actually
distinguish between a half-brane solution and its conjugate in practice.
The energy and closed string overlap, computed in section
\ref{sec:observables}, are real observables for all
half-brane solutions. Finding an observable which can detect the failure of
the reality condition would give much insight into the physical significance of
half-brane solutions, as well as their topological structure.
\section{Regularization and Phantom Piece}
\label{sec:phantom}
In this section we discuss the regularization and phantom term for the
half-brane solution. For clarity we focus on the Schnabl-like half-brane
solution, though the discussion can be extended to solutions based on more
general choices of $f_\pm$.
The Schnabl-like solution \eq{Schnabl} can be written in the form
\begin{equation}
{\Psi_\mathrm{Sch}} = \left[c\frac{KB}{1-(1+iaG)\Omega}c
+B\gamma^2\right](1+ia G)\Omega.\label{eq:Schnabl2}\end{equation}
It is useful\cite{Erler} to replace the factor between the $c$ insertions
by the partial sum of a geometric series, with the appropriate ``error term'':
\begin{equation}\frac{K}{1-(1+iaG)\Omega} = \sum_{n=0}^N K[(1+iaG)\Omega]^n
+\frac{K}{1-(1+iaG)\Omega}[(1+iaG)\Omega]^{N+1}.\end{equation}
With this substitution, the solution can be written in the form
\begin{equation}{\Psi_\mathrm{Sch}} = \Psi_{N+1}-\sum_{n=0}^N\psi_n'+\Gamma,
\label{eq:sum_phantom}\end{equation}
where, reflecting the notation of Schnabl\cite{Schnabl}, we have defined
the fields
\begin{eqnarray}
\psi_n' \!\!\!\!\!\!\!\!&& = -cKB\Big[(1+iaG)\Omega\Big]^n c\Big[(1+iaG)\Omega
\Big]\\
\Gamma \!\!\!\!\!\!\!\!&& = B\gamma^2\Big[(1+iaG)\Omega\Big] \\
\Psi_{N+1}\!\!\!\!\!\!\!\!&&
= c\left(\frac{KB}{1-(1+iaG)\Omega}\Big[(1+iaG)\Omega\Big]^{N+1}
\right)c\Big[(1+iaG)\Omega\Big]\label{eq:phantom}
\end{eqnarray}
Since we have just made a trivial substitution, \eq{sum_phantom} is equal
to the Schnabl-like solution for all $N$. However this expression is most
useful in the $N\to\infty$ limit. In this limit $\Psi_{N+1}$ becomes the
so-called ``phantom term,'' and vanishes in the Fock space.
Compared with phantom terms for the tachyon vacuum
solution\cite{Erler,Schnabl,SSF2}, the large $N$ limit of $\Psi_{N+1}$ is
novel and requires careful treatment.
To understand the phantom term we should study the large $N$ behavior of
the string field
\begin{equation} \Big[(1+iaG)\Omega\Big]^N. \end{equation}
It is helpful to decompose this into GSO($\pm$) components as follows:
\begin{equation}\Big[(1+iaG)\Omega\Big]^N= X_N + (ia N G) Y_N,\label{eq:XY}
\end{equation}
where
\begin{eqnarray}
X_N \!\!\!\!\!\!\!\!&& =
\frac{1}{2}\Big[(1+ia\sqrt{K})^N+(1-ia\sqrt{K})^N\Big]\Omega^N,\nonumber\\
Y_N\!\!\!\!\!\!\!\!&& = \frac{1}{2iaN\sqrt{K}}
\Big[(1+ia\sqrt{K})^N-(1-ia\sqrt{K})^N\Big]
\Omega^N .\label{eq:XYnosum}\end{eqnarray}
Above we introduced $\sqrt{K}$ formally in order to write closed form
expressions for the sums:
\begin{eqnarray}
X_N \!\!\!\!\!\!\!\!&& =
\sum_{0\leq k\leq N/2}{N \choose 2k}a^{2k}(-K)^k
\Omega^N,\nonumber\\
Y_N\!\!\!\!\!\!\!\!&& = \frac{1}{N}\sum_{0\leq k\leq \frac{N-1}{2}}{N \choose 2k+1}
a^{2k}(-K)^k \Omega^N. \label{eq:ph_sums}\end{eqnarray}
A naive argument suggests that the large $N$ limit of $X_N$ and $Y_N$ should be
divergent. Note that while $(-K)^k\Omega^N$ vanishes as a power
$\frac{1}{N^{2+k}}$ in the Fock space, the binomial coefficients grow very
rapidly, so for fixed $k$ and large $N$ a term in the sum for $X_N$ diverges
as a power
\begin{equation}{N\choose 2k}(-K)^k\Omega^N\sim N^{k-2},
\end{equation}
Generically a sum of such terms would diverge faster than any
power of $N$ in the Fock space.
Miraculously, however, for a certain range of the parameter $a$
$X_N$ and $Y_N$ converge to the sliver state:
\begin{equation}\lim_{N\to\infty}X_N =\lim_{N\to\infty}Y_N
=\Omega^\infty.\label{eq:fNpm}
\end{equation}
To see how this happens, consider the Fock space expansion for the wedge
state $\Omega^\alpha$. We can write the expansion in the form
\begin{equation}|\Omega^\alpha\rangle =
\sum_{\vec{n}}P_{\vec{n}}\left(\frac{1}{\alpha+1}\right)\,L_{-n_q}...
L_{-n_2}L_{-n_1}|0\rangle,
\label{eq:fock_wedge}\end{equation}
where $\vec{n}=(n_q,...n_2,n_1)$ is a list of integers of arbitrary length
satisfying
\begin{equation}n_q\geq... \geq n_2\geq n_1\geq 2,\end{equation}
and $P_{\vec{n}}(x)$ are a collection of polynomials in $x$ which determine the
coefficients of $|0\rangle$ and its descendents. For example, up to level
$4$ the nonvanishing polynomials are
\begin{eqnarray}P_{()}(x) \!\!\!\!\!\!\!\!&& = 1,
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \,
P_{(2)}(x) = -\frac{1}{3} + \frac{4}{3}x^2,\nonumber\\
P_{(2,2)}(x) \!\!\!\!\!\!\!\!&& = \frac{1}{9} - \frac{8}{9}x^2+\frac{16}{9}x^4,
\ \ \ \ \ \
P_{(4)}(x) = \frac{1}{30}-\frac{16}{30}x^4.
\end{eqnarray}
To compute $X_N,Y_N$, replace the factors of $K$ multiplying $\Omega^N$ in
\eq{ph_sums} with derivatives via the formula,
\begin{equation}(-K)^k\Omega^N =
\left.\frac{d^k}{d\alpha^k}\Omega^\alpha\right|_{\alpha=N},\end{equation}
and plug in the Fock space expansion \eq{fock_wedge}. The coefficient of the
state labeled by $\vec{n}$ will then be
\begin{eqnarray}\sum_{0\leq k\leq N/2}{N \choose 2k}a^{2k}
\left.\frac{d^k}{d\alpha^k}
P_{\vec{n}}\left(\frac{1}{\alpha+1}\right)\right|_{\alpha=N}\!\!\!\!\!\!\!\!&&
\ \ \ \ \mathrm{for}\ \
X_N,\nonumber\\
\frac{1}{N} \sum_{0\leq k\leq \frac{N-1}{2}}{N \choose 2k+1}
a^{2k}\left.\frac{d^k}{d\alpha^k}P_{\vec{n}}\left(\frac{1}{\alpha+1}\right)
\right|_{\alpha=N}\!\!\!\!\!\!\!\!&&\ \ \ \ \mathrm{for}\ \ \, Y_N.
\label{eq:fpm_coef}\end{eqnarray}
The miracle of convergence as $N\to\infty$ is due to the following
identities:
\begin{eqnarray}\lim_{N\to\infty}\left[
\sum_{0\leq k\leq N/2}{N \choose 2k}a^{2k}
\left.\frac{d^k}{d\alpha^k}\frac{1}{(\alpha+1)^h}\right|_{\alpha=N}\right]
\!\!\!\!\!\!\!\!&& = 0, \label{eq:limits1}\\
\lim_{N\to\infty}\left[\sum_{0\leq k\leq \frac{N-1}{2}}{N \choose 2k+1}
a^{2k}\left.\frac{d^k}{d\alpha^k}\frac{1}{(\alpha+1)^h}\right|_{\alpha=N}
\right]\!\!\!\!\!\!\!\!&& =0,\label{eq:limits2}\end{eqnarray}
which assume
\begin{equation}h>0,\ \ \ a\in[-\sqrt{2},\sqrt{2}].\end{equation}
Thus taking the $N\to\infty$ of equation \eq{fpm_coef}, all nonzero powers
of $\frac{1}{1+\alpha}$ are killed, leaving
\begin{eqnarray}
P_{\vec{n}}(0)\!\!\!\!\!\!\!\!&&\ \ \ \ \mathrm{for}\ \
\lim_{N\to\infty}X_N,\nonumber\\
P_{\vec{n}}(0)\!\!\!\!\!\!\!\!&&\ \ \ \ \mathrm{for}\ \
\lim_{N\to\infty}Y_N.
\end{eqnarray}
These are exactly the coefficients of the sliver state.
To prove the identities \eq{limits1} and \eq{limits2}, it is helpful to
represent the ratio $\frac{1}{1+\alpha}$ as an integral:
\begin{equation}\frac{1}{(\alpha+1)^h} = \frac{1}{(h-1)!}\int_0^\infty dt\,
t^{h-1} e^{-(\alpha+1)t}.\end{equation}
Substituting in \eq{limits1} converts the sum into a simple integral:
\begin{eqnarray}\!\!\!\!\!\!\!\!&& \sum_{0\leq k\leq N/2}{N \choose 2k}a^{2k}
\left.\frac{d^k}{d\alpha^k}\frac{1}{(\alpha+1)^h}\right|_{\alpha=N}\nonumber\\
\!\!\!\!\!\!\!\!&&\ \ \ \ \ \ \ \ \
= \frac{1}{(h-1)!}\int_0^\infty dt\, t^{h-1}e^{-t}\frac{1}{2}\left(
\Big[(1+ia\sqrt{t})e^{-t}\Big]^N+\Big[(1-ia\sqrt{t})e^{-t}\Big]^N\right).
\label{eq:int_limit}\end{eqnarray}
Note the similarity of the integrand with \eq{XYnosum}. Let us assume that the two terms in the integrand should be bounded in
absolute value in the $N\to\infty$ limit. This requires
\begin{equation}|(1\pm ia\sqrt{t})e^{-t}|^2\leq 1,\end{equation}
which implies that $a$ must be a real number in the range
$[\sqrt{2},-\sqrt{2}]$.\footnote{Numerical computations
suggest that that \eq{limits1} may hold even in a limited region of the
complex plane around the line segment $-\sqrt{2}\leq a\leq\sqrt{2}$. We do not
have an analytic proof, however.} Now we can compute the $N\to\infty$
limit in the integrand by simply noting
\begin{equation}\lim_{N\to\infty}\Big[(1\pm ia\sqrt{t})e^{-t}\Big]^N =
\left\{\begin{matrix}1 &\ \ \mathrm{at}\ t=0\\
0 & \ \ \mathrm{otherwise}\end{matrix}\right..
\end{equation}
Since the integrand vanishes almost everywhere, the sum \eq{limits1} must
vanish. This completes the proof that $X_N$ and $Y_N$ approach this sliver
state in the large $N$ limit.
\begin{figure}
\begin{center}
\resizebox{3.9in}{2.7in}{\includegraphics{exotic_phantom.eps}}
\end{center}
\caption{\label{fig:exotic_phantom} Plots of $[(1+ i\sqrt{K})e^{-K}]^N$ as a
function of $K$ for $N=1,3,5,...31$. The
vertical axis is the real part, the forward axis is the imaginary part, and
the horizontal axis is $K$.}
\end{figure}
Perhaps this result is not surprising, since the sliver state is the
only projector which could have emerged from the $N\to\infty$ limit.
What is more interesting is the manner in which $X_N$ and $Y_N$ approach the
sliver. Recall that the sliver state corresponds to a function of $K$
which takes the value $1$ at $K=0$ and vanishes everywhere else.
$X_N$ and $Y_N$ approach the sliver by a sequence of functions
\begin{equation}[(1+ ia\sqrt{K})e^{-K}]^N.\label{eq:XYmod}\end{equation}
A plot of the real and imaginary parts of these functions is shown in figure
\ref{fig:exotic_phantom}. The plot reveals a spiral, which as $N$
increases, winds increasingly many times around the $K$ axis and
becomes increasingly damped away from $K=0$. As $N\to\infty$, the function
$[(1+ ia\sqrt{K})e^{-K}]^N$ vanishes for $K>0$, but for infinitesimal $K$
winds so many times that it fills the whole unit disk in the complex plane.
Now compare this to how wedge states approach the sliver at large wedge
angle, corresponding to the sequence of functions $e^{-NK}$. The large $N$
limit of $e^{-NK}$ reveals none of the highly oscillatory behavior seen in
figure \ref{fig:exotic_phantom}. We can formalize this qualitative
observation as follows. Assume, following a proposal of
Rastelli\cite{Rastelli}, that convergence in the wedge algebra is determined
by the norm
\begin{equation}||f(K)|| = \sup|f(K)|.\label{eq:Rast_norm}\end{equation}
In the large $N$ limit, $X_N$ and $Y_N$ could be considered ``close'' to
a wedge state if there were a set of numbers $\sigma(N),\rho(N)$ such that the
sequence of norms
\begin{equation}||X_N - \Omega^{\sigma(N)}||,\ \ \ \
||Y_N - \Omega^{\rho(N)}||\label{eq:bad_seq}\end{equation}
converged to zero. However, this is impossible; $\Omega^{\sigma(N)}$
only takes values between $0$ and $1$, whereas $X_N$ and $Y_N$ take all
values in the interval $[-1,1]$ for sufficiently large $N$. We can make a
similar observation by looking at the Fock space expansion. In appendix
\ref{app:largeN} we compute the leading order corrections to the sums
\eq{limits1} and \eq{limits2}:
\begin{eqnarray}\sum_{0\leq k\leq N/2}{N \choose 2k}a^{2k}
\left.\frac{d^k}{d\alpha^k}\frac{1}{(\alpha+1)^h}\right|_{\alpha=N}
\!\!\!\!\!\!\!\!&& = (-1)^h \frac{2(2h-1)!}{(h-1)!}\frac{1}{(aN)^{2h}} + ... ,\nonumber\\
\sum_{0\leq k\leq \frac{N-1}{2}}{N \choose 2k+1}
a^{2k}\left.\frac{d^k}{d\alpha^k}\frac{1}{(\alpha+1)^h}\right|_{\alpha=N}
\!\!\!\!\!\!\!\!&& =(-1)^{h+1}
\frac{4N(2h-3)!}{(h-2)!}\frac{1}{(aN)^{2h}}+....\label{eq:reg_limits}
\end{eqnarray}
Plugging these into \eq{fpm_coef}, we find that the leading order
correction to $X_N$ and $Y_N$ for large $N$ is numerically quite different
from that of a wedge state with large wedge angle, especially due to the
$h$-dependent factors in \eq{reg_limits}. In principle these differences
could have an important effect on calculations involving the phantom term.
Therefore, though $X_N$ and $Y_N$ approach the sliver, it is not
the ``same'' sliver as $\Omega^N$ for large
$N$.\footnote{Note that the analog of $X_N,Y_N$ for the
tachyon vacuum solutions of \cite{SSF2} is the infinite power $f(K)^N$,
where $f$ is some function of $K$ satisfying the constraints
$f(0)=1,f'(0)=-\gamma<0$ and $|f(K)|\leq 1$. We can show that
the leading correction to the Fock space coefficients for any such state
match those of a wedge state $\Omega^{\gamma N}$. Moreover, the sequence of
norms $||f^N - \Omega^{\gamma N}||$ converges to zero. In this sense,
$f(K)^N$ is approximately equal to the wedge state $\Omega^{\gamma N}$ for
large $N$.}
This motivates us to introduce a new class of states, more general than
wedge states, which could describe the large $N$ behavior of $X_N$ and $Y_N$:
\begin{equation}e^{-\alpha K}e^{i\beta G}=
\Omega^\alpha\left[\cos(\beta\sqrt{K})+iG\,
\frac{\sin(\beta\sqrt{K})}{\sqrt{K}}\right].\label{eq:ext_wedge}\end{equation}
We will call these {\it super-wedge states}. It turns out that super-wedge
states cannot be easily described as a superposition of wedge surfaces. We will
say more about how these states can be constructed in the next section
and appendix \ref{app:largeN}. Consider the large $N$ limit of the
super-wedge state
\begin{equation}\Big(e^{iaG}\Omega^{1-\frac{a^2}{2}}
\Big)^N=\hat{X}_N+(iaNG)\hat{Y}_N.
\label{eq:Gwedge}\end{equation}
One can check that the sequence of norms
\begin{equation}||X_N - \hat{X}_N||,\ \ \ \ \
||Y_N - \hat{Y}_N||\end{equation}
converges to zero, and moreover the leading large $N$ correction to the
Fock space coefficients of $(\hat{X}_N,\hat{Y}_N)$ match those of
$(X_N,Y_N)$. We can therefore simplify the phantom piece for large $N$
by replacing
\begin{equation}\Big[(1+iaG)\Omega\Big]^N\ \
\rightarrow\ \ \left(e^{ia G}\Omega^{1-\frac{a^2}{2}}\right)^N,
\end{equation}
and
\begin{equation} \frac{K}{1-(1+iaG)\Omega}\ \ \rightarrow\ \ \frac{i}{a}G,
\end{equation}
which is the leading term in the $\mathcal{L}^-$ level
expansion of this factor.\footnote{Sometimes it is
necessary to include more than the leading term in the $\mathcal{L}^-$ level
expansion\cite{Erler}, but ignore this possibility for simplicity.} The phantom
term therefore simplifies to
\begin{equation}\hat{\Psi}_N =
\frac{i}{a}c\left[ GB \left(e^{iaG}\Omega^{1-\frac{a^2}{2}}\right)^N \right]
c\,(1+iaG)\Omega,
\label{eq:reg_phantom}\end{equation}
and we can express the Schnabl-like solution in regularized form,
\begin{equation}{\Psi_\mathrm{Sch}} = \lim_{N\to\infty}\left[\hat{\Psi}_N
-\sum_{n=0}^N\psi_n'+\Gamma\right].\label{eq:reg_sum_phantom}
\end{equation}
Unlike \eq{sum_phantom}, this expression is only valid in the $N\to\infty$
limit.
A few comments about the parameter $a$. Note that the phantom
term \eq{reg_phantom} is manifestly singular as $a$ approaches zero. This
corresponds to the fact that at $a=0$ ${\Psi_\mathrm{Sch}}$ is actually a solution for the
the tachyon vacuum, which requires a different phantom piece.
Also note we needed to fix $a$ to lie within a restricted range
$-\sqrt{2}\leq a\leq\sqrt{2}$. We do not know whether this bound
reflects a limitation of the regularization \eq{reg_sum_phantom} or a
deeper problem with the solution when $a$ sits outside the restricted range.
\subsection{An Aside: General States in the Wedge Algebra}
\label{subsec:gen_f}
Up to now, most states in the wedge algebra have been assumed to
take the form
\begin{equation}f(K) = \int_0^\infty dt \tilde{f}(t)\Omega^t,
\label{eq:laplace_f}\end{equation}
and thus are a ``linear combination'' of wedge states. When this
integral converges, the function $f(K)$ can be identified with the Laplace
transform of $\tilde{f}(t)$. However, the set of functions which can be
represented as a Laplace transform in this sense is limited. Here we
would like to give a more general construction of $f(K)$ motivated by our
analysis of the phantom term.
Consider the space of polynomials over a variable $x$, which we denote
$\mathbb{R}[x]$. Here, $x$ will be identified with the ratio
$\frac{1}{1+\alpha}$ in the Fock space expansion of the wedge state
$\Omega^\alpha$. Given a suitable function $f(t)$, we define a linear
functional $L_f$ on $\mathbb{R}[x]$ as follows:
\begin{eqnarray}L_f(x^0) \!\!\!\!\!\!\!\!&& = f(0),\nonumber\\
L_f(x^1) \!\!\!\!\!\!\!\!&& = \int_0^\infty dt\, f(t)e^{-t},\nonumber\\
\!\!\!\!\!\!\!\!&&\vdots\nonumber\\
L_f(x^h)\!\!\!\!\!\!\!\!&& = \frac{1}{(h-1)!}\int_0^\infty dt\, t^{h-1}f(t)e^{-t}.
\label{eq:Lf}\end{eqnarray}
With the help of this functional, we define the state,
\begin{equation}f(K) = L_f\left(\Omega^\alpha\right),\ \ \ \ \
x=\frac{1}{1+\alpha}.
\label{eq:gen_f}\end{equation}
This should be understood as a definition of $f(K)$ in the Fock
space. One can check that \eq{gen_f} and \eq{laplace_f} give exactly the same
expressions for the coefficients of $f(K)$ in the domain
where both formulas are defined. However \eq{gen_f} is much more general.
For example, \eq{gen_f} allows one to construct a string field for any $f(K)$
in the algebra of bounded, continuous functions on the positive real line.
The existence of such string fields is implied by Rastelli's proposal for the
definition of the algebra of wedge states\cite{Rastelli}.
The simplest example of a state which we can construct from \eq{gen_f},
but not \eq{laplace_f}, is a wedge state with complex wedge angle
\begin{equation}\Omega^{\alpha+i\beta}.\end{equation}
The linear functional \eq{Lf} is
\begin{equation}L_{\Omega^{\alpha+i\beta}}(x^h) =
\frac{1}{(h-1)!}\int_0^\infty dt\, t^{h-1}e^{-(\alpha+1)t+i\beta t}
=\frac{1}{(1+\alpha+i\beta)^h},\ \ \ \ \ \alpha>-1.\label{eq:complex_wedge}
\end{equation}
This is the obvious analytic continuation of the usual Fock space expansion
of a wedge state to complex wedge angle. But note that the linear functional
only converges if $\alpha>-1$. This ``explains'' the curious relation between
the Fock space coefficients of wedge states with positive and negative wedge
angle:
\begin{equation}\Omega^\alpha=\Omega^{-2-\alpha}\ \ \ ?\end{equation}
In reality an inverse wedge state for $\alpha<-1$ is divergent in the Fock
space, and not the analytic continuation of \eq{complex_wedge}.
Another example is the state
\begin{equation}f(K) = \frac{\lambda K\Omega}{1-\lambda\Omega}.\label{eq:prgg}
\end{equation}
This is the pure gauge solution of Schnabl\cite{Schnabl},
either in the ghost number zero toy model or in the ghost number one case
after ignoring the $B,c$ insertions. For $|\lambda|<1$ we can express this
as a Laplace transform by making a geometric series
expansion of the denominator, but for $|\lambda|>1$ this series is divergent.
Still we can define the linear functional
\begin{equation}L_f(x^h) = \frac{1}{(h-1)!}\int_0^\infty dt
\frac{\lambda t^h e^{-2t}}{1-\lambda e^{-t}} = \lambda h\,\Phi(\lambda,h,2),
\ \ \ \lambda\ngtr 1,\end{equation}
where $\Phi(z,s,v)$ is the Lerch function. The integral is absolutely
convergent as long as $\lambda$ is not a real number greater than one.
This suggests that the pure gauge solutions are defined even for
$\lambda<-1$, though the geometric series is divergent. This would bring
Schnabl's pure gauge solutions into line with the pure gauge solutions of
\cite{simple}, which are also nonsingular for $\lambda<-1$.
As a final (somewhat peculiar) example, consider the characteristic
function on the interval $[a,b]>0$:
\begin{equation}{\bf 1}_{[a,b]}(K)= \left\{\begin{matrix}1 & \ \mathrm{for}\
a\leq K\leq b\\ 0 & \ \mathrm{otherwise}\end{matrix}\right..\end{equation}
Formally these are all projectors in the wedge algebra, and if the interval
does not include $0$, the projectors are orthogonal to the sliver state.
There is no hope of representing such states as a Laplace transform
\eq{laplace_f}, but we can still define them using the functional
\begin{equation}L_{{\bf 1}_{[a,b]}}(x^h) = \sum_{n=0}^{h-1}\frac{a^n}{n!}e^{-a}
-\sum_{n=0}^{h-1}\frac{b^n}{n!}e^{-b}.
\end{equation}
Note that these projectors are infinite rank. Unlike the sliver, they are
difficult to reach by taking the infinite power of a ``reasonable'' $f(K)$
in the wedge algebra. While one can apparently define such an $f(K)$
using \eq{gen_f}, it would have to satisfy the awkward constraint of being
exactly equal to unity on the interval $[a,b]$, and strictly less than unity
in absolute value outside that interval.
We have not confirmed whether any of these states behave as expected
under star multiplication. Since they are not linear combinations of surfaces,
one cannot study their star products using the usual gluing rules of
conformal field theory. Nevertheless, these states are concrete
constructions which could be interesting for future study.
\section{Observables}
\label{sec:observables}
\subsection{Gauge Invariant Overlap for Simple Half-Brane Solution}
We start with the simplest computation, that of the gauge invariant
overlap for the simple half-brane solution \eq{simple}:
\begin{equation}W({\Psi_\mathrm{simp}},\mathcal{V}) =
\langle\!\langle{\Psi_\mathrm{simp}}\rangle\!\rangle_\mathcal{V}.\label{eq:simp_ov}
\end{equation}
Here the bracket $\langle\!\langle\cdot\rangle\!\rangle_\mathcal{V}$ is defined
in the same way as the vertex $\langle\!\langle\cdot\rangle\!\rangle$ (see
\eq{left_vertex}) except the picture changing operator $Y_{-2}$ is replaced
by an on shell closed string vertex operator $\mathcal{V}(i)$ inserted at the
midpoint. We assume $\mathcal{V}$ is an NS-NS closed string vertex
operator of the form,
\begin{equation}\mathcal{V}(z) = c\tilde{c}e^{-\phi}e^{-\tilde{\phi}}
\mathcal{O}^{(\frac{1}{2},\frac{1}{2})}(z,\bar{z}),\label{eq:V}\end{equation}
where $\mathcal{O}^{(\frac{1}{2},\frac{1}{2})}$ is a weight $(\fraction{1}{2},\fraction{1}{2})$
superconformal matter primary. We work in the small Hilbert space, so the
$\xi$ zero mode is absent. If the interpretation of Ellwood\cite{Ellwood} is
correct, the gauge invariant overlap should represent the shift in the closed
string tadpole of the solution relative to the perturbative vacuum.
Plugging in the simple solution \eq{simple} into the overlap, the BRST
exact term does not contribute since $\mathcal{V}$ is on-shell. Furthermore
the GSO($-$) component vanishes in the correlator. This leaves
\begin{equation}W({\Psi_\mathrm{simp}},\mathcal{V}) =
-\left\langle\!\!\!\left\langle cGBc\frac{G}{1+K}\right\rangle\!\!\!\right\rangle_\mathcal{V}.
\end{equation}
Now expand $\frac{1}{1+K}$ in terms of wedge states
\begin{equation}-\left\langle\!\!\!\left\langle cGBc\frac{G}{1+K}\right\rangle\!\!\!\right\rangle_\mathcal{V} =
-\int_0^\infty dt\, e^{-t}\Big\langle\!\!\Big\langle cGBcG \Omega^t\Big\rangle\!\!\Big\rangle_\mathcal{V}.
\end{equation}
Note
\begin{equation}cGBcG\Omega^t = t^{\frac{1}{2}\mathcal{L}^-}(cGBcG\Omega).
\end{equation}
Since the operator $t^{\frac{1}{2}\mathcal{L}^-}$ is a reparameterization,
it leaves the bracket invariant. We can then easily evaluate the integral
over $t$ to find
\begin{equation}W({\Psi_\mathrm{simp}},\mathcal{V})
=-\langle\!\langle cGBcG\,\Omega\rangle\!\rangle_\mathcal{V}.\end{equation}
Now commute the leftmost $G$ insertion towards the other $G$ insertion:
\begin{eqnarray}W({\Psi_\mathrm{simp}},\mathcal{V})
\!\!\!\!\!\!\!\!&&= -\langle\!\langle cB(cG +\delta c)G\,\Omega\rangle\!\rangle_\mathcal{V}\nonumber\\
\!\!\!\!\!\!\!\!&& =
-\langle\!\langle cK\,\Omega\rangle\!\rangle_\mathcal{V}+
2i\langle\!\langle c\gamma BG\,\Omega \rangle\!\rangle_\mathcal{V}. \label{eq:1step_ov}
\end{eqnarray}
To compute the first term note
\begin{equation}-cK\Omega = \fraction{1}{2}\mathcal{L}^- (c\Omega) +c\Omega .
\label{eq:trick1}\end{equation}
Since $\mathcal{L}^-$ kills the bracket, this leaves
\begin{equation}W({\Psi_\mathrm{simp}},\mathcal{V})=\langle\!\langle c\Omega\rangle\!\rangle_\mathcal{V}
+2i\langle\!\langle c\gamma BG\,\Omega \rangle\!\rangle_\mathcal{V}.\label{eq:1step_ovf}
\end{equation}
Now focus on the second term. Using cyclicity we can rewrite it in the form,
\begin{equation}
2i\langle\!\langle c\gamma BG\Omega\rangle\!\rangle_\mathcal{V} =
i \Big\langle\!\!\Big\langle\, G (c\gamma B\Omega) + (c\gamma B\Omega)G\,
\Big\rangle\!\!\Big\rangle_\mathcal{V}. \label{eq:2step_ov}\end{equation}
Since $\gamma$ carries odd worldsheet spinor number, the $G$
anticommutator above is exactly the worldsheet supersymmetry
variation $\delta$:
\begin{equation}\delta(c\gamma B\Omega) = G (c\gamma B\Omega)
+ (c\gamma B\Omega) G .
\end{equation}
Therefore we can explicitly eliminate $G$:
\begin{eqnarray}2i\langle\!\langle c\gamma BG\,\Omega\rangle\!\rangle_\mathcal{V}
\!\!\!\!\!\!\!\!&& =i\langle\!\langle \delta(c\gamma B\Omega)\rangle\!\rangle_\mathcal{V}\nonumber\\
\!\!\!\!\!\!\!\!&& =\Big\langle\!\!\Big\langle -2\gamma^2 B\Omega+ \frac{1}{2}c\partial cB\Omega
\Big\rangle\!\!\Big\rangle_\mathcal{V},\end{eqnarray}
where we used the derivation property of $\delta$ and the explicit variations
\eq{dother}. To eliminate the remaining $B$ insertion we use the
derivation $\mathcal{B}^-$, which satisfies
\begin{equation}\fraction{1}{2}\mathcal{B}^-K = B,\ \ \ \ \ \
\fraction{1}{2}\mathcal{B}^-\{B,\, c,\,\, \mathrm{or}\, \gamma\} =0,
\end{equation}
and leaves the vertex invariant. This allows us to express
\begin{equation}
-2\gamma^2 B\Omega+ \frac{1}{2}c\partial cB\Omega =
\mathcal{B^-}\left[-2\gamma^2
\Omega -\frac{1}{2}cKc\Omega\right]-\frac{1}{2}c\Omega.
\end{equation}
The $\mathcal{B}^-$ term kills the bracket leaving
\begin{equation}2i\langle\!\langle c\gamma B G\,\Omega\rangle\!\rangle_\mathcal{V}
= -\frac{1}{2}\langle\!\langle c\Omega\rangle\!\rangle_\mathcal{V}.
\label{eq:2step_ovf}\end{equation}
Plugging into \eq{1step_ovf} and gives the final answer:
\begin{equation}W({\Psi_\mathrm{simp}},\mathcal{V}) =
\frac{1}{2}\langle\!\langle c\Omega\rangle\!\rangle_\mathcal{V}.
\label{eq:ov}\end{equation}
This is exactly one-half the value of the overlap at the tachyon
vacuum\cite{Ellwood}. This confirms that the half-brane
solutions are not gauge equivalent to either the tachyon vacuum or the
perturbative vacuum (where the overlap vanishes identically). It also
indicates that half-brane solutions must source closed strings with half
the strength of a non-BPS D-brane.
\subsection{Gauge Invariant Overlap for Schnabl-Like Half-Brane Solution}
As a check on the consistency of our results, we would like to compute
the gauge invariant overlap for the Schnabl-like solution.
Plugging in \eq{sum_phantom} we find
\begin{eqnarray}W({\Psi_\mathrm{Sch}},\mathcal{V})\!\!\!\!\!\!\!\!&& =
\langle\!\langle{\Psi_\mathrm{Sch}}\rangle\!\rangle_\mathcal{V},\nonumber\\
\!\!\!\!\!\!\!\!&& = \langle\!\langle\Psi_{N+1}\rangle\!\rangle_\mathcal{V} -
\sum_{n=0}^N\langle\!\langle \psi_n'\rangle\!\rangle_\mathcal{V} + \langle\!\langle
\Gamma\rangle\!\rangle_\mathcal{V}.
\label{eq:sch_ov}\end{eqnarray}
The second two terms do not contribute to the overlap. The
easiest way to see this is to note that the overlap for the
pure gauge solution
\begin{equation}\Psi_\lambda = -\sum_{n=0}^\infty \lambda^n \psi_n'
+\lambda\Gamma\label{eq:sch_gauge}\end{equation}
must vanish order by order in $\lambda$ by gauge invariance.
Therefore \eq{sch_ov} simplifies to
\begin{equation}W({\Psi_\mathrm{Sch}},\mathcal{V}) =
\langle\!\langle \Psi_N\rangle\!\rangle. \end{equation}
This equation holds for any $N$, but we will be interested in the limit
$N\to\infty$.
Plugging in the \eq{phantom} for $\Psi_{N}$ the overlap reduces to
a sum of four terms:
\begin{eqnarray}W({\Psi_\mathrm{Sch}},\mathcal{V})
\!\!\!\!\!\!\!\!&& = \ \ \
\left\langle\!\!\!\left\langle c\,\frac{KB(1-\Omega)}{(1-\Omega)^2+a^2K\Omega^2}
\,X_N\,c\,\Omega\right\rangle\!\!\!\right\rangle_\mathcal{V}\nonumber\\
\!\!\!\!\!\!\!\!&& \ \ - Na^2 \left\langle\!\!\!\left\langle c\,
\frac{K^2B\Omega}{(1-\Omega)^2+a^2K\Omega^2}\,
Y_N\,c\,\Omega\right\rangle\!\!\!\right\rangle_\mathcal{V}\nonumber\\
\!\!\!\!\!\!\!\!&& \ \ \ -a^2 \left\langle\!\!\!\left\langle c\,\frac{KGB\Omega}{(1-\Omega)^2+a^2K\Omega^2}
\,X_N\,c\,G\,\Omega\right\rangle\!\!\!\right\rangle_\mathcal{V}\nonumber\\
\!\!\!\!\!\!\!\!&& -Na^2 \left\langle\!\!\!\left\langle c\,\frac{KGB(1-\Omega)}{(1-\Omega)^2+a^2K\Omega^2}
\,Y_N\,c\,G\,\Omega\right\rangle\!\!\!\right\rangle_\mathcal{V}.
\label{eq:4terms}\end{eqnarray}
Now in each of the four terms expand all of the wedge state factors besides
$X_N$ and $Y_N$ in powers of $K$:
\begin{eqnarray}
\left\langle\!\!\!\left\langle c\,\frac{KB(1-\Omega)}{(1-\Omega)^2+a^2K\Omega^2}
\,X_N\,c\,\Omega\right\rangle\!\!\!\right\rangle_\mathcal{V}
\!\!\!\!\!\!\!\!&& = \sum_{m\geq1,n\geq1} C^{(1)}_{mn}\langle\!\langle c\,B\,K^m\, X_N\,c\,K^n
\rangle\!\rangle_\mathcal{V},
\nonumber\\\label{eq:ds1}\\
-Na^2 \left\langle\!\!\!\left\langle c\, \frac{K^2B\Omega}{(1-\Omega)^2+a^2K\Omega^2}\,
Y_N\,c\,\Omega\right\rangle\!\!\!\right\rangle_\mathcal{V}
\!\!\!\!\!\!\!\!&& = \sum_{m\geq1,n\geq1}C^{(2)}_{mn}\langle\!\langle c\,B\,K^m\, Y_N\,c\,K^n
\rangle\!\rangle_\mathcal{V},
\nonumber\\\label{eq:ds2}\\
-a^2 \left\langle\!\!\!\left\langle c\,\frac{KGB\Omega}{(1-\Omega)^2+a^2K\Omega^2}
\,X_N\,c\,G\,\Omega\right\rangle\!\!\!\right\rangle_\mathcal{V}
\!\!\!\!\!\!\!\!&&= \sum_{m\geq0,n\geq0}
C^{(3)}_{mn}\langle\!\langle c\,B\,G\,K^m\, X_N\,c\,G\,K^n\rangle\!\rangle_\mathcal{V},
\nonumber\\ \label{eq:ds3}\\
-Na^2 \left\langle\!\!\!\left\langle c\,\frac{KGB(1-\Omega)}{(1-\Omega)^2+a^2K\Omega^2}
\,Y_N\,c\,G\,\Omega\right\rangle\!\!\!\right\rangle_\mathcal{V}
\!\!\!\!\!\!\!\!&&= \sum_{m\geq1,n\geq0}
C^{(4)}_{mn}\langle\!\langle c\,B\,G\,K^m\, Y_N\,c\,G\,K^n
\rangle\!\rangle_\mathcal{V},
\nonumber\\ \label{eq:ds4}
\end{eqnarray}
where $C_{m,n}^{(a)}$ are constants. Let us focus on \eq{ds3}. To calculate
the double sum, we should compute the traces
\begin{equation}
\langle\!\langle c\,B\,G\,K^m\, X_N\,c\,G\,K^n\rangle\!\rangle_\mathcal{V}.
\end{equation}
Plugging in \eq{ph_sums} for $X_N$ this becomes
\begin{equation}\langle\!\langle cBGK^m\, X_N\,cGK^n\rangle\!\rangle_\mathcal{V} =
\sum_{0\leq K\leq N/2}{N \choose 2k}a^{2k}\left.\frac{d^k}{d\alpha^k}
\langle\!\langle cBG K^m\Omega^\alpha cG K^n\rangle\!\rangle_\mathcal{V}\right|_{\alpha=N}.
\end{equation}
Now reparameterize the bracket with $\mathcal{L}^-$ to factor the $\alpha$
dependence:
\begin{equation} \langle\!\langle cBGK^m\, X_N\,cGK^n\rangle\!\rangle_\mathcal{V} =
\langle\!\langle cBG K^m\Omega cG K^n\rangle\!\rangle_\mathcal{V}
\sum_{0\leq K\leq N/2}{N \choose 2k}a^{2k}\left.\frac{d^k}{d\alpha^k}
\frac{1}{\alpha^{m+n}}
\right|_{\alpha=N}.\end{equation}
We recognize the sum on the right hand side from the identity \eq{limits1}.
Provided $a\in[-\sqrt{2},\sqrt{2}]$ and $m+n>0$, this vanishes in the
$N\to \infty$ limit. Therefore if we take $N\to\infty$ only the $m=n=0$
term in \eq{ds3} contributes to the overlap:
\begin{equation}
-a^2 \lim_{N\to\infty}
\left\langle\!\!\!\left\langle c\,\frac{KGB\Omega}{(1-\Omega)^2+a^2K\Omega^2}
\,X_N\,c\,G\,\Omega\right\rangle\!\!\!\right\rangle_\mathcal{V}
= -\langle\!\langle cBG \Omega cG\rangle\!\rangle_\mathcal{V}.\label{eq:sole_cont}
\end{equation}
Now repeat this argument for equations \eq{ds1}, \eq{ds2} and \eq{ds4}.
However, this time the range of summation over $m,n$ excludes all traces
which could make a nonzero contribution in the $N\to\infty$ limit. So,
in fact, \eq{sole_cont} is the only contribution to the overlap in the
$N\to\infty$ limit, and we find
\begin{equation}W({\Psi_\mathrm{Sch}},\mathcal{V})=
-\langle\!\langle cBG \Omega cG\rangle\!\rangle_\mathcal{V}.\end{equation}
With a few manipulations this can be rewritten,
\begin{equation}W({\Psi_\mathrm{Sch}},\mathcal{V})=
-2i\langle\!\langle c\gamma BG\Omega\rangle\!\rangle_\mathcal{V}.\end{equation}
From here on the derivation follows the steps given in
\eq{2step_ov}-\eq{2step_ovf}, with an extra minus sign, to yield
\begin{equation}W({\Psi_\mathrm{Sch}},\mathcal{V})=
\frac{1}{2}\langle\!\langle c \Omega\rangle\!\rangle_\mathcal{V}.\end{equation}
in agreement with the overlap for the simple half-brane solution, \eq{ov}.
\subsection{Energy}
Let us now calculate the energy. The energy can be computed from the
on-shell action:
\begin{equation}E = -S[\Psi] = -\frac{1}{6}\langle\!\langle \Psi Q\Psi\rangle\!\rangle.
\label{eq:energy}\end{equation}
We have only attempted this calculation for the simple half-brane
solution \eq{simple}, though the final answer should be the same
for any sufficiently regular half-brane solution. Plugging in ${\Psi_\mathrm{simp}}$
we find the expression
\begin{equation}E =
-\frac{1}{6}\left[-\left\langle\!\!\!\left\langle cGBc\frac{1}{1+K}Q(cGBc)\frac{1}{1+K}
\right\rangle\!\!\!\right\rangle
+\left\langle\!\!\!\left\langle cGBc\frac{G}{1+K}Q(cGBc)\frac{G}{1+K}\right\rangle\!\!\!\right\rangle\right].
\end{equation}
To simplify we eliminate the $G$ insertions by repeated
use of the identity
\begin{equation}\langle\!\langle G\Phi \rangle\!\rangle =\fraction{1}{2}\langle\!\langle \delta\Phi\rangle\!\rangle.
\label{eq:Gdelta}\end{equation}
The calculation is straightforward, but tedious; the repeated supersymmetry
variations generate dozens of terms. We give some details in appendix
\ref{app:energy}. In the end, all of the inner products can be evaluated
with the correlation function
\begin{equation}\left\langle Y_{-2}\,c(x_1)c(x_2)\gamma(y_1)\gamma(y_2)
\int_{-i\infty}^{i\infty}\frac{dz}{2\pi i}b(z)\right\rangle_{C_L}
=-\frac{L}{2\pi^2}(x_1-x_2)\cos\frac{\pi(y_1-y_2)}{L}
\label{eq:corr}\end{equation}
evaluated on a cylinder of circumference $L$. Adding
the terms up, the energy turns out to be
\begin{equation}E=-\frac{1}{4\pi^2},\end{equation}
which is precisely $-1/2$ times the tension of the D-brane. Remarkably, this
is consistent with the computation of the overlap.
\section{Concluding Remarks}
In this paper we have presented a new class of nonperturbative analytic
solutions of cubic superstring field theory on a non-BPS D-brane. The
nature of these solutions is fundamentally mysterious; they violate the reality
condition, and appear not to exist in Berkovits string field theory.
Probably they are only formal artifacts of the cubic equations of motion.
However, their existence is nontrivial and seems significant. We hope that
further study will shed light into what these solutions represent and why
they exist.
One immediate consequence of our analysis is that the cubic and Berkovits
equations of motion are not equivalent. The fact that half-brane solutions
appear to come in two topologically distinct varieties, and the existence of a
``tachyon vacuum'' on a BPS D-brane, suggest that the failure of this
equivalence is related to topological charge. A microscopic understanding
of D-brane charges is one of
the longstanding goals of string field theory. We hope that continued
development along these lines will give further insight.
\bigskip
\bigskip
\noindent {\bf Acknowledgments}
\bigskip
\noindent The author thanks M. Schnabl and C. Maccaferri for interesting
conversations, and M. Schnabl and Y. Okawa for critical reading of the
manuscript. The author also thanks C. Maccaferri
hospitality during a visit to ULB Brussells, and D. Gross for hospitality
at the Kavli Institute for Theoretical Physics. This research was supported
by the EURYI grant GACR EYI/07/E010 from EUROHORC and ESF.
\begin{appendix}
\section{Vertices, Reality and Twist conjugation}
\label{app:conventions}
In this appendix we discuss some important signs connected with the vertices
and the reality condition in the GSO($-$) sector. Our discussion extends the
classic analysis of Ohmori\cite{Ohmori} to our preferred ``left handed''
star product convention\cite{simple}.
\bigskip
\noindent{\bf Definition of vertices:} In the left handed convention,
the $N$-string vertex of open string
fields $\Phi_k = \sigma_{i_k}\phi_k(0)|0\rangle$ is defined as a correlator
on the upper half plane as follows:
\begin{equation}\langle\!\langle \Phi_1,\Phi_2,...,\Phi_N\rangle\!\rangle =
\frac{1}{2}\mathrm{tr}(\sigma_3\sigma_{i_1}\sigma_{i_2}...\sigma_{i_N})
\Big\langle Y_{-2}\ f_{1,N}\circ\phi_1(0)\,f_{2,N}\circ\phi_2(0)\,...
\,f_{N,N}\circ\phi_N(0)\Big\rangle,\label{eq:left_vertex}\end{equation}
where
\begin{equation}Y_{-2} = Y(i)\tilde{Y}(i),\ \ \
Y(z) = -\partial\xi e^{-2\phi}c(z), \end{equation}
and the conformal maps defining the vertex are,
\begin{equation}f_{k,N}(z) = -\cot\left[\frac{2}{N}\tan^{-1}z -\frac{\pi}{N}
\left(k-\frac{1}{2}\right)\right].\end{equation}
Let us define
\begin{equation}
x_{k,N} = \cot \frac{\pi}{N}\left(k-\frac{1}{2}\right),\ \ \ \ \
y_{k,N}=-\sqrt{\frac{2}{N}}\csc\frac{\pi}{N}\left(k-\frac{1}{2}\right).
\end{equation}
Then $f_{k,N}$ acts on a primary of weight $h$ explicitly as
\begin{equation}f_{k,N}\circ\phi(0)
= \left(y_{k,N}\right)^{(2h)} \phi(x_{k,N}),
\end{equation}
where the parentheses around $2h$ implies that $h$ must be multiplied by two
{\it before} the power of $y_{k,N}$ is taken. Note that
\begin{equation}x_{1,N} > x_{2,N} > ... >x_{N,N},\end{equation}
so the position of the vertex operator on the real axis {\it decreases} as we
increase the string label $k$ in the vertex. This is the hallmark of the left
handed star product convention.
By contrast, in the right handed convention the $N$-string
vertex would be defined as
\begin{equation}
\langle\!\langle \Phi_1,\Phi_2,...,\Phi_N\rangle\!\rangle_R =
\frac{1}{2}\mathrm{tr}(\sigma_3\sigma_{i_1}\sigma_{i_2}...\sigma_{i_N})
\Big\langle Y_{-2}\ \tilde{f}_{1,N}\circ\phi_1(0)\,
\tilde{f}_{2,N}\circ\phi_2(0)\,...
\,\tilde{f}_{N,N}\circ\phi_N(0)\Big\rangle,\label{eq:right_vertex}
\end{equation}
where the superscript $R$ reminds us that the vertex is defined in the right
handed convention. The conformal maps $\tilde{f}_{k,N}$ are related to
$f_{k,N}$ simply as,
\begin{equation}\tilde{f}_{k,N}(z) = f_{N+1-k,N}(z).\end{equation}
The positions of the vertex operators on the real axis are
\begin{equation}\tilde{x}_{k,N} = x_{N+1-k,N},\end{equation}
so the position {\it increases} as we increase the string label. This is
the hallmark of the right handed star product convention.
The vertices \eq{left_vertex} and \eq{right_vertex} implicitly define the
open string star product. The left handed star product $\Psi\Phi$ and right
handed star products $[\Psi\Phi]_R$ are related by the equation,
\begin{equation}[\Psi\Phi]_R = (-1)^{E(\Psi)E(\Phi)+F(\Psi)F(\Phi)}\Phi\Psi.
\end{equation}
The sign appears from anticommuting vertex operators and internal CP factors.
Let us consider the 2-string vertex. Following \cite{Ohmori}, we define
the action of the BPZ conformal map $I(z) = -\frac{1}{z}$ on a primary of
weight $h$
\begin{equation}I\circ\phi(z) = \phi(z)^\star
= \frac{1}{z^{(2h)}}\phi\left(-\frac{1}{z}\right).\end{equation}
By an $SL(2,\mathbb{R})$ transformation, one can then show that the
2-string vertex in the left and right handed conventions is given by,
\begin{eqnarray}\langle\!\langle \Phi_1,\Phi_2\rangle\!\rangle \!\!\!\!\!\!\!\!&& =
\frac{1}{2}\mathrm{tr}(\sigma_3\sigma_{i_1}\sigma_{i_2})
\Big\langle Y_{-2}\ \phi_1(0)\,\phi_2(0)^\star\Big\rangle,\nonumber\\
\langle\!\langle \Phi_1,\Phi_2\rangle\!\rangle_R \!\!\!\!\!\!\!\!&& =
\frac{1}{2}\mathrm{tr}(\sigma_3\sigma_{i_1}\sigma_{i_2})
\Big\langle Y_{-2}\ \phi_1(0)^\star\,\phi_2(0)\Big\rangle.\end{eqnarray}
Now transform the left handed vertex with $I(z)$ and note
\begin{equation}^{\star\star}=(-1)^F.\end{equation}
Then
\begin{equation}\langle\!\langle \Phi_1,\Phi_2\rangle\!\rangle=
(-1)^F\langle\!\langle \Phi_1,\Phi_2\rangle\!\rangle_R.
\end{equation}
So the 2-vertex differs by a sign in the GSO($-$) sector between the two
star product conventions. This sign plays an important role in fixing the
string field reality condition.
\bigskip
\noindent{\bf Reality conjugation:} To formulate the string field reality
condition, we need to define reality conjugation. To do this it is
helpful to express the string field in the operator formalism.
We can define Hermitian and BPZ conjugation of a state $|\Psi\rangle$ or dual
state $\langle \Psi|$ using the following rules:
\begin{eqnarray}\Big[\,\mathcal{O}|0\rangle\,\Big]^\dag \!\!\!\!\!\!\!\!&&
= \langle 0|\mathcal{O}^\dag,\ \ \ \ \ \
\Big[\,\langle 0|\mathcal{O}\,\Big]^\dag =\mathcal{O}^\dag|0\rangle,
\nonumber\\
\Big[\,\mathcal{O}|0\rangle\,\Big]^\star \!\!\!\!\!\!\!\!&&
= \langle 0|\mathcal{O}^\star,\ \ \ \ \ \
\Big[\,\langle 0|\mathcal{O}\,\Big]^\star =\mathcal{O}^\star|0\rangle,
\end{eqnarray}
and\footnote{We use the six pointed
star $^*$ to denote complex conjugation and the five pointed star $^\star$ to
denote BPZ conjugation, hopefully without confusion.}
\begin{eqnarray}
(\mathcal{O}_1\mathcal{O}_2)^\dag \!\!\!\!\!\!\!\!&& =
\mathcal{O}_2^\dag\mathcal{O}_1^\dag,\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\ \ \,
(\mathcal{O}_1\mathcal{O}_2)^\star =
(-1)^{\mathcal{O}_1\mathcal{O}_2}\mathcal{O}_2^\star\mathcal{O}_1^\star,
\nonumber\\
(a\mathcal{O}_1 + b\mathcal{O}_2)^\dag \!\!\!\!\!\!\!\!&& = a^*\mathcal{O}_1^\dag +
b^*\mathcal{O}_2^\dag\nonumber,
\ \ \ \ \ \ (a\mathcal{O}_1 + b\mathcal{O}_2)^\star = a\mathcal{O}_1^\star +
b\mathcal{O}_2^\star,\ \ \ \ \ \ \ \ a,b\in\mathbb{C}
\end{eqnarray}
and
\begin{equation}\phi(z)^\dag = \frac{(-1)^\nu}{\bar{z}^{(2h)}}
\phi\left(\frac{1}{\bar{z}}\right),\ \ \ \ \phi(z)^\star = \frac{1}{z^{(2h)}}
\phi\left(-\frac{1}{z}\right).\end{equation}
Here $\mathcal{O}$ are general CFT operators, $a,b$ are complex constants
and $\phi(z)$ is a primary of weight $h$. The sign $(-1)^\nu$ is needed to
distinguish between Hermitian and antihermitian fields. We will take the
$\beta$ ghost to be antihermitian, so that the $\gamma$ ghost
and the worldsheet supercurrent are Hermitian. Also, we define BPZ conjugation
to leave internal CP factors invariant, whereas Hermitian conjugation takes
their conjugate transpose. With these definitions we define
{\it reality conjugation} of a string field $\Psi$ as
\begin{equation}\Psi^\ddag = \Psi^{\dag\star}.
\label{eq:real_conj}\end{equation}
Note that the order matters: first we perform Hermitian and {\it then} BPZ
conjugation to compute the reality conjugate. In particular,
\begin{equation}^{\dag\star} =\, ^{\star\dag}(-1)^F.\end{equation}
Since $^{\star\star}=(-1)^F$ this implies that reality conjugation is
idempotent,
\begin{equation}^{\ddag\ddag} = 1,\end{equation}
and therefore analogous to complex conjugation. Reality conjugation
satisfies the important properties
\begin{equation}
(\Psi\Phi)^\ddag = \Phi^\ddag\Psi^\ddag,\ \ \ \ \ \ \
(a\Psi+b\Phi)^\ddag = a^*\Psi^\ddag+b^*\Phi^\ddag,\ \ \ a,b\in\mathbb{C},
\label{eq:real_conj_prop1}\end{equation}
and
\begin{equation}\langle\!\langle \Psi\rangle\!\rangle^* =
\langle\!\langle\Psi^\ddag\rangle\!\rangle.\label{eq:real_conj_prop2}\end{equation}
Equations \eq{real_conj}-\eq{real_conj_prop2} hold with the appropriate
CP factors attached to the string field.
\bigskip
\noindent {\bf Twist conjugation:}
We define {\it twist conjugation}
\begin{equation}\Psi^\S = e^{i\pi N}\Psi, \end{equation}
where $N$ is the number operator (the zero momentum component of $L_0$).
For the bosonic string, twist conjugation is related to the
twist operator $\Omega$ of \cite{Zwiebach,Review} by a
sign:
\begin{equation}\Psi^\S = -\Omega\Psi.\end{equation}
Twist conjugation satisfies
\begin{eqnarray}(\Psi\Phi)^\S \!\!\!\!\!\!\!\!&&
= (-1)^{E(\Psi)E(\Phi)+F(\Psi)F(\Phi)}\Phi^\S
\Psi^\S,\nonumber\\
(a\Psi+b\Phi)^\S \!\!\!\!\!\!\!\!&&
= a\Psi^\S+b\Phi^\S\ \ \ \ a,b\in \mathbb{C},\nonumber\\
^{\S\S}\!\!\!\!\!\!\!\!&& = (-1)^F.\end{eqnarray}
and
\begin{eqnarray}\langle\!\langle\Psi^\S\rangle\!\rangle = \langle\!\langle\Psi
\rangle\!\rangle.\end{eqnarray}
Twist conjugation gives a way to map between string fields in the left and
right-handed star product conventions\cite{simple}. Suppose $\Phi$ is a
string field in a theory with left handed star product. The equivalent
field in the right handed theory is
\begin{equation}\Phi'=\Phi^\S.\label{eq:lr_map}\end{equation}
In the right handed convention, the string field reality
condition is\cite{Ohmori}:
\begin{equation}(\Psi')^\ddag=(-1)^F\Psi'.\end{equation}
Using \eq{lr_map} it follows that the reality condition for the left handed
theory is
\begin{equation}\Psi^\ddag=\Psi.\end{equation}
Note that this means that real string fields in the two conventions differ by
a factor of $i$ in the GSO($-$) sector. This factor of $i$ corrects the sign
discrepancy between the 2-vertices, so the tachyon field has the correct sign
kinetic term in either convention.
\section{Superconformal Generator in the Sliver Frame}
\label{app:G}
The string field $G$ is closely related to the superconformal generator
$G_{-1/2}$ in the sliver conformal frame:
\begin{equation}\mathcal{G} = f_S^{-1}\circ G_{-1/2} =
\oint \frac{d\xi}{2\pi i}\left(\sqrt{\frac{\pi}{2}}\sqrt{1+\xi^2}\right)
G(\xi).\end{equation}
Since this operator is crucial to the construction of the half-brane
solution, it is worth understanding in more detail.
Consider an operator of the form
\begin{equation}{\bm \phi}[f]=\oint \frac{d\xi}{2\pi i}f(\xi)\phi(\xi),
\end{equation}
where $\phi$ is a primary of weight $h$, $f(\xi)$ is a function,
and the contour passes inside an annulus of analyticity of $f(\xi)$ around
the unit circle. We define Hermitian, BPZ, and dual conjugation\cite{RZ}
of this operator, respectively,
\begin{eqnarray}
{\bm \phi}[f]^\dag \!\!\!\!\!\!\!\!&& = {\bm \phi}[f^\dag],\ \ \ \ \ \ \ \ \ \ \
f^\dag(\xi) = (-1)^\nu\xi^{(2h-2)}f^*\left(\xi^{-1}\right),\nonumber\\
{\bm \phi}[f]^\star \!\!\!\!\!\!\!\!&& = {\bm \phi}[f^\star],
\ \ \ \ \ \ \ \ \ \ \,
f^\star(\xi) = -(-\xi)^{(2h-2)}f\left(-\xi^{-1}\right),\nonumber\\
{\widetilde{\bm \phi}}[f]\ \!\!\!\!\!\!\!\!&&= {\bm \phi}[\tilde{f}],
\ \ \ \ \ \ \ \ \ \ \ \ {\tilde f}(\xi)\ = \epsilon(\xi)f(\xi).
\end{eqnarray}
Here $\epsilon(\xi)$ is the step function\footnote{$\epsilon(\xi)$ has a
branch cut extending across the
entire imaginary axis. To define dual conjugation carefully, one should
represent $\epsilon(\xi)$ as the limit of a sequence of functions which are
analytic in some (vanishingly thin) annulus containing the unit
circle\cite{RZ}.}
\begin{equation}\epsilon(\xi) = \left\{ { 1\ \mathrm{for}\ \mathrm{Re}(\xi)>0
\atop -1\ \mathrm{for}\ \mathrm{Re}(\xi)<0}\right..\end{equation}
We also define the combinations
\begin{eqnarray}
{\bm \phi}[f]^+ \!\!\!\!\!\!\!\!&& = {\bm \phi}[f]+{\bm \phi}[f]^\star,
\ \ \ \ \,\ \ \ \ \
{\bm \phi}[f]^-={\bm \phi}[f]-{\bm \phi}[f]^\star,\nonumber\\
{\bm \phi}[f]_L \!\!\!\!\!\!\!\!&& = \frac{1}{2}\Big({\bm \phi}[f]
+{\widetilde {\bm \phi}}[f]\Big),\ \ \ \ \
{\bm \phi}[f]_R=\frac{1}{2}\Big({\bm \phi}[f]-{\widetilde {\bm \phi}}[f]
\Big).\end{eqnarray}
The subscripts $L$ and $R$ denote the left and right halves of the
charge ${\bm \phi}[f]$. In some cases, the action of ${\bm \phi}[f]_L$ and
${\bm \phi}[f]_R$ on a state can be described by left or right
star multiplication with the appropriate string field. When this is possible,
we say that ${\bm \phi}[f]$ has a {\it non-anomalous} left/right decomposition.
Consider the operators
\begin{eqnarray}\mathcal{L} \!\!\!\!\!\!\!\!&&
= {\bf T}[\ell],\ \ \ \ \ \ \
\ell(\xi)=(1+\xi^2)\tan^{-1}\xi ,\nonumber\\
\mathcal{G} \!\!\!\!\!\!\!\!&& = {\bf G}[g],
\ \ \ \ \ \ \ g(\xi) = \sqrt{\frac{\pi}{2}}\sqrt{1+\xi^2}.
\end{eqnarray}
where
\begin{equation}{\bf T}[v]=\oint \frac{d\xi}{2\pi i}v(\xi)T(\xi),\ \ \ \ \ \
{\bf G}[s] = \oint \frac{d\xi}{2\pi i} s(\xi)G(\xi).\end{equation}
The first is the familiar $\mathcal{L}_0$ of Schnabl\cite{Schnabl}, and the
second is the operator $G_{-1/2}$ in the sliver conformal frame. The
functions $\ell(\xi)$ and $g(\xi)$ have branch points at $+i$
and $-i$, connected by a branch cut on the imaginary axis passing through
infinity. The branch points of $\ell(\xi)$ takes the form $x\ln x$ for small
$x=\xi\pm i$, whereas those of $g(\xi)$ take the form $\sqrt{x}$. The BPZ
conjugate operators are
\begin{eqnarray}
\mathcal{L}^\star \!\!\!\!\!\!\!\!&& = {\bf T}[\ell^\star],\ \ \ \ \ \ \ell^\star(\xi)=
(1+\xi^2)\tan^{-1}\frac{1}{\xi},\nonumber\\
\mathcal{G}^\star \!\!\!\!\!\!\!\!&& = {\bf G}[g^\star],\ \ \ \ \ \ g^\star(\xi)^=
\sqrt{\frac{\pi}{2}}\xi\sqrt{1+\frac{1}{\xi^2}}.\label{eq:LsGs}
\end{eqnarray}
$\ell^\star,g^\star$ also have branch points at $\pm i$, but the cuts now
extend on the imaginary axis through the origin. Note that by factoring
$\xi$ into the square root in \eq{LsGs}, $g^\star$ formally appears to be the
same as $g$. In fact, they are equal up to a sign:
\begin{equation}g^\star(\xi)=\epsilon(\xi)g(\xi).\end{equation}
This means that the BPZ conjugate of $\mathcal{G}$ is equal to
its dual conjugate:
\begin{equation}\mathcal{G}^\star=\widetilde{\mathcal{G}}.
\label{eq:BPZ_dual}
\end{equation}
It is also useful to consider the operators
\begin{eqnarray}
\mathcal{L}^+ \!\!\!\!\!\!\!\!&& = \mathcal{L}+\mathcal{L}^\star={\bf T}[\ell^+],
\ \ \ \ \ \ \ell^+(\xi)=
\frac{\pi}{2}(1+\xi^2)\epsilon(\xi),\nonumber\\
\tilde{\mathcal{L}}^+ \!\!\!\!\!\!\!\!&& = {\bf T}[\tilde{\ell}^+],\ \ \ \ \ \ \ \ \ \ \
\ \ \ \ \ \ \ \ \tilde{\ell}^+(\xi)=\frac{\pi}{2}(1+\xi^2).
\end{eqnarray}
The Hermitian conjugates of $\mathcal{L},\mathcal{L}^*$ and $\mathcal{G}$
are equal to their BPZ conjugates. For $\mathcal{G}^\star$ and
$\tilde{\mathcal{L}}^+$ there is a sign difference:
$\mathcal{G}^{\star\dag} = \mathcal{G} =
-\mathcal{G}^{\star\star}$ and $\tilde{\mathcal{L}}^{+\dag}
=\tilde{\mathcal{L}}^+=-\tilde{\mathcal{L}}^{+\star}$.
The string fields $K$ and $G$ can be defined through the action of
$\mathcal{L}^+,\mathcal{G}$, and their dual conjugates on a test state:
\begin{eqnarray}\mathcal{L}^+\Phi \!\!\!\!\!\!\!\!&& = K\Phi+\Phi K,
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\widetilde{\mathcal{L}}^+\Phi = K\Phi-\Phi K = \partial\Phi,\nonumber\\
\sigma_1\mathcal{G}^\star\Phi\!\!\!\!\!\!\!\!&& = G\Phi + (-1)^{F(\Phi)}\Phi G,\ \ \ \,
\ \ \ \ \
\sigma_1\mathcal{G}\Phi
= G\Phi-(-1)^{F(\Phi)}\Phi G = \delta\Phi.\label{eq:KGop}
\end{eqnarray}
This definition implies three consistency conditions:
\begin{eqnarray}
\!\!\!\!\!\!\!\!&& {\bf 1)}\ \ \tilde{\mathcal{L}}^+|I\rangle = 0
\ \ \ \ \mathcal{G}|I\rangle=0\nonumber\\
\!\!\!\!\!\!\!\!&& {\bf 2)}\ \ \mathcal{L}^+_L (\Phi\Psi) = (\mathcal{L}^+_L\Phi)\Psi
\ \ \ \ \mathcal{L}^+_R (\Phi\Psi) = \Phi(\mathcal{L}^+_R\Psi)\nonumber\\
\!\!\!\!\!\!\!\!&& \ \ \ \ \ \mathcal{G}_L (\Phi\Psi) = (\mathcal{G}_L\Phi)\Psi
\ \ \ \ \ \,
\mathcal{G}_R (\Phi\Psi) = (-1)^{\epsilon(\Phi)}\Phi(\mathcal{G}_R\Psi)
\nonumber\\
\!\!\!\!\!\!\!\!&& {\bf 3)}\ \ \{\mathcal{G}_L,\mathcal{G}_R\}=0,\ \ \ \ \ \
[\mathcal{G}_L,\mathcal{L}^+_R]=0,\ \ \ \ \ \
[\mathcal{L}^+_L,\mathcal{L}^+_R]=0
\end{eqnarray}
The first condition follows from setting $\Phi=|I\rangle$ in \eq{KGop}.
The second follows from associativity of the star product. The third
condition follows from the assumption that $\mathcal{L}^+,\mathcal{G}$,
and their dual conjugates should have well defined action on $K$ and $G$;
in other words, $K$ and $G$ can be consistently star multiplied
among themselves. If these three conditions are satisfied, $\mathcal{L}^+$ and
$\mathcal{G}$ have a non-anomalous left/right decomposition.
It is not difficult to verify conditions {\bf 1)} and {\bf 2)} by contracting
with ``reasonable'' test states (for example, wedge states of positive width
with insertions placed away from the midpoint) and mapping to the upper half
plane. Condition {\bf 3)} is more subtle and is worth checking explicitly.
We can compute the commutators using the superconformal algebra expressed
in the form
\begin{eqnarray}\Big[{\bf T}[v_1],{\bf T}[v_2]\Big]\, \!\!\!\!\!\!\!\!&& =\,
{\bf T}[v_2\partial v_1-v_1\partial v_2],\nonumber\\
\Big[{\bf T}[v],{\bf G}[s]\Big]\ \, \!\!\!\!\!\!\!\!&& =\, {\bf G}[\fraction{1}{2} s\partial v
- v\partial s],\nonumber\\
\Big\{{\bf G}[s_1],{\bf G}[s_2]\Big\}\, \!\!\!\!\!\!\!\!&& =\,
{\bf T}[2s_1s_2].\label{eq:superconf_alg}
\end{eqnarray}
To be careful about singularities at the midpoint, we regulate
$\mathcal{G}$ and $\mathcal{L}^+$ by replacing
\begin{eqnarray} g(\xi)\!\!\!\!\!\!\!\!&& \rightarrow g(\lambda\xi),\ \ \ \ \ \ \,
g(\xi)^\star\rightarrow g(\xi/\lambda)^\star, \nonumber\\
\ell(\xi)\!\!\!\!\!\!\!\!&& \rightarrow \ell(\lambda\xi),\ \ \ \ \ \ \
\ell(\xi)^\star\rightarrow \ell(\xi/\lambda)^\star.
\end{eqnarray}
Condition {\bf 3)} is satisfied in the limit $\lambda\to 1^-$.
Another check is to expand the operators in modes
\begin{eqnarray}
\mathcal{L}\ \!\!\!\!\!\!\!\!&& =L_0+\frac{2}{3}L_2-\frac{2}{15}L_4+
\ \ \ \ \ \ \ \ .\, .\, .\ \ \ \ \ \ \ \ \
= L_0+2\sum_{n=1}^\infty \frac{(-1)^{n+1}}{4n^2-1}L_{2n},
\nonumber\\
\mathcal{L}^\star\, \!\!\!\!\!\!\!\!&&
=L_0+\frac{2}{3}L_{-2}-\frac{2}{15}L_{-4}+
\ \ \ \ \ \ .\, .\, .\ \ \ \ \ \ \ \
= L_0+2\sum_{n=1}^\infty \frac{(-1)^{n+1}}{4n^2-1}L_{-2n},\nonumber\\
\widetilde{\mathcal{L}}^+ \!\!\!\!\!\!\!\!&& = \frac{\pi}{2}(L_1+L_{-1}),\nonumber\\
\mathcal{G}\ \!\!\!\!\!\!\!\!&& = \sqrt{\frac{\pi}{2}}\left(G_{-1/2}
+\frac{1}{2}G_{3/2}-\frac{1}{8}G_{7/2}+...\right)\ \,
= \sqrt{\frac{\pi}{2}}\sum_{n=0}^\infty
{1/2 \choose n} G_{2n-\frac{1}{2}}, \nonumber\\
\mathcal{G}^\star \!\!\!\!\!\!\!\!&& = \sqrt{\frac{\pi}{2}}\left(G_{1/2}
+\frac{1}{2}G_{-3/2}-\frac{1}{8}G_{-7/2}+...\right)
= \sqrt{\frac{\pi}{2}}\sum_{n=0}^\infty
{1/2 \choose n} G_{\frac{1}{2}-2n},\end{eqnarray}
and calculate using the usual mode commutators of the
superconformal algebra. Again we have found that {\bf 3)} is
satisfied, and the infinite sums needed in the
computation are absolutely convergent.\footnote{One further subtlety is that
the vanishing of left/right commutators is not always sufficient to
guarantee that nonpolynomial combinations of left and right charges commute.
Splitting $\mathcal{L}$ into left and right halves we find
$[\mathcal{L}_L,\mathcal{L}_R]=0$, but $\mathcal{L}_L$ actually
does not commute with $e^{-s\mathcal{L}_R}$. This is crucial for recovering
closed string moduli in Schnabl gauge amplitudes\cite{KZ,boundary}. We have
found no evidence for similar anomalies when splitting
$\mathcal{L}^+$ and $\mathcal{G}$.} Given these and other checks, we believe
that $\mathcal{L}^+,\mathcal{G}$ have a non-anomalous left/right
decomposition.
The operators $\mathcal{L},\mathcal{L}^\star,\tilde{\mathcal{L}}^+$ and
$\mathcal{G},\mathcal{G}^\star$ form a super-Lie algebra with commutators,
\begin{eqnarray}[\mathcal{L},\mathcal{L}^\star]\!\!\!\!\!\!\!\!&& =\mathcal{L}^+,
\ \ \ \ \ \ \ \
[\mathcal{L},\widetilde{\mathcal{L}}^+]=\widetilde{\mathcal{L}}^+,
\ \ \ \ \ \ \ \ \ \
[\mathcal{L}^\star,\widetilde{\mathcal{L}}^+]=-\widetilde{\mathcal{L}}^+,
\nonumber\\
\ \{\mathcal{G},\mathcal{G}^\star\}\!\!\!\!\!\!\!\!&& =2\mathcal{L}^+,
\ \ \ \ \ \ \
\{\mathcal{G},\mathcal{G}\}=2\widetilde{\mathcal{L}}^+,
\ \ \ \ \ \ \ \
\{\mathcal{G}^\star,\mathcal{G}^\star\}=2\widetilde{\mathcal{L}}^+,
\nonumber\\
\ [\mathcal{L},\mathcal{G}]\!\!\!\!\!\!\!\!&& =\fraction{1}{2} \mathcal{G},
\ \ \ \ \ \ \ \ \
[\mathcal{L}^\star,\mathcal{G}] =-\fraction{1}{2}\mathcal{G},
\ \ \ \ \ \ \ \
\ [\widetilde{\mathcal{L}}^+,\mathcal{G}] =0,
\nonumber\\
\ [\mathcal{L},\mathcal{G}^\star]\!\!\!\!\!\!\!\!&& =\fraction{1}{2}\mathcal{G}^\star,
\ \ \ \ \ \ \,
[\mathcal{L}^\star,\mathcal{G}^\star]=-\fraction{1}{2}\mathcal{G}^\star,
\ \ \ \ \ \ \
[\widetilde{\mathcal{L}}^+,\mathcal{G}^\star]=0.
\end{eqnarray}
This can be thought of as a supersymmetric extension of the
special projector algebra\cite{Schnabl,RZ}. Assuming
\eq{KGop}, this algebra can be compactly summarized by the relations
\begin{equation}G^2=K\ \ \ \ \ [K,G]=0\ \ \ \ \ \fraction{1}{2}\mathcal{L}^-K=K
\ \ \ \ \
\fraction{1}{2}\mathcal{L}^-G = \frac{1}{2}G\end{equation}
\section{Splitting Charges and Midpoint Insertions}
When computing the action and gauge invariant overlap, we implicitly assumed
cyclicity of the vertices $\langle\!\langle\cdot\rangle\!\rangle$ and
$\langle\!\langle\cdot\rangle\!\rangle_\mathcal{V}$. However, the presence of midpoint
insertions makes this subtle. In the $K,B,c,G$ subalgebra, cyclicity of
$\langle\!\langle\cdot\rangle\!\rangle$ and $\langle\!\langle\cdot\rangle\!\rangle_\mathcal{V}$ requires
\begin{equation}
[K,Y_{-2}]=0,\ \ \ \ \ \ \ [K,\mathcal{V}]=0,\end{equation}
and likewise for $B$ and $G$. (The cyclicity of $c$ appears
unproblematic since the $c$ insertion is far from the midpoint.)
While the geometry of the Witten vertex appears to guarantee that midpoint
insertions commute, this expectation fails in at least some
examples\cite{Horowitz}.\footnote{A
related question is whether the derivations $\mathcal{L}^-$ and
$\mathcal{B}^-$ annihilate $\langle\!\langle\cdot\rangle\!\rangle$ and
$\langle\!\langle\cdot\rangle\!\rangle_\mathcal{V}$. This can be shown along similar lines
to the argument presented here.}
To keep the discussion general, consider a string field $\Phi$
corresponding to a vertical line integral insertion of a primary
$\phi(z)$ of weight $h>0$ in the cylinder coordinate frame:
\begin{equation}\Phi\ \rightarrow\
\int_{-i\infty}^{i\infty}\frac{dz}{2\pi i}\phi(z).\end{equation}
Explicitly we can write
\begin{equation}\Phi = {\bm \phi}_L|I\rangle,\ \ \ \mathrm{where}\ \ \ \
{\bm\phi}_L = \int_L \frac{d\xi}{2\pi i}\left(\sqrt{\frac{\pi}{2}}
\sqrt{1+\xi^2}\right)^{2(h-1)}\phi(\xi),\end{equation}
and the contour $L$ is over the positive half of the unit circle. Let
$m=m(i)|I\rangle$ correspond to an insertion of a dimension zero primary $m(z)$
at the midpoint. Then we can show that $[\Phi,m]=0$ if and only if
\begin{equation}\lim_{\sigma\to\frac{\pi}{2}}[\,{\bm\phi}_L\,,
\,m(e^{i\sigma})\,]=0.
\label{eq:vanish_com}\end{equation}
Suppose $\phi(z)$ and $m(z)$ have an OPE of the form,
\begin{equation}\phi(z+w)m(z)\sim \sum_{n=1}^\infty \frac{1}{w^n}V_n(z),
\end{equation}
where $V_n(z)$ are local operators of dimension $h-n$. Computing the
commutator \eq{vanish_com} we can prove the following:
\begin{claim2} The limit of the commutator
\eq{vanish_com} vanishes if and only if one of the two following criteria are
satisfied:
\begin{description}
\item{\bf a)} If $h\in \mathbb{Z}+\frac{1}{2}$, then $V_n(z)=0$ for all $n>h$.
\item{\bf b)} If $h\in \mathbb{Z}$, then $V_n(z)=0$ for all $n$ in the range
$2h> n\geq h$.
\end{description}
\end{claim2}
\noindent In the current context, the role of $\Phi$ is played by $K,B$, and
$G$ and the role of $m$ is played by $Y_{-2}$ and $\mathcal{V}$. According
to the above claim, $K,B$ and $G$ commute with $Y_{-2}$ and $\mathcal{V}$
if and only if the OPEs between $T,G,b$, and $Y_{-2},\mathcal{V}$ take
the following form:
\begin{eqnarray}T(z+w)Y_{-2}(z,\bar{z})\!\!\!\!\!\!\!\!&&\sim\mathcal{O}(w^{-1}),
\ \ \ \ \ \
T(z+w)\mathcal{V}(z,\bar{z})\sim\mathcal{O}(w^{-1}),\nonumber\\
G(z+w)Y_{-2}(z,\bar{z})\!\!\!\!\!\!\!\!&&\sim\mathcal{O}(w^{-1}),
\ \ \ \ \ \
G(z+w)\mathcal{V}(z,\bar{z})\sim\mathcal{O}(w^{-1}),\nonumber\\
b(z+w)Y_{-2}(z,\bar{z})\!\!\!\!\!\!\!\!&&\sim\mathcal{O}(w^{-1}),
\ \ \ \ \ \
b(z+w)\mathcal{V}(z,\bar{z})\sim\mathcal{O}(w^{-1}),
\end{eqnarray}
and likewise for the antiholomorphic currents $\tilde{T},\tilde{G},\tilde{b}$.
Let us assume that $Y_{-2}$ and $\mathcal{V}$ take the explicit forms given in
\eq{Ym2} and \eq{V}. The OPEs with $T$ follow from the fact that $Y_{-2}$
and $\mathcal{V}$ are dimension $(0,0)$ primaries. The OPEs with $G$
follow from the fact that
$Y_{-2}$ and $\mathcal{V}$ are superconformal primaries. Finally the OPEs
with $b$ follow from the fact that in the $bc$ CFT $Y_{-2}$ and $\mathcal{V}$
are proportional to $c\tilde{c}$, which produces only a single pole in the
OPE with $b$. Therefore the vertices $\langle\!\langle\cdot\rangle\!\rangle$ and
$\langle\!\langle\cdot\rangle\!\rangle_\mathcal{V}$ are expected to be cyclic when evaluated
on fields in the $K,B,c,G$ subalgebra.
\section{Phantom Piece and Super-Wedge States}
\label{app:largeN}
In this appendix we prove that the phantom term \eq{phantom} can be described
by a super-wedge state \eq{ext_wedge} in the large $N$ limit. First we give
an explicit definition of super-wedge states in the Fock space. Write
\begin{equation}e^{-\alpha K}e^{i\beta G} = f_1(\alpha,\beta)+iG\,
f_2(\alpha,\beta),
\end{equation}
where
\begin{eqnarray}
f_1(\alpha,\beta)\!\!\!\!\!\!\!\!&& =\Omega^\alpha\cos(\beta\sqrt{K}),\nonumber\\
f_2(\alpha,\beta)\!\!\!\!\!\!\!\!&& =\Omega^\alpha\frac{\sin(\beta\sqrt{K})}{\sqrt{K}}.
\end{eqnarray}
We can compute the Fock space coefficients of $(f_1,f_2)$ using the
linear functional \eq{gen_f}:
\begin{eqnarray}L_{f_1}(x^h)\!\!\!\!\!\!\!\!&&
=\frac{1}{(h-1)!}\int_0^\infty dt\, t^{h-1}\cos(\beta\sqrt{t})e^{-(\alpha+1)t}
\nonumber\\
\!\!\!\!\!\!\!\!&& = \frac{1}{(1+\alpha)^h}
\,_1 F_1\left[h,\frac{1}{2},-\frac{\beta^2}{4(1+\alpha)}\right],
\label{eq:f1_fock}\\
L_{f_2}(x^h)\!\!\!\!\!\!\!\!&&
=\frac{1}{(h-1)!}\int_0^\infty dt\, t^{h-1}\frac{\sin(\beta\sqrt{t})}{\sqrt{t}}
e^{-(\alpha+1)t}\nonumber\\
\!\!\!\!\!\!\!\!&& = \frac{\beta}{(1+\alpha)^h}
\,_1 F_1\left[h,\frac{3}{2},-\frac{\beta^2}{4(1+\alpha)}\right],
\label{eq:f2_fock}
\end{eqnarray}
where $_1 F_1$ is the confluent hypergeometric function.
Consider the states $(X_N,Y_N)$ appearing in the phantom piece through
equation \eq{XY}. We can also define these states using the linear functional
\eq{gen_f}:
\begin{eqnarray}L_{X_N}(x^h)\!\!\!\!\!\!\!\!&& =
\frac{1}{(h-1)!}\int_0^\infty dt\, t^{h-1}
\frac{(1+ia\sqrt{t})^N+(1-ia\sqrt{t})^N}{2}e^{-(N+1)t},\nonumber\\
L_{Y_N}(x^h)\!\!\!\!\!\!\!\!&& = \frac{1}{(h-1)!}\int_0^\infty dt\,t^{h-1}
\frac{(1+ia\sqrt{t})^N-(1-ia\sqrt{t})^N}{2i aN\sqrt{t}}e^{-(N+1)t}.
\label{eq:XY_linfunc}\end{eqnarray}
To compute the large $N$ limit, substitute $s=(N+1)t$ in the integrand so that
for example
\begin{equation}L_{X_N}(x^h) = \frac{1}{2(h-1)!}\frac{1}{(N+1)^h}
\int_0^\infty ds\, s^{h-1}
\left[\left(1+ia\sqrt{\frac{s}{N+1}}\right)^N+\left(1-ia\sqrt{\frac{s}{N+1}}
\right)^N\right]e^{-s}.\end{equation}
Now approximate
\begin{eqnarray}\left(1\pm ia\sqrt{\frac{s}{N+1}}\right)^N\!\!\!\!\!\!\!\!&& =
\exp\left[N\ln\left(1\pm ia\sqrt{\frac{s}{N+1}}\right)\right]\nonumber\\
\!\!\!\!\!\!\!\!&& = \exp\left[N\left(\pm ia\sqrt{\frac{s}{N}}+\frac{a^2}{2}\frac{s}{N}
+\mathcal{O}(N^{-1/2})...\right)\right]\nonumber\\
\!\!\!\!\!\!\!\!&& = e^{\pm ia\sqrt{Ns}}e^{a^2s/2}[1+\mathcal{O}(N^{-1/2})]
,\end{eqnarray}
so that
\begin{eqnarray}L_{X_N}(x^h)\!\!\!\!\!\!\!\!&& = \frac{1}{(h-1)!}\frac{1}{N^h}
\int_0^\infty ds\, s^{h-1} \cos(a\sqrt{Ns}) e^{-(1-\frac{a^2}{2})s}
[1+\mathcal{O}(N^{-1/2})]\nonumber\\
\!\!\!\!\!\!\!\!&& = \frac{1}{N^h}\left(\frac{2}{2-a^2}\right)^h\,
_1 F_1\left(h,\frac{1}{2},-\frac{1}{4}\frac{2}{2-a^2}a^2 N\right)
[1+\mathcal{O}(N^{-1/2})].
\end{eqnarray}
Similarly,
\begin{equation}L_{Y_N}(x^h) = \frac{1}{N^{h}}\left(\frac{2}{2-a^2}\right)^h
\,_1F_1\left(h,\frac{3}{2},-\frac{1}{4}\frac{2}{2-a^2}a^2N\right)
[1+\mathcal{O}(N^{-1/2})].
\end{equation}
Comparing with equations \eq{f1_fock} and \eq{f2_fock}, this precisely
corresponds to the large $N$ behavior of the super-wedge state
$e^{iN a G}\Omega^{N(1-\frac{a^2}{2})}$, as claimed in equation \eq{Gwedge}.
To simplify the large $N$ limit further we use the asymptotic formula
\begin{equation}_1 F_1(a,b,z)=\frac{\Gamma(a)}{\Gamma(b-a)}e^{i\pi a}
\frac{1}{z^a}[1+\mathcal{O}(z^{-1})],\ \ \ \ \ (\mathrm{large}\ |z|,\ \
\mathrm{Re}(z)<0)\label{eq:hyper_ass}.\end{equation}
Thus,
\begin{eqnarray}L_{X_N}(x^h)\!\!\!\!\!\!\!\!&& =
\frac{2(-1)^h}{(aN)^{2h}}\frac{(2h-1)!}{(h-1)!}
[1+\mathcal{O}(N^{-1/2})],\\
L_{Y_N}(x^h) \!\!\!\!\!\!\!\!&& = \frac{4(-1)^h}{(aN)^{2h}}\frac{(2h-3)!}{(h-2)!}
[1+\mathcal{O}(N^{-1/2})].
\end{eqnarray}
This agrees with the large $N$ behavior of the sums quoted in
\eq{reg_limits}. We have verified this behavior numerically.
\section{Details of Energy Computation}
\label{app:energy}
In this appendix we give some details of the computation of the action for
the simple half-brane solution. To avoid cluttered formulas, it is helpful to
introduce the notation,
\begin{equation}(\Phi_1,\Phi_2) = \left\langle\!\!\!\left\langle
\Phi_1\frac{1}{1+K}\Phi_2\frac{1}{1+K}\right\rangle\!\!\!\right\rangle.\end{equation}
The kinetic term of the action can be expressed as the sum of
two terms:
\begin{equation}\langle\!\langle \Psi Q\Psi\rangle\!\rangle = -(1)+(2),\end{equation}
where
\begin{eqnarray}
(1) = \Big(cGBc,Q(cGBc)\Big),\ \ \ \ \ \
(2) = \Big(cGBcG, Q(cGBc)G\Big).
\end{eqnarray}
Now replace the $G$ insertions with supersymmetry variations $\delta$ acting
inside the vertex, following \eq{Gdelta}. This generates many terms,
some of which vanish by $\phi$-momentum conservation or by $\mathcal{L}^-$ or
$\mathcal{B}^-$ invariance of the vertex. In the end the answer simplifies
to \begin{equation}(1) = -(cK,\gamma^2)+5(B\gamma^2, c\partial c)+2(\gamma,
\partial\gamma c)-4(cB\gamma, \partial\gamma c)-4(cB\gamma,\gamma Kc),
\end{equation}
and
\begin{eqnarray}(2) =\!\!\!\!\!\!\!\!&& -(cK,\gamma^2 K)-4(cB\gamma, c\partial\gamma K)
+2(cB\gamma, \partial c\gamma K)+(B\gamma^2,Kc\partial c)
-2(cB\gamma,K\partial \gamma c)\nonumber\\
\!\!\!\!\!\!\!\!&&+4(cB\gamma,K\gamma Kc)+(cB\gamma,\partial\gamma\partial c)
+2(cB\gamma,\partial^2\gamma c)-(cB\gamma,\gamma\partial^2 c)
-\frac{1}{2}(B\gamma^2,c\partial^2 c).\nonumber\\
\end{eqnarray}
We compute the inner products $(,)$ by mapping them to the appropriate
correlation function on the cylinder, evaluating the correlator with
\eq{corr}, and performing the Schwinger integrals. For $(1)$ the inner
products turn out to be
\begin{eqnarray}(cK,\gamma^2) \!\!\!\!\!\!\!\!&& = \frac{2}{\pi^2},
\ \ \ \ \ \ \ \
(B\gamma^2,c\partial c) = \frac{1}{\pi^2},
\ \ \ \ \
(\gamma,\partial\gamma c)=-\frac{2}{\pi^2},
\nonumber\\
(cB\gamma,\partial\gamma c)\!\!\!\!\!\!\!\!&& = -\frac{1}{\pi^2},
\ \ \ \
(cB\gamma,\gamma Kc)= \frac{6}{\pi^4},\end{eqnarray}
giving
\begin{equation}(1) = -\frac{2}{\pi^2}+\frac{5}{\pi^2}-\frac{4}{\pi^2}
+\frac{4}{\pi^2}-\frac{24}{\pi^4}=\frac{3}{\pi^2}-\frac{24}{\pi^4}
.\end{equation}
For $(2)$ we have the inner products
\begin{eqnarray}
(cK,\gamma^2 K)\!\!\!\!\!\!\!\!&& =-\frac{1}{\pi^2},
\ \, \ \ \ \
(cB\gamma, c\partial\gamma K)=-\frac{1}{\pi^2},
\ \ \ \ \,
(cB\gamma, \partial c\gamma K)= -\frac{1}{2\pi^2},
\nonumber\\
(B\gamma^2,Kc\partial c)\!\!\!\!\!\!\!\!&& =-\frac{1}{2\pi^2},
\ \ \ \
(cB\gamma,K\partial \gamma c) = \frac{1}{\pi^2},
\ \ \ \ \ \
(cB\gamma,K\gamma Kc)=\frac{1}{2\pi^2}-\frac{6}{\pi^4},
\nonumber\\
(cB\gamma,\partial\gamma\partial c)\!\!\!\!\!\!\!\!&& =-\frac{1}{\pi^2},
\ \ \ \ \ \ \
(cB\gamma,\partial^2\gamma c)=\frac{1}{\pi^2},
\ \ \ \ \ \ \ \ \,
(cB\gamma,\gamma\partial^2 c) =
(B\gamma^2,c\partial^2 c)= 0,\nonumber\\
\end{eqnarray}
giving
\begin{eqnarray}(2)\!\!\!\!\!\!\!\!&& = \frac{1}{\pi^2}+\frac{4}{\pi^2}-\frac{1}{\pi^2}
-\frac{1}{2\pi^2}-\frac{2}{\pi^2}+\frac{2}{\pi^2}-\frac{24}{\pi^4}
-\frac{1}{\pi^2}+\frac{2}{\pi^2}+0+0\nonumber\\
\!\!\!\!\!\!\!\!&& = \frac{5}{\pi^2}-\frac{1}{2\pi^2}-\frac{24}{\pi^4}.
\end{eqnarray}
Adding things up
\begin{eqnarray}\langle\!\langle \Psi Q\Psi\rangle\!\rangle \!\!\!\!\!\!\!\!&& = -(1)+(2) \nonumber\\
\!\!\!\!\!\!\!\!&& = -\frac{3}{\pi^2}
+\frac{24}{\pi^4}+\frac{5}{\pi^2}-\frac{1}{2\pi^2}-\frac{24}{\pi^4}\nonumber\\
\!\!\!\!\!\!\!\!&& = \frac{3}{2\pi^2}.
\end{eqnarray}
The energy is
\begin{equation}E=-\frac{1}{6}\langle\!\langle \Psi Q\Psi\rangle\!\rangle = -\frac{1}{4\pi^2},
\end{equation}
which is precisely $-1/2$ times the tension of the D-brane.
\section{Auxiliary Tachyon Coefficient}
\label{app:level}
In this appendix we compute the coefficient of the auxiliary tachyon state
$c_1|0\rangle$ for the Schnabl-like half-brane solution in the $L_0$ level
expansion. To achieve this we write the Schnabl-like solution in
the form
\begin{eqnarray}{\Psi_\mathrm{Sch}} \!\!\!\!\!\!\!\!&& = -\sum_{n=0}^\infty\psi_n' +\Gamma\nonumber\\
\!\!\!\!\!\!\!\!&& =
-\sum_{n=0}^\infty\sum_{0\leq k\leq n/2}{n\choose 2k}a^{2k}
\left.\frac{d^{k+1}}{dr^{k+1}}\right|_{r=0}cB\Omega^{n+r}c(1+iaG)\Omega
\nonumber\\ \!\!\!\!\!\!\!\!&&\ \ \ \ \
-i\sum_{n=0}^\infty \sum_{0\leq k\leq \frac{n-1}{2}}{n\choose 2k+1}a^{2k+1}
\left.\frac{d^{k+1}}{dr^{k+1}}\right|_{r=0} cBG\Omega^{n+r}c(1+iaG)\Omega
\nonumber\\ \!\!\!\!\!\!\!\!&&\ \ \ \ \
+B\gamma^2(1+iaG)\Omega. \label{eq:Psch_Fock}
\end{eqnarray}
We can drop the phantom term since it vanishes in the Fock space. The states
inside the sums can be expressed using the operator formalism of
Schnabl\cite{Schnabl,Schnabl_wedge}, which yields an expression for the
solution in terms of a canonically ordered set of mode operators acting on
the $SL(2,\mathbb{R})$ vacuum. Using \eq{reg_limits} one can argue that
the infinite sums above converge for any coefficient in the Fock space as
long as the parameter $a$ is restricted to the range
$-\sqrt{2}\leq a\leq\sqrt{2}$.
Expanding \eq{Psch_Fock} in the Fock space we can extract the coefficient
of the auxiliary tachyon. Define two functions
\begin{eqnarray}\phi_1(r) \!\!\!\!\!\!\!\!&& = \frac{1}{\pi X^2}\left[\frac{1}{\pi}
\cos^2\left(\frac{\pi}{2}X_+\right)\sin(\pi X_-)
- \frac{1}{\pi}\sin(\pi X_+)\cos^2\left(\frac{\pi}{2}X_-\right)\right.
\nonumber\\ \!\!\!\!\!\!\!\!&& \ \ \ \ \ \ \ \ \ \ \ \left.
-(X_+-1)\cos^2\left(\frac{\pi}{2}X_-\right)
+(X_-+1)\cos^2\left(\frac{\pi}{2}X_+\right)\right],\nonumber\\
\phi_2(r)\!\!\!\!\!\!\!\!&& = -\frac{d}{dr}\phi_1(r)+\frac{1}{X}
\left[-\frac{1}{2\pi} \sin(\pi X_+)\cos\left(\frac{\pi}{2}X_-\right)
-\frac{1}{2}(X_+ - 1)\cos\left(\frac{\pi}{2}X_-\right)\right.\nonumber\\
\!\!\!\!\!\!\!\!&& \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\left.-\frac{1}{4\pi}\sin(\pi X_+)\sin(\pi X_-)
-\frac{1}{2\pi}\cos^2\left(\frac{\pi}{2}X_+\right)(\cos(\pi X_-)+1)\right.
\nonumber\\
\!\!\!\!\!\!\!\!&& \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\left.-\frac{1}{4}(X_+-1)\sin(\pi X_-)\right],
\end{eqnarray}
where for short we have denoted
\begin{equation}X=\frac{2}{r+2},\ \ \ \ \ X_+ = \frac{r+1}{r+2},\ \ \ \ \
X_- =\frac{-r+1}{r+2}.\end{equation}
The auxiliary coefficient is then
\begin{eqnarray}\phi\!\!\!\!\!\!\!\!&& =
\sum_{n=0}^\infty\left[-\sum_{0\leq k\leq n/2}{n\choose 2k}a^{2k}
\frac{d^{k+1}\phi_1(n)}{dn^{k+1}}
+\sum_{0\leq k\leq \frac{n-1}{2}}{n\choose 2k+1}a^{2k+2}
\frac{d^{k+1}\phi_2(n)}{dn^{k+1}}\right].\nonumber\\
\label{eq:aux_tach}\end{eqnarray}
Since $\phi_1$ and $\phi_2$ vanish as $1/r^3$ for large $r$, \eq{reg_limits}
implies that the terms in the summand vanish as $1/n^8$ for sufficiently
large $n$. We have checked this behavior numerically. Therefore
\eq{aux_tach} is a convergent sum if $-\sqrt{2}\leq a\leq\sqrt{2}$.
Unfortunately, the multiple derivatives of $\phi_1$ and $\phi_2$ make a
direct numerical evaluation of \eq{aux_tach} very time-consuming.
To evaluate \eq{aux_tach} with sufficient precision, we found it necessary
to expand $\phi_1$ and $\phi_2$ in powers of $\frac{1}{r+2}$ out to
$\frac{1}{(r+2)^{40}}$, which simplifies the numerical computation of
derivatives. For $a=1$ we found the auxiliary tachyon coefficient to be
\begin{equation}\phi = -.0599156.\end{equation}
More interesting is the plot of the auxiliary tachyon coefficient as a
function of $a$, shown in figure \ref{fig:aux_tach}. At $a=0$ the coefficient
corresponds to that of a tachyon vacuum solution, and has positive expectation
value, as we would expect from the usual picture of the cubic potential
in bosonic string field theory. However, as $a$ becomes large, the
expectation value becomes zero and even negative. This suggests that the
negative energy of the half-brane solution is not principally due to the
condensation of the auxiliary tachyon. This is one way to see that the
Schnabl-like solution must not satisfy the reality condition.
\begin{figure}
\begin{center}
\resizebox{3.2in}{1.9in}{\includegraphics{exotic_auxtach.eps}}
\end{center}
\caption{\label{fig:aux_tach} Coefficient of the auxiliary tachyon
$c_1|0\rangle$ in the Schnabl-like solution, as a function of
$a\in[-\sqrt{2},\sqrt{2}]$.}
\end{figure}
We have also computed the coefficients for a few descendents of the auxiliary
tachyon. Let us denote coefficients of the states
\begin{equation}(L_{-2})^n c_1|0\rangle,\ \ \ (L_{-4})^n c_1|0\rangle
\end{equation}
by $x_n$ and $y_n$ respectively for $n\geq 1$. At $a=1$ we have found the
explicit values
\begin{eqnarray}x_1\!\!\!\!\!\!\!\!&& = .067747,
\ \ \ \ \ \ \ \ \ \ \
y_1 = -.019133,
\nonumber\\
x_2\!\!\!\!\!\!\!\!&& = .0060976,
\ \ \ \ \ \ \ \ \ \,
y_2 = .000064506,
\nonumber\\
x_3\!\!\!\!\!\!\!\!&& = -.000042514,
\ \ \ \ y_3 = 7.9488\times 10^{-7}.
\end{eqnarray}
We have computed $x_n$ and $y_n$ out to $n=60$ and found that they decay
significantly faster then the corresponding coefficients of
$(L_{-2})^n|0\rangle$ and $(L_{-4})^n|0\rangle$ of the sliver state. We
therefore believe that the Schnabl-like solution is a regular
state in the $L_0$ level expansion.
\end{appendix}
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 6,489
|
Variability in the pediatric intensivists' threshold for withdrawal/limitation of life support as perceived by bedside nurses: a multicenter survey study
Colleen S Gresiuk1 &
Ari R Joffe1,2
We hypothesized that bedside nurses perceive significant variability in the pediatric intensivist thresholds for approaching a family about withdrawal/limitation of life-sustaining therapy.
All nurses working in four university-affiliated medical-surgical pediatric intensive care units staffed by 11, 7, 6, and 5 intensivists with 36, 18, 10, and 8 beds were sent three mailings of a survey asking questions about intensivist decisions for withdrawal/limitation of life-sustaining therapy. Responses were tabulated; chi-square compared results among centers; a p < 0.05 after Bonferroni correction was significant.
The response rate was 205 of 415 (49%); 152 of 205 (74%) disagreed with the statement that each of the intensivists had the same threshold for approaching a family to suggest withdrawal/limitation of life-sustaining therapy, with no significant difference between centers. Also, 110 of 205 (54%) and 119 of 205 (58%) disagreed with the statement that each intensivist has the same threshold of the patient's chance for survival or projected quality of life when making a decision to withdraw/limit life-sustaining therapy with no significant difference between centers. The threshold to suggest withdraw/limit life-sustaining therapy based on chance of survival or projected quality of life differs between intensivists by at least 10% according to 113 of 184 (61%) and 121 of 184 (66%) nurses; the two larger centers had significantly higher difference among intensivists for projected quality of life. Fifty-five of 200 (27%) disagreed with the statement that they would have equal confidence in each intensivist accepting a recommendation for withdrawal/limitation of life-sustaining therapy for their own child, with no difference between centers.
Bedside pediatric intensive care unit nurses in this multicenter Canadian study perceive wide variability in intensivist thresholds for approaching a family to suggest withdrawal/limitation of life-sustaining therapy.
Most deaths in Pediatric Intensive Care Units (PICU) follow withdrawal or limitation of life-sustaining treatments (W/L) [1], and the number has increased in recent years [2–5]. This is considered ethically permissible in the context of legally incompetent minors when the proportionately greatest benefits of treatment are outweighed by the harms of treatment [6]. The principle of patient autonomy allows a competent patient or their surrogate to refuse or stop any lifesaving treatment when it is what the patient would wish for their care [6, 7]. In contrast, for children, the decision is usually made in discussion with the parents to arrive at a treatment plan that is in the best interests of the child.
There is limited research to determine how these decisions are made. When presented with hypothetical patient scenarios, the thresholds for W/L vary significantly between intensivists in both adult [8] and pediatric [9] intensive care units (ICU). One study showed that medical residents perceive a difference in thresholds between ICU attending physicians when making these decisions [10]. The bedside nurses' perception is another and, arguably, potentially a more realistic independent observer-based reflection of these decisions than asking intensivists to respond to hypothetical scenarios. To our knowledge, there is no multicenter study that evaluates the threshold of pediatric intensivists in decisions to W/L in everyday practice.
We previously reported a single-center survey of PICU nurses where respondents perceived significant variability in intensivist thresholds for approaching a family to suggest W/L [11]. Our objective was to determine whether this is a reproducible finding in Canadian PICUs. Wide decisional variability is not desirable; it raises significant concern considering the implication of W/L decisions for patient mortality. We find that bedside PICU nurses in this multicenter Canadian study perceive wide variability in intensivist thresholds for approaching a family to suggest W/L.
Questionnaire administration
This study was a survey of PICU nurses' opinions regarding their experience with W/L and do not resuscitate (DNR) orders. Each staff nurse in four university-affiliated, tertiary, medical-surgical Canadian PICUs was delivered the survey in 2009. The four PICUs were in Alberta and Ontario, each staffed by 11, 7, 6, and 5 full-time intensivists with 36, 18, 10, and 8 beds. A cover letter was included asking the nurse to fill in the survey and return it in the self-addressed envelope. A second and third delivery of the survey was done at 4- to 8-week intervals to nonresponders, and all responses were considered received by the end of 2009.
The cover letter stated:
"As you know, unfortunately, children may die from critical illness in the PICU. Often the decision is made to limit therapy (including a DNR or "do not resuscitate" order), or withdraw therapy to allow a patient to die when the harms of treatment outweigh any potential benefits. This is a value-laden decision, based on an assessment of prognosis and quality of life (QOL), and heavily influenced by religious and personal beliefs. You are one of the 'front line' workers in the PICU who sees these decisions made and the effect of them on staff, patient, and families. We have designed this survey as an attempt to determine your perspective on this process of 'ethical decision making in the PICU' as a bedside nurse.... Your responses are voluntary and confidential...return of the survey implies consent to participate."
The study was approved by our university health research ethics board.
The development of the survey has been described previously [11]. We searched MEDLINE from 1966 to 2004 for articles on W/L from children in a PICU, using search terms, including "withholding treatment," "resuscitation orders," "pediatrics," and "child," and found no reports of a similarly designed study. We wanted the three-page survey (Additional File 1) to be simple and focused. Therefore, to generate the items for inclusion in the questionnaire, we focused the questions specifically on any perceived differences in the threshold to suggest W/L and DNR by each pediatric intensivist. There are no written guidelines for end-of-life decision-making in the PICUs apart from those published by the American Academy of Pediatrics and the Canadian Pediatric Society, which intensivists in Canada are expected to follow [6, 12]. In general, these decisions are made jointly by the parents and attending intensivist at the time, with meetings that include the bedside nurse, and commonly pastoral care and social work.
Questions utilized a five-point Likert scale: strongly agree (SA), agree (A), neutral, disagree (D), or strongly disagree (SD). Two questions did not use this Likert scale; these questions stated that "the threshold to suggest W/L based on chance of survival (or based on projected QOL) differs among intensivists by: < 1%, 5%, 10%, 15%, or ≥ 20%.
To ensure clarity, perceived reality of the situations presented, validity, and ease of completion of the questionnaire, initial pilot testing was done by having the survey completed by six PICU nurses, followed by a semistructured interview. All found the survey to be understandable, easy to follow, not difficult or confusing, and were confident that their responses reflected their intended answers. The survey asked nurses to make a subjective judgment about intensivist thresholds to suggest W/L. We chose this subjective measure to describe nursing perception of practice variations. There is no standard threshold defined; indeed, a set standard is not desirable, because it could not be expected to take into account all of the myriad of considerations of each individual case, and whose (patient, parent, physician, nurse, ethicist, etc.) values would define it is problematic. The pilot testing indicated that the respondents understood this subjective concept of "threshold."
Anonymous data were entered into a computer database (Microsoft® Excel; Microsoft Corp, Redmond, WA). The proportion of respondents with different answers was tabulated. We compared responses between the four PICUs by grouping responses into SA/A, neutral, and D/SD. In addition, two subgroups in the pooled data were identified before survey distribution to reflect the level of PICU experience: those working in PICU < 5 years vs. ≥ 5 years, and those who had attended < 5 vs. ≥ 5 family meetings. The responses in the PICUs and in these subgroups were compared using the Chi-square statistic, with p < 0.05 accepted as significant after Bonferroni adjustment for multiple comparisons.
Of 415 surveys delivered, 205 (49%) were returned. The respondents' demographics are shown in Table 1.
Table 1 Demographics of the survey respondents
The intensivist's role in decisions
The responses to questions about the threshold to approach a family, the family contribution to decisions, and unilateral decisions are shown in Table 2. Most respondents (152/205, 74%) did not believe that each of the intensivists have the same threshold for approaching a family to suggest a W/L or DNR order, and many (68/205; 33%) did not believe that each intensivist allows the same amount of family contribution to these decisions.
Table 2 Responses to the questions about the intensivist's role in making withdrawal/limitation of therapy decisions
In two questions, it was stated that "each intensivist has the same threshold of the patient's chance for survival [or, projected QOL] when making a decision to W/L." Of respondents, the majority disagreed with these statements (Figure 1). The next two questions stated: "The threshold to suggest to W/L based on chance of survival [or, projected QOL] differs between intensivists by: < 1%, 5%, 10%, 15%, or ≥ 20%." For the question based on chance of survival, respondents answered < 1% in 20 (10%), 5% in 52 (25%), 10% in 61 (30%), 15% in 28 (14%), and ≥ 20% in 24 (12%). For the question based on projected QOL, respondents answered < 1% in 18 (9%), 5% in 45 (22%), 10% in 56 (27%), 15% in 32 (16%), and ≥ 20% in 33 (16%). The threshold to suggest to W/L based on chance of survival differed between intensivists by at least 10% for 113 of 185 (61%) and by at least 15% for 52 of 185 (28%) of respondents. The threshold to suggest to W/L based on QOL differed between intensivists by at least 10% for 121 of 184 (66%) and by at least 15% for 65 of 184 (35%) respondents. Although subjective, we believe that these may be meaningful differences in outcome for some patients.
Response to the statement: "Each intensivist has the same threshold of the patient's chance for survival [or, projected quality of life] when making a decision to limit/withdraw therapy." Chance for survival: hatched bars; quality of life: solid bars.
Few statistically significant differences between PICUs were found in the responses to questions (Table 3). In each case, the two smaller PICUs suggested more consistency in not allowing too much family contribution, not often having unilateral decisions, and the threshold of projected QOL in making decisions.
Table 3 Questions where there were statistically significant differences in responses between the four pediatric intensive care units
The hypothetical "nurse's own child" scenario
A significant minority of respondents (55/205, 27%) disagreed with the statement that, for their own child in the PICU, they would have equal confidence in accepting a recommendation for W/L or DNR from each intensivist (Figure 2). Similarly, 93 of 205 (45%) responded that they would have confidence in the intensivist's opinion to W/L only in certain situations (Additional File 2). For these questions, there were no statistically significant differences between centers.
Response to the question: "Assume your child was in the pediatric intensive care unit and the intensivist on service approached you to recommend a limiting/withdrawing life support or 'do not resuscitate' order. You would have equal confidence accepting this recommendation from each intensivist."
With this last question, the survey asked to "please explain." Comments were written by only 77 of 205 (38%) respondents. On review of the written comments, we determined that all could be classified into two themes: the confidence level in the intensivists' recommendations for W/L (n = 57), and general comments on how these decisions should be made (n = 35). Of the 57 comments regarding confidence level in the intensivist recommendations, comments suggested total confidence in 24 (42%), confidence in certain situations only in 23 (40%) [some wanted to review all the tests done for themselves (n = 5, 22%), and some suggested that the intensivist and the nurse's view of QOL may differ (n = 10, 43%)], and lack of confidence in 10 (17%) [some suggested that they would have confidence in an individual intensivist only (n = 6, 60%), and some claimed that they have seen disagreement amongst intensivists leading them to doubt individual opinions (n = 3, 30%)]. Of the 33 general comments, the comments included a need for multidisciplinary team and consultant involvement in these decisions in 9 (27%), the decision to suggest W/L is delayed too long in 16 (48%), and other comments in 9 (27%). Examples of written comments are shown in Additional File 3.
Differences between respondent subgroups
Only one question had statistically significant differences in responses in the two subgroups: more experienced nurses--with ≥ 5 years of PICU experience or ≥ 5 family meetings--were more likely to respond D/SD that intensivists have W/L without discussion with the family (Table 4). There were remarkably similar results between the subgroups in the questions asking for the threshold for W/L based on chance of survival or QOL (all p > 0.9).
Table 4 Response to the survey questions in the prespecified subgroups of nurses
To our knowledge, this is the first published multicenter report to examine bedside PICU nurses' perception of variability in the pediatric intensivists' thresholds for approaching a family about W/L or DNR decisions. We found that these nurses do perceive significant variability among the intensivists. Of respondents, only 21 (10%) agreed with the statement that each of the intensivists had the same threshold for approaching a family to suggest W/L. Only 52 (25%) and 46 (22%) respondents agreed with the statement that each intensivist has the same threshold of the patient's chance for survival or projected QOL when making a decision to W/L. Most nurses perceived that the difference between intensivists in the threshold to suggest W/L based on chance of survival (n = 113, 61%) or projected QOL (n = 121, 66%) differs by at least 10%. This perception may explain the response to the question of the confidence that the nurse would have in the intensivist's opinion if it concerned the nurse's own child: only 121 (59%) agreed with the statement "you would have equal confidence accepting this recommendation [for W/L] from each intensivist." The nurses' written comments about confidence level (n = 57) support this conclusion: 42% suggested total confidence in the recommendation of each intensivist, 40% suggested confidence in only certain situations, and 17% suggested a lack of confidence. These results confirm and generalize our previous single-center study using the same survey instrument [11].
Our results are compatible with other studies that used different methodologies. The thresholds for W/L vary significantly between intensivists in both adult and pediatric studies where hypothetical patient scenarios are presented [8, 9]. Medical residents perceive a difference in thresholds among their attending physicians in making W/L decisions [10]. Hospital characteristics are associated with the use of DNR orders, even after accounting for differences in patient characteristics; indeed, a tenfold difference in standardized rates of DNR across counties in California may reflect different institutional cultures [13]. In Europe, the frequency of DNR and W/L decisions varies markedly between and within countries [14, 15].
The nurse-perceived variability in physician thresholds to suggest W/L in our study is of significant concern considering the implication of W/L for patient mortality. A multicenter Canadian study found that of the 341 adult patients who were assessed by a physician on at least one occasion to have a probability of ICU survival of < 10%, 99 (29%) survived the ICU [16]. Even for those where this prediction was made on at least three occasions, the actual survival was 27 of 120 (22.5%). For patients with clinician predicted survival of 10-40%, the actual survival was 79.3%. When the physician predicted a survival of < 10%, patients were more likely to have withdrawal of life support (including ventilation, inotropes, and dialysis), and this prediction more powerfully predicted ICU mortality than illness severity, evolving or resolving organ dysfunction, and use of inotropes or vasopressors, and predicted mortality more strongly for patients who had no stated preferences regarding W/L and who had less severe organ dysfunction [16]. The withdrawal of mechanical ventilation was predicted by the physician's prediction of the likelihood of patient survival in ICU of < 10%, and not by patient age, prior functional status, severity of illness, or severity of organ dysfunction [17]. Other studies have found that there is large variability in the accuracy of prognostication by intensivists [18]. This can and does lead to self-fulfilling prophesies in predicting outcomes [19]. In the Canadian multicenter study, 3.6% of patients who had withdrawal of mechanical ventilation in anticipation of death were discharged home [17], and in an international ICU adult study the proportion of hospital survivors who had W/L decisions ranged from 2.4-30.3% [20].
The Task Force on Values, Ethics, and Rationing in Critical Care (VERICC) suggests that rationing decisions based on clinician judgment are "particularly susceptible to unethical subjectivity and bias" [21]. Examples cited include rationing decisions made based on: age, pre-illness employment status, the political power of the surgical services, race, iatrogenic complications, families who the physician knows or likes particularly well, and families who are more demanding [21]. They suggest that "when clinicians withhold interventions based on their interpretation of the standard of care...it becomes clear that a potentially beneficial intervention is being withheld for reasons other than the best interests of the patient [21]." Although we did not ask what factors the nurses believed influenced the variability among intensivists in our survey, it is likely that some of the conscious or unconscious [22] biases mentioned influence the individual intensivist's judgments about chance of survival and QOL. The nurses' comments suggested that some have confidence in only individual intensivists and some have a different view on QOL than certain intensivists. A "shared decision-making" model has been suggested for end-of-life decisions [23]. In this model, one component is a physician's recommendation. Concerns with this model include that the power differential between physician and patient and physician's personal biases may both unduly influence the decision [23, 24]. This study suggests that the concern about personal bias may be very real.
There has been debate about whether physicians can make unilateral decisions about W/L and DNR orders [25–30]. These decisions involve value judgments about the chance of survival and the worth of differences in QOL without any uniform consensus [29–31]. There are very few circumstances where one can invoke the "futility" argument [29–32]. The recently recommended shared decision-making model explicitly suggests that clinicians discuss prognostic uncertainty with family decision makers, and empiric data show that most surrogates desire this be acknowledged [33, 34]. There are "no objective incontrovertible metrics" for prognostication and "no clinician is omniscient; no clinician is infallible; and the clinician [should not] prioritize his [or his perception of the patient's] values...." [28]. If there are varying knowledge, biases, values, and recent experience among intensivists, the end-of-life decision making "must not depend on luck of the draw: who is in the emergency department or intensive care unit that night" [28]. Our survey found that many nurses perceive that intensivists have different judgments regarding what chance of survival or QOL is worth pursuing.
Strengths of this study include the multicenter representation, the reasonable response rate (204/415 subjects; 49%), the survey development methods, including the simple focused nature of the questions, and the favorable pilot testing. Most respondents were highly experienced, having been in practice for ≥ 5 years in 63% and working in the PICU for ≥ 5 years in 48%. The similar results compared with studies that used different methodologies and our previous single-center study enhanced the generalizability of the findings.
Limitations of this study include: lack of open-ended questions allowing respondents to expand on their intended answers, and the possible discrepancies between perceived and actual practice of the intensivists. In addition, factors that may explain the heterogeneity of responses were not available. It is unclear whether all of the intensivists differ in their threshold or whether a subgroup of intensivists are perceived to be more homogenous in their threshold. The survey required subjective recalled perceptions for responses, even in questions that attempted to quantify the amount of variability. It is difficult to know exactly what, for example, a 10% difference in QOL means to the respondents. Finally, testing for statistical differences in responses among institutions should be interpreted with caution given our small sample size and inadequate power to rule out differences. However, we believe that these limitations would not affect the main conclusion of this study.
This multicenter study demonstrates that bedside PICU nurses perceive wide variability in the intensivist's threshold for approaching a family to suggest W/L; this variability includes different thresholds of the chance of survival and projected QOL. This finding has significant implications for how end-of-life decisions, particularly unilateral decisions, are made in a PICU. We suggest that intensivists need to be aware of this nursing perception and to consider seriously its implications for their own decision making. The intensive care that a child is offered may depend to a large degree on the physician in charge, which may affect a patient's mortality and palliative care decisions. Further study is required to determine ways to improve consistency in end-of-life practice.
A/SA:
agree or strongly agree
D/SD:
disagree or strongly disagree
DNR:
do not resuscitation
PICU:
pediatric intensive care unit
QOL:
W/L:
withdrawal/limitation of life-sustaining therapy.
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Ryan AC, Byrne P, Kuhn S, Tyebkhan J: No resuscitation and withdrawal of therapy in a neonatal and a pediatric intensive care unit in Canada. J Pediatr 1993, 123: 534–538. 10.1016/S0022-3476(05)80946-1
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Manthous CA: Counterpoint: is it ethical to order "do not resuscitate" without patient consent? Chest 2007, 132: 751–754. 10.1378/chest.07-0912
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Helft PR, Siegler M, Lantos J: The rise and fall of the futility movement. N Engl J Med 2000, 343: 293–296. 10.1056/NEJM200007273430411
Curtis JR, White DB: Practical guidance for evidence-based ICU family conferences. Chest 2008, 134: 835–843. 10.1378/chest.08-0235
Evans LR, Boyd EA, Malvar G, Apatira L, Luce JM, Lo B, White DB: Surrogate decision-makers' perspectives on discussing prognosis in the face of uncertainty. Am J Respir Crit Care Med 2009, 179: 48–53.
There was no source of funding for this project. AJ had full access to all the data in the study and takes responsibility for the integrity of the data and the accuracy of the data analysis. Preliminary results of this study were presented as a poster at the 39th Critical Care Congress of the Society of Critical Care Medicine, Miami, Florida, USA, in January 2010. We sincerely thank our collaborators at the four PICUs where the survey was conducted, who volunteered their help in survey distribution and institutional ethics approval.
University of Alberta, Stollery Children's Hospital, 8440 112 Street, Edmonton, Alberta, T6G 2B7, Canada
Colleen S Gresiuk & Ari R Joffe
The John Dossetor Health Ethics Center, 8440 112 Street, Edmonton, Alberta, T6G 2B7, Canada
Ari R Joffe
Colleen S Gresiuk
Correspondence to Ari R Joffe.
AJ drafted the first version of the manuscript. CG and AJ made substantial contributions to conception and design of the study, acquisition of data, and analysis and interpretation of data, have been involved in revising the manuscript critically for important intellectual content, and have given final approval of the version to be published.
The survey instrument
Additional file 1: . Table (DOC 52 KB)
Response to the question: For your own child, "you would have confidence in the intensivist's opinion to limit/withdraw life support only in certain situations
Additional file 2: ." Figure (TIFF 229 KB)
Representative written comments to the instruction "please explain" after questions about confidence in intensivist decisions for the nurse's hypothetical own child
Gresiuk, C.S., Joffe, A.R. Variability in the pediatric intensivists' threshold for withdrawal/limitation of life support as perceived by bedside nurses: a multicenter survey study. Ann. Intensive Care 1, 31 (2011). https://doi.org/10.1186/2110-5820-1-31
Intensive Care Unit Survival
Family Contribution
Bedside Nurse
Unilateral Decision
|
{
"redpajama_set_name": "RedPajamaCommonCrawl"
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| 1,785
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Q: Fastest serialize data format form PHP reading I have a PHP frontend and a C++ backend, and I need to be able to send groups of names to the frontend. What serialized format would be the most efficient/fastest for the PHP to read?
Example data
group1:
name1 3923
name2 9879
name3 8944
group2:
name5 9823
group3:
name9 9822
name1 4894
What would be the fastest for PHP to read?
*
*XML
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*YAML
*Protocol Buffer
*Comma/Space Delimited our own system
*Anything else? other?
A: PHP's own serialized format will probably be the fastest. unserialize() is the function PHP uses to convert this data back to its own types. This post has various links to other languages' implementations of PHP's serialized format, I'm sure you could convert one of those easily.
A: I've used PHP's serialize() and unserialize() on large text files, and it performed miserably (that was a couple of years ago - maybe it's better now). Anyway, I devised a little trick to overcome this, it simply involves generating a PHP array declaration from the data you're exporting straight into a text file, e.g.:
<?php
$groups = array('groups' => array( array('jeff' => 2343,
'tom' => 8477),
array('baal' => 2873,
'scorpio' => 3210),
array('jeff' => 2343,
'tom' => 8477)
)
)
);
?>
...and then unserializing it by simply calling:
include 'groups.php';//makes $groups available
Worked nicely back then.
A: JSON would be pretty easy using json_decode. I'm not sure about speed, but unless you plan on transferring megabytes of this data between the systems it should be irrelevant which one you go with.
A: As Paolo pointed out you can use json_decode which is very fast. On the C++ backend these are some of your options ( taken directly from the json.org website ):
C++:
*
*TinyJSON.
*jsoncpp.
*zoolib.
*Jaula.
*JOST.
*JSON Spirit.
*CAJUN.
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 3,299
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Subscribe To Star Wars: Rogue One May Have Found Its Male Leads Updates
Star Wars: Rogue One May Have Found Its Male Leads
There's a fair amount of mystery still swirling around Star Wars: Rogue One, even after the panel hosted by Gareth Edwards at Star Wars Celebration. We know that Edwards is directing the heist story about a team retrieving Death Star plans. At the moment, Felicity Jones has been cast, and the project – from what we hear – might have circled her co-stars.
According to Variety, the Star Wars: Rogue One team is looking at Nightcrawler co-star Riz Ahmed to play the lead male role in the Star Wars spinoff movie. At the same time, The Wrap adds that The Hunger Games standout Sam Claflin (Finnick Odair) also is up for a role in the growing ensemble. Sources couldn't tell the trade site who Ahmed or Claflin would play, though Variety did note that Ahmed "likely" would play a rebel soldier, which is what we believe Jones would play, as well.
Ahmed's most recent part was that of Rick, an entrepreneurial camera man who got in with the wrong people – well, person – when he took a job combing the streets of Los Angeles with the predatory Louis Bloom (played by Jake Gyllenhaal) in Nightcrawler. It was a striking turn in a dark and somber movie, and Ahmed held his own opposite Gyllenhaal and against the cool and stylish scene design by writer/director Dan Gilroy.
Claflin, meanwhile, has dabbled in large-scale blockbuster fare. In addition to the Hunger Games movies, he played a part in the fourth Pirates of the Caribbean films, and acted in Snow White and the Huntsman.
We learned a good amount regarding Star Wars: Rogue One at the Star Wars Celebration in Anaheim. Gareth Edwards took to the stage to fill fans in on the plans to expand Star Wars beyond the normal Saga, which will be contained to J.J. Abrams and Rian Johnson's films. Set in the years between Star Wars: Episode III – Revenge of the Sith and Star Wars: Episode IV – A New Hope, this films is being positioned as an action-packed heist film. It's also believed to be an ensemble film, so we expect to hear more names added to the cast sheet before too long.
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Why Diego Luna Was 'So Disappointed' At The End Of Rogue One
Riz Ahmed Reveals He Missed Star Wars Celebration Due To Homeland Security
Upcoming Star Wars Movies: List Of Titles And Release Dates
Bachelorette Spoilers: Did Tyler C.'s Mom Reveal The Next Bachelor 2020?
|
{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
| 742
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Sierra Joy
Writing and Photography based in travel and ecology
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International Escapades
Idea Exchange at the UN Climate Change Conference
Posted bySierra Joy
January 15, 2015 May 28, 2015 — Leave a Comment
Lima welcomed us with a pale sky, bustling streets and a promise of being unforgettable. Through the taxi window, flat topped rectangular buildings of tan brick blurred together as our driver shifted into a higher gear, accelerating us down the bumpy road, weaving freely between a towering blue bus and a dented sedan. The building edges crumbled slightly, bare rebar extended from the walls, reaching for the sky as if waiting for another level to be added. The neighborhood shifted with added wood facades painted only the brightest colors—lavender, pink and yellow, green with orange trim. We passed under an arching pedestrian bridge with a sign reading: Peru, lleno de creatividad. A man with dark eyes and rounded waistline weaved through stationary traffic selling newspapers. Sophisticated apartments with high gates sat adjacent to gaudy casinos—royal blue with a golden sun, black with cards drawn on it and a lighthouse protruding from the front. A grassy median featured bush sculptures of panda bears intricately trained and trimmed to feature the bicolored bear in alternating green and white foliage. Horns honked and traffic inched, fighting towards the heart of this expansive city of 12 million. Overhead a banner featured a wreath of colored word bubbles and the slogan, Voces por el Clima with the insignia for COP20.
COP20 held the international spotlight as the world looked on to see what the world's leaders would agree on in regards to combating climate change. Presidents, prime ministers, delegates, non-governmental organizations (including ours, the UC Revelle Program on Climate Science and Policy), businessmen and women and so many other thinkers and leaders gathered to share ideas. This December, five other Scripps graduate students and I went to Lima to represent Scripps, communicate ocean science, and gain insight into the world of international policy.
Noah, Kate, Chris and I are the COP
As graduate students interested in conducting science that can be used by policymakers for a better tomorrow, attending such an event was a great opportunity. It also provided a chance to be immersed in Peruvian culture and consider how climate change is affecting the host country.
"These types of experiences are invaluable and greatly enhance my understanding of how the science we do on a daily basis is translated into international policy decisions," said Noah Ben-Aderet, a PhD candidate in marine biology at Scripps.
Change can, and does occur on many levels—individual, community, country, region, world. Each of these levels is important to solve widespread issues like climate change, but there is not one solution that will fit at each level or region. The overarching goal of the COP process is to form international agreements toward climate mitigation. But the COP is more than a place for policy debates; it is a platform for idea-sharing.
In partnership with the U.K.'s Plymouth Marine Laboratories (PML) and several other international agencies, Scripps hosted an educational exhibit on oceans. It provided a platform for us to discuss how climate change is affecting the oceans with interested delegates and attendees. The educational conversations are aimed to remind negotiators about the oceans, the largest ecosystem on earth, as they make climate policy. It also provides an opportunity to hear from participants what concerns they have for our oceans.
"The exhibition booth was very popular with a near-constant flow of delegates keen to learn more about the effects of climate change on the ocean," wrote Carol Turley, senior scientist at PML in a post-COP20 report. "Compared to our first foray to the UNFCCC COP in Copenhagen in 2009, most delegates have now heard of ocean acidification, in addition to the other two stressors (higher temperature, lower oxygen). Nevertheless, they were not always aware of its potential effects on ecosystems, aquaculture, and human society and were pleased to receive the summaries for policy makers and other outreach material."
Filming the Fisheries Video at WWF Peru
At the heart of idea-sharing is education—the spread of knowledge gained through experience. As student scientists increasing the communication of our science is extremely important. We formed Ocean Scientists for Informed Policy, a group dedicated to understanding policy and to communicating science to policy makers and to a lay audience. The effort includes blogging from the COP and creating short films on ocean issues raised during the event. During our time in Lima, I worked with fellow marine biologists Kathryn Furby, Noah Ben-Aderet, and filmmaker Chris Neighbors to produce several videos.
"The goal of the videos is to communicate our experiences and the important issues at COP to a broad audience," said Furby, a Scripps PhD candidate in marine biology. "It was important to me to communicate how global policy and science intersect on a large scale. I also wanted to personalize a large, sometimes overwhelming, conference. Grad student scientists have a great voice for communicating complex ideas. As an unexpected consequence, the videos gave us a concrete reason to approach important players at the COP. In Peru we were excited to find that people were engaged in science and deeply concerned about climate change. As ocean scientists, our perspective seemed encouraged and appreciated. Through these videos we interviewed interesting people from all over the world."
Some of our videos connected climate issues specifically with Peru, which allowed us to connect with the host country at a more meaningful level. In the last month since publication, the videos have reached almost 20,000 viewers.
For me, the richest aspect of the week stemmed from the diversity of people I exchanged ideas with. Scripps students attended a dinner hosted by the United Nation's Reducing Emissions from Deforestation and Forest Degradation program and spoke to people working on the ground in Indonesia, the Democratic Republic of Congo and Cameroon about the challenges of stopping deforestation and poaching. I met a U.N. event organizer who discussed the changes he'd seen in the last 15 COPs and his hopes for the next COP in Paris in December, an event anticipated to produce a treaty guiding international action in the future. I spoke about climate change with taxi drivers who wondered why the U.S. uses gasoline in cars instead of natural gas like they use in Peru. We spoke with chefs who were proud of their Peruvian cuisine and hopeful that a locavore and sustainable agriculture movement might come to Peru. We interviewed a U.S. delegate who specialized in wetland conservation and blue carbon. We met with Peruvian marine biology students and discussed the similarities between coastal waters in Peru and California—both rich temperate upwelling systems. We attended the high-level plenary opening ceremony and debates and heard leaders press for immediate and drastic action.
Prime Minister Enele Sopoaga addressing the audience at Opening of the High Level Segment
"The fossil fuels we are burning today are made from extinct plants and animals. Fossil fuels signify extinction," said Enele Sopoaga, prime minister of Tuvalu at the outset of high-level negotiations. "We must not condemn ourselves to extinction, by riding on the back of the extinct. We must strive for renewal… I want everyone to look into [a] child's eyes and imagine what they will see in 10 or 20 years. Will they see Hell? Or will they see a sustainable planet?… Let us make 2015 the year we saved the earth."
Our delegation, in partnership with PML and other science agencies, succeeded in working ocean climate issues into the COP20 media coverage and educating delegates about the importance and urgency of ocean changes. We also had successful presentations about ocean acidification at side events. At an event hosted by the U.S. State Department, Ph.D. student Natalya Gallo participated in a panel for the event "What goes up in the air, also goes into the sea."
Scripps marine biologist Lisa Levin presented in a pavilion hosted by Peru, "Ocean acidification interaction with multiple stressors and their impact on marine organisms." Levin and Gallo also presented at a public event, "Voces por el Clima," aimed at educating the people of Lima about climate change science and solutions. These presentations are key in spreading the importance of climate effects on the ocean and getting the oceans into policy legislation.
Everyone we met came from different backgrounds but they shared an urgency to committing to change. When it comes to combating climate change as well as other pressing environmental issues, the solutions will come at many levels, but the key is to keep sharing ideas and to keep working for that better future.
You can read the original publication of this story in Explorations Now
Giant plastic display at Voces por el Clima pushes the public to consider how much single use plastic they use
Dancers perform at the Opening of the High Level Segment
Chris Neighbors, video extraordinaire
Tourist trip to the ruins at Pachacamac with Dr Carol Turley, Natasha Gallo, Dr. Phil Williamson and James Johnson
Mr. Manuel Pulgar-Vidal, Minister of the Environment of Peru and President-Designate of COP 20
Display at the COP on Peru's habitat: Peru has 84 of 104 identified life zones on the planet, 27 of the world's climates, the world's longest tropical mountain range, the second largest portion of the Amazon and 71% of all tropical glaciers.
Kate, Noah and Chris at the COP
The COP venue this year was featured outdoor walkways, lawn and plazas as well as indoor meeting rooms
#climate change #Conference of Parties #COP #COP20 #UNFCCC #United Nations
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After a very white winter (yay powder!) it was fun to go chase the colors of spring for a week. What is your favorite time of year?
Turned 32 today. As I get ready to bring another life into the world, reflecting on what my mom did for me and what it means to create another being. PC: @kielrucker 📷 Hat and Bikini: @tahoenative👙 🎩
My husband is well trained. A friend spotted this bobcat on our drive home from shooting around the lake. I of course freaked out and called "big lens!" My husband had my 400mm out of the bag in the back, and in my hands in seconds! Always a fleeting and special to share a glimpse into the lives of wild animals, and as a photographer its even more special when you can snag a shot.
I've always loved watching storms—rain, snow, thunder and lightning. But rain at a distance is truly magical. My favorite part of this picture is the fact that the mountains on the left are blurred by an almost invisible sheet of rain we can just see falling into the lake.
Five minutes prior we were getting blown sideways in a snowstorm...
It got COLD this winter. Water is an amazing substance, it expands when frozen causing it to float, allowing live to thrive below.
Carter Niemeyer: Meeting the Man Himself
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{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
| 4,533
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Free exchange | German recovery
Still upbeat
Germans remains confident about their economy
By R.A. | WASHINGTON
TWO weeks ago, Germany released data indicating a blockbuster second quarter for the economy:
Figures released on August 13th showed that the German economy grew by 2.2% (an annualised rate of close to 9%) in the three months to the end of June, well above even the most optimistic forecasts. The German figures, the best since reunification almost two decades ago, meant that the euro-area economy had a good quarter, too. GDP in the 16-country block rose at an annualised rate of 4%—much faster than in America and only a bit shy of surprisingly strong growth figures in Britain.
The German case is an unusual bright spot in the developed world. As Paul Krugman points out, real German output remains below its previous peak as it does in America. But there is a key difference between the two—Germans remain upbeat while Americans grow ever more dour:
The Munich-based Ifo institute said its business climate index had risen to the highest since June 2007, defying expectations of a modest correction. That suggested the country's industry-led upswing had lost little momentum and could lift prospects across Europe...
Although slower growth is inevitable in the second half of the year, the Ifo survey indicated the pace of expansion would remain brisk. Germany's economy was on a "stable summer high," said Hans-Werner Sinn, Ifo's president.
Of course, after the summer comes the fall:
Details of the Ifo survey showed the improvement in business confidence in August was driven by a more optimistic assessment of current conditions. When asked about expectations for the next six months, however, the companies surveyed were less optimistic than in July, which was consistent with the slower but still robust growth expected by economists.
That lines up with other surveys showing reduced expectations for second half growth. Which, on its own, is not that big a deal; it's unreasonable to expect the German economy to grow at a 9% annual pace for more than a quarter or two. The trouble is that the German economy is driving European recovery, which continues to suffer an enormous drag from ongoing contraction in southern Europe. Things may look very ugly in the euro zone once the German recovery cools.
|
{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
| 5,837
|
def test_dummy():
assert True
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 4,619
|
using System;
using System.Collections.ObjectModel;
using XamarinFormsTester.Infrastructure.ReduxVVM;
using System.Linq;
using System.Windows.Input;
using Xamarin.Forms;
namespace XamarinFormsTester.ViewModels
{
public class DeviceSelectedAction : XamarinFormsTester.Infrastructure.ReduxVVM.Action {
public string deviceId;
}
public class StoreActionCommand<State> : ICommand where State : new(){
Store<State> store;
Func<XamarinFormsTester.Infrastructure.ReduxVVM.Action> execute;
public StoreActionCommand (Store<State> store, Func<XamarinFormsTester.Infrastructure.ReduxVVM.Action> execute)
{
this.execute = execute;
this.store = store;
}
public event EventHandler CanExecuteChanged;
public bool CanExecute (object parameter)
{
return true;
}
public void Execute (object parameter)
{
store.Dispatch (execute ());
}
}
public static class StoreMVVMExtensions {
public static ICommand createActionCommand<State>(this Store<State> store, Func<XamarinFormsTester.Infrastructure.ReduxVVM.Action> actionMaker) where State : new() {
return new StoreActionCommand<State> (store, actionMaker);
}
}
public class DeviceListPageViewModel : ViewModel
{
public ObservableCollection<DeviceSummary> Devices { get; set;}
public Boolean Pulling { get; set;}
public ICommand Clicked;
public DeviceListPageViewModel (Store<AppState> store)
{
Clicked = store.createActionCommand(() => new DeviceSelectedAction{});
store.Subscribe ((s) => {
Pulling = s.DevicePage.inProgress;
Devices = new ObservableCollection<DeviceSummary>(s.DevicePage.Devices.Select(d => new DeviceSummary{Name = d.Name, Location = d.Location}));
});
}
}
}
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 96
|
\section{Introduction}
Many statistical measures have been developed to distinguish between
the various cosmological models, ranging from direct measures of the
power spectrum and correlation function from redshift surveys, to
measures of velocity dispersion, bulk flows, and Mach number from
peculiar velocity surveys (cf., \cite{Strauss95} for a review). Each of
these measures is designed to be sensitive to different aspects of
various models, such as $\Omega$, the initial power spectrum, or the
Gaussian character of the initial phase distribution.
One can get an intuitive understanding of the dependencies of
different statistics by considering the following example. Compare
N-body simulations of four different cosmological models: standard CDM,
open CDM, CDM + $\Lambda$, and a pure HDM model, the details of which
are given in \S3. Each simulation is normalized to have the same value
of $\sigma_8$, the standard deviation of the mass fluctuations within an
8 ${h^{-1} \mpc}$ sphere, where $h$ is the Hubble parameter in units of 100
${\rm km~s^{-1}}~{\rm Mpc}^{-1}$. The two point correlation function, $\xi(r)$, for
each model is presented in Figure 1a. Despite their differences in
initial power spectra, the final correlation functions are similar in
all models. In general, the correlation function does not depend
strongly on $\Omega$ if one retains the freedom to set the $\sigma_8$
normalization. In contrast, Figure 1b shows the pairwise velocity
dispersion, $\avg{v^2}(r)$, as a function of separation for the same
set of models. A strong differentiation between the low
and high $\Omega$ models is apparent.
The velocity dispersion is easy to compute in simulations, but is
difficult to measure in the geometric projection of phase space that is
measured by redshift surveys. The traditional way to measure the
small-scale velocity dispersion of galaxies is via the anisotropy it
introduces in the redshift-space two-point correlation function
(\cite{Davis83}; \cite{Fisher94}; \cite{Marzke95}; \cite{Loveday96}).
However, as it is a pair-weighted statistic, it is dominated by the
densest regions, which are necessarily rare. Thus one finds large
variance between estimates of the small-scale velocity dispersion
between different samples (\cite{Mo93}; \cite{Zurek94}; \cite{Guzzo95};
\cite{Somerville97}), indicating that the small-scale velocity
dispersion measured in this way is not very robust. Attempts have been
made to use direct measures of peculiar velocities of galaxies as a
constraint on $\avg{v^2}$ (e.g., \cite{Strauss93}, \cite{Willick97}),
but other than the nearby universe, where the velocity field is
observed to be very quiet (\cite{Sandage86}; \cite{Brown87};
\cite{Burstein90}), the errors on the individual peculiar velocity
measurements swamp the signal from $\avg{v^2}$.
The primary goal of this paper is to present a statistic---the
redshift dispersion (${\sigma_z}$)---that captures ${\avg{v^2}}$ in a way which is
naturally applied to volume limited samples drawn from redshift surveys.
In the subsequent sections we motivate and present our redshift
dispersion statistic. Again, because it is a pair-weighted statistic,
computing ${\avg{v^2}}$ by averaging over all densities results in a value
heavily weighted by the densest regions. However, this problem can be
alleviated if the dispersion can be calculated {\it as a function of
density}. The theoretical motivation for our statistic stems from the
Cosmic Virial Theorem (CVT), derived in Peebles (1976a, hereafter P76),
which relates ${\avg{v^2}}$ to ${\Omega}$ and $\xi$. In \S2 we show that the
CVT can be applied to subsets of particles in a system which correspond
to surfaces of constant density. Many assumptions are necessary to
obtain the results shown in \S2, not all of which are obvious. In \S3 we
explore the relationships derived in \S2 with simple N-body simulations,
suggesting that the results of \S2 hold over a wide range of scales. The
redshift dispersion is entirely independent of the results of \S2 and
\S3, which provide a context for the redshift dispersion that would
otherwise be an unmotivated simulation based statistic. \S4 describes
how to compute the redshift dispersion from a redshift survey. \S5
contains our conclusions and remarks on future work.
\section{Velocity Dispersion on Surfaces of Constant Density}
As will be shown in \S4, the redshift dispersion probes the pairwise
peculiar velocity dispersion in regions of different density and is
designed for comparing simulated and observed redshift surveys. In this
section we attempt to provide a theoretical context and some motivation.
One of the main challenges of working with the pairwise velocity
dispersion arises from its strong density dependence. Intuitively, both
the number of galaxies and the velocity dispersion should be highest in
the densest regions. Thus, averaging over all densities will give
results dominated by rare, high density peaks. This problem can be
eliminated if the dispersion is calculated as a function of density.
In the case of clustered, gravitationally interacting particles there
exists a scaling relation between the velocity dispersion on small
scales ${\avg{v^2}} (r)$ averaged over pairs separated by a distance $r$,
and the two point correlation function $\xi(r)$, the excess fractional
probability of finding two particles with separation $r$. This result
is contained in the CVT derived in P76 (see also \cite{Peebles76b};
\cite{Davis77}; \cite{Peebles80}), which can be written
\begin{equation}
{\avg{v^2}}(r) \propto {\Omega} \xi(r) r^2 .
\end{equation}
Several assumptions are used to obtain Eq.~(1) (see P76), including
that $\xi$ is given by a power law $\xi(r) = (r_0/r)^\gamma$; $r \ll
r_0$, implying $\xi(r) \gg 1$; the three-point correlation function
$\zeta$ is given by a symmetrized product of the two-point correlation
function (see Eq.~[A23]); and the mass of each galaxy is concentrated on
scales smaller than their separation. The validity of these assumptions
is not obvious (\cite{Fisher94}). For an excellent discussion of the
implications of extended dark matter halos on the CVT, see Bartlett \&
Blanchard (1996). In addition, we are assuming that the {\it galaxy\/}
velocity field is unbiased with respect to that of the dark matter.
While theoretical investigations have given strong suggestions that a
velocity bias of order 20--30\% may exist (\cite{Couchman92},
\cite{Evrard94}, \cite{Gelb94}), the difficulty in reliably tracing
galaxies in simulations has prevented a good estimate of its magnitude
(\cite{Summers95}). For the motivational purposes of this paper,
let us keep these assumptions as we work to derive the density
dependence of the CVT.
To rewrite Eq.~(1) in terms of the density requires some new
notation. Let $\xi_{a \times b}(r)$ denote the cross-correlation of
particle sets $a$ and $b$. The two-point (auto) correlation function
of all particles in a system ${\cal P}$ can be written $\xi_{{\sP \! \times \! \sP}}(r)$. The
correlation function can also be written as the average of the
individual cross correlations
\begin{equation}
\xi(r) = {1 \over N_{{\cal P}}} \sum_{i \in {\cal P}} \xi_{{i \! \times \! \sP}}(r),
\end{equation}
where $N_{{\cal P}}$ is the total number of particles. We can write down a
similar expression for the velocity dispersion
\begin{equation}
{\avg{v^2}}(r) = {1 \over N_{{\cal P}}} \sum_{i \in {\cal P}} {\avg{v_{\icP}^2}} (r),
\end{equation}
where ${\avg{v_{\icP}^2}} (r)$ is the variance of the pairwise velocity between
particle $i$ and all particles in ${\cal P}$ which lie a distance $r$ from
$i$. We show in Appendix A that these expressions lead to a
generalization of the CVT that holds for each particle:
\begin{equation}
{\avg{v_{\icP}^2}}(r) \propto {\Omega} \xi_{{i \! \times \! \sP}}(r) r^2.
\end{equation}
If the CVT holds for each particle, then it holds for any subset, ${\cal S}
\subset {\cal P}$:
\begin{equation}
{\avg{v_{\ScP}^2}}(r) \propto {\Omega} \xi_{{\sS \! \times \! \sP}}(r) r^2,
\end{equation}
where
\begin{equation}
{\avg{v_{\ScP}^2}}(r) \equiv {1 \over N_{\cal S}}
\sum_{i \in {\cal S} \subset {\cal P}} {\avg{v_{\icP}^2}}(r),
\end{equation}
and
\begin{equation}
\xi_{{\sS \! \times \! \sP}}(r) \equiv {1 \over N_{\cal S}}
\sum_{i \in {\cal S} \subset {\cal P}} \xi_{{i \! \times \! \sP}}(r).
\end{equation}
Let $N_i(r)$ be the number of particles within a radius $r$ of the $i$th
particle, and ${\cal S}_n$ the set of all particles for which $N_i = n$.
For this subset, the CVT is
\begin{equation}
{\avg{v^2}}(n,r) = C_1 {\Omega} \xi(n,r) r^2 ,
\end{equation}
where ${\avg{v^2}}(n,r) \equiv {\avg{v_{\SncP}^2}}(r)$, $\xi(n,r) \equiv
\xi_{{\sS_n \! \times \! \sP}}(r)$ and $C_1$ is given by Eq.~(A32). The quantity
$\xi(n,r)$ is related to the expected number of particles within a
radius $r$ around any particle by
\begin{equation}
{\bar{\nu}} \int_0^r [1 + \xi(n,r')] 4 \pi r'^2 dr' = n,
\end{equation}
where ${\bar{\nu}}$ is the mean number of particles per unit volume. If
$\xi(n,r) \propto r^{-\gamma}$ and $\xi(n,r) \gg 1$ (as assumed in the
original P76 derivation), the above expression yields
\begin{equation}
{3 \over {3 - \gamma}} \xi(n,r) {\bar{n}}(r) = n,
\end{equation}
where ${\bar{n}}(r) \equiv {\bar{\nu}} \frac{4}{3} \pi r^3$.
Inserting the above expression into Eq.~(8) yields
\begin{equation}
{\avg{v^2}}(n,r) = C_1 C_2 {\Omega} n/{\bar{n}}(r)^{1/3},
\end{equation}
where $C_2 = (3-\gamma)/3(4\pi{\bar{\nu}}/3)^{2/3}$. The above expression
shows that the pairwise velocity dispersion is proportional to ${\Omega}$ and
the local density (through $n$) smoothed on a scale $r$. Finally,
since we are working with $N_i(r)$, which is the number of particles
{\it within} a volume of radius $r$ centered on a particle, it is
convenient to work with a similar velocity dispersion. Let ${\sigma_v^2}(r)$
be the average pairwise velocity dispersion of all the particles {\it
within} a volume of radius $r$. ${\sigma_v^2}(r)$ can be obtained by
integrating ${\avg{v^2}}(r)$ under the same assumptions used to derive
Eq.~(1)
\begin{equation}
{\sigma_v^2}(r) \equiv \frac{
\int_0^r {\avg{v^2}}(r') [1 + \xi(r')] 4 \pi r'^2 dr'}
{\int_0^r [1 + \xi(r')] 4 \pi r'^2 dr'}
= C_1 C_3 \frac{{\Omega} \xi(r)^2 r^5}{r^3 \xi(r)}
= C_3 {\avg{v^2}}(r),
\end{equation}
where $C_3 = (3-\gamma)/(5-2\gamma)$. Substituting Eq.~(12) into
(11) results in
\begin{equation}
{\sigma_v^2}(n,r) \propto {\Omega} n/{\bar{n}}(r)^{1/3}
\end{equation}
where the constant of proportionality is equal to
\begin{equation}
C_1 C_2 C_3
= \frac{3 {\Omega} Q M_\gamma
(3-\gamma)^2 }
{4(\gamma - 1)(2 - \gamma)(4 - \gamma)
(5-2\gamma)(4\pi{\bar{\nu}}/3)^{2/3}},
\end{equation}
and $Q$ and $M_\gamma$ are defined in Appendix A.
The above derivation of Eq.~(13) holds whether or not the quantities
are averaged over one or many particles with the same density. The
above velocity dispersion is computed with respect to the velocity of
the central particle, ${\bf{v}}$. The velocity dispersion can also be
computed with respect to the mean velocity of the particles in a cell,
${\bf{u}}$. In a given cell, these two values of the velocity dispersion will
differ by $|{\bf{u}}|^2 - |{\bf{v}}|^2$. The distribution of these differences
for all cells in ${\cal S}_n$ will peak at zero and have a width on the order
of ${\sigma_v^2}$. Subsequent averaging over many cells with the same density
will result in zero net difference. Thus, Eq.~(13) also applies for
velocity dispersions computed with respect to the mean velocity in the
cell if the results are averaged over many cells in ${\cal S}_n$.
Eq.~(13) conveniently relates two readily computable
quantities: the velocity dispersion with respect to the mean velocity in
a cell of radius $r$ to the number of particles in the cell, which is
proportional to the density. In the next section, we explore the above
form of the CVT with N-body simulations.
\section{N-Body Results}
The previous section presented a derivation for a relationship linking
the velocity dispersion to the local density. In this section we
explore the range over which Eq.~(13) holds using N-body
simulations of specific cosmological models with different initial power
spectra. We are particularly interested in the dependence on ${\Omega}$ when
the number of objects is of the same order as expected from volume
limited redshift surveys.
The simulations we consider are designed to probe a variety of popular
cosmological models. The four models are: standard CDM with $\Omega =
1$, $h=0.5$; HDM with $\Omega = 1$, $h=1.0$; open CDM with $\Omega =
0.35$, $h=0.7$; and CDM + $\Lambda$ with $\Omega_{CDM} = 0.35$,
$\Omega_{\Lambda} = 0.65$, $h=0.7$. The open CDM and CDM + $\Lambda$
models provide two alternatives to standard CDM that increase the ratio
of large scale to small scale power. They differ in evolution in that
structure formation ``freezes out'' at an earlier epoch in the open
model as the expansion rate exceeds the rate of gravitational collapse.
Thus, to achieve the same level of structure today, collapse must begin
earliest in the open CDM model, later in the CDM + $\Lambda$ model, and
latest in standard CDM model. While HDM is not generally considered a
viable theory, it provides a significantly different power spectrum
shape with which to explore our ideas.
The initial conditions are designed to treat the models, as much as
possible, on an even footing. All models assume a Harrison-Zel'dovich
primordial power spectrum, and use the same random phases for the
Fourier modes to generate the initial density field from their
respective power spectra. The CDM transfer functions are taken from
Efstathiou, Bond, \& White (1992, Eq.~[7]) with the parameter
$\Gamma\equiv\Omega_{CDM}\,h$. Although this function was not intended
for use in open models, it fits more detailed calculations to within 5\%
(D. N. Spergel 1995, private communication). The HDM transfer function is
taken from Holtzman (1989, Table 2A, line 52). As stated in \S1, each
model is normalized to have the same linear value of $\sigma_8=0.67$,
so as to provide similar correlation strengths and to isolate out the
velocity dependencies. Although this normalization does not match that
predicted from the observed fluctuations in the Cosmic Microwave
Background for some or all of the models (c.f., \cite{Stompor95};
\cite{Gorski95}), it roughly equalizes the amount of power on the
scales where this paper is focused.
Each of the simulations follows $32^3=32,768$ dark matter particles
within a periodic cube of comoving size 100 ${h^{-1} \mpc}$ (10,000 ${\rm km~s^{-1}}$) on a
side. The P3MG3A code (\cite{Brieu95}), which implements the P$^3$M
algorithm (\cite{Hockney81}; \cite{Efstathiou85}) on the GRAPE-3A
hardware board (\cite{Okumura93}), is used to evolve the simulations
from redshift $z=23$ to $z=0$ using 1200 time steps. A Plummer force
law with softening parameter of 156 ${h^{-1} \kpc}$ is used for the
gravitational interactions and the mass per particle is
$8.5\times 10^{12}~\Omega\,h^{-1}~M_{\sun}$. Each simulation took
approximately 2 hours to run on a Sun Sparc 10/51 workstation with a 4
chip GRAPE-3A board.
The ideal way to construct an artificial galaxy catalog is by
identifying concentrations of gaseous and stellar material in high
resolution N-body/hydrodynamic simulations. Current computer technology
and algorithms now permit such simulations on the scale of small groups
of galaxies (\cite{Evrard94}). However, at the present time the realm
of large volumes remain the domain of strictly gravitational N-body
codes. Identifying galaxies from dark matter halos is a significant
problem that may not be solvable (\cite{Summers95}); although a variety
of of impressive methods have been developed (\cite{Efstathiou88};
\cite{Bertschinger91}). However, to keep things as simple as possible
we choose a model in which mass traces light, and each particle is
assumed to be a galaxy. This approach neglects important processes,
such as mass and velocity bias, the dependence of bias on cosmological
models, and the interaction of dark matter halos. Nevertheless this
approach should be sufficient for our purely motivational purposes.
Future simulations which can both cover large volumes and resolve
galaxies hydrodynamically will hopefully clarify the nature of the bias.
To extract the velocity dispersion of a particular surface of constant
density, consider a set of particles where ${\bf{x}}_i$ and ${\bf{v}}_i$ are the
position and velocity of the $i$th particle. Place spherical cells of
radius $r$ on a uniform grid over the entire domain. Let $N_j$ be the
number of particles in cell $j$. The correct way to compute the velocity
dispersion in a cell is with respect to the mean motion of the
particles. As was argued in \S2, Eq.~(13) will apply if the results are
averaged over many cells with the same density. Denote the mean
velocity in the $j$th cell by ${\bf{u}}_j$; the velocity variance,
$\sigma_j^2$, is then $|{\bf{v}}_i - {\bf{u}}_j|^2$ averaged over all particles
in the cell. If ${\cal S}_n$ is the set of all cells having $n$ particles
({i{.}e{.}}, $N_j = n$), then the average velocity and variance as a function
of $n$ is
\begin{equation}
\mu_v(n,r) \equiv {1 \over N_{{\cal S}_n}} \sum_{j \in {\cal S}_n} |{\bf{u}}_j|.
\end{equation}
and
\begin{equation}
\sigma^2_v(n,r) \equiv
{1 \over N_{{\cal S}_n}} \sum_{j \in {\cal S}_n} \sigma^2_j,
\end{equation}
where $N_{{\cal S}_n}$ is the number of particles in the set ${\cal S}_n$ (Note:
$N_{{\cal S}_n} \neq n$). These equations provide a specific prescription for
computing the density dependence of the mean velocity and the velocity
dispersion, which can readily be applied to the simulations.
Plots of $\mu_v(n,r)$ and $\sigma_v(n,r)$ are shown in Figure~2 for
the four models. The cell size is $r = 194~{\rm km~s^{-1}}$, corresponding to
${\bar{n}}(r) = 1$. All distances are expressed in units of ${\rm km~s^{-1}}$. Figure~2
demonstrates three important points: both $\mu_v$ and $\sigma_v^2$ are
independent of the shape of the initial power spectrum; $\mu_v$ is
independent of the local density, while $\sigma_v^2$ is proportional to
the density; and $\mu_v$ and $\sigma_v^2$ have the same strong ${\Omega}$
dependence that ${\avg{v^2}}$ demonstrated in Figure~1b.
Formally, the power spectrum independence is explained by the
derivation of the CVT (see Appendix A). The velocity dispersion
depends upon the evolved power spectrum, which is essentially
indistinguishable between models (Figure~1). Moreover, any remaining
difference between models is encoded in the density distribution
function, which is not apparent when density is the {\it dependent\/}
variable in Figure~2.
To explore the range over which Eq.~(13) is valid, $\mu_v$ and ${\sigma_v^2}$
were calculated over the range of cell sizes $77~{\rm km~s^{-1}} \leq r \leq
488~{\rm km~s^{-1}}$, corresponding to $2^{-4} \leq {\bar{n}}(r) \leq 2^4$. We obtained
the following empirical fit for the mean velocity magnitude
\begin{equation}
\mu_v \propto {\Omega}^{1/2} /{\bar{n}}^{\alpha} .
\end{equation}
where $\alpha \sim 0.05$. As $\mu_v$ already contains the desired
${\Omega}$ dependence, it is convenient to represent $\sigma_v^2$ in
terms of the normalized velocity dispersion $\sigma_v^2/\mu_v^2$,
with the resulting fit
\begin{equation}
\sigma_v^2/\mu_v^2 \propto n/{\bar{n}}^{1/3} ,
\end{equation}
which agrees with the theory to within the small factor ${\bar{n}}^{2\alpha}$.
The quality of the fits can be observed by plotting $\mu_v$ and
${\sigma_v^2}/\mu_v^2$ against the scaled density ($n/{\bar{n}}^{1/3}$), for each
value of $r$. Figure~3a shows the ratio of $\mu_v$ to the fitted value
computed from Eq.~(17) for the ${\Omega}=1$ and ${\Omega}=0.35$ CDM models. Each
line of data corresponds to a different value of $r$, and has been
offset from the next by one unit. Figure~3b shows a similar plot for the
normalized velocity dispersion ${\sigma_z^2}/\mu_v^2$. The solid lines are
the best fits given by Eq.~(18); this has not been divided out. The
larger scatter at higher densities and smaller scales is due to the
small number of cells contributing to the calculation at these values.
The key point to observe from Figure~3a is how well the data points
follow the horizontal lines corresponding to their fitted values.
The fits to the data shown in Figure~3 are remarkable,
considering that the range includes cells with underdensities ($n/{\bar{n}} -
1$) of $-0.8$ on scales of $488~{\rm km~s^{-1}}$ and cells with overdensities of
nearly 200 on scales of $77~{\rm km~s^{-1}}$. Furthermore, the scaling of
$\sigma_v^2$ is almost exactly that derived from the CVT, indicating
that Eq.~(13) holds over a large range of scales and densities, even
when $\xi(r) \sim 1$.
\section{Redshift Dispersion}
The purpose of \S2 was to theoretically motivate the local density and
${\Omega}$ dependence of ${\sigma_v^2}$. In \S3 the scaling relations in \S2 were
explored with simple N-body simulations. In addition, \S\S2 and 3 have
introduced the ideas which allow us to describe the main point of this
paper---the redshift dispersion.
The formalism we have developed so far cannot be directly applied to
observations due to the geometric projection of a six dimensional phase
space into a three dimensional redshift survey. The redshift of
galaxies represents the only probe, albeit indirect, of the peculiar
velocity. Thus, any statistic that desires to take advantage of the
properties of ${\sigma_v^2}$ must be defined with the specific geometry of
redshift space in mind. In this section we now describe the redshift
dispersion, a statistic with the aim of being readily measurable from
volume limited samples taken from redshift surveys and which captures
the essence of ${\sigma_v^2}$.
Now let us define quantities analogous to those in \S3, but for
redshift space. Consider a volume limited sample from a survey out to a
maximum redshift of $Z$. Each data point consists of two angular
coordinates on the celestial sphere and a redshift. Let us define cells
within which to measure density and velocity dispersion in projection
on the celestial sphere. The cell $j$ consists of a cone emanating from
the origin with solid angle $\pi \theta^2$ around the cell center.
The number of points in the cone $j$ is $N_j$ and is proportional to the
projected density on the sky. The mean and the variance of the
redshifts in the cone are denoted $u_j$ and $\sigma_j^2$, which are
depicted schematically in Figure 4. If ${\cal S}_n$ is the set of all cones
with $N_j = n$, then the average and variance of the redshift as a
function of $n$ is
\begin{equation}
\mu_z(n,\theta) \equiv
{1 \over N_{{\cal S}_n}} \sum_{j \in {\cal S}_n} u_j.
\end{equation}
and
\begin{equation}
\sigma^2_z(n,\theta) \equiv
{1 \over N_{{\cal S}_n}} \sum_{j \in {\cal S}_n} \sigma^2_j.
\end{equation}
in analogy with Eqs.~(15) and (16).
The efficacy with which the ${\sigma_z}$ statistic might distinguish between
models is examined with the simulations discussed in \S3. The
simulations were transformed into redshift-space using $Z = 5000~{\rm km~s^{-1}}$,
which is equal to one half of the simulation box size. Since the data
are periodic, the origin can be placed at any point, allowing multiple
perspectives to be drawn from a single simulation. We computed
$\sigma_z$ for ${\Omega} = 1.0$ and ${\Omega} = 0.35$ CDM models. The angular
cell size was $\theta =1.8 \arcdeg$ (${\bar{n}}(\theta) = 2$), and the cell
centers were computed by creating a pseudo-uniform grid of points
across the celestial sphere (Baumgardner \& Frederickson 1985). The
data have been averaged over 27 different origins regularly distributed
throughout the domain, and error bars computed from the standard
deviation of this averaging.
The ${\sigma_z}$ curves are plotted in Figure~5. As with the velocity
dispersion in Figure~1b, the redshift dispersion shows a strong
separation between the low and high ${\Omega}$ models. The differences
become apparent at moderate angular over-densities ($\delta\sim 5$). The
shape of the curves in Figure~5 is due to the combined spatial and
peculiar velocity contributions to ${\sigma_z}$, as is illustrated in Figure~6
for the $\Omega = 1$ CDM simulation. We know the full six-dimensional
position of each galaxy in phase space in the N-body simulation, and
thus can separate these two contributions. At low overdensities, ${\sigma_z}$
is dominated by the spatial separation of the particles, as is indicated
by the solid points in Figure~6; the spatial component scales
approximately as $n^{-1/2}$, due to the more tightly bound nature of
denser systems. At higher overdensities the peculiar velocity dominates,
scaling approximately as $n^{1/2}$, which is consistent with results of
\S2 and \S3. For smaller values of ${\Omega}$ the spatial component behaves
the same, but the peculiar velocity component is down by a factor of
${\Omega}^{1/2}$.
These results indicate that the greatest separation in the dispersion
between models with different values of ${\Omega}$ will occur in the denser
regions; so it is important that we sample many modestly dense regions,
which requires a large volume. In addition, to minimize projection
effects the sample should not be too deep (i.e., $Z$ not too large).
Therefore, to apply the ${\sigma_z}$ statistic requires a dense sample over a
wide field. If believable simulations of galaxies can be developed it
might be possible to constrain models with the same final correlation
function by varying $\Omega$ in the simulations and finding the best fit
to the observations. For any one point, Figure~5 indicates an error of
about 0.15 in $\Omega$. Using many points along the curve, the errors
may be small enough to significantly constrain the value of $\Omega$.
The current redshift surveys do not have enough data to attempt such a
comparison. To get the level of separation shown in Figure~5 required
${\cal O}(10^4)$ particles. We have calculated $\sigma_z$ from the IRAS
1.2 Jansky survey (\cite{Fisher95}), but a volume-limited sample taken
from this survey contains only 800 galaxies at best. The error bars
from an 800 point sample in the simulations were too large to be able to
distinguish between high and low ${\Omega}$ models.
Fortunately, a substantial increase in the amount of data available
will be brought about by the Sloan Digital Sky Survey (SDSS). The SDSS
will obtain spectra and measure the redshifts of the $10^6$ galaxies
down to $r' \sim 18$ (\cite{Gunn95}). We can estimate the size of a
volume-limited sample taken from the SDSS using the Schechter luminosity
function $\phi_s$ fitted to the Stromlo-APM survey (\cite{Loveday92})
\begin{equation}
\phi_s(L) dL = \phi^{\star} y^{\alpha} e^{-y} dy, ~~~
y = L/L^{\star},
\end{equation}
where $\phi^{\star} \simeq 1.4\times 10^{-8}\,{\rm km^{-3}\,s^3}$,
$M^{\star} \simeq -19.5$, and $\alpha \simeq -0.97$ are parameters
obtained from the fit. The number of galaxies in a given volume,
$V$, brighter than $L_0$, $N(V,L_0)$, can be computed by
integrating the luminosity function
\begin{equation}
N(V,L_0) = V \int_{L_0}^{\infty} \phi(L) dL
= V \int_{y_0}^{\infty}
\phi^{\star} y^{\alpha} e^{-y} dy
= V \phi^{\star} \Gamma(\alpha + 1,y_0),
\end{equation}
where $y_0 = L_0/L^{\star}$. For a volume limited survey, $L_0$ is the
luminosity an object would have if it had an apparent magnitude $m_0$
and was located at the volume edge $Z$ ($H_0 = 100\,{\rm km~s^{-1}}\,{\rm Mpc}^{-1}$).
From this definition it follows that
\begin{equation}
y_0 = Z^2_{{\rm km~s^{-1}}}\,10^{0.4 (M^{\star} - m_0 +15)},
\end{equation}
where $m_0$ is the apparent magnitude limit of the survey. The
Stromlo-APM survey was taken in the $b_j$ band, while the Sloan will
use the $r'$ band. The two can be equated by the approximate relation
$b_j \sim r' + 1$. However, we set $m_0 = 18.7$ which gives a better
value for the estimated total number of galaxies in the survey
($10^6$). Inserting $Z = 5000\,{\rm km~s^{-1}}$ into Eq.~(23) gives $y_0 \approx
0.013$. Setting $V_{SDSS} = \pi Z^3/3$ results in $N_{SDSS} \approx
7000$. This number might in fact be appreciably larger if there are
many more faint galaxies than Eq.~(22) implies, and has been suggested
by recent surveys for low-surface brightness galaxies (cf.,
\cite{Dalcanton95}). However, these galaxies are unlikely to have
redshifts measured as part of the SDSS.
One could substantially increase the number of galaxies in a
volume-limited survey of fixed depth by dropping the requirement that it
include the origin. Figure~7 plots the expected number of galaxies in a
series of volume limited shells for SDSS and shows that a redshift shell
between $25,000\,{\rm km~s^{-1}}$ and $30,000\,{\rm km~s^{-1}}$ would include almost $10^5$
galaxies!
Ultimately, applying the redshift dispersion statistic to the SDSS
could provide many data points for comparing with simulations and
perhaps constraining $\Omega$. The ${\sigma_z}$ curves can be evaluated for
many angular sizes and redshift shells for both the SDSS and for several
next generation, high resolution, N-body/hydrodynamic simulations with
different values of ${\Omega}$. Since the number of estimated objects in the
SDSS is the same order or more as the simulations shown in Figure~5, one
might expect to obtain an equivalent separation between models for each
value. It should be interesting to compare ${\sigma_z}$ measured in different
simulations with the SDSS as a function of $n$, $\theta$ and shell
geometry.
\section{Conclusions}
Our goal has been to present a new statistic---the redshift dispersion
(${\sigma_z}$)---that is sensitive to ${\Omega}$ and is well suited for comparing
real and simulated volume limited samples from redshift surveys. Given
the proper data, ${\sigma_z}$ is easy to compute and can be applied on many
scales without applying arbitrary assumptions. We have used low
resolution simulations, which are sufficient for our motivational
purposes, to do a simple exploration of ${\sigma_z}$, which suggests that
applying it to the SDSS may be worthwhile. In addition, with the right
simulations, it might be possible to constrain ${\Omega}$ in models where the
simulations match the observed final correlation function.
We have shown that the pairwise velocity dispersion is intrinsically
related to the local density and ${\Omega}$. In \S2 and Appendix A it is
shown that the CVT holds for each particle in a system and subsequently
for any subset of particles, if we are careful to define the velocity
dispersion and correlation function for these subsets appropriately.
The density dependence of the velocity dispersion can be extracted by
looking at subsets of particles of a given local density (see Eq.~13).
Knowing how ${\sigma_v^2}$ depends upon the local density gives an indication
as to why the standard approach of averaging ${\avg{v^2}}$ over all
densities is highly sensitive to the presence of rare density peaks.
Exploring Eq.~(13) with N-body simulations of several cosmological
models indicates that it holds over a wide range of length scales,
$77~{\rm km~s^{-1}} \leq r \leq 488~{\rm km~s^{-1}}$. Our redshift dispersion statistic is
simply the redshift-space analog of the quantity ${\sigma_v}$.
We have treated our galaxies as equal mass particles containing all
the mass, ignoring the hypothesized global stochastic mapping from the
dark matter mass and velocity distribution to the distribution of
galaxies (i.e. the mass and velocity bias functions), which may differ
among the models. A more general treatment would estimate the mass and
velocity bias in each model and normalize the models such that the
galaxy correlations, not the dark matter correlations, are similar. At
best, reliable estimates of these bias functions await the next
generation of simulations where galaxies can be resolved within
statistically meaningful volumes. However, there are several arguments
as to why the differences in the biases between models may not strongly
affect our work. First, the initial power spectra of currently favored
hierarchical models all have similar slopes on galaxy formation scales.
Hence, the initial conditions of galaxy formation---and the resulting
bias---may be similar. In addition, the HDM model, which has a very
different initial power spectrum, has a similar final correlation
function, which suggests that the final power on small scales is
dominated by non-linear processes which erase initial differences. Any
velocity bias which arises through dynamical friction should be similar
for models evolved to similar clustering levels. If velocity bias is
related to galaxy formation sites preferentially near potential wells,
then velocity bias could depend on mass density and the efficacy of this
measure could be diminished. However, both types of bias appear to be
strong only in the most non-linear regions ($\delta\gtrsim200$)
(\cite{Summers95}), while our study focuses on the mildly non-linear
regimes ($\delta\sim10$).
High resolution, hydrodynamic simulations that include detailed galaxy
formation should improve our understanding of bias and how it changes
from model to model. Intuitively, any bias, by eliminating objects in
the underdense regions, should have the overall effect of shifting the
data points in Figure~5 to the left. Higher resolution will also
incorporate the interactions of dark matter halos, which may
significantly effect the three point correlation function on galaxy
scales (see \cite{Bartlett96}). All of these effects indicate more
detailed simulations, beyond what is currently available, may be
necessary for the actual application of the redshift dispersion.
The next logical step is to attempt to apply ${\sigma_z}$ to denser redshift
surveys than the IRAS 1.2 Jansky survey. In the mean time, additional
studies on larger volume, higher resolution N-body simulations would be
useful, but perhaps overkill until a suitable redshift survey becomes
available. Also, it is not clear that using dark matter halos without
the proper means for identifying galaxies would add to these results.
Further exploration should also be done on a wide variety of survey
geometries. Here, we only looked at a single small sphere. It is quite
possible that a larger sphere or a shell might be the optimal shape to
balance the tradeoff between high density and large numbers of clusters
that make ${\sigma_z}$ work best.
\acknowledgments We would like to thank David Spergel and our
editor Ed Bertschinger for their helpful comments, and John
Baumgardner for the use of his icosahedral mesh code. We gratefully
acknowledge the Grand Challenge Cosmology Consortium and grants of
computer time at the San Diego and Pittsburgh Supercomputer Centers.
MAS acknowledges the support of the WM Keck Foundation, the Alfred
P. Sloan Foundation, and NASA
Astrophysical Theory Grant NAG5-2882. This work was
supported by NSF grants ASC 93--18185 and AST 91--08103.
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 8,689
|
{% extends 'base.html' %}
{% block content %}
<div class="alert alert-error">
<strong>Not found!</strong>
</div>
<p>What you are looking for is not here, sorry.</p>
{% endblock %}
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 4,371
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Carriera
Con la ha disputato i Giochi olimpici di Rio de Janeiro 2016, due edizioni dei Campionati mondiali (2014, 2018) e cinque dei Campionati europei (2011, 2013, 2015, 2017, 2019).
Collegamenti esterni
Vincitori di medaglia d'argento olimpica per la Spagna
|
{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 1,318
|
module Vips
# The type of access an operation has to supply.
#
# * `:random` means requests can come in any order.
#
# * `:sequential` means requests will be top-to-bottom, but with some
# amount of buffering behind the read point for small non-local
# accesses.
#
# * `:sequential_unbuffered` means requests will be strictly
# top-to-bottom with no read-behind. This can save some memory.
class Access
end
end
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 8,137
|
namespace base {
namespace {
// The message loops on which each waitable event timer should be tested.
const MessageLoop::Type testing_message_loops[] = {
MessageLoop::TYPE_DEFAULT,
MessageLoop::TYPE_IO,
#if !defined(OS_IOS) // iOS does not allow direct running of the UI loop.
MessageLoop::TYPE_UI,
#endif
};
const int kNumTestingMessageLoops = arraysize(testing_message_loops);
void QuitWhenSignaled(WaitableEvent* event) {
MessageLoop::current()->QuitWhenIdle();
}
class DecrementCountContainer {
public:
explicit DecrementCountContainer(int* counter) : counter_(counter) {
}
void OnWaitableEventSignaled(WaitableEvent* object) {
--(*counter_);
}
private:
int* counter_;
};
void RunTest_BasicSignal(MessageLoop::Type message_loop_type) {
MessageLoop message_loop(message_loop_type);
// A manual-reset event that is not yet signaled.
WaitableEvent event(true, false);
WaitableEventWatcher watcher;
EXPECT_TRUE(watcher.GetWatchedEvent() == NULL);
watcher.StartWatching(&event, Bind(&QuitWhenSignaled));
EXPECT_EQ(&event, watcher.GetWatchedEvent());
event.Signal();
MessageLoop::current()->Run();
EXPECT_TRUE(watcher.GetWatchedEvent() == NULL);
}
void RunTest_BasicCancel(MessageLoop::Type message_loop_type) {
MessageLoop message_loop(message_loop_type);
// A manual-reset event that is not yet signaled.
WaitableEvent event(true, false);
WaitableEventWatcher watcher;
watcher.StartWatching(&event, Bind(&QuitWhenSignaled));
watcher.StopWatching();
}
void RunTest_CancelAfterSet(MessageLoop::Type message_loop_type) {
MessageLoop message_loop(message_loop_type);
// A manual-reset event that is not yet signaled.
WaitableEvent event(true, false);
WaitableEventWatcher watcher;
int counter = 1;
DecrementCountContainer delegate(&counter);
WaitableEventWatcher::EventCallback callback =
Bind(&DecrementCountContainer::OnWaitableEventSignaled,
Unretained(&delegate));
watcher.StartWatching(&event, callback);
event.Signal();
// Let the background thread do its business
base::PlatformThread::Sleep(base::TimeDelta::FromMilliseconds(30));
watcher.StopWatching();
RunLoop().RunUntilIdle();
// Our delegate should not have fired.
EXPECT_EQ(1, counter);
}
void RunTest_OutlivesMessageLoop(MessageLoop::Type message_loop_type) {
// Simulate a MessageLoop that dies before an WaitableEventWatcher. This
// ordinarily doesn't happen when people use the Thread class, but it can
// happen when people use the Singleton pattern or atexit.
WaitableEvent event(true, false);
{
WaitableEventWatcher watcher;
{
MessageLoop message_loop(message_loop_type);
watcher.StartWatching(&event, Bind(&QuitWhenSignaled));
}
}
}
void RunTest_DeleteUnder(MessageLoop::Type message_loop_type) {
// Delete the WaitableEvent out from under the Watcher. This is explictly
// allowed by the interface.
MessageLoop message_loop(message_loop_type);
{
WaitableEventWatcher watcher;
WaitableEvent* event = new WaitableEvent(false, false);
watcher.StartWatching(event, Bind(&QuitWhenSignaled));
delete event;
}
}
} // namespace
//-----------------------------------------------------------------------------
TEST(WaitableEventWatcherTest, BasicSignal) {
for (int i = 0; i < kNumTestingMessageLoops; i++) {
RunTest_BasicSignal(testing_message_loops[i]);
}
}
TEST(WaitableEventWatcherTest, BasicCancel) {
for (int i = 0; i < kNumTestingMessageLoops; i++) {
RunTest_BasicCancel(testing_message_loops[i]);
}
}
TEST(WaitableEventWatcherTest, CancelAfterSet) {
for (int i = 0; i < kNumTestingMessageLoops; i++) {
RunTest_CancelAfterSet(testing_message_loops[i]);
}
}
TEST(WaitableEventWatcherTest, OutlivesMessageLoop) {
for (int i = 0; i < kNumTestingMessageLoops; i++) {
RunTest_OutlivesMessageLoop(testing_message_loops[i]);
}
}
#if defined(OS_WIN)
// Crashes sometimes on vista. http://crbug.com/62119
#define MAYBE_DeleteUnder DISABLED_DeleteUnder
#else
#define MAYBE_DeleteUnder DeleteUnder
#endif
TEST(WaitableEventWatcherTest, MAYBE_DeleteUnder) {
for (int i = 0; i < kNumTestingMessageLoops; i++) {
RunTest_DeleteUnder(testing_message_loops[i]);
}
}
} // namespace base
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 6,587
|
class OwnershipsController < ApplicationController
before_action :logged_in_user
# HaveまたはWantボタンが押されたとき
def create
# render text: params
# createアクションにpostされたときのparams
# {"_method"=>"post", "authenticity_token"=>"bWM+d61ejMW1ZDCv56XbUGxVwikwalvNPBlnsJdAMqFylZ7TELg+oCcIWIygPTUqkZ9EdkjbYuzu2LSFBDEkDw==", "asin"=>"B00777SXPS", "type"=>"Want", "controller"=>"ownerships", "action"=>"create"}
if params[:asin]
# Itemをasinという値で検索して、存在する場合はそのデータを返し、それ以外は与えた値(今回はasin)でItem.newした状態のItemモデルを返します
@item = Item.find_or_initialize_by(asin: params[:asin])
else
@item = Item.find(params[:item_id])
end
# itemsテーブルに存在しない場合はAmazonのデータを登録する。
if @item.new_record?
begin
# TODO 商品情報の取得 Amazon::Ecs.item_lookupを用いてください
# response = {}
response = Amazon::Ecs.item_lookup(params[:asin], :response_group => 'Medium', :country => 'jp')
# render text: response.inspect
rescue Amazon::RequestError => e
return render :js => "alert('#{e.message}')"
end
amazon_item = response.items.first
@item.title = amazon_item.get('ItemAttributes/Title')
@item.small_image = amazon_item.get("SmallImage/URL")
@item.medium_image = amazon_item.get("MediumImage/URL")
@item.large_image = amazon_item.get("LargeImage/URL")
@item.detail_page_url = amazon_item.get("DetailPageURL")
@item.raw_info = amazon_item.get_hash
@item.save!
end
# TODO ユーザにwant or haveを設定する
# params[:type]の値ににHaveボタンが押された時には「Have」,
# Wantボタンがされた時には「Want」が設定されています。
type = params[:type]
if type == "Have"
current_user.have(@item)
# flash[:success] = "アイテムをHaveしました" # Ajaxだとflashメッセージが表示されない
end
if type == "Want"
current_user.want(@item)
# flash[:success] = "アイテムをWantしました" # Ajaxだとflashメッセージが表示されない
end
# redirect_to request.referrer # remote true するときは redirectは必要ない
end
# Wantedまたは Havedボタンが押されたとき
def destroy
@item = Item.find(params[:item_id])
# TODO 紐付けの解除。
# params[:type]の値ににHavedボタンが押された時には「Have」,
# Wantedボタンがされた時には「Want」が設定されています。
type = params[:type]
if type == "Want"
current_user.unwant(@item)
# flash[:success] = "アイテムをunWantしました" # Ajaxだとflashメッセージが表示されない
end
if type == "Have"
current_user.unhave(@item)
# flash[:success] = "アイテムをunHaveしました" # Ajaxだとflashメッセージが表示されない
end
# redirect_to request.referrer # remote true するときは redirectは必要ない
end
end
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 3,921
|
Adolfo John Canepa (ur. 17 grudnia 1940 w Londynie) – gibraltarski polityk. Były przewodniczący Stowarzyszenia na Rzecz Rozwoju Praw Obywatelskich, szef ministrów Gibraltaru od 8 grudnia 1987 do 25 marca 1988 roku, lider opozycji.
Od 2012 roku spiker Parlamentu Gibraltaru. 1 kwietnia 2014 roku mianowany na urząd burmistrza Gibraltaru.
Przypisy
Urodzeni w 1940
Szefowie ministrów Gibraltaru
Brytyjscy samorządowcy
Ludzie urodzeni w Londynie
|
{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 3,912
|
{"url":"http:\/\/nlab-pages.s3.us-east-2.amazonaws.com\/nlab\/show\/Ashoke+Sen","text":"nLab Ashoke Sen\n\nSelected writings\n\nOn microscopic explanation of Bekenstein-Hawking entropy via geometric engineering of black holes in string theory as bound states of D-branes:\n\nIntroducing Sen's conjecture on open string tachyon condensation and the decay of D-brane\/anti-D-brane pairs in superstring theory via open superstring tachyon condensation:\n\nwith review in:\n\nDiscussion of gravitational waves in relation to the soft graviton theorem:\n\n\u2022 Arnab Priya Saha, Biswajit Sahoo, Ashoke Sen, Proof of the Classical Soft Graviton Theorem in $D=4$ (arXiv:1912.06413)\ncategory: people","date":"2022-01-28 23:06:05","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 2, \"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\": 0.4785343110561371, \"perplexity\": 7923.651551287382}, \"config\": {\"markdown_headings\": false, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2022-05\/segments\/1642320306346.64\/warc\/CC-MAIN-20220128212503-20220129002503-00365.warc.gz\"}"}
| null | null |
What Makes Irish Whiskey...Irish?
What makes Irish whiskey so different from its counterparts in the whiskey world? Perhaps its unique smoothness and complex fruit flavors, with evident cereal grain notes, set it in a class all its own. Some will argue for it to truly be "Irish" it must be tripled distilled. Alas, this is not one of the legal requirements for it to be a true Irish whiskey. To be a whiskey which bears the name of The Emerald Isle it must meet the following legal requirements:
Must be made from mash of malted barley, plus other cereal grain.
Be mashed, fermented, and distilled to no more than 94.8% ABV.
Matured in wooden casks not exceeding 700 liters.
Must be a minimum age of 3 years and be aged in the Republic/Northern Ireland.
Contains no additives other then water and caramel coloring (e150a).
Retain the characteristics of its raw materials (in other words smell and taste like whiskey).
Be bottled at no less than 40% ABV.
The final product may be bottled outside of Ireland, but no whiskey can leave the island in wooden casks that impart additional aging.
If an age statement is on the bottle, the youngest whiskey that is added must be the age statement on the bottle (i.e., if the blend is a 14 year old whiskey and a 19 year old whiskey, the bottle must be labeled as a 14 year old whiskey).
The Spirits Act of 1880 breaks down the complex and profound art of Irish whiskey making. It is a trove of legal language defining, clarifying, and solidifying what makes the Irish spirit different from its counterparts.
Those bold enough to get through the entire Spirits Act, without making their eyes cross, will be among the rare whiskey drinkers. The few who can raise their glass, say "sláinte mhaith", and truly understand what it means to enjoy a glass of whiskey from Éire (that's Gaelic for Ireland)!
Is that enough though? Is a legal requirement all that makes Irish whiskey unique? If any credence is given to the notion of terroir (the idea that a spirits' place of origin, its environment, contributes to its uniqueness and flavor), then surely the Irish whiskey cannot simply be a product of laws and treaties.
There is a Colorado distillery that makes an "Irish Style" whiskey. Perhaps this provides the best opportunity to decide if terroir makes a difference.
Cliffs of Moher. Photo: Deposit
Raising a Glass to the man we call Rabbie Burns
Whiskey or Whisky?
An Introduction to Whiskey Barrels
|
{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
| 2,199
|
Q: Stop wrapping text around font awesome icons Hi actually i tried to prevent the text that wrapping around the font awesome icons. I tried so many methods. but those didn't work in this case. Please help me to fix this.
body {
font-family: "Montserrat";
}
.fa {
margin-right: 10px;
font-size: 20px !important;
}
<link href="https://cdn.paperindex.com/bootstrap/css/font-awesome.min.css" rel="stylesheet">
<link href="https://fonts.googleapis.com/css?family=Lato|Montserrat" rel="stylesheet" type="text/css">
<div class="dashboard-profile-wrp dashboard-todo">
<p><i class="fa fa-rocket mob-mnu-icon" aria-hidden="true" data-original-title="" title=""></i><a href="/dashboard/boost-your-sales.html" class="link" data-original-title="" title="">Upgrade Membership</a> to <b>connect with buyers like never before</b> and grow your sales beyond your local and national boundaries.and grow your sales beyond your local and national</p>
<p><i class="fa fa-shopping-basket fa-fw" aria-hidden="true"></i><a href="/dashboard/boost-your-sales.html" class="link" data-original-title="" title="">Upgrade Membership</a> to <b>connect with buyers like never before</b> and grow your sales beyond your
local and national boundaries.and grow your salesgrow your sales </p>
<p><i class="fa fa-camera fa-fw" aria-hidden="true"></i><a href="/dashboard/boost-your-sales.html" class="link" data-original-title="" title="">Upgrade Membership</a> to <b>connect with buyers like never before</b> and grow your sales beyond your local
and national boundaries.and grow your sales beyond your local and national boundaries.</p>
</div>
A:
body {
font-family: "Montserrat";
}
.fa {
margin-right: 10px;
font-size: 20px !important;
}
<link href="https://cdn.paperindex.com/bootstrap/css/font-awesome.min.css" rel="stylesheet">
<link href="https://fonts.googleapis.com/css?family=Lato|Montserrat" rel="stylesheet" type="text/css">
<div class="dashboard-profile-wrp dashboard-todo">
<p>
<p><i class="fa fa-rocket mob-mnu-icon" aria-hidden="true" data-original-title="" title=""></i></p>
<a href="/dashboard/boost-your-sales.html" class="link" data-original-title="" title="">Upgrade Membership</a> to <b>connect with buyers like never before</b> and grow your sales beyond your local and national boundaries.and grow your sales beyond your local and national</p>
<p><p><i class="fa fa-shopping-basket fa-fw" aria-hidden="true"></i></p><a href="/dashboard/boost-your-sales.html" class="link" data-original-title="" title="">Upgrade Membership</a> to <b>connect with buyers like never before</b> and grow your sales beyond your
local and national boundaries.and grow your salesgrow your sales </p>
<p><p><i class="fa fa-camera fa-fw" aria-hidden="true"></i><a href="/dashboard/boost-your-sales.html" class="link" data-original-title="" title=""></p>Upgrade Membership</a> to <b>connect with buyers like never before</b> and grow your sales beyond your local
and national boundaries.and grow your sales beyond your local and national boundaries.</p>
</div>
A: Give the paragraphs a left padding, and move the icon left using text-indent:
body {
font-family: "Montserrat";
}
p {
padding-left: 1.5em;
}
.fa {
text-indent: -1.3em;
font-size: 20px !important;
}
<link href="https://cdn.paperindex.com/bootstrap/css/font-awesome.min.css" rel="stylesheet">
<link href="https://fonts.googleapis.com/css?family=Lato|Montserrat" rel="stylesheet" type="text/css">
<div class="dashboard-profile-wrp dashboard-todo">
<p><i class="fa fa-rocket mob-mnu-icon" aria-hidden="true" data-original-title="" title=""></i><a href="/dashboard/boost-your-sales.html" class="link" data-original-title="" title="">Upgrade Membership</a> to <b>connect with buyers like never before</b> and grow your sales beyond your local and national boundaries.and grow your sales beyond your local and national</p>
<p><i class="fa fa-shopping-basket" aria-hidden="true"></i><a href="/dashboard/boost-your-sales.html" class="link" data-original-title="" title="">Upgrade Membership</a> to <b>connect with buyers like never before</b> and grow your sales beyond your
local and national boundaries.and grow your salesgrow your sales </p>
<p><i class="fa fa-camera" aria-hidden="true"></i><a href="/dashboard/boost-your-sales.html" class="link" data-original-title="" title="">Upgrade Membership</a> to <b>connect with buyers like never before</b> and grow your sales beyond your local
and national boundaries.and grow your sales beyond your local and national boundaries.</p>
</div>
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 6,587
|
Q: Terraform-Cloudformation- aws instance provider: Provided Arn is not in correct format I am creating a cloudformation stack to generate aws instance scheduler in aws gov cloud via TF. The goal is to start/stop ec2 based on tags. Many way to achieve it but I have to use terraform and cloudformation. Here is the repo --> https://github.com/Vinod1908/TestTerraform/blob/master/instanceScheduler.tf
Below is the part of the code where I think I am blocked:
"InstanceSchedulerEncryptionKey": {
"Type": "AWS::KMS::Key",
"Properties": {
"Description": "Key for SNS",
"Enabled": true,
"EnableKeyRotation": true,
"KeyPolicy": {
"Statement": [
{
"Sid": "default",
"Effect": "Allow",
"Principal": {
"AWS": {
"Fn::Sub": "arn:$${AWS::Partition}:iam::$${AWS::AccountId}:root"
}
},
"Action": "kms:*",
"Resource": "*"
},
{
"Sid": "Allows use of key",
"Effect": "Allow",
"Principal": {
"AWS": {
"Fn::GetAtt": [
"SchedulerRole",
"Arn"
]
}
},
"Action": [
"kms:GenerateDataKey*",
"kms:Decrypt"
],
"Resource": "*"
}
]
}
}
},
"Code": {
"S3Bucket": {
"Fn::Join": [
"-",
[
"solutions",
{
"Ref": "AWS::Region"
}
]
]
},
"S3Key": "aws-instance-scheduler/v1.3.1/instance-scheduler.zip"
The error :
Error: error waiting for CloudFormation Stack creation: failed to create CloudFormation stack, rollback requested (ROLLBACK_COMPLETE): ["The following resource(s) failed to create: [InstanceSchedulerEncryptionKey, SchedulerRule]. Rollback requested by user."
"Resource creation cancelled" "Parameter arn:aws:lambda:us-gov-west-1:###########..:function:Schedule-InstanceSchedulerMain is not valid. Reason: Provided Arn is not in correct format. (Service: AmazonCloudWatchEvents; Status Code: 400; Error Code: ValidationException; Request ID: 37adac0c-6758-4b4f-ac86-0d0140742c80; Proxy: null)"]
Not sure if it's doable in gov cloud but I am looking for potential solutions and found this https://github.com/awslabs/aws-instance-scheduler/issues/11. I am testing it but no success yet.. please help !!
Adding a new line:
Thank you all for the response. My issue was using the correct arn arn:aws-us-gov
I just apply the code and it's going through. Now I am getting this below and I am sure it's related to the policy/role on my s3. Please let me know what is wrong in my code below. Any thoughts?
the s3 code part:
"SchedulerPolicy": {
"Type": "AWS::IAM::Policy",
"Metadata": {
"cfn_nag": {
"rules_to_suppress": [
{
"id": "W12",
"reason": "All policies have been scoped to be as restrictive as possible. This solution needs to access ec2/rds resources across all regions."
}
]
}
},
"Properties": {
"PolicyName": "SchedulerPolicy",
"Roles": [
{
"Ref": "SchedulerRole"
}
],
"PolicyDocument": {
"Version": "2012-10-17",
"Statement": [
{
"Effect": "Allow",
"Action": [
"logs:CreateLogGroup",
"logs:CreateLogStream",
"logs:PutLogEvents",
"logs:PutRetentionPolicy"
],
"Resource": [
{
"Fn::Join": [
":",
[
"arn:aws-us-gov:logs",
{
"Ref": "AWS::Region"
},
{
"Ref": "AWS::AccountId"
},
"log-group",
{
"Ref": "SchedulerLogGroup"
},
"*"
]
]
},
{
"Fn::Join": [
":",
[
"arn:aws-us-gov:logs",
{
"Ref": "AWS::Region"
},
{
"Ref": "AWS::AccountId"
},
"log-group:/aws/lambda/*"
]
]
}
]
},
{
"Effect": "Allow",
"Action": [
"s3:GetObject",
"s3:PutObject",
"s3:*"
],
"Resource": {
"Fn::Join": [
":",
[
"arn:aws-us-gov:s3:::instanceschedulertest",
"arn:aws-us-gov:s3:::instanceschedulertest/*"
]
]
}
},
{
"Effect": "Allow",
"Action": [
"rds:DeleteDBSnapshot",
"rds:DescribeDBSnapshots",
"rds:StopDBInstance"
],
"Resource": {
"Fn::Join": [
":",
[
"arn:aws-us-gov:rds:*",
{
"Ref": "AWS::AccountId"
},
"snapshot:*"
]
]
}
},
{
"Effect": "Allow",
"Action": [
"rds:AddTagsToResource",
"rds:RemoveTagsFromResource",
"rds:DescribeDBSnapshots",
"rds:StartDBInstance",
"rds:StopDBInstance"
The error:
Error: error waiting for CloudFormation Stack creation: failed to create CloudFormation stack, rollback requested (ROLLBACK_COMPLETE): ["The following resource(s) failed to create: [Main]. Rollback requested by user." "Your access has been denied by S3, please make sure your request credentials have permission to GetObject for solutions-us-gov-west-1/aws-instance-scheduler/v1.3.1/instance-scheduler.zip. S3 Error Code: AccessDenied. S3 Error Message: Access Denied (Service: AWSLambdaInternal; Status Code: 403; Error Code: AccessDeniedException; Request ID: 95db6874-d4ad-4499-95f7-f73777a6d4db; Proxy: null)"]
Thank you all for all the pointers I really appreciate your input.
A: The reason why it is failing is because you are forming the wrong ARN in your Terraform Code.
In your repo,
link
Replace these following lines with respect to Lambda: 1047, 1358, 1420 as "arn:aws-us-gov:lambda" instead of "arn:aws:lambda".
As per the documentation of aws: The ARN should be in this format arn:aws-us-gov:lambda:account-id:function:function-name.
The answer to your question is to update the above-mentioned line. But I am sure you will get errors with respect to other resources as all resources which you are creating are in the Us-Region. So please update all the necessary Joining Function Arn lines which your forming in your code. :)
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 3,662
|
{"url":"https:\/\/www.physicsforums.com\/threads\/is-there-a-problem-with-the-latex.308798\/","text":"Bug Is there a problem with the latex?\n\n1. Apr 20, 2009\n\nMatterwave\n\nA lot of threads I've been looking at doesn't display the Latex code. It just says: Latex code: blahblahblah. All underlined. What's wrong?\n\n2. Apr 21, 2009\n\nIvan Seeking\n\nStaff Emeritus\nWe have a notice up at the top of the page.\n\n3. Apr 21, 2009\n\nMatterwave\n\nAh, shows me for not reading notices. :P\n\nThanks.\n\n4. Apr 22, 2009\n\nSparky_\n\nI no longer see the announcement but I can't post Latex code and have it look correct.\n\nI've even tried to cut and paste older Latex lines that look fine but the new \"preview\" do not look correct.\n\nAs far as you know - is Latex working now? is the problem on me?\n\n5. Apr 22, 2009\n\nGreg Bernhardt\n\nlatex is still down, sorry\n\n6. Apr 22, 2009\n\nSparky_\n\nno problem\n\nThanks\n\n7. Apr 23, 2009\n\ncaseyjay\n\nIs Latex still down? I can't seem to produce right Latex despite I thought what I typed was compliant to Latex code.\n\n8. Apr 23, 2009\n\nHootenanny\n\nStaff Emeritus\nYes, unfortunately, LaTeX is still down. We are aware of the problem and hopefully chroot will be able to resolve it soon.\n\n9. Apr 23, 2009\n\nCompuChip\n\nlol @ people keeping asking if LaTeX is still down.\n\nIf you type something correct between tex-tags and it doesn't show, it's still down. If it does show, it's up again :)\n\n10. Apr 23, 2009\n\ntiny-tim\n\nwhere are we?\n\nI thought the bronx was up, and the battery was down.\n\n11. Apr 24, 2009\n\nsnoopies622\n\n12. Apr 24, 2009\n\ntiny-tim\n\nneed to re-type?\n\nHi snoopies622!\n\nI'm fairly sure you'll need to re-type it \u2026\n\nI think the server looks for a thingy created at the time of posting, and of course it won't be able to find one.\n\nAlternatively, edit now, using extra symbols and the X2 and X2 tags.\n\n13. Apr 24, 2009\n\nsnoopies622\n\nRe: need to re-type?\n\nI'm sorry; I'm not sure what this means. Not using LaTeX? Can one make a decent looking fraction without it?\n\n14. Apr 24, 2009\n\nMoonbear\n\nStaff Emeritus\nRe: need to re-type?\n\nDecent enough, yes. I just go for plain and simple, like: 1\/2\nBut what tiny tim means is: 1\/2\n\n15. Apr 24, 2009\n\ntiny-tim\n\nhow vulgar!\n\nHi Moonbear!\nYou call it a plain and simple fraction \u2026\n\nsome people call it vulgar!!\nHonestly, I never thought of that!\n\nThat's quite neat!\n\n16. Apr 24, 2009\n\nMoonbear\n\nStaff Emeritus\nRe: how vulgar!\n\nWell, I never thought of it until I read your post here. :rofl: We'll call it a collaborative effort, then!\n\n17. Apr 25, 2009\n\nCompuChip\n\nRe: how vulgar!\n\nPatent pending... ?\n\n18. Apr 25, 2009\n\nsnoopies622\n\nThanks all, but I think I'll just wait for the LaTeX to return. I hope the problem is fixed soon.\n\n19. Apr 27, 2009\n\nchroot\n\nStaff Emeritus\nTesting...\n\n$E=mc^2$\n\n20. Apr 27, 2009\n\nchroot\n\nStaff Emeritus\n$\\LaTeX$ is back up. I'm still not a fan of the font, though, so I'm going to try tweaking a few more things.\n\n- Warren\n\n21. Apr 27, 2009\n\nKurdt\n\nStaff Emeritus\nThanks chroot!\n\n22. Apr 30, 2009\n\nCompuChip\n\n$$\\mathrm{Y^A{}_Y}, \\text{ it } = \\cup p \\text{ again!}$$\n\n23. May 10, 2009\n\nnyrychvantel\n\nMust we use LateX? Can't we switch to other language such as Wikipedia's one?\nWell i notice some restrictions in LateX,\nfirstly, the mathematics symbols are sometimes too small and may cause confusion,\n\nsuch as this, $$\\mathop {\\lim }\\limits_{x \\to \\infty } \\sum\\limits_{x = 1}^N {\\left( {{a_1}{b_{n + 1}}} \\right)\\frac{{{\\partial ^2}}}{{\\sqrt {\\frac{{n!}}{{r!\\left( {n - r} \\right)!}}} }}}$$\n\nIs that x-1 or x=1 ?\n\nsecondly, LateX doesn't know how to position itself in a suitable manner.\nfrom the above example, the sentence \"such as this\" should position at the same line as lim x->$$\\infty$$ , instead of above it. Another example $${x^2}\\frac{\\partial }{x}$$\nIn wikipedia, all equations are automatically adjusted to fit into the middle of the sentence for better viewing experience.\n\nSorry, these are just my random grumble....\n\nok i feel better now after my grumble.\n\n24. May 11, 2009\n\nCompuChip\n\nAFAIK, Wikipedia is using LaTeX.\nIn your first expression, it is x = 1. From the context it makes sense, if you have any doubt just click on it and check it.\nEven inline, you can write lim x -> $\\infty$ and $x^2 \\frac{\\partial}{\\partial x}$ very well if you just use itex tags instead of tex.\n\nThat said, if it takes a lot of server resources and - agreed - doesn't always look great, I am noticing that many sites, such as http:\/\/planetmath.org\/encyclopedia\/PropertiesOfRiemannStieltjesIntegral.html [Broken] which seems to work rather nicely except for some messages at the top when not all recommended fonts are installed.\n\nLast edited by a moderator: May 4, 2017\n25. May 11, 2009\n\nfluppocinonys\n\nSince PF is one of the most renowned science forums in the web currently, I thought the standard required would be higher.\n\nI hardly use LateX now unless necessary.\nEven if I have to use mathematical language to better illustrate my question, I normally use MathType\u00ae and save it as .GIF, then upload it to tinypic.com\nSuch as this,\n\nAttached Files:\n\n\u2022 2nl5t1x.gif\nFile size:\n9 KB\nViews:\n128\nLast edited: May 11, 2009","date":"2018-07-22 22:55:54","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 1, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.683626651763916, \"perplexity\": 3988.569601570584}, \"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-2018-30\/segments\/1531676594018.55\/warc\/CC-MAIN-20180722213610-20180722233610-00299.warc.gz\"}"}
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{"url":"https:\/\/www.ideals.illinois.edu\/handle\/2142\/97005","text":"## Files in this item\n\nFilesDescriptionFormat\n\napplication\/vnd.openxmlformats-officedocument.presentationml.presentation\n\n972183.pptx (2MB)\nPresentationMicrosoft PowerPoint 2007\n\napplication\/pdf\n\n2649.pdf (19kB)\nAbstractPDF\n\n## Description\n\n Title: DETERMINING THE CONCENTRATIONS AND TEMPERATURES OF PRODUCTS IN A CF4\/CHF3\/N2 PLASMA VIA SUBMILLIMETER ABSORPTION SPECTROSCOPY Author(s): Helal, Yaser H. Contributor(s): Armacost, Michael D.; Stout, Phillip J.; Craver, Barry; Agarwal, Ankur; Ewing, Paul R.; De Lucia, Frank C.; Neese, Christopher F. Subject(s): Instrument\/Technique Demonstration Abstract: Plasmas used for the manufacturing of semiconductor devices are similar in pressure and temperature to those used in the laboratory for the study of astrophysical species in the submillimeter (SMM) spectral region. The methods and technology developed in the SMM for these laboratory studies are directly applicable for diagnostic measurements in the semiconductor manufacturing industry. Many of the molecular neutrals, radicals, and ions present in processing plasmas have been studied and their spectra have been cataloged or are in the literature. In this work, a continuous wave, intensity calibrated SMM absorption spectrometer was developed as a remote sensor of gas and plasma species. A major advantage of intensity calibrated rotational absorption spectroscopy is its ability to determine absolute concentrations and temperatures of plasma species from first principles without altering the plasma environment. An important part of this work was the design of the optical components which couple $500-750$ GHz radiation through a commercial inductively coupled plasma chamber. The measurement of transmission spectra was simultaneously fit for background and absorption signal. The measured absorption was used to calculate absolute densities and temperatures of polar species. Measurements for chem{CHF_3}, chem{CF_2}, FCN, HCN, and CN made in a chem{CF_4}\/chem{CHF_3}\/chem{N_2} plasma will be presented. Temperature equilibrium among species will be shown and the common temperature is leveraged to obtain accurate density measurements for simultaneously observed species. The densities and temperatures of plasma species are studied as a function of plasma parameters, including flow rate, pressure, and discharge power. Issue Date: 6\/20\/2017 Publisher: International Symposium on Molecular Spectroscopy Citation Info: APS Genre: Conference Paper \/ Presentation Type: Text Language: English URI: http:\/\/hdl.handle.net\/2142\/97005 DOI: 10.15278\/isms.2017.TH11 Date Available in IDEALS: 2017-07-272018-01-29\n\ufeff","date":"2020-06-01 06:29:28","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 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\": 0.507533609867096, \"perplexity\": 5036.933848146851}, \"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-2020-24\/segments\/1590347414057.54\/warc\/CC-MAIN-20200601040052-20200601070052-00009.warc.gz\"}"}
| null | null |
\section{Introduction}
Let $\Omega$ be an open bounded domain in $\mathbb{R}^3$ with boundary $\partial\Omega$. We denote the space-time cylinder by $Q=\Omega\times (0,T)$, where $T>0$ is given. In this paper we prove some error estimates for the numerical approximation of a distributed optimal control problem governed by the evolution Navier-Stokes-Voigt equations with pointwise control constraints. More precisely, we consider the following problem
\begin{equation*}
(P)\hspace{0.5cm}
\begin{cases}
\min\, J(y,u)\\
u\in U_{\alpha,\beta},
\end{cases}
\end{equation*}
where
\begin{multline*}
J(y,u)=\dfrac{\alpha_T}{2}\int_{\Omega}|y(x,T)-y_T(x)|^2dx + \dfrac{\alpha_Q}{2}\iint_{Q}|y(x,t)-y_Q(x,t)|^2dxdt\\
+\dfrac{\gamma}{2}\iint_{Q}|u(x,t)|^2dxdt,
\end{multline*}
and $U_{\alpha,\beta}$ is the set of admissible controls defined for given real constants $\alpha_j,\beta_j, j=1,2,3$, by
\begin{equation*}
\label{AdSet}
U_{\alpha,\beta}=\{u\in \mathbb{L}^2(Q): \alpha_j\le u_j(x,t)\le \beta_j \text{ a.e. } (x,t)\in Q, j=1,2,3\}.
\end{equation*}
Here the free variables - state $y$ and control $u$ - have to fulfill the following 3D Navier-Stokes-Voigt (sometimes written Voight) equations
\begin{equation}
\text{}
\begin{cases}\label{NSV1}
y_t- \nu\Delta y -\alpha^2\Delta y_t +(y\cdot\nabla)y+\nabla p&=u, \;x\in \Omega, t>0,\\
\hfil \nabla \cdot y&=0,\; x\in \Omega, t>0,\\
\hfil y(x,t)&=0, \; x\in\partial\Omega, t>0,\\
\hfil y(x,0)&=y_0(x),\; x\in\Omega.\\
\end{cases}
\end{equation}
In system \eqref{NSV1}, $y=y(x,t)=(y_1(x,t),y_2(x,t),y_3(x,t))$ is the velocity, $y_0=y_0(x)$ is the initial velocity, $p=p(x,t)$ is the pressure, $\nu >0$ is the kinematic viscosity coefficient, and $\alpha\ne 0$ is the length-scale parameter characterizing the elasticity of the fluid.
To study the above optimal control problem we assume that
\begin{itemize}
\item The domain $\Omega$ is an open bounded subset of $\mathbb{R}^3$ with $C^2$ boundary $\partial \Omega.$
\item The initial value $y_0$ is a given function in $D(A)$. The desired states have to satisfy $y_T\in V$ and $y_Q\in \mathbb{L}^2(Q).$
\item The coefficients $\alpha_T,\alpha_Q$ are non-negative real numbers, where at least one of them is positive to get a non-trivial objective functional. The regularization parameter $\gamma$, which measures the cost of the control, is also a positive number.
\end{itemize}
The Navier-Stokes-Voigt equations was introduced by Oskolkov in \cite{Oskolkov} as a model of motion of certain linear viscoelastic incompressible fluids. This system was also proposed by Cao, Lunasin and Titi in \cite{Cao} as a regularization, for small values of $\alpha$, of the 3D Navier-Stokes equations for the sake of direct numerical simulations. The presence of the regularizing term $-\alpha^2 \Delta u_t$ in \eqref{NSV1} has two important consequences. First, it leads to the global well-posedness of \eqref{NSV1} both forward and backward in time, even in the case of three dimensions. Second, it changes the parabolic character of the limit Navier-Stokes equations, so the Navier-Stokes-Voigt system behaves like a damped hyperbolic system. In fact, the Navier-Stokes-Voigt system is perhaps the newest model in the the so-called $\alpha$-models in fluid mechanics (see e.g. \cite{Holst2010}), but it has attractive advantage over other $\alpha$-models in that one does not need to impose any additional artificial boundary condition (besides the Dirichlet boundary conditions) to get the global well-posedness. We also refer the reader to \cite{EbrahimiHolstLunasin} for an interesting application of the Navier-Stokes-Voigt equations in image inpainting.
In the past years, the existence and long-time behavior of solutions to the Navier-Stokes-Voigt equations has attracted the attention of many mathematicians. In bounded domains or unbounded domains satisfying the Poincar\'{e} inequality, there are many results on the existence and long-time behavior of solutions in terms of existence of attractors, see e.g. \cite{AT2013, Gal2015, Damazio2016, GMR2012, KT2009, Qin2012, YueZhong}. In the whole space, the existence and time decay rates of solutions have been studied in \cite{AT2016, Niche2016, ZZ2015}. However, to the best of our knowledge, there are few works on optimal control problems of Navier-Stokes-Voigt equations, except two recent works \cite{AN2016, AN2017} where the quadratic optimal control and time optimal control problems with distributed controls were investigated. In this paper we continue studying a numerical scheme for the distributed optimal control problem of this system in both time and space variables.
Since the pioneering work \cite{AT1990} in 1990 of Abergel and Temam, optimal control problems for Navier-Stokes equations have been studied by many authors in the last three decades. The analysis of these control problems is well understood, see e.g. \cite{G2003, HK2001, Sritharan1998, TW2006, Wachsmuth} and references therein, where various aspects including first and second order necessary conditions are developed and analyzed. To the contrary, numerical analysis of such optimal control problems is quite limited. This is due to the fact that the restricted regularity of solutions of the evolutionary Navier-Stokes equations, as well as the divergence free condition, and the convective nature of the adjoint equation of the first order necessary condition, pose significant difficulties when analyzing numerical schemes. Standard techniques developed for the numerical analysis of the uncontrolled Navier-Stokes equations cannot be directly applied in the optimal control setting. Furthermore, the presence of control constraints, create many additional difficulties and hence require special techniques involving both first and second order necessary and sufficient conditions. In the literature not many contributions to numerical analysis for control problems with time-dependent Navier-Stokes equations can be found. In \cite{GM2000, GM2000a} Gunzburger and Manservisi presente a numerical approach to control of the instationary Navier-Stokes equations using the first discretize then optimize approach. The first optimize then discretize approach applied to the same problem class is discussed by Hinze in \cite{Hinze2000}. Deckelnick and Hinze provide numerical analysis for a general class of control problems with the instationary Navier-Stokes system in \cite{DH2004}.
In \cite{Casas2012} Casas and Chrysafinos proposed a numerical scheme which is based on the discontinuous time-stepping Galerkin scheme for the piecewise constant time discretization combined with standard conforming finite element subspaces for the discretization in space. They presented space-time error estimates of order $O(h)$, under suitable regularity assumptions on the data, when the controls are discretized by piecewise constants in space and time. Two parameters $\tau$ and $h$ are associated to the numerical scheme (here $\tau$ and $h$, indicating the size of the grids in time and space) and they were needed to satisfy the usual technical assumption $\tau \leq Ch^2$ in order to prove that the discrete equation has a unique solution, and the estimate was optimal in $L^2(0, T; \mathbb{H}^1(\Omega))$ norms for the state and adjoint. The key idea of \cite{Casas2012} was to utilize ideas from \cite{CMR2007} developed for the stationary Navier-Stokes equations, together with a detailed error analysis of the uncontrolled state and adjoint equations of the underlying scheme. Later in \cite{Casas2015}, Casas and Chrysafinos continued their work in \cite{Casas2012} in the sense that error estimates in $L^2(0, T; \mathbb{L}^2(\Omega))$ of order $O(h^2)$ and $O(h^{3/2-2/p})$ with $p > 3$ depending on the regularity of the target and the initial velocity were proved. We also refer the interested reader to \cite{Casas2016, Casas2017} for some very close related results.
In this paper following the general lines of the approach in \cite{Casas2012, Chrys} we will study a numerical scheme for the optimal control problem $(P)$ of 3D Navier-Stokes-Voigt equations based on the discontinuous time-stepping Galerkin scheme for the piecewise constant time combined with standard conforming finite element subspaces for the discretization in space. It is noticed that in \cite{AN2016} we proved the existence of optimal solutions and established the first and second order optimality conditions for problem $(P)$, where the admissible set is an arbitrary non-empty, convex, closed subset in $\mathbb{L}^2(Q)$. Our main result in the present paper is to derive space-time error estimates under suitable regularity assumptions on the data together with a detailed error analysis of the uncontrolled state and adjoint equations of the underlying scheme. Here the presence of control constraints prevents a direct analysis of the system of state and adjoint state equations. To overcome this difficulty, as in \cite{Casas2012}, we need to use the second order conditions for optimality. The box control constraint assumption ensures that the set of discrete control designed in the paper is convex and closed, which implies that the discrete control problem has a solution. Besides, in our work, since the solution of the Navier-Stokes-Voigt equations is more regular, we are able to prove the uniqueness of the discrete equation without the widely used technical assumption $\tau \leq Ch^2$ as in \cite{Casas2012}, and we obtain that the order of error estimates is $O(\sqrt{\tau}+h)$ instead of $O(h)$ as in \cite{Casas2012}. It is noticed that, under the assumption $\tau\le Ch^2$ as in \cite{Casas2012}, these two orders of errors are the same. To prove the main results, we modify the techniques used in \cite{Casas2012, Chrys} with appropriate adjusts while contributing project operators from the spaces of states and pressure terms to the respective discrete spaces. Because of some technical reasons as in \cite{Casas2012}, the pressure terms must belong to the space $L^2(0,T;H^1(\Omega))$, so we need to assume that the initial state $y_0$ is in $D(A)$ instead of $V$ as in \cite{AN2016}, that is, we consider the strong solutions. It is worthy noticing that, by using a variational discretization in \cite{Hinze2005}, Casas and Chrysafinos can also prove error estimates (in $L^2(0, T; \mathbb{L}^2(\Omega))$) of order $O(h^2)$ for the distributed optimal control problem of 2D Navier-Stokes equations in \cite{Casas2015}. The proof of such an error estimates for the variational discretization of the optimal control problem $(P)$ will be the goal of a forthcoming paper currently in preparation.
The paper is organized as follows. In Section 2, for convenience of the reader, we recall some auxiliary results on the existence and unique of weak and strong solutions to the system \eqref{NSV1}. We also restate the optimality conditions of the optimal control problem obtained in \cite{AN2016}, however, in this paper these results are transferred into the box control constraint case. The main results of the paper are presented in Section 3, where we analyze the discrete state equations, the discrete adjoint state equations, and we prove the convergence of the discrete control problem and derive the space-time error estimates.
\section{Preliminaries and auxiliary results }
\subsection{Function spaces and inequalities for the nonlinear terms}
Let $\Omega$ be an open bounded domain in $\mathbb{R}^3$ with $C^1$ boundary $\partial\Omega$. For convenience, we set
$$L_0^2(\Omega):=\left\{f\in L^2(\Omega): \int_\Omega f(x)dx=0\right\}, \quad\mathbb{L}^2(\Omega):=L^2(\Omega)^3, $$
$$\mathbb{H}^1(\Omega):=H^1(\Omega)^3,\;\; \mathbb{H}^1_0(\Omega):=H^1_0(\Omega)^3.$$
We denote by $\mathbb{H}^{-1}(\Omega)$ the dual spaces of $\mathbb{H}^1_0(\Omega)$. Define
$$
(u,v):=\int\limits_{\Omega} \sum_{j=1}^3 u_jv_j\, dx, \; \; \; u=(u_1,u_2,u_3),v=(v_1,v_2,v_3)\in \mathbb{L}^2(\Omega),
$$
$$
((u,v)):=\int\limits_{\Omega}\sum_{j=1}^3 \nabla u_j \cdot \nabla v_j\, dx, \; \;\; u=(u_1,u_2,u_3), v=(v_1,v_2,v_3)\in \mathbb{H}^1_0(\Omega),$$
and the associated norms $|u|^2:=(u,u), \|u\|^2:=((u,u)).$
Set
$$ \mathcal{V}=\big\{u\in(C_0^\infty(\Omega))^3\, :\ \nabla\cdot u=0\big\},$$
and denote by $H$ and $V$ the closure of $\mathcal{V}$ in $\mathbb{L}^2(\Omega$ and $\mathbb{H}^1_0(\Omega)$, respectively. Then $H,\,V$ are Hilbert spaces with scalar products $(.,.),\,((.,.))$ respectively.
Let $X$ be a real Banach space with the norm $\|.\|_X$. We denote by $L^p(0,T;X)$ the standard Banach space of all functions from $(0,T)$ to $X$, endowed with the norm
\begin{align*}
\| y \|_{L^p(0,T;X)}&:= \left( \int^{T}_{0}\|y(t)\|_X^p dt \right)^{1/p},\quad 1\le p <\infty,\\
\| y \|_{L^\infty(0,T; X)}&:= \underset{t\in (0,T)}{{\rm ess sup}}\;\|y(t)\|_X.
\end{align*}
When $X$ is a Banach space with the dual space $X'$, we will use $\|.\|_{X'}$ for the norm in $X'$, $\langle .,.\rangle_{X',X}$ for the duality pairing between $X'$ and $X$. In this case, $L^p(0,T;X)$ is also a Banach space, with the dual space being $L^{p'}(0,T;X')$, where $1/p+1/p'=1$. The pairing between $u\in L^{p'}(0,T;X')$ and $v\in L^{p}(0,T;X)$ is
$$\langle u,v\rangle_{L^{p'}(0,T; X'),L^p(0,T; X)}=\int^{T}_{0}\langle u(t),v(t)\rangle_{X',X}dt.$$
To deal with the time derivative in the state equation, we introduce the common space of functions $y$ whose time derivatives $y_t$ exist as abstract functions
$$W^{1,2}(0,T; X):= \{ y\in L^2(0,T;X): y_t \in L^2(0,T;X) \},$$
endowed with the norm
$$ \| y \|_{W^{1,2}(0,T; X)}:=\left( \|y\|^2_{L^2(0,T; X)}+\|y_t\|^2_{L^2(0,T; X)}\right)^{1/2}.$$
When $X$ is a Hilbert space, $L^2(0,T;X)$ and $W^{1,2}(0,T;X)$ are also Hilbert spaces. We will use the following embedding results:
\begin{align*}
&W^{1,2}(0,T; X)\hookrightarrow C([0,T];X) \text{ is continuous (see \cite[p. 190]{Robinson})},\\
&W^{1,2}(0,T;\mathbb{H}^1(\Omega))\hookrightarrow \mathbb{L}^2(Q) \text{
is compact (see \cite{Simon})},\\
&W^{1,2}(0,T;\mathbb{H}^1(\Omega))\hookrightarrow C([0,T];\mathbb{L}^2(\Omega)) \text{
is compact (see \cite{Simon})}.
\end{align*}
We now define the bilinear and trilinear forms $a:\mathbb{H}^1(\Omega)\times\mathbb{H}^1(\Omega)\to \mathbb{R}$ and $c:\mathbb{H}^1(\Omega)\times\mathbb{H}^1(\Omega)\times\mathbb{H}^1(\Omega)\to \mathbb{R}$ by
\begin{align*}
a(u,v)&=\sum_{i,j=1}^{3}\int_\Omega \partial_{x_i}u_j\partial_{x_i}v_jdx,\\
c(u,v,w)&=\dfrac{1}{2}\left[b(u,v,w)-b(u,w,v)\right]\;\text{with } b(u,v,w)=\sum_{i,j=1}^3\int\limits_\Omega u_i\dfrac{\partial v_j}{\partial x_i}w_j \,dx.
\end{align*}
It is easy to check that if $u\in V, v,w\in \mathbb{H}^1_0(\Omega)$ then $b(u,v,w)=-b(u,w,v)$. Hence
\begin{equation*}
b(u,v,v)=0,\;\forall\, u\in V, v\in\mathbb{H}^1_0(\Omega).
\end{equation*}
We also define an operator $A:V\to V'$ by $\langle Au,v\rangle=((u,v)),\, u,v\in V.$ Denote by $D(A)$ the domain of $A$, $D(A):=\{u\in H:Au\in H\}$. We have $D(A)=\mathbb{H}^2(\Omega)\cap V.$ The norm in $D(A)$ is defined by $\|u\|_{D(A)}:=|Au|.$
\begin{lemma} \label{trilinear} \cite{ConstantinFoias, Temam} We have
\begin{align}
&b(u,v,w)=c(u,v,w)=-c(u,w,v),\quad\forall\; u\in V,\forall\, v,w\in \mathbb{H}^1_0(\Omega),\notag\\
&|b(u,v,w)|\le C\|u\|\|v\|^{1/2}|Av|^{1/2}|w|,\quad\forall u\in V,v\in D(A),w\in H,\label{b4}\\
&|b(u,v,w)|\le C|u|^{1/4}\|u\|^{3/4}\|v\||w|^{1/4}\|w\|^{3/4},\quad \forall \,u,\,v,\,w\,\in\, \mathbb{H}^1_0(\Omega),\notag\\
&|b(u,v,w)|\le C\|u\|\|v\|\|w\|, \quad \forall \,u,\,v,\,w\,\in\, \mathbb{H}^1_0(\Omega),\notag\\
&|b(u,v,u)|\leq C|u|^{1/2}\|u\|^{3/2}\|v\|, \quad \forall\, u,v\,\in\, \mathbb{H}^1_0(\Omega).\notag\\
&c(u,v,w)=-c(u,w,v), \quad \forall\, u,v,w\in \mathbb{H}^1_0(\Omega),\notag\\
&c(u,v,v)=0, \quad\forall\, u,v\in \mathbb{H}^1_0(\Omega),\notag\\
&|c(u,v,w)|\le C\|u\|\|v\|\|w\|, \quad \forall\, u,v,w\in \mathbb{H}^1_0(\Omega),\notag\\
&|c(u,v,w)|\le C|u|^{1/4}\|u\|^{3/4}\|v\|\|w\|, \quad \forall\, u,v,w\in \mathbb{H}^1_0(\Omega).\notag
\end{align}
\end{lemma}
\subsection{Existence and uniqueness of solutions to the Navier-Stokes-Voigt equations}
\begin{definition} Let $u\in L^2(0,T;\mathbb{L}^2(\Omega))$ be given. A pair $(y,p)\in W^{1,2}(0,T; V)\times L^2(0,T;L^2_0(\Omega))$ is called a weak solution to the system \eqref{NSV1} on the interval $(0,T)$ if it fulfills
\begin{equation}\label{ST1}
\begin{cases}
(y_t(s),w)+\nu a(y(s),w) +\alpha^2 a(y_t(s),w) +c(y(s),y(s),w)+(p(s),{\rm div}\,w)\\
\hspace{4cm} =(u(s),w), \;\;\forall w\in \mathbb{H}^1_0(\Omega), \text{ for a.e. } s\in [0,T],\\
y(0)=y_0.
\end{cases}
\end{equation}
\end{definition}
\begin{remark} {\rm
It is clear that if $(y,p)\in W^{1,2}(0,T; V)\times L^2(0,T;L^2_0(\Omega))$ satisfies \eqref{ST1} then $y$ satisfies equations
\begin{equation}
\label{ST2}
\begin{cases}
(y_t(s),w)+\nu a(y(s),w) +\alpha^2 a(y_t(s),w) +c(y(s),y(s),w)=(u(s),w),\\
\hspace{6cm} \forall w\in V,\;\text{ for a.e. } s\in [0,T],\\
y(0)=y_0.
\end{cases}
\end{equation}
Conversely, if $y\in W^{1,2}(0,T;V)$ satisfies \eqref{ST2} then there exists a unique $p\in L^2(0,T;L^2_0(\Omega))$ such that \eqref{ST1} holds.}
\end{remark}
\begin{theorem}\label{tontaiNSV} \cite{AT2013}
For any $y_0\in V$ and $u\in L^2(0,T;\mathbb{L}^2(\Omega))$ given, problem \eqref{ST2} has a unique weak solution $(y,p)$ belonging to $W^{1,2}(0,T;V)\times L^2(0,T;L_0^2(\Omega))$. Furthermore, if $\|u\|_{L^2(0,T;\mathbb{L}^2(\Omega))}\le M$
then
\begin{equation}
\label{ST2.1}
\|y\|_{W^{1,2}(0,T;V)}+\|p\|_{L^2(0,T;L^2(\Omega))}\le C_M,
\end{equation}
where $C_M$ is a constant depending on $M$.
\end{theorem}
The existence of strong solutions to problem \eqref{NSV1} is given by the following theorem.
\begin{theorem}
\label{THR2.2}
If $y_0\in D(A)$ and $u\in L^2(0,T;\mathbb{L}^2(\Omega)$ then the unique weak solution $(y,p)$ of \eqref{NSV1} belongs to $W^{1,2}(0,T;D(A))\times L^2(0,T;H^1(\Omega)\cap L^2_0(\Omega))$. Moreover, if $\|u\|_{L^2(0,T;\mathbb{L}^2(\Omega))}\le M$ then
\begin{equation}\label{Pre1.0}
\|y\|_{W^{1,2}(0,T;D(A))}+\|p\|_{L^2(0,T;H^1(\Omega))}\le C_M,\end{equation}
where $C_M$ is a constant depending on $M$.
\end{theorem}
\begin{proof}
The proof is standard by using the Galerkin method, so we only present here some {\it a priori} estimates. Taking $w=y(s)$ in the first equation of \eqref{ST2}, then integrating from $0$ to $t$ we get
\begin{multline}
\label{Pre1.5}
\int^{t}_{0}(y_t(s),y(s))ds + \nu\int^{t}_{0}\|y(s)\|^2ds +\alpha^2\int^{t}_{0}((y_t(s),y(s)))ds\\
=\int^{t}_{0}(u(s),y(s))ds.
\end{multline}
Here we have used the identity $c(y(s),y(s),y(s))=0$. The right-hand side can be estimated by
\begin{align*}
\left |\int_0^t (u(s),y(s))ds\right | &\le \int_0^t |u(s)|.|y(s)|ds\le C\int_0^t\|y(s)\|.|u(s)|ds\\
&\le \dfrac{\nu}{2}\int_0^t \|y(s)\|^2ds +\dfrac{C^2}{2\nu}\int_0^t |u(s)|^2ds,
\end{align*}
where $C$ depends only on $\Omega$. By integrating by parts, we get from \eqref{Pre1.5} that
\begin{equation*}
\begin{aligned}
|y(t)|^2+\nu \int^{t}_{0}\|y(s)\|^2ds+\alpha^2\|y(t)\|^2
&\le |y(0)|^2+\alpha^2\|y(0)\|^2+\dfrac{C^2}{\nu}\int^{t}_{0}|u(s)|^2ds\\
&\le |y_0|^2+\alpha^2\|y_0\|^2+\dfrac{C^2}{\nu}\|u\|^2_{\mathbb{L}^2(Q)}.
\end{aligned}
\end{equation*}
So, $y$ belongs to $L^\infty(0,T; V)$ and
\begin{equation}\label{reg1.2}
\|y\|^2_{L^\infty(0,T;V)}\le C(\|y_0\|^2+\|u\|^2_{\mathbb{L}^2(Q)}).
\end{equation}
Now, taking $w=Ay(s)$ in the first equation of \eqref{ST2}, then integrating from $0$ to $t$ we have
\begin{multline}\label{reg1}
\dfrac{1}{2}\|y(t)\|^2-\dfrac{1}{2}\|y_0\|^2 + \nu\int_0^t|A y(s)|^2ds+\dfrac{\alpha^2}{2}|A y(t)|^2-\dfrac{\alpha^2}{2}|A y_0|^2\\
=-\int_0^tb(y(s),y(s),A y(s))ds + \int_0^t(u(s),A y(s))ds.
\end{multline}
We have
\begin{equation}\label{reg1.1}
\left|\int_0^t(u(s),A y(s))ds\right|\le \int_0^t|u(s)||A y(s)|ds \le \dfrac{\nu}{2}\int_0^t|A y(s)|^2ds + C\int_0^t|u(s)|^2ds.
\end{equation}
From inequality \eqref{b4} and the fact that $y\in L^\infty(0,T; V)$ we deduce that
\begin{equation}\label{reg1.0}
\begin{aligned}
\left|\int_0^t b(y(s),y(s),A y(s))ds \right|&\le C\int_0^t\|y(s)\|\|y(s)\|^{1/2}|A y(s)|^{1/2}|A y(s)| ds.\\
&\le C\int_0^t\|y(s)\|^{1/2} |A y(s)|^{3/2}ds\\
&\le \dfrac{\nu}{2}\int_0^t|A y(s)|^2ds + C\int_0^t\|y(s)\|^2ds\\
&\le \dfrac{\nu}{2}\int_0^t|A y(s)|^2ds + C\|y\|^2_{L^2(0,T;V)}.
\end{aligned}
\end{equation}
Using the estimates above, \eqref{reg1} leads to
\begin{equation*}\label{reg2}
|A y(t)|^2\le C\left(|A y_0|^2+\int_0^T |u(s)|^2ds\right)\quad \text{for every } t\in (0,T).
\end{equation*}
Thus we obtain $y\in L^\infty (0,T;D(A))$ and
\begin{equation}\label{reg2.0}
\|y\|^2_{L^\infty(0,T;D(A))}\le C\left(|Ay_0|^2+\int_0^T |u(s)|^2ds\right).
\end{equation}
Now, taking $w=Ay_t(s)$ in the first equation of \eqref{ST2}, then integrating from $0$ to $T$ we have
\begin{multline}
\label{reg3}
\int_0^T\|y_t(s)\|^2ds +\dfrac{\nu}{2}(|Ay(T)|^2-|Ay_0|^2)+\alpha^2\int_0^T|Ay_t(s)|^2ds\\
=\int_0^T(u(s),Ay_t(s))ds - \int_0^Tb(y(s),y(s),Ay_t(s))ds.
\end{multline}
Using the same arguments as in \eqref{reg1.1} and \eqref{reg1.0} we get
\begin{align*}
\left|\int_0^T(u(s),A y_t(s))ds\right| &\le \dfrac{\alpha^2}{3}\int_0^T|A y_t(s)|^2ds + C\int_0^T|u(s)|^2ds,\\
\left|\int_0^T b(y(s),y(s),A y_t(s))ds \right| &\le \dfrac{\alpha^2}{3}\int_0^T|A y_t(s)|^2ds + C\|y\|^2_{L^2(0,T;V)}.
\end{align*}
From these estimates and \eqref{reg1.2}, \eqref{reg3} we obtain
$$\int_0^T |Ay_t(s)|^2ds \le C\left(|Ay_0|^2 + \int_0^T|u(s)|^2ds\right).$$
This means that $y_t\in L^2(0,T;D(A))$. Using this estimate and \eqref{reg2.0} we get that
$$\|y\|_{W^{1,2}(0,T;D(A))}\le C_M.$$
The conclusion that $p\in L^2(0,T;H^1(\Omega))$ and that $p$ is bounded in $L^2(0,T;H^1(\Omega))$ by a constant depending on $M$ are direct consequences of the regular properties of the equation $\nabla p =f.$
\end{proof}
Denote by $G$ the control-to-state mapping:
\begin{align*}
G:L^2(0,T;\mathbb{L}^2(\Omega))&\to W^{1,2}(0,T;V)\\
u&\mapsto G(u):=y_u, \text{ the unique solution of \eqref{ST2}.}
\end{align*}
From now on, for convenience, we sometimes write $J$ as a functional of control variable $u$ as follows $J(u):=J(G(u), u)$.
\begin{theorem}
\label{THR2.3}
If $y_0\in D(A)$ and $u_n\rightharpoonup u$ in the space $L^2(0,T;\mathbb{L}^2(\Omega))$, then from the sequence $\{y_{u_n}\}_n$ we can extract a subsequence, denoted in the same way, such that $y_{u_n}\rightharpoonup y_u$ in the space $W^{1,2}(0,T;D(A))$.
\end{theorem}
\begin{proof}
It follows from Theorem \ref{THR2.2} that the sequence $\{y_{u_n}\}_n$ is bounded in the space $W^{1,2}(0,T;D(A))$. Hence we can extract a subsequence converging weakly to some $y\in W^{1,2}(0,T;D(A))$. Then we can easily pass to the limit in \eqref{ST2} to show that $y\equiv y_u$.
\end{proof}
\subsection{Optimality conditions}
Now, we restate the results obtained in \cite{AN2016} when the set of admissible controls is the box $U_{\alpha,\beta}$.
\begin{theorem}
The mapping $G$ is of class $C^2$. If we set $z_v:=G'(u)v$, $z_{vv}=G''(u)v^2$ then $z_v,z_{vv}$ are respectively the unique solutions of the following equations:
\begin{equation*}
\begin{cases}
(z_{vt},w)+\nu a(z_v,w)+\alpha^2a(z_{vt},w)+c(z_v,y_u,w)+c(y_u,z_v,w)=(v,w),\quad\forall w\in V,\\
z_v(0)=0,
\end{cases}
\end{equation*}
\begin{equation*}
\begin{cases}
(z_{vvt},w)+\nu a(z_{vv},w)+\alpha^2a(z_{vvt},w)+c(z_{vv},y_u,w)+c(y_u,z_{vv},w)\\
\hspace{6cm} =-2c(z_v,z_v,w),\quad\forall w\in V,\\
z_{vv}(0)=0.
\end{cases}
\end{equation*}
Furthermore, if $\|y_u\|_{W^{1,2}(0,T;V)}\le M$ then
\begin{equation}
\label{DGE2.1}
\|z_v\|_{W^{1,2}(0,T;V)}\le C_M\|v\|_{L^2(0,T;\mathbb{L}^2(\Omega))},
\end{equation}
where $C_M$ is a constant depending on $M$.
\end{theorem}
\begin{theorem}
The cost functional $J:L^2(0,T;\mathbb{L}^2(\Omega))\to \mathbb{R}$ is of class $C^2$. The first and second order derivatives of $J$ are given by
\begin{align*}
J'(u)v= &\int_0^T\int_\Omega (\lambda+\gamma u)vdxdt,\\
J''(u)v^2=&\alpha_T\int_\Omega|z_v(T)|^2dx+\alpha_Q\int_0^T\int_\Omega|z_v|^2dxdt+\gamma\int_0^T\int_\Omega|v|^2dxdt\\
&-2\int_0^Tc(z,z,\lambda)dt,
\end{align*}
where $\lambda$ is a weak solution of the following system
\begin{equation}
\label{DGE3}
\begin{cases}
-\lambda_t- \nu\Delta\lambda +\alpha^2\Delta\lambda _t-(y_u\cdot\nabla)\lambda + (\nabla y_u)^T\lambda+\nabla q= \alpha _Q(y_u-y_Q), \;x\in \Omega, t>0,\\
\hfil \nabla \cdot \lambda=0,\; x\in \Omega, t>0,\\
\hfil \lambda(x,t)=0, \; x\in\partial\Omega, t>0,\\
\hfil \lambda(T)-\alpha^2\Delta \lambda(T)+\nabla r=\alpha _T(y_u(T)-y_T),\; x\in\Omega.
\end{cases}
\end{equation}
\end{theorem}
\begin{remark}{\rm
\begin{itemize}
\item[i.] A triplet $(\lambda,q,r)\in W^{1,2}(0,T;V)\times L^2(0,T;L^2_0(\Omega))\times L^2_0(\Omega)$ is called a weak solution to the system \eqref{DGE3} on the interval $(0,T)$ if for every $w\in \mathbb{H}^1_0(\Omega)$, it fulfills
\begin{equation}
\label{DGE4}
\begin{cases}
-(\lambda_t,w)+\nu a(\lambda,w)-\alpha^2a(\lambda_t,w)+c(y_u,w,\lambda)+c(w,y_u,\lambda)+(q,{\rm div}\, w)\\
\hspace{9cm}=\alpha_Q(y_u-y_Q,w),\\
(\lambda(T),w)+\alpha^2a(\lambda(T),w)+(r,{\rm div}\,w)=\alpha_T(y_u(T)-y_T,w).
\end{cases}
\end{equation}
\item[ii.] If $(\lambda,q,r)\in W^{1,2}(0,T;V)\times L^2(0,T;L^2_0(\Omega))\times L^2_0(\Omega)$ is a weak solution of the system \eqref{DGE3} then for every $w\in V$, $\lambda$ satisfies the following equations
\begin{equation}
\label{DGE5}
\begin{cases}
-(\lambda_t,w)+\nu a(\lambda,w)-\alpha^2a(\lambda_t,w)+c(y_u,w,\lambda)+c(w,y_u,\lambda)=\alpha_Q(y_u-y_Q,w),\\
(\lambda(T),w)+\alpha^2a(\lambda(T),w)=\alpha_T(y_u(T)-y_T,w).
\end{cases}
\end{equation}
Conversely, if $\lambda\in W^{1,2}(0,T;V)$ satisfies \eqref{DGE5} then there exists a unique pair $(q,r)\in L^2(0,T;L^2_0(\Omega))\times L^2_0(\Omega)$ such that \eqref{DGE4} holds for every $w\in \mathbb{H}^1_0(\Omega)$.
\item[iii.] The system \eqref{DGE3} has a unique weak solution $(\lambda,q,r)\in W^{1,2}(0,T;V)\times L^2(0,T;L^2_0(\Omega))\times L^2_0(\Omega)$. One can prove that this solution belongs to the space $W^{1,2}(0,T;D(A))\times L^2(0,T;H^1(\Omega)\cap L^2_0(\Omega))\times (H^1(\Omega)\cap L^2_0(\Omega)).$ Furthermore, if $\|u\|_{L^2(0,T;\mathbb{L}^2(\Omega))}\le M$ then
$$\|\lambda\|_{W^{1,2}(0,T;D(A))}+\|q\|_{L^2(0,T;H^1(\Omega))}+\|r\|_{H^1(\Omega)}\le C_M,$$
where $C_M$ is a constant depending on $M$.
\end{itemize}}
\end{remark}
\begin{theorem}
Let $\bar{u}\in L^2(0,T;\mathbb{L}^2(\Omega))$ be a local optimal control with associated state $\bar{y}\in W^{1,2}(0,T;D(A))$. Then
\begin{equation*}
\int_0^T\int_\Omega (\bar{\lambda}+\gamma\bar{u})\cdot (v-\bar{u})dxdt\ge 0,\quad \forall v\in U_{\alpha, \beta},
\end{equation*}
where $\bar{\lambda}$ is the adjoint state, i.e. $(\bar{\lambda},\bar{q},\bar{r})$ is the unique weak solution of the following system on the interval $(0,T)$
\begin{equation*}
\begin{cases}
-\bar{\lambda}_t- \nu\Delta\bar{\lambda} +\alpha^2\Delta\bar{\lambda} _t-(\bar{y}\cdot\nabla)\bar{\lambda} + (\nabla \bar{y})^T\bar{\lambda}+\nabla \bar{q}= \alpha _Q(\bar{y}-y_Q), \;x\in \Omega, t>0,\\
\hfil \nabla \cdot \bar{\lambda}=0,\; x\in \Omega, t>0,\\
\hfil \bar{\lambda}(x,t)=0, \; x\in\partial\Omega, t>0,\\
\hfil \bar{\lambda}(T)-\alpha^2\Delta \bar{\lambda}(T)+\nabla \bar{r}=\alpha _T(\bar{y}(T)-y_T),\; x\in\Omega.
\end{cases}
\end{equation*}
\end{theorem}
To state the second-order optimality conditions we define the cone of critical directions as follows
$$\mathcal{C}_{\bar{u}}=\left\{v\in L^2(0,T;\mathbb{L}^2(\Omega)): v \text{ satisfies } \eqref{DGE8}, \eqref{DGE9}, \eqref{DGE10} \right\}.$$
\begin{align}
&v_j(t,x)\ge 0 \text{ if } -\infty < \alpha_j=\bar{u}_j(t,x),\;j=1,2,3,\label{DGE8}\\
&v_j(t,x)\le 0 \text{ if } \bar{u}_j(t,x)=\beta_j<+\infty,\;j=1,2,3, \label{DGE9}\\
&v_j(t,x)=0 \text{ if } \bar{\lambda}+\gamma\bar{u}\ne 0,\;j=1,2,3.\label{DGE10}
\end{align}
The following theorem gives the second-order necessary and sufficient optimality conditions.
\begin{theorem}
If $\bar{u}$ be a local solution of problem $(P)$ then
$$J''(\bar{u})v^2\ge 0\;\;\forall v\in \mathcal{C}_{\bar{u}}.$$
Conversely, if $\bar{u}\in U_{\alpha, \beta}$ satisfies
\begin{align}
&J'(\bar{u})(u-\bar{u})\ge 0\;\; \forall u\in U_{\alpha, \beta},\notag\\
&J''(\bar{u})v^2>0\;\;\forall v\in \mathcal{C}_{\bar{u}}\backslash \{0\},\label{DGE12}
\end{align}
then there exist $\varepsilon>0$ and $\rho>0$ such that
\begin{equation*}
J(u)\ge J(\bar{u})+\varepsilon\|u-\bar{u}\|^2_{L^2(0,T;\mathbb{L}^2(\Omega))}\;\;\forall u\in U_{\alpha, \beta}\cap \bar{B}_\rho(\bar{u}).
\end{equation*}
\end{theorem}
\section{Numerical approximation of the optimal control problem}
Let $\{\mathcal{T}_h\}_{h>0}$ be a family of triangulation of $\overline{\Omega}$, defined in the standard way. To each element $T\in\mathcal{T}_h$, we denote by $h_T$ and $\rho_T$ the diameter of the set $T$ and diameter of the largest ball contained in $T$. Define the size of the mesh by $h$, i.e. $h=\max_{T\in \mathcal{T}_h}h_T.$ We also assume that the following standard regularity assumptions on the triangulation hold:
\begin{itemize}
\item[(i)] There exist two positive constants $\rho_\mathcal{T},\delta_\mathcal{T}$ such that $\dfrac{h_T}{\rho_T}\le \rho_\mathcal{T}$ and $\dfrac{h}{h_T}\le \delta_\mathcal{T}$ for every $T\in\mathcal{T}_h$ and for every $h>0.$
\item[(ii)] Set $\overline{\Omega}_h=\cup_{T\in\mathcal{T}_h}T$ and denote by $\Omega_h$ and $\Gamma_h$ its interior and its boundary, respectively. We assume that the vertices of $\mathcal{T}_h$ placed on the boundary $\Gamma_h$ are points of $\Gamma$.
\end{itemize}
Since $\Omega$ is convex, from the last assumption we have that $\Omega_h$ is also convex. Moreover, we assume that
\begin{equation}
\label{Ap1}
|\Omega\backslash \Omega_h|\le Ch^2.
\end{equation}
On the mesh $\mathcal{T}_h$ we consider two finite dimensional spaces $Z_h\subset \mathbb{H}^1_0(\Omega)$ and $Q_h\subset L^2_0(\Omega)$ formed by piecewise polynomials in $\Omega_h$ and vanishing in $\Omega\backslash \Omega_h.$ We make the following assumptions on these spaces:
\begin{itemize}
\item[(A1)] If $z\in \mathbb{H}^{1+l}(\Omega)\cap \mathbb{H}_0^1(\Omega)$ then
\begin{equation*}
\inf_{z\in Z_h}\|z-z_h\|_{{\mathbb{H}}^s(\Omega)}\le Ch^{l+1-s}\|z\|_{\mathbb{H}^{1+l}(\Omega)} \text{ for } 0\le l\le 1 \text{ and } s=0,1.
\end{equation*}
\item[(A2)] If $q\in H^1(\Omega)\cap L^2_0(\Omega)$ then
\begin{equation*}
\inf_{q_h\in Q_h}|q-q_h|\le Ch\|q\|_{H^1(\Omega)}.
\end{equation*}
\item[(A3)] The subspaces $Z_h$ and $Q_h$ satisfy the inf-sup condition: $\exists \beta>0$ such that
\begin{equation*}
\inf_{q_h\in Q_h}\sup_{z_h\in Z_h}\dfrac{\mathfrak{b}(z_h,q_h)}{\|z_h\|_{{\mathbb{ H}}^1(\Omega)}\|q_h\|_{L^2(\Omega)}}\ge \beta,
\end{equation*}
where $\mathfrak{b}:\mathbb{H}^1(\Omega)\times L^2(\Omega)\to \mathbb{R}$ is defined by
$$\mathfrak{b}(z,q)=\int_{\Omega} q(x)\text{div}z(x)dx.$$
\end{itemize}
These assumptions are satisfied by the usual finite elements considered in the discretization of Navier-Stokes equations, see \cite[Chapter 2]{Girault}.
We also consider a subspace $V_h$ of $Z_h$ defined by
$$V_h=\{y_h\in Z_h:\mathfrak{b}(y_h,q_h)=0\;\;\forall q_h\in Q_h\},$$
and set
$$U_h=\{u_h\in \mathbb{L}^2(\Omega_h):u_h|_T\equiv u_T\in \mathbb{R}^3\;\;\forall T\in \mathcal{T}_h\}.$$
Now, we consider the discretization in time. Let $0=t_0<t_1<\cdots <t_{N_\tau}=T$ be a partition of interval $[0,T]$. We denote $\tau_n=t_n-t_{n-1}.$ We make the following assumption:
\begin{equation*}
\exists\, \rho_0>0 \text{ such that } \tau = \max_{1\le n\le N_\tau} \tau_n<\rho_0\tau_n\;\;\forall\, 1\le n\le N_\tau \text{ and } \forall \tau > 0.
\end{equation*}
Given a triangulation $\mathcal{T}_h$ of $\Omega$ and a grid of points $\{t_n\}_{n=1}^{N_\tau}$ of $[0,T]$, we set $\sigma=(\tau,h)$. We consider the subspace of functions that are piecewise constant in time
$$U_\sigma=\{u_\sigma\in L^2(0,T;U_h): u_\sigma|_{(t_{n-1},t_n)}\in U_h\, \text{ for } 1\le n\le N_{\tau}\}.$$
We seek for the discrete controls in the space $U_\sigma$. An element of this space can be written in the form
\begin{equation*}
u_\sigma = \Sigma_{n=1}^{N_\tau}\Sigma_{T\in \mathcal{T}_h}u_{n,T}\chi_n\chi_T, \,\, \text{ with } u_{n,T}\in \mathbb{R}^3,
\end{equation*}
where $\chi_n$ and $\chi_T$ are the characteristic functions of $(t_{n-1},t_n)$ and $T$, respectively. Therefore, the dimension of $U_\sigma$ is $3N_\tau N_h$, where $N_h$ is the number of elements in $\mathcal{T}_h$. In $U_\sigma$ we consider the convex subset
$$U_{\sigma,ad}=U_\sigma\cap U_{\alpha,\beta}=\{u_\sigma\in U_\sigma:u_{n,T}\in \Pi_{i=1}^3[\alpha_i,\beta_i]\}.$$
For each given sequence $(y_{0,h},y_{1,h},\ldots,y_{N_\tau,h})\in V_h^{N_\tau+1}$, we define a function $y_\sigma:[0,T]\to V_h$ by
\begin{equation*}
\begin{cases}
y_\sigma(t_n)=y_{n,h},\;n=0,1,\ldots,N_\tau,\\
y_\sigma(t)=y_{n,h}\;\forall t\in (t_{n-1},t_n).
\end{cases}
\end{equation*}
We denote by $V_\sigma$ the set of all functions that are defined by this way.
Now, we consider the numerical discretization of the state equations \eqref{ST1}. We will use a discontinuous time-stepping Galerkin method, with piecewise constants in time and conforming finite element spaces in space. For $u\in \mathbb{L}^2(Q)$, the discrete state equation is given by
\begin{equation}
\label{DSE1}
\begin{cases}
\text{ For } n=1,2,\ldots,N_\tau,\\
\left(\dfrac{y_{n,h}-y_{n-1,h}}{\tau_n},w_h\right)+\nu a(y_{n,h},w_h)+\alpha^2 a\left(\dfrac{y_{n,h}-y_{n-1,h}}{\tau_n},w_h\right)\\
\hspace{5cm}+c(y_{n,h},y_{n,h},w_h)=(u_n,w_h),\;\;\forall w_h\in V_h,\\
y_{0,h}=P_hy_0,
\end{cases}
\end{equation}
where $(u_n,w_h)=\dfrac{1}{\tau_n}\int\limits_{t_{n-1}}^{t_n}(u(t),w_h)dt$ and $P_hy_0$ is defined in Definition \ref{Def1}.
We will prove later that for every $u\in\mathbb{L}^2(Q)$, problem \eqref{DSE1} has a unique solution $y_\sigma(u)\in V_\sigma$. Now we can define the discrete control problem as follows
\begin{equation*}
(P_\sigma)\hspace{0.5cm}
\begin{cases}
\min J_\sigma (u_\sigma)\\
u_\sigma\in U_{\sigma,ad},
\end{cases}
\end{equation*}
with
\begin{multline*}
J_\sigma(u_\sigma)=\dfrac{\alpha_T}{2}\int_{\Omega_h}|y_\sigma(u_\sigma)(T)-y_T^h|^2dx+\dfrac{\alpha_Q}{2}\int_0^T\int_{\Omega_h}|y_\sigma(u_\sigma)-y_Q|^2dxdt\\
+\dfrac{\gamma}{2}\int_0^T\int_{\Omega_h}|u_\sigma|^2dxdt,
\end{multline*}
where $y_T^h\in V_h$ satisfies the following condition
\begin{equation*}
\exists C>0 \text{ such that } \|y_T-y_T^h\|_{\mathbb{L}^2(\Omega_h)}\le Ch \text{ and } \|y_T^h\|_{\mathbb{H}^1(\Omega_h)}\le C, \;\forall h>0.
\end{equation*}
The outline of this section is as follows. In Subsection 3.1, we analyze the discrete state equations \eqref{DSE1}. Then we study the discrete adjoint state equations in Subsection 3.2. Finally, we prove the convergence of solutions to problem $(P_\sigma)$ and derive the error estimates for the discretization in the last subsection.
\subsection{Analysis of the discrete state equations}
By a standard argument, using the identify $c(u,v,v)=0 \;\forall u,v\in \mathbb{H}^1_0(\Omega)$ and Brouwer's fixed-point theorem, one can easily prove that system \eqref{DSE1} has at least one solution. In this subsection, we will prove that the solution is unique. According to an abstract approximation result (see \cite{Girault}), for given $y\in V$, $p\in L^2_0(\Omega)$, the following problems have unique solutions.
\begin{align*}
(Pr_{1})&\quad\text{Find } y_h\in V_h \text{ satisfying: }
a_\alpha(y_h,v_h)=a_\alpha(y,v_h)\;\;\forall v_h\in V_h,\\
(Pr_{2})&\quad \text{Find a pair } (y_h,p_h)\in V_h\times Q_h \text{ satisfying:}\\
&\quad a_\alpha(y_h,v_h)+\mathfrak{b}(v_h,p_h)=\mathfrak{b}(v_h,p)\;\; \forall v_h\in Z_h.
\end{align*}
Here, $a_\alpha(u,v)=(u,v)+\alpha^2(\nabla u,\nabla v).$ These results allow us to give the following definition.
\begin{definition}
\label{Def1}
We define operators
\begin{tabular}{rcrl}
$P_h:$& $V$&$\to$ &$V_h$\\
&$y$&$\mapsto$& $y_h,\text{ which is the unique solution of the problem } (Pr_1),$
\end{tabular}
\begin{tabular}{rcrl}
$R_h:$&$L^2_0(\Omega)$&$\to$&$ Q_h$\\
&$p$&$\mapsto$&$ p_h, \text{ which is the second component of the solution of } (Pr_2)$.
\end{tabular}
We also define $P_\sigma:C([0,T];V)\to V_\sigma$ by $(P_\sigma y)_{n,h}=P_hy(t_n)$ for $0\le n\le N_\tau.$
\end{definition}
Obviously, there exists a constant $C$ depending only on $\alpha$ such that $\|P_hy\|_{\mathbb{H}^1(\Omega)}\le C\|y\|_{\mathbb{H}^1(\Omega)},\, \forall y\in V$. Using Theorem 1.1 in \cite[Chapter 2]{Girault}, we can easily get the following lemma from assumptions $(A1)-(A3)$.
\begin{lemma}
\label{LM1}
For every $u\in V\cap \mathbb{H}^2(\Omega)$, $p\in L^2_0(\Omega)\cap H^1(\Omega)$, we have
\begin{align*}
\|u-P_hu\|_{\mathbb{H}^1(\Omega)}&\le Ch\|u\|_{\mathbb{H}^2(\Omega)},\\
|p-R_hp|&\le Ch\|p\|_{H^1(\Omega)},
\end{align*}
where $C$ is a constant independent of $h$.
\end{lemma}
\begin{lemma}
\label{LM2}
There exists a constant $C>0$ independent of $\sigma$ such that for every $y\in W^{1,2}(0,T;D(A))$ we have
$$\|y-P_\sigma y\|_{L^2(0,T;\mathbb{H}^1(\Omega))}\le C\left(h\|y\|_{L^2(0,T;\mathbb{H}^2(\Omega))}+\tau\|y'\|_{L^2(0,T;\mathbb{H}^1(\Omega))}\right).$$
\end{lemma}
\begin{proof} We have
\begin{align*}
&\|y-P_\sigma y\|_{L^2(0,T;\mathbb{H}^1(\Omega))}
=\left\{\sum_{n=1}^{N_\tau}\int_{t_{n-1}}^{t_n}\|y(t)-P_h y(t_n)\|_{\mathbb{H}^1(\Omega)}^2dt\right\}^{1/2}\\
&\le \left\{\sum_{n=1}^{N_\tau}\int_{t_{n-1}}^{t_n}\|y(t)-P_h y(t)\|_{\mathbb{H}^1(\Omega)}^2dt\right\}^{1/2}\\
&\hspace{5cm} + \left\{\sum_{n=1}^{N_\tau}\int_{t_{n-1}}^{t_n}\|P_hy(t)-P_h y(t_n)\|_{\mathbb{H}^1(\Omega)}^2dt\right\}^{1/2}\\
&\le Ch\left\{\sum_{n=1}^{N_\tau}\int_{t_{n-1}}^{t_n}\|y(t)\|_{\mathbb{H}^2(\Omega)}^2dt\right\}^{1/2}+C\left\{\sum_{n=1}^{N_\tau}\int_{t_{n-1}}^{t_n}\|y(t)-y(t_n)\|_{\mathbb{H}^1(\Omega)}^2dt\right\}^{1/2}\\
&= Ch\left\{\sum_{n=1}^{N_\tau}\int_{t_{n-1}}^{t_n}\|y(t)\|_{\mathbb{H}^2(\Omega)}^2dt\right\}^{1/2}+C\left\{\sum_{n=1}^{N_\tau}\int_{t_{n-1}}^{t_n}\|\int_{t}^{t_n}y'(s)ds\|_{\mathbb{H}^1(\Omega)}^2dt\right\}^{1/2}\\
&\le Ch\left\{\sum_{n=1}^{N_\tau}\int_{t_{n-1}}^{t_n}\|y(t)\|_{\mathbb{H}^2(\Omega)}^2dt\right\}^{1/2}\\
&\hspace{4cm}+C\left\{\sum_{n=1}^{N_\tau}\int_{t_{n-1}}^{t_n}(t_n-t)\int_{t_{n-1}}^{t_n}\|y'(s)\|^2_{\mathbb{H}^1(\Omega)}dsdt\right\}^{1/2}\\
&\le C(h\|y\|_{L^2(0,T;\mathbb{H}^2(\Omega))}+\tau \|y'\|_{L^2(0,T;\mathbb{H}^1(\Omega))}).
\end{align*}
\end{proof}
\begin{lemma}
\label{LM53}
Let $y\in W^{1,2}(0,T;D(A))$ be the unique solution of \eqref{ST2}. We consider the following system
\begin{equation}
\label{DS2}
\begin{cases}
\text{For } n=1,\ldots,N_\tau,\\
\left (\dfrac{\hat{y}_{n,h}-\hat{y}_{n-1,h}}{\tau_n},w_h\right )+\nu a(\hat{y}_{n,h},w_h)+\alpha^2a(\dfrac{\hat{y}_{n,h}-\hat{y}_{n-1,h}}{\tau_n},w_h)\\
\hspace{8.5cm}=(f_n,w_h)\;\;\forall\, w_h\in V_h,\\
\hat{y}_{0,h}=y_{0h},
\end{cases}
\end{equation}
where
$$(f_n,w_h)=\dfrac{1}{\tau_n}\int_{t_{n-1}}^{t_n}\{\nu a(y(t),w_h)+\alpha^2a(y'(t),w_h)+(y'(t),w_h)\}dt.$$
This system has a unique solution $\hat{y}_\sigma\in V_\sigma$. Moreover, we have the following properties:
\begin{enumerate}
\item $\{\hat{y}_\sigma\}_\sigma$ is bounded in $L^\infty(0,T;\mathbb{H}^1(\Omega)).$
\item There exists a constant $C>0$ independent of $\sigma$ such that
\begin{multline}
\label{DS3}
\max_{1\le n\le N_\tau}\|y(t_n)-\hat{y}_\sigma(t_n)\|_{\mathbb{H}^1(\Omega)} +
\|y-\hat{y}_\sigma\|_{L^2(0,T;\mathbb{H}^1(\Omega))}\\
\le C(h\|y\|_{C([0,T];\mathbb{H}^2(\Omega))}+\tau\|y'\|_{L^2(0,T;\mathbb{H}^1(\Omega))}),
\end{multline}
\begin{multline}
\label{DS3.1}
\|y-\hat{y}_\sigma\|_{L^\infty(0,T;\mathbb{H}^1(\Omega))}
\le C\{(\tau+\sqrt{\tau})\|y'\|_{L^2(0,T;\mathbb{H}^1(\Omega))}+h\|y\|_{C([0,T];\mathbb{H}^2(\Omega))}\}.
\end{multline}
\end{enumerate}
\end{lemma}
\begin{proof}
The existence and uniqueness of the solution $\hat{y}_\sigma$ is easily proved by using the Lax-Milgram theorem. We are going to prove the boundedness of $\hat{y}_\sigma$. Taking $w_h=\hat{y}_{n,h}$ in \eqref{DS2} we have
\begin{multline}
\label{DS4}
|\hat{y}_{n,h}|^2+\nu\tau_n|\nabla\hat{y}_{n,h}|^2+\alpha^2|\nabla\hat{y}_{n,h}|^2\\
=(\hat{y}_{n-1,h},\hat{y}_{n,h})+\alpha^2(\nabla\hat{y}_{n-1,h},\nabla \hat{y}_{n,h})+\tau_n(f_n,\hat{y}_{n,h}).
\end{multline}
It follows from H\"older's inequality that
\begin{equation}
\label{DS5}
(\hat{y}_{n-1,h},\hat{y}_{n,h})\le \dfrac{1}{2} |\hat{y}_{n-1,h} |^2 +\dfrac{1}{2} |\hat{y}_{n,h} |^2 ,
\end{equation}
\begin{equation}
\label{DS6}
\alpha^2(\nabla\hat{y}_{n-1,h},\nabla\hat{y}_{n,h})\le \dfrac{\alpha^2}{2} |\nabla\hat{y}_{n-1,h} |^2 +\dfrac{\alpha^2}{2} |\nabla\hat{y}_{n,h} |^2,
\end{equation}
$$
\int_{t_{n-1}}^{t_n}\nu a(y(t),\hat{y}_{n,h})dt\le \dfrac{\nu}{4}\tau_n |\nabla\hat{y}_{n,h} |^2 +C_{\nu} \int_{t_{n-1}}^{t_n} |\nabla y(t) |^2dt,
$$
$$
\int_{t_{n-1}}^{t_n}\alpha^2a(y'(t),\hat{y}_{n,h})dt\le
\dfrac{\nu}{4}\tau_n |\nabla\hat{y}_{n,h} |^2 +C_{\nu,\alpha} \int_{t_{n-1}}^{t_n} |\nabla y'(t) |^2 dt,
$$
$$
\int_{t_{n-1}}^{t_n}(y'(t),\hat{y}_{n,h})dt\le\dfrac{\nu}{4}\tau_n |\nabla\hat{y}_{n,h} |^2 +C_{\nu,\Omega} \int_{t_{n-1}}^{t_n} |y'(t)|^2dt.
$$
In the last estimate, we have used the fact that
$|\hat{y}_{n,h}| \le C_\Omega |\nabla\hat{y}_{n-1,h}|,$
by Poincar\'e's inequality, since $\hat{y}_{n,h}\in \mathbb{H}^1_0(\Omega)$. Summarizing the last three estimates leads to
\begin{equation}
\label{DS7}
\tau_n(f_n,\hat{y}_{n,h})\le \dfrac{3\nu}{4}\tau_n|\nabla\hat{y}_{n,h}|^2 +C_{\nu,\alpha,\Omega}\|y\|^2_{W^{1,2}(t_{n-1},t_n;\mathbb{H}^1(\Omega))}.
\end{equation}
Combining \eqref{DS4} with \eqref{DS5}, \eqref{DS6}, \eqref{DS7} gives
\begin{align*}
&\dfrac{1}{2} |\hat{y}_{n,h}|^2 +\dfrac{\nu}{4}\tau_n |\nabla\hat{y}_{n,h} |^2 +\dfrac{\alpha^2}{2} |\nabla\hat{y}_{n,h} |^2 \\
\le & \dfrac{1}{2} |\hat{y}_{n-1,h} |^2+\dfrac{\alpha^2}{2} |\nabla\hat{y}_{n-1,h}|^2 +C_{\nu,\alpha,\Omega}\|y\|^2_{W^{1,2}(t_{n-1},t_n;\mathbb{H}^1(\Omega))}.
\end{align*}
Summarizing these estimates from $n=1$ to $n=k$ ($k$ is an arbitrary integer in the set $\{1,2,\ldots,N_\tau\}$) we obtain
\begin{equation}
\label{DS7.1}
\|\hat{y}_\sigma\|_{L^\infty(0,T;\mathbb{H}^1(\Omega))}\le C(\|y_{0h}\|_{\mathbb{H}^1(\Omega)}+\|y\|_{W^{1,2}(0,T;\mathbb{H}^1(\Omega))}),
\end{equation}
which gives the first statement. To prove the second statement, we set
$$e=y-\hat{y}_\sigma,\quad e_h=P_\sigma y -\hat{y}_\sigma, \quad e_p= y-P_\sigma y.$$
Since $\hat{y}_{n,h}=y(t_n)-e(t_n)$, \eqref{DS2} gives
\begin{multline*}
\left(\dfrac{e(t_n)-e(t_{n-1})}{\tau_n},w_h\right)+\dfrac{1}{\tau_n}\int_{t_{n-1}}^{t_n}\nu a(e(t),w_h) + \alpha^2a\left(\dfrac{e(t_n)-e(t_{n-1})}{\tau_n},w_h\right)=0.
\end{multline*}
Replacing $e$ by $e_p+e_h$ and using the definition of $P_h$ yield
\begin{multline*}
(e_h(t_n)-e_h(t_{n-1}),w_h)+\alpha^2a(e_h(t_n)-e_h(t_{n-1}),w_h)+\int_{t_{n-1}}^{t_n}\nu a(e_h(t),w_h)dt\\
+\int_{t_{n-1}}^{t_n}\nu a(e_p(t),w_h)dt=0.
\end{multline*}
Taking $w_h=e_h(t_n)$ we have
\begin{multline*}
\dfrac{1}{2}|e_h(t_n)|^2-\dfrac{1}{2}|e_h(t_{n-1})|^2+\dfrac{1}{2}|e_h(t_n)-e_h(t_{n-1})|^2
+ \dfrac{\alpha^2}{2}|\nabla e_h(t_n)|^2\\
-\dfrac{\alpha^2}{2}|\nabla e_h(t_{n-1})|^2+\dfrac{\alpha^2}{2}|\nabla e_h(t_n)-\nabla e_h(t_{n-1})|^2
+\nu \int_{t_{n-1}}^{t_n}|\nabla e_h|^2dt\\
\le \dfrac{\nu}{2}\|y-P_\sigma y\|^2_{L^2(t_{n-1},t_n;\mathbb{H}^1(\Omega))}
+\dfrac{\nu}{2} \int_{t_{n-1}}^{t_n}|\nabla e_h|^2dt.
\end{multline*}
Therefore,
\begin{multline*}
\dfrac{1}{2}|e_h(t_n)|^2+\dfrac{1}{2}|e_h(t_n)-e_h(t_{n-1})|^2
+ \dfrac{\alpha^2}{2}|\nabla e_h(t_n)|^2
+\dfrac{\alpha^2}{2}|\nabla e_h(t_n)-\nabla e_h(t_{n-1})|^2\\
+\dfrac{\nu}{2} \int_{t_{n-1}}^{t_n}|\nabla e_h|^2dt
\le\dfrac{1}{2}|e_h(t_{n-1})|^2+\dfrac{\alpha^2}{2}|\nabla e_h(t_{n-1})|^2
+ \dfrac{\nu}{2}\|y-P_\sigma y\|^2_{L^2(t_{n-1},t_n;\mathbb{H}^1(\Omega))}.
\end{multline*}
Adding these inequalities for $n=1,\ldots,k$, and noticing that $e_h(0)=0$, we have
\begin{multline*}
\dfrac{1}{2}|e_h(t_k)|^2+\dfrac{\alpha^2}{2}|\nabla e_h(t_k)|^2+\dfrac{\nu}{2}\int_{0}^{t_k}|\nabla e_h|^2dt
\le \dfrac{\nu}{2}\|y-P_\sigma y\|^2_{L^2(0,T;\mathbb{H}^1(\Omega))}.
\end{multline*}
This implies that, for every $k\in\{1,2, \ldots,N_\tau\}$
\begin{align}
\|e_h(t_k)\|_{\mathbb{H}^1(\Omega)}&\le C\{h\|y\|_{L^2(0,T;\mathbb{H}^2(\Omega))}+\tau\|y'\|_{L^2(0,T;\mathbb{H}^1(\Omega))}\},\label{DS8}\\
\|e_h\|_{L^2(0,T;\mathbb{H}^1(\Omega))}&\le C\{h\|y\|_{L^2(0,T;\mathbb{H}^2(\Omega))}+\tau\|y'\|_{L^2(0,T;\mathbb{H}^1(\Omega))}\},\label{DS9}
\end{align}
by Lemma \ref{LM2}. Now, we are going to estimate $\|e_p(t_k)\|_{\mathbb{H}^1(\Omega)}$. We have
$e_p(t_k)=y(t_k)-P_hy(t_k)$. Then,
\begin{equation}
\label{DS10}
\|e_p(t_k)\|_{\mathbb{H}^1(\Omega)}\le Ch\|y(t_k)\|_{\mathbb{H}^2(\Omega)}\le Ch\|y\|_{C([0,T];\mathbb{H}^2(\Omega))}.
\end{equation}
From \eqref{DS8}, \eqref{DS10} we have
\begin{align*}
\|e(t_k)\|_{\mathbb{H}^1(\Omega)}&\le \|e_h(t_k)\|_{\mathbb{H}^1(\Omega)}+\|e_p(t_k)\|_{\mathbb{H}^1(\Omega)}\\
&\le C(h\|y\|_{C([0,T];\mathbb{H}^2(\Omega))}+\tau\|y'\|_{L^2(0,T;\mathbb{H}^1(\Omega))}),
\end{align*}
for every $k=1,\ldots,N_\tau$. This estimate combining with Lemma \ref{LM2}, \eqref{DS9} imply \eqref{DS3}. Finally, we prove \eqref{DS3.1}. Assume that $t\in (t_{n-1},t_n)$ for some $n\in \{1,\ldots,N_\tau\}.$ Then
$$\|y(t)-\hat{y}_\sigma(t)\|_{\mathbb{H}^1(\Omega)}\le \|y(t)-y(t_n)\|_{\mathbb{H}^1(\Omega)}+\|y(t_n)-\hat{y}_\sigma(t_n)\|_{\mathbb{H}^1(\Omega)}.$$
The first term can be estimated as follows
\begin{align*}
\|y(t)-y(t_n)\|_{\mathbb{H}^1(\Omega)}&=\|\int_t^{t_n}y'(s)ds\|_{\mathbb{H}^1(\Omega)}\\
&\le \int_t^{t_n}\|y'(s)\|_{\mathbb{H}^1(\Omega)}ds\le \sqrt{\tau}\|y'\|_{L^2(0,T;\mathbb{H}^1(\Omega))}.
\end{align*}
This combining with \eqref{DS3} implies \eqref{DS3.1}.
\end{proof}
\begin{theorem}
For every given $u\in \mathbb{L}^2(Q)$, \eqref{DSE1} has a unique solution $y_\sigma\in V_\sigma$. Moreover, there exists a constant $C>0$ independent of $\sigma$ such that
\begin{multline}
\label{DS11}
\max_{1\le n\le N_\tau}\|y(t_n)-y_\sigma(t_n)\|_{\mathbb{H}^1(\Omega)} +
\|y-y_\sigma\|_{L^2(0,T;\mathbb{H}^1(\Omega))}\\
\le C(\tau\|y'\|_{L^2(0,T;\mathbb{H}^1(\Omega))}+h\|y\|_{C([0,T];\mathbb{H}^2(\Omega))}+h\|p\|_{L^2(0,T;H^1(\Omega))}),
\end{multline}
\begin{multline}
\label{DS12}
|y-y_\sigma\|_{L^\infty(0,T;\mathbb{H}^1(\Omega))}\le
C\{(\tau+\sqrt{\tau})\|y'\|_{L^2(0,T;\mathbb{H}^1(\Omega))}\\
+h\|y\|_{C([0,T];\mathbb{H}^2(\Omega))}+h\|p\|_{L^2(0,T;H^1(\Omega))}\},
\end{multline}
where $y\in W^{1,2}(0,T;D(A))$ is the unique solution of \eqref{ST2}.
\end{theorem}
\begin{proof}
Set $\varepsilon=y-y_\sigma,\,e=y-\hat{y}_\sigma,\,e_\sigma=\hat{y}_\sigma-y_\sigma,$ where $\hat{y}_\sigma$ is the solution of \eqref{DS2}. Replacing $y_{n,h}$ by $\hat{y}_{n,h}-e_{n,h}$ in \eqref{DSE1} gives
\begin{multline*}
\left(\dfrac{\hat{y}_{n,h}-\hat{y}_{n-1,h}}{\tau_n},w_h\right)+\nu a(\hat{y}_{n,h},w_h)+\alpha^2 a\left(\dfrac{\hat{y}_{n,h}-\hat{y}_{n-1,h}}{\tau_n},w_h\right)\\
-\left(\dfrac{e_{n,h}-e_{n-1,h}}{\tau_n},w_h\right)-\nu a(e_{n,h},w_h)-\alpha^2 a\left(\dfrac{e_{n,h}-e_{n-1,h}}{\tau_n},w_h\right)\\
+c(y_{n,h},y_{n,h},w_h)=\dfrac{1}{\tau_n}\int_{t_{n-1}}^{t_n}(u(t),w_h)dt.
\end{multline*}
Using \eqref{DS2} we have
\begin{multline*}
\left(\dfrac{e_{n,h}-e_{n-1,h}}{\tau_n},w_h\right)+\nu a(e_{n,h},w_h)+\alpha^2 a\left(\dfrac{e_{n,h}-e_{n-1,h}}{\tau_n},w_h\right)=\\
\dfrac{1}{\tau_n}\int_{t_{n-1}}^{t_n}\{\nu a(y(t),w_h)+\alpha^2a(y'(t),w_h)+(y'(t),w_h)-(u(t),w_h)\}dt\\
+c(y_{n,h},y_{n,h},w_h).
\end{multline*}
Combining with \eqref{ST1} we get
\begin{multline*}
\left(\dfrac{e_{n,h}-e_{n-1,h}}{\tau_n},w_h\right)+\nu a(e_{n,h},w_h)+\alpha^2 a\left(\dfrac{e_{n,h}-e_{n-1,h}}{\tau_n},w_h\right)=\\
c(y_{n,h},y_{n,h},w_h)-\dfrac{1}{\tau_n}\int_{t_{n-1}}^{t_n}c(y(t),y(t),w_h)dt-\dfrac{1}{\tau_n}\int_{t_{n-1}}^{t_n}(p(t),{\rm div}\, w_h)dt.
\end{multline*}
Therefore,
\begin{multline*}
\left(e_{n,h}-e_{n-1,h},w_h\right)+\nu \int_{t_{n-1}}^{t_n}a(e_{n,h},w_h)dt+\alpha^2 a\left(e_{n,h}-e_{n-1,h},w_h\right)=\\
\int_{t_{n-1}}^{t_n}\{c(y_{n,h},y_{n,h},w_h)-c(y(t),y(t),w_h)\}dt-\int_{t_{n-1}}^{t_n}(p(t),{\rm div}\, w_h)dt.
\end{multline*}
Setting $w_h=e_{n,h}$ we obtain
\begin{align}
&\dfrac{1}{2}|e_{n,h}|^2-\dfrac{1}{2}|e_{n-1,h}|^2+\dfrac{1}{2}|e_{n,h}-e_{n-1,h}|^2+\nu\int_{t_{n-1}}^{t_n}|\nabla e_{n,h}|^2dt\notag\\
&+\dfrac{\alpha^2}{2}|\nabla e_{n,h}|^2-\dfrac{\alpha^2}{2}|\nabla e_{n-1,h}|^2+\dfrac{\alpha^2}{2}|\nabla(e_{n,h}-e_{n-1,h})|^2\notag\\
=&\int_{t_{n-1}}^{t_n}\{c(y_{n,h},y_{n,h},e_{n,h})-c(y(t),y(t),e_{n,h})\}dt-\int_{t_{n-1}}^{t_n}(p(t),{\rm div}\, e_{n,h})dt\notag\\
=&\int_{t_{n-1}}^{t_n}\{c(y_{n,h},y_{n,h},e_{n,h})-c(y(t),y(t),e_{n,h})\}dt-\int_{t_{n-1}}^{t_n}(p(t)-R_hp(t),{\rm div}\, e_{n,h})dt.\label{DS13}
\end{align}
After some algebraic computations we get for every $t\in (t_{n-1},t_n)$ that
\begin{align*}
&c(y(t),y(t),e_{n,h})-c(y_{n,h},y_{n,h},e_{n,h})\\
=&c(e(t),y(t),e_{n,h})+c(\hat{y}_{n,h},e(t),e_{n,h})+c(e_{n,h},\hat{y}_{n,h},e_{n,h})+c(y_{n,h},e_{n,h},e_{n,h})\\
=&c(e(t),y(t),e_{n,h})+c(\hat{y}_{n,h},e(t),e_{n,h})+c(e_{n,h},\hat{y}_{n,h},e_{n,h}).
\end{align*}
Hence, we get from \eqref{DS13} that
\begin{align}
&\dfrac{1}{2}|e_{n,h}|^2-\dfrac{1}{2}|e_{n-1,h}|^2+\dfrac{1}{2}|e_{n,h}-e_{n-1,h}|^2+\nu\int_{t_{n-1}}^{t_n}|\nabla e_{n,h}|^2dt\notag\\
&+\dfrac{\alpha^2}{2}|\nabla e_{n,h}|^2-\dfrac{\alpha^2}{2}|\nabla e_{n-1,h}|^2+\dfrac{\alpha^2}{2}|\nabla(e_{n,h}-e_{n-1,h})|^2\notag\\
\le\; & \int_{t_{n-1}}^{t_n}\{|c(e(t),y(t),e_{n,h})|+|c(\hat{y}_{n,h},e(t),e_{n,h})|+|c(e_{n,h},\hat{y}_{n,h},e_{n,h})|\}dt\notag\\
&+\int_{t_{n-1}}^{t_n}|(p(t)-R_hp(t),{\rm div}\,e_{n,h})|dt.\label{DS14}
\end{align}
Since $y\in W^{1,2}(0,T;V)$ and $\{\hat{y}_\sigma\}_\sigma$ is bounded in $L^\infty(0,T;\mathbb{H}^1(\Omega))$, we have
\begin{align*}
\int_{t_{n-1}}^{t_n}|c(e(t),y(t),e_{n,h})|dt \le & C\int_{t_{n-1}}^{t_n}|\nabla e(t)||\nabla e_{n,h}|dt\\
\le &C\int_{t_{n-1}}^{t_n}|\nabla e(t)|^2dt+\dfrac{\nu}{8}\int_{t_{n-1}}^{t_n}|\nabla e_{n,h}|^2dt,\\
\int_{t_{n-1}}^{t_n}|c(\hat{y}_{n,h},e(t),e_{n,h})|dt \le & C\int_{t_{n-1}}^{t_n}|\nabla e(t)||\nabla e_{n,h}|dt\\
\le & C\int_{t_{n-1}}^{t_n}|\nabla e(t)|^2dt+\dfrac{\nu}{8}\int_{t_{n-1}}^{t_n}|\nabla e_{n,h}|^2dt,\\
\int_{t_{n-1}}^{t_n}|c(e_{n,h},\hat{y}_{n,h},e_{n,h})|dt\le & C\int_{t_{n-1}}^{t_n}|e_{n,h}|^{1/4}|\nabla e_{n,h}|^{7/4}dt\\
\le & C\tau_n|e_{n,h}|^2+\dfrac{\nu}{8}\int_{t_{n-1}}^{t_n}|\nabla e_{n,h}|^2dt,\\
\int_{t_{n-1}}^{t_n}|(p(t)-R_hp(t),{\rm div}\, e_{n,h})|dt\le & C\int_{t_{n-1}}^{t_n}|p(t)-R_hp(t)|^2dt+\dfrac{\nu}{8}|\nabla e_{n,h}|^2dt.
\end{align*}
Putting all these estimates in \eqref{DS14} we obtain
\begin{multline*}
(1-C\tau_{n})|e_{n,h}|^2+|e_{n,h}-e_{n-1,h}|^2+\nu\int_{t_{n-1}}^{t_n}|\nabla e_{n,h}|^2dt\\
+\alpha^2|\nabla e_{n,h}|^2+\alpha^2|\nabla(e_{n,h}-e_{n-1,h})|^2\\
\le |e_{n-1,h}|^2+\alpha^2|\nabla e_{n-1,h}|^2+ C\int_{t_{n-1}}^{t_n}|\nabla e(t)|^2dt+C\int_{t_{n-1}}^{t_n}|p(t)-R_hp(t)|^2dt.
\end{multline*}
Adding these inequalities for $n=1,2,\ldots,k$, and noticing that $e_{0,h}=0$, we get
\begin{multline*}
(1-C\tau)|e_{k,h}|^2+\alpha^2|\nabla e_{k,h}|^2+\nu\int_0^{t_k}|\nabla e_\sigma(t)|^2dt\\
\le \sum_{n=1}^{k-1}C\tau_n|e_{n,h}|^2+ C\int_{0}^{t_k}|\nabla e(t)|^2dt+C\int_{0}^{t_k}|p(t)-R_hp(t)|^2dt.
\end{multline*}
Using the discrete Gronwall inequality we have, for every $k=1,2,\ldots,N_\tau$,
\begin{align*}
\|e_{k,h}\|^2_{\mathbb{H}^1(\Omega)}+\int_0^{t_k}|\nabla e_\sigma(t)|^2dt
\le &C\int_0^T|\nabla e(t)|^2dt+C\int_0^T|p(t)-R_hp(t)|^2dt\\
\le &C\|e\|^2_{L^2(0,T;\mathbb{H}^1(\Omega))}+Ch^2\|p\|^2_{L^2(0,T;H^1(\Omega))}.
\end{align*}
This together with \eqref{DS3} imply \eqref{DS11}. By a similar argument as in the proof of \eqref{DS3.1} we get \eqref{DS12} from \eqref{DS11}.
To finish the proof, we have to prove the uniqueness of a solution to \eqref{DSE1}. Assume that $y_\sigma^1, y_\sigma^2\in V_\sigma$ are two solutions of \eqref{DSE1}. Setting $y_\sigma=y_\sigma^2-y_\sigma^1$, then we need to prove that $y_\sigma=0$. Subtracting \eqref{DSE1} for $y_\sigma^2-y_\sigma^1$ and taking $w_h=y_{n,h}$ we get
\begin{multline}
\label{DS15}
\left(\dfrac{y_{n,h}-y_{n-1,h}}{\tau_n},y_{n,h}\right)+\nu a(y_{n,h},y_{n,h})+\alpha^2 a\left(\dfrac{y_{n,h}-y_{n-1,h}}{\tau_n},y_{n,h}\right)\\
=-c(y_{n,h},y_{n,h}^1,y_{n,h}).
\end{multline}
Here, we have used the fact that
$$c(y^1_{n,h},y^1_{n,h},y_{n,h})-c(y^2_{n,h},y^2_{n,h},y_{n,h})=-c(y_{n,h},y_{n,h}^1,y_{n,h}).$$
It follows from \eqref{DS12} that $\|y^1_{\sigma}\|_{L^\infty(0,T;\mathbb{H}^1(\Omega))}\le C$, where $C$ is a constant depending only on $y,\sigma$. Hence, we can get from \eqref{DS15} that
\begin{multline*}
\dfrac{1}{2}|y_{n,h}|^2-\dfrac{1}{2}|y_{n-1,h}|^2+\dfrac{1}{2}|y_{n,h}-y_{n-1,h}|^2+\nu\tau_n|\nabla y_{n,h}|^2\\
+\dfrac{\alpha^2}{2}|\nabla y_{n,h}|^2-\dfrac{\alpha^2}{2}|\nabla y_{n-1,h}|^2+\dfrac{\alpha^2}{2}|\nabla (y_{n,h}-y_{n-1,h})|^2\\
\le C\tau_n|y_{n,h}|^{1/4}|\nabla y_{n,h}|^{7/4}\le C\tau_n|y_{n,h}|^2+\dfrac{\nu}{2}\tau_n|\nabla y_{n,h}|^2.
\end{multline*}
Hence,
\begin{equation*}
(1-C\tau_n)|y_{n,h}|^2+\alpha^2|\nabla y_{n,h}|^2\le |y_{n-1,h}|^2+\alpha^2|\nabla y_{n-1,h}|^2.
\end{equation*}
Adding these estimates for $n=1,2,\ldots,k$, and noticing that $y_{0,h}=0$, we get
$$(1-C\tau)|y_{k,h}|^2+\alpha^2|\nabla y_{k,h}|^2\le \sum_{n=1}^{k-1}C\tau_n|y_{n,h}|^2.$$
Using once again the discrete Gronwall inequality we conclude that $y_\sigma=0.$
\end{proof}
\begin{remark}{\rm
\label{rm1}
According to the proof above, the constants $C$ in \eqref{DS11} and \eqref{DS12} are dependent on $\|y\|_{W^{1,2}(0,T;V)}$, $\|\hat{y}_\sigma\|_{L^\infty(0,T;\mathbb{H}^1(\Omega))}$. However, by using Theorem \ref{tontaiNSV} and \eqref{DS7.1} we see that if $\|u\|_{L^2(0,T;\mathbb{L}^2(\Omega))}\le M$ then these constants depend only on $M$, not on $y,u$.}
\end{remark}
\begin{corollary}
\label{Cor4.1}
Assume that $\max\{\|u\|_{L^2(0,T;\mathbb{L}^2(\Omega))},\|v\|_{L^2(0,T;\mathbb{L}^2(\Omega))}\}\le M$. Denote by $y_u\in W^{1,2}(0,T;D(A))$ the unique solution of \eqref{ST2} and by $y_\sigma(v)\in V_\sigma$ the unique solution of \eqref{DSE1} corresponding to the control $v$. Then there exists a constant $C_M>0$ such that
\begin{equation}
\label{DS15.2}
\|y_u-y_\sigma(v)\|_{L^\infty(0,T;\mathbb{H}^1(\Omega))}\le C_M\{h+\tau+\sqrt{\tau}+\|u-v\|_{L^2(0,T;\mathbb{L}^2(\Omega))}\}.
\end{equation}
Moreover, if $u_\sigma\in U_\sigma$ and $u_\sigma\rightharpoonup u$ in $L^2(0,T;\mathbb{L}^2(\Omega_h))$ as $\sigma\to 0$ then
\begin{equation}
\label{DS15.3}
\|y_u-y_\sigma(u_\sigma)\|_{L^\infty(0,T;\mathbb{H}^1(\Omega))}\to 0 \text{ as } \sigma\to 0,
\end{equation}
\begin{equation}
\label{DS15.3.0}
\|y_u(T)-y_\sigma(u_\sigma)(T)\|_{\mathbb{H}^1(\Omega)}\to 0 \text{ as } \sigma\to 0.
\end{equation}
\end{corollary}
\begin{proof}
We have
\begin{align*}
\|y_u-y_\sigma(v)\|_{L^\infty(0,T;\mathbb{H}^1(\Omega))}&\le \|y_u-y_v\|_{L^\infty(0,T;\mathbb{H}^1(\Omega))}+\|y_v-y_\sigma(v)\|_{L^\infty(0,T;\mathbb{H}^1(\Omega))}\\
&= \|G(u)-G(v)\|_{L^\infty(0,T;\mathbb{H}^1(\Omega))} + \|y_v-y_\sigma(v)\|_{L^\infty(0,T;\mathbb{H}^1(\Omega))}.
\end{align*}
From \eqref{DS12}, \eqref{Pre1.0} and Remark \ref{rm1} we have
\begin{equation}
\label{DS15.3.1}
\|y_v-y_\sigma(v)\|_{L^\infty(0,T;\mathbb{H}^1(\Omega))}\le C_M(h+\tau+\sqrt{\tau}).
\end{equation}
In addition, the control-to-state mapping $G$ is of class $C^2$, so we can use mean value theorem, \eqref{DGE2.1}, and \eqref{ST2.1} to get \eqref{DS15.2}.
Next, we are going to prove \eqref{DS15.3}. We have
\begin{equation}
\label{DS15.4}
\|y_u-y_\sigma(u_\sigma)\|_{L^\infty(0,T;\mathbb{H}^1(\Omega))}\le \|y_u-y_{u_\sigma}\|_{L^\infty(0,T;\mathbb{H}^1(\Omega))} + \|y_{u_\sigma}-y_\sigma(u_\sigma)\|_{L^\infty(0,T;\mathbb{H}^1(\Omega))}.
\end{equation}
Since $u_\sigma\rightharpoonup u$ in $L^2(0,T;\mathbb{L}^2(\Omega_h))$ as $\sigma\to 0$ we get the boundedness of the sequence $\{u_\sigma\}_\sigma$ in the space $L^2(0,T;\mathbb{L}^2(\Omega_h))$. Then, from \eqref{DS15.3.1} we get that the second term in the right-hand side of \eqref{DS15.4} tends to $0$ as $\sigma\to 0$. By Theorem \ref{THR2.3}, we can extract from sequence $\{y_{u_\sigma}\}_\sigma$ a subsequence denoted in the same way such that
\begin{equation}
\label{DS15.5}
y_{u_\sigma}\rightharpoonup y_u \text{ in } W^{1,2}(0,T;D(A)).
\end{equation}
Since the embedding $W^{1,2}(0,T;D(A))\hookrightarrow L^\infty(0,T;\mathbb{H}^1(\Omega))$ is compact, we get
$$\|y_u-y_{u_\sigma}\|_{L^\infty(0,T;\mathbb{H}^1(\Omega))}\to 0 \text{ as } \sigma\to 0,$$
and \eqref{DS15.3} is proved. It follows from \eqref{DS15.5} that $y_{u_\sigma}(T)\rightharpoonup y_u(T)$ in $D(A)$. On the other hand, since $D(A)$ is compactly embedded in $\mathbb{H}^1(\Omega)$, then we have
$$\|y_{u_\sigma}(T)-y_u(T)\|_{\mathbb{H}^1(\Omega)}\to 0.$$
This together with \eqref{DS11} imply \eqref{DS15.3.0}. The proof is complete.
\end{proof}
\begin{theorem}
\label{THRDiffG}
The mapping $G_\alpha: L^2(0,T;\mathbb{L}^2(\Omega))\to V_\sigma$ defined by $G_\sigma(u)=y_\sigma(u)$, the solution of \eqref{DSE1}, is of class $C^\infty$. Moreover, $z_\sigma(v)=G'_\sigma(u)v$ is the unique solution of the following problem
\begin{equation}
\label{DifDSE}
\begin{cases}
\left(\dfrac{z_{n,h}-z_{n-1,h}}{\tau_n},w_h\right) + \nu a(z_{n,h},w_h) +\alpha^2 a\left(\dfrac{z_{n,h}-z_{n-1,h}}{\tau_n},w_h\right)\\
\hspace{1cm}+c(z_{n,h},y_{n,h},w_h)+c(y_{n,h},z_{n,h},w_h)=\dfrac{1}{\tau_n}\int_{t_{n-1}}^{t_n}(v(t),w_h)dt,\\
\hspace{6cm}\forall w_h\in V_h,\;\forall n=1,2,\ldots,N_\tau,\\
z_{0,h}=0,
\end{cases}
\end{equation}
where we have set $y_\sigma=y_\sigma(u).$
\end{theorem}
\begin{proof}
We consider the mapping $F_\sigma: V_\sigma\times L^2(0,T;\mathbb{L}^2(\Omega))\to {V'_h}^{N_\tau}\times V_h$ defined by $F_\sigma(y_\sigma,u)=(g_1,\ldots,g_{N_{\tau}},y_{0,h}-P_hy_0)$, where
\begin{align*}
\langle g_n,w_h\rangle=&(y_{n,h}-y_{n-1,h},w_h)+\nu\tau_na(y_{n,h},w_h)+\alpha^2a(y_{n,h}-y_{n-1,h},w_h)\\
&+\tau_nc(y_{n,h},y_{n,h},w_h)
-\int_{t_{n-1}}^{t_n}(u(t),w_h)dt,\;\;\;\forall w_h\in V_h,\;\forall n=1,2,\ldots,N_\tau.
\end{align*}
We can easily check that $F_\sigma$ is of class $C^\infty$ and for an arbitrary $e_\sigma\in V_\sigma$ we have
\begin{equation}
\label{DS15.1}
\dfrac{\partial F_\sigma}{\partial y_\sigma}(y_\sigma,u)e_\sigma
= (f_1,\ldots,f_{N_\tau},e_{0,h}),
\end{equation}
where $f_n\in V'_h$ is defined by
\begin{multline}
\label{DS16}
\langle f_n,w_h\rangle =(e_{n,h}-e_{n-1,h},w_h)+\nu\tau_na(e_{n,h},w_h)
+\alpha^2a(e_{n,h}-e_{n-1,h},w_h)\\
+\tau_nc(y_{n,h},e_{n,h},w_h)+\tau_nc(e_{n,h},y_{n,h},w_h),\quad\forall w_h\in V_h,\;\forall n= 1,2, \ldots,N_\tau.
\end{multline}
On the other hand, $F_\sigma(G_\sigma(u),u)=F_\sigma(y_\sigma(u),u)=0$ for every $u\in L^2(0,T;\mathbb{L}^2(\Omega))$. The proof is a consequence of the implicit function; we need to prove that $\dfrac{\partial F_\sigma}{\partial y_\sigma}(y_\sigma,u):V_\sigma\to {V'_h}^{N_\tau}\times V_h$ is an isomorphism for every $(y_\sigma,u)\in V_\sigma\times L^2(0,T;\mathbb{L}^2(\Omega))$. Since $V_\sigma$ and ${V'_h}^{N_\tau}\times V_h$ are spaces with the same finite dimension, we only need to prove that $\dfrac{\partial F_\sigma}{\partial y_\sigma}(y_\sigma,u)$ is injective. Suppose that $\dfrac{\partial F_\sigma}{\partial y_\sigma}(y_\sigma,u)e_\sigma=0$ for some $e_\sigma\in V_\sigma$, then \eqref{DS15.1} implies that $e_{0,h}=0$. Using \eqref{DS16} with $w_h=e_{n,h}$ we have
\begin{multline*}
(e_{n,h}-e_{n-1,h},e_{n,h})+\nu\tau_na(e_{n,h},e_{n,h})
+\alpha^2a(e_{n,h}-e_{n-1,h},e_{n,h})\\+\tau_nc(e_{n,h},y_{n,h},e_{n,h})=0.
\end{multline*}
Hence,
\begin{multline*}
\dfrac{1}{2}|e_{n,h}|^2-\dfrac{1}{2}|e_{n-1,h}|^2+\dfrac{1}{2}|e_{n,h}-e_{n-1,h}|^2+\nu\tau_n|\nabla e_{n,h}|^2
+\dfrac{\alpha^2}{2}|\nabla e_{n,h}|^2\\
-\dfrac{\alpha^2}{2}|\nabla e_{n-1,h}|^2+\dfrac{\alpha^2}{2}|\nabla (e_{n,h}-e_{n-1,h})|^2
\le C\tau_n|e_{n,h}|^{1/4}|\nabla e_{n,h}|^{7/4}|\nabla y_{n,h}|\\
\le \dfrac{\nu}{2}\tau_n|\nabla e_{n,h}|^2 + C\tau_n |e_{n,h}|^2.
\end{multline*}
Here, the constant $C$ depends on $\nu$ and $\|y_\sigma\|_{L^\infty(0,T;\mathbb{H}^1(\Omega))}$. Again, by using the discrete Gronwall inequality and noticing that $e_{0,h}=0$, we obtain $e_\sigma=0$.
\end{proof}
\subsection{Analysis of the discrete adjoint state equations}
It follows from Theorem \ref{THRDiffG} and the chain rule that the functional $J_\sigma: L^2(0,T;\mathbb{L}^2(\Omega))\to \mathbb{R}$ is of class $C^\infty$, and its derivative is given by
\begin{multline*}
J'_\sigma(u)v=\alpha_T\int_{\Omega_h}(y_\sigma(T)-y^h_T)z_\sigma(T)dx + \alpha_Q\int_0^T\int_{\Omega_h}(y_\sigma-y_Q)z_\sigma dxdt\\
+\gamma\int_0^T\int_{\Omega_h}uvdxdt,
\end{multline*}
where $y_\sigma=y_\sigma(u)=G_\sigma(u)$ and $z_\sigma=G'_\sigma(u)v$ is the solution of \eqref{DifDSE}.
To study the discrete adjoint state equation, we are going to introduce the space $V_\sigma^r$. For each given sequence $(\lambda_{1,h},\lambda_{2,h}, \ldots,\lambda_{N_\tau+1,h})\in V_h^{N_\tau+1}$, we define a function $\lambda_\sigma:[0,T]\to V_h$ by
\begin{equation*}
\begin{cases}
\lambda_\sigma(t_n)=\lambda_{n+1,h},\;n=0,1,\ldots,N_\tau,\\
\lambda_\sigma(t)=\lambda_{n,h}, \;\forall t\in (t_{n-1},t_n).
\end{cases}
\end{equation*}
We denote by $V_\sigma^r$ the set of all functions that are defined by this way.
Now, we consider the discrete adjoint state equation:
{\it Find} $\lambda_\sigma\in V_\sigma^r$ {\it such that}
\begin{equation}
\label{ADE1}
\begin{cases}
\left(\dfrac{\lambda_{n,h}-\lambda_{n+1,h}}{\tau_n},w_h\right) + \nu a(\lambda_{n,h},w_h)+\alpha^2a\left(\dfrac{\lambda_{n,h}-\lambda_{n+1,h}}{\tau_n},w_h\right)\\
+c(w_h,y_{n,h},\lambda_{n,h})+c(y_{n,h},w_h,\lambda_{n,h})=\dfrac{\alpha_Q}{\tau_n}\int_{t_{n-1}}^{t_n}(y_{n,h}-y_Q(t),w_h)dt,\\
\hspace{6cm}\forall w_h\in V_h,\;\forall n=1,2,\ldots,N_\tau,\\
(\lambda_{N_\tau+1,h},w_h)+\alpha^2a(\lambda_{N_\tau+1,h},w_h)=\alpha_T(y_{N_\tau,h}-y_T^h,w_h),\quad\forall w_h\in V_h.
\end{cases}
\end{equation}
In this system, first we compute $\lambda_{N_\tau+1,h}$ from the last equation in \eqref{ADE1}, then we descend in $n$ until $n=1$. We can prove the existence and uniqueness of a solution to \eqref{ADE1} by a similar way that we did for the system \eqref{DSE1}. Now, we are going to check that \eqref{ADE1} is actually the discrete adjoint state equation. Indeed, it follows from \eqref{DifDSE} and \eqref{ADE1} that
\begin{align*}
&\alpha_Q\int_0^T\int_{\Omega_h}(y_\sigma-y_Q)z_\sigma dxdt\\
&=\sum_{n=1}^{N_\tau}\alpha_Q\int_{t_{n-1}}^{t_n}(y_{n,h}-y_Q(t),z_{n,h})dt=\sum_{n=1}^{N_\tau}[(\lambda_{n,h}-\lambda_{n+1,h},z_{n,h})+\nu\tau_na(\lambda_{n,h},z_{n,h})]\\
&+\sum_{n=1}^{N_\tau}[\alpha^2a(\lambda_{n,h}-\lambda_{n+1,h},z_{n,h})+\tau_nc(z_{n,h},y_{n,h},\lambda_{n,h})+\tau_nc(y_{n,h},z_{n,h},\lambda_{n,h})]\\
&=\sum_{n=1}^{N_\tau}[(z_{n,h}-z_{n-1,h},\lambda_{n,h})+\nu\tau_na(z_{n,h},\lambda_{n,h})+\alpha^2a(z_{n,h}-z_{n-1,h},\lambda_{n,h})]\\
&+\sum_{n=1}^{N_\tau}[\tau_nc(z_{n,h},y_{n,h},\lambda_{n,h})+\tau_nc(y_{n,h},z_{n,h},\lambda_{n,h})]\\
&\hspace{6cm}-(\lambda_{N_\tau+1,h},z_{N_\tau,h})-\alpha^2a(\lambda_{N_\tau+1,h},z_{N_\tau,h})\\
&=\int_{0}^T\int_{\Omega_h}v\lambda_\sigma dxdt -\alpha_T\int_{\Omega_h}(y_{N_{\tau},h}-y_T^h)z_{N_\tau,h}dx\\
&=\int_{0}^T\int_{\Omega_h}v\lambda_\sigma dxdt -\alpha_T\int_{\Omega_h}(y_\sigma(T)-y_T^h)z_\sigma(T)dx.
\end{align*}
Here, we have used the fact that $z_{0,h}=0$. Hence,
\begin{equation*}
J'_\sigma(u)v=\int_0^T\int_{\Omega_h}(\lambda_\sigma + \gamma u)v dxdt.
\end{equation*}
The next theorem gives us the error estimates when approximating the adjoint state equation.
\begin{theorem}
Given $u\in L^2(0,T;\mathbb{L}^2(\Omega))$, let $(y,p)$ be the solution of \eqref{ST1}, $(\lambda,q,r)$ be the unique weak solution of \eqref{DGE3}, $y_\sigma=y_\sigma(u)$ be the associated discrete state, solution of \eqref{DSE1}, and $\lambda_\sigma$ be the associated discrete adjoint state, solution of \eqref{ADE1}. Then $\{\lambda_\sigma\}_\sigma$ is bounded in $L^\infty(0,T;\mathbb{H}^1(\Omega))$ and there exists a constant $C>0$ independent of $\sigma$ such that
\begin{equation}\label{ADE3}
\begin{aligned}
\|\lambda-\lambda_\sigma\|_{L^\infty(0,T;\mathbb{H}^1(\Omega))}\le &C\big\{\tau\|y'\|_{L^2(0,T;\mathbb{H}^1(\Omega))}+h\|y\|_{C([0,T];\mathbb{H}^2(\Omega))}\\\
&+h\|p\|_{L^2(0,T;H^1(\Omega))}+h\|r\|_{H^1(\Omega)}+h\|\lambda\|_{C([0,T];\mathbb{H}^2(\Omega))}\\
&+(\tau+\sqrt{\tau})\|\lambda'\|_{L^2(0,T;\mathbb{H}^1(\Omega))}
+h\|q\|_{L^2(0,T;H^1(\Omega))}+h\big\}.
\end{aligned}
\end{equation}
\end{theorem}
\begin{proof}
Define the operator $P'_\sigma: C([0,T];V)\to V_\sigma^r$ by
\begin{equation}
\label{ADE4}
(P'_\sigma w)_{n,h}=P_hw(t_{n-1}),\;n=1,2, \ldots,N_\tau+1,
\end{equation}
where $P_h$ is given in Definition \ref{Def1}. Analogously to Lemma \ref{LM2} we have
\begin{equation}
\label{ADE4.1}
\|w-P'_\sigma w\|_{L^2(0,T;\mathbb{H}^1(\Omega))}\le C(h\|w\|_{L^2(0,T;\mathbb{H}^2(\Omega))}+\tau\|w'\|_{L^2(0,T;\mathbb{H}^1(\Omega))}),
\end{equation}
for every $w\in W^{1,2}(0,T;D(A))$.
Set
$$\epsilon=\lambda-\lambda_\sigma=(\lambda-P'_\sigma\lambda)+(P'_\sigma\lambda-\lambda_\sigma)= \psi +\epsilon_\sigma.$$
From \eqref{ADE4} we have
$$\psi(t_n)=\lambda(t_n)- (P'_\sigma\lambda)(t_n)=\lambda(t_n)-P_h\lambda(t_n),\quad n=0,1,\ldots,N_\tau.$$
Also we have $\epsilon_\sigma(t_n)=\epsilon_{n+1,h}$. Since $\epsilon(t_n)=\lambda(t_n)-\lambda_{n+1,h}$, it follows from \eqref{DGE3} and \eqref{ADE1} that
\begin{multline*}
(\epsilon(t_{n-1})-\epsilon(t_n),w_h)+\nu\int_{t_{n-1}}^{t_n}a(\epsilon(t),w_h)dt+\alpha^2a(\epsilon(t_{n-1})-\epsilon(t_n),w_h)\\
+\int_{t_{n-1}}^{t_n}c(w_h,y(t),\lambda(t))dt +\int_{t_{n-1}}^{t_n}c(y(t),w_h,\lambda(t))dt\\
-\int_{t_{n-1}}^{t_n}c(w_h,y_{n,h},\lambda_{n,h})dt-\int_{t_{n-1}}^{t_n}c(y_{n,h},w_h,\lambda_{n,h})dt\\
+\int_{t_{n-1}}^{t_n}(q(t)-R_hq(t),{\rm div}\, w_h)dt=\alpha_Q\int_{t_{n-1}}^{t_n}(y(t)-y_{n,h},w_h)dt,\quad\forall w_h\in V_h.
\end{multline*}
Here, we have used the fact that $(R_hq(t),{\rm div}\, w_h)=0\;\forall w_h\in V_h$. Now, replacing $\epsilon$ by $\psi+\epsilon_\sigma$, $w_h$ by $\epsilon_{n,h}$ and taking into account that
\begin{multline*}
(\psi(t_n),w_h)+\alpha^2a(\psi(t_n),w_h)=(\lambda(t_n)-P_h\lambda(t_n),w_h)+\alpha^2a(\lambda(t_n)-P_h\lambda(t_n),w_h)\\
=(\lambda(t_n),w_h)+\alpha^2a(\lambda(t_n),w_h)-[(P_h\lambda(t_n),w_h)+\alpha^2a(P_h\lambda(t_n),w_h)]=0,\;\;\forall w_h\in V_h,
\end{multline*}
we obtain
\begin{multline*}
(\epsilon_{n,h}-\epsilon_{n+1,h},\epsilon_{n,h}) + \alpha^2a(\epsilon_{n,h}-\epsilon_{n+1,h},\epsilon_{n,h})+\nu\int_{t_{n-1}}^{t_n}a(\epsilon_{n,h},\epsilon_{n,h})dt\\
=\alpha_Q\int_{t_{n-1}}^{t_n}(y(t)-y_{n,h},\epsilon_{n,h})dt-\nu\int_{t_{n-1}}^{t_n}a(\psi(t),\epsilon_{n,h})dt-\int_{t_{n-1}}^{t_n}c(\epsilon_{n,h},y(t),\lambda(t))dt\\
-\int_{t_{n-1}}^{t_n}c(y(t),\epsilon_{n,h},\lambda(t))dt
+\int_{t_{n-1}}^{t_n}c(\epsilon_{n,h},y_{n,h},\lambda_{n,h})dt+\int_{t_{n-1}}^{t_n}c(y_{n,h},\epsilon_{n,h},\lambda_{n,h})dt\\
-\int_{t_{n-1}}^{t_n}(q(t)-R_hq(t),{\rm div}\, \epsilon_{n,h})dt.
\end{multline*}
Hence,
\begin{align}
&\dfrac{1}{2}|\epsilon_{n,h}|^2-\dfrac{1}{2}|\epsilon_{n+1,h}|^2
+\dfrac{1}{2}|\epsilon_{n,h}-\epsilon_{n+1,h}|^2+\nu \int_{t_{n-1}}^{t_n}|\nabla \epsilon_{n,h}|^2dt+\dfrac{\alpha^2}{2}|\nabla\epsilon_{n,h}|^2\notag\\
&-\dfrac{\alpha^2}{2}|\nabla\epsilon_{n+1,h}|^2+\dfrac{\alpha^2}{2}|\nabla(\epsilon_{n,h}-\epsilon_{n+1,h})|^2\le \alpha_Q\int_{t_{n-1}}^{t_n}|y(t)-y_\sigma(t)||\epsilon_{n,h}|dt\notag\\
&+\nu\int_{t_{n-1}}^{t_n}|\nabla \psi(t)||\nabla\epsilon_{n,h}|dt
+\int_{t_{n-1}}^{t_n}|q(t)-R_hq(t)||{\rm div}\,\epsilon_{n,h}|dt\notag\\
&+\int_{t_{n-1}}^{t_n}|c(y_{n,h},\epsilon_{n,h},\lambda_{n,h})-c(y(t),\epsilon_{n,h},\lambda(t))|dt\notag\\
&+\int_{t_{n-1}}^{t_n}|c(\epsilon_{n,h},y_{n,h},\lambda_{n,h})-c(\epsilon_{n,h},y(t),\lambda(t))|dt.\label{ADE5}
\end{align}
The right-hand side of \eqref{ADE5} can be estimated as follows
\begin{multline*}
\alpha_Q\int_{t_{n-1}}^{t_n}|y(t)-y_\sigma(t)||\epsilon_{n,h}|dt
+\nu\int_{t_{n-1}}^{t_n}|\nabla \psi(t)||\nabla\epsilon_{n,h}|dt
\le \dfrac{\tau_n}{2}|\epsilon_{n,h}|^2\\
+C\int_{t_{n-1}}^{t_n}|y(t)-y_\sigma(t)|^2dt+\dfrac{\nu}{8}\int_{t_{n-1}}^{t_n}|\nabla \epsilon_{n,h}|^2dt+C\int_{t_{n-1}}^{t_n}|\nabla\psi(t)|^2dt,
\end{multline*}
\begin{multline*}
\int_{t_{n-1}}^{t_n}|q(t)-R_hq(t)||{\rm div}\,\epsilon_{n,h}|dt\le \dfrac{\nu}{8}\int_{t_{n-1}}^{t_n}|\nabla\epsilon_{n,h}|^2dt+C\int_{t_{n-1}}^{t_n}|q(t)-R_hq(t)|^2dt,
\end{multline*}
\begin{align}
&\;\;\;\left|\int_{t_{n-1}}^{t_n}|c(y_{n,h},\epsilon_{n,h},\lambda_{n,h})-c(y(t),\epsilon_{n,h},\lambda(t))|dt\right|\notag\\
&=\left|\int_{t_{n-1}}^{t_n}[c(y_\sigma(t)-y(t),\epsilon_{n,h},\lambda(t))-c(y_\sigma(t),\epsilon_{n,h},\epsilon(t))]dt\right|\notag\\
&=\left|\int_{t_{n-1}}^{t_n}[c(y_\sigma(t)-y(t),\epsilon_{n,h},\lambda(t))-c(y_\sigma(t),\epsilon_{n,h},\psi(t))]dt\right|\notag\\
&\le C\|\lambda\|_{L^\infty(0,T;\mathbb{H}^1(\Omega))}\int_{t_{n-1}}^{t_n}\|y(t)-y_\sigma(t)\|_{\mathbb{H}^1(\Omega)}|\nabla \epsilon_{n,h}|dt\notag\\
&\;\;\;+C\|y_\sigma\|_{L^\infty(0,T;\mathbb{H}^1(\Omega))}\int_{t_{n-1}}^{t_n}\|\psi(t)\|_{\mathbb{H}^1(\Omega)}|\nabla \epsilon_{n,h}|dt\notag\\
&\le C\int_{t_{n-1}}^{t_n}\big[\|y(t)-y_\sigma(t)\|^2_{\mathbb{H}^1(\Omega)}+\|\psi(t)\|_{\mathbb{H}^1(\Omega)}^2\big]dt+\dfrac{\nu}{4}\int_{t_{n-1}}^{t_n}|\nabla \epsilon_{n,h}|^2dt.\label{ADE5.1}
\end{align}
For the last term in the right-hand side of \eqref{ADE5}, we first see that
\begin{equation}
\begin{aligned}
\label{ADE6}
&c(\epsilon_{n,h},y_{n,h},\lambda_{n,h})-c(\epsilon_{n,h},y(t),\lambda(t))\\
= &-[c(\epsilon_{n,h},y(t)-y_\sigma(t),\lambda(t))+c(\epsilon_{n,h},y_\sigma(t),\psi(t))+c(\epsilon_{n,h},y_\sigma(t),\epsilon_{n,h})].
\end{aligned}
\end{equation}
The first two terms in \eqref{ADE6} can be treated analogously as in \eqref{ADE5.1}. For the last we have
\begin{align}
\left|\int_{t_{n-1}}^{t_n}c(\epsilon_{n,h},y_\sigma(t),\epsilon_{n,h})dt\right|&\le C\int_{t_{n-1}}^{t_n}|\epsilon_{n,h}|^{1/4}|\nabla\epsilon_{n,h}|^{7/4}\|y_\sigma(t)\|_{\mathbb{H}^1(\Omega)}dt\notag\\
&\le C\|y_\sigma\|_{L^\infty(0,T;\mathbb{H}^1(\Omega))}\int_{t_{n-1}}^{t_n}|\epsilon_{n,h}|^{1/4}|\nabla\epsilon_{n,h}|^{7/4}dt\notag\\
&\le \dfrac{\nu}{8}\int_{t_{n-1}}^{t_n}|\nabla\epsilon_{n,h}|^2dt+C\tau_n|\epsilon_{n,h}|^2.\label{ADE7}
\end{align}
From \eqref{ADE5}-\eqref{ADE7} we get that
\begin{multline}
\label{ADE8}
(1-C\tau_n)|\epsilon_{n,h}|^2+|\epsilon_{n,h}-\epsilon_{n+1,h}|^2+\alpha^2|\nabla\epsilon_{n,h}|^2+\alpha^2|\nabla(\epsilon_{n,h}-\epsilon_{n+1,h})|^2\\
+\dfrac{\nu}{4}\int_{t_{n-1}}^{t_n}|\nabla\epsilon_{n,h}|^2dt\le |\epsilon_{n+1,h}|^2+\alpha^2|\nabla\epsilon_{n+1,h}|^2\\
+C\left\{\int_{t_{n-1}}^{t_n}\|y(t)-y_\sigma(t)\|^2_{\mathbb{H}^1(\Omega)}dt+\int_{t_{n-1}}^{t_n}\|\psi(t)\|^2_{\mathbb{H}^1(\Omega)}dt+\int_{t_{n-1}}^{t_n}|q(t)-R_hq(t)|^2dt\right\}.
\end{multline}
Set $\tilde{\epsilon}_{k,h}:=\epsilon_{N_\tau+1-k,h},\;\tilde{\tau}_k:=\tau_{N_\tau+1-k}$, $k=0,1,\ldots,N_\tau$, then \eqref{ADE8} gives
\begin{align*}
&(1-C\tilde{\tau}_k)|\tilde{\epsilon}_{k,h}|^2+|\tilde{\epsilon}_{k,h}-\tilde{\epsilon}_{k-1,h}|^2+\alpha^2|\nabla\tilde{\epsilon}_{k,h}|^2+\alpha^2|\nabla(\tilde{\epsilon}_{k,h}-\tilde{\epsilon}_{k-1,h})|^2\\
&+\dfrac{\nu}{4}\int_{t_{N_\tau-k}}^{t_{N_\tau+1-k}}|\nabla\tilde{\epsilon}_{k,h}|^2dt\le |\tilde{\epsilon}_{k-1,h}|^2+\alpha^2|\nabla\tilde{\epsilon}_{k-1,h}|^2\\
&+C\left\{\int_{t_{N_\tau-k}}^{t_{N_\tau+1-k}}\|y(t)-y_\sigma(t)\|^2_{\mathbb{H}^1(\Omega)}dt+\int_{t_{N_\tau-k}}^{t_{N_\tau+1-k}}\|\psi(t)\|^2_{\mathbb{H}^1(\Omega)}dt\right.\\
&\left.
+\int_{t_{N_\tau-k}}^{t_{N_\tau+1-k}}|q(t)-R_hq(t)|^2dt\right\}.
\end{align*}
Adding this inequality from $k=1$ to $k=n$ we have
\begin{align*}
&(1-C\tau)|\tilde{\epsilon}_{n,h}|^2+\alpha^2|\nabla \tilde{\epsilon}_{n,h}|^2+\dfrac{\nu}{4}\int_{t_{N_\tau-n}}^T|\nabla \epsilon_\sigma(t)|^2dt\\
&\le\sum_{k=1}^{n-1}C\tilde{\tau}_k|\tilde{\epsilon}_{k,h}|^2+ |\tilde{\epsilon}_{0,h}|^2+\alpha^2|\tilde{\epsilon}_{0,h}|^2+C\left\{\int_{t_{N_\tau-n}}^{T}\|y(t)-y_\sigma(t)\|^2_{\mathbb{H}^1(\Omega)}dt\right.\\&+\int_{t_{N_\tau-n}}^{T}\|\psi(t)\|^2_{\mathbb{H}^1(\Omega)}dt
\left.
+\int_{t_{N_\tau-n}}^{T}|q(t)-R_hq(t)|^2dt\right\}.
\end{align*}
Using the discrete Gronwall inequality we have, for every $n=0,1,\ldots,N_\tau$,
\begin{multline*}
\|\tilde{\epsilon}_{n,h}\|^2_{\mathbb{H}^1(\Omega)}\le \|\tilde{\epsilon}_{0,h}\|^2_{\mathbb{H}^1(\Omega)}
+C\left\{\int_0^{T}\|y(t)-y_\sigma(t)\|^2_{\mathbb{H}^1(\Omega)}dt\right.\\+\int_0^{T}\|\psi(t)\|^2_{\mathbb{H}^1(\Omega)}dt
\left.
+\int_0^{T}|q(t)-R_hq(t)|^2dt\right\}.
\end{multline*}
Hence,
\begin{multline}
\label{ADE8.1}
\|\epsilon_{n,h}\|^2_{\mathbb{H}^1(\Omega)}\le \|P_h\lambda(T)-\lambda_{N_\tau+1,h}\|^2_{\mathbb{H}^1(\Omega)}
+C\left\{\int_0^{T}\|y(t)-y_\sigma(t)\|^2_{\mathbb{H}^1(\Omega)}dt\right.\\+\int_0^{T}\|\psi(t)\|^2_{\mathbb{H}^1(\Omega)}dt
\left.
+\int_0^{T}|q(t)-R_hq(t)|^2dt\right\},\quad\forall n=1,2,\ldots,N_\tau+1.
\end{multline}
Now, we are going to estimate $\|P_h\lambda(T)-\lambda_{N_\tau+1,h}\|_{\mathbb{H}^1(\Omega)}$. Set
$$\varepsilon=\lambda(T)-\lambda_{N_\tau+1,h}=(\lambda(T)-P_h\lambda(T))+(P_h\lambda(T)-\lambda_{N_\tau+1,h})=\varepsilon_1+\varepsilon_2.$$
Replacing $\lambda_{N_\tau+1,h}$ by $\lambda(T)-\varepsilon$ in the last equation in \eqref{ADE1} we have
\begin{multline*}
(\varepsilon,w_h)+\alpha^2a(\varepsilon,w_h)=(\lambda(T),w_h)+\alpha^2a(\lambda(T),w_h)-\alpha_T(y_{N_\tau,h}-y_T^h,w_h).
\end{multline*}
Replacing $\varepsilon$ by $\varepsilon_1+\varepsilon_2$, $w_h$ by $\varepsilon_2$ and notice that $(\varepsilon_1,w_h)+\alpha^2a(\varepsilon_1,w_h)=0\;\forall w_h\in V_h$, we get
\begin{align*}
|\varepsilon_2|^2+\alpha^2|\nabla\varepsilon_2|^2
&=(\lambda(T),\varepsilon_2)+\alpha^2a(\lambda(T),\varepsilon_2)-\alpha_T(y_{N_\tau,h}-y_T^h,\varepsilon_2)\\
&=\alpha_T(y(T)-y_T,\varepsilon_2)-(r,{\rm div}\,\varepsilon_2)-\alpha_T(y_{N_\tau,h}-y_T^h,\varepsilon_2)\\
&=\alpha_T(y(T)-y_{N_\tau,h},\varepsilon_2)-\alpha_T(y_T-y_T^h,\varepsilon_2)-(r,{\rm div}\,\varepsilon_2)\\
&=\alpha_T(y(T)-y_{N_\tau,h},\varepsilon_2)-\alpha_T(y_T-y_T^h,\varepsilon_2)-(r-R_hr,{\rm div}\,\varepsilon_2).
\end{align*}
This implies that
$$\|\varepsilon_2\|_{\mathbb{H}^1(\Omega)}\le C(|y(T)-y_{N_\tau,h}|+|y_T-y_T^h|+|r-R_hr|).$$
Hence,
\begin{multline*}
\|\varepsilon_2\|_{\mathbb{H}^1(\Omega)}\le C(\tau\|y'\|_{L^2(0,T;\mathbb{H}^1(\Omega))}+h\|y\|_{C([0,T];\mathbb{H}^2(\Omega))}+h\|p\|_{L^2(0,T;H^1(\Omega))}\\
+h\|r\|_{H^1(\Omega)}+h).
\end{multline*}
This combining with \eqref{DS11}, \eqref{ADE4.1}, \eqref{ADE8.1} imply that
\begin{multline*}
\|\epsilon_{n,h\|_{\mathbb{H}^1(\Omega)}}\le C\big\{\tau\|y'\|_{L^2(0,T;\mathbb{H}^1(\Omega))}+h\|y\|_{C([0,T];\mathbb{H}^2(\Omega))}+h\|p\|_{L^2(0,T;H^1(\Omega))}\\
+h\|r\|_{H^1(\Omega)}+h\|\lambda\|_{L^2(0,T;\mathbb{H}^2(\Omega))}+\tau\|\lambda'\|_{L^2(0,T;\mathbb{H}^1(\Omega))} +h\|q\|_{L^2(0,T;H^1(\Omega))}+h\big\}\\
\forall n =1,2,\ldots,N_\tau+1.
\end{multline*}
We have
$$\|\psi(t_n)\|_{\mathbb{H}^1(\Omega)}=\|\lambda(t_n)-P_h\lambda(t_n)\|_{\mathbb{H}^1(\Omega)}\le Ch\|\lambda(t_n)\|_{\mathbb{H}^2(\Omega)}\le Ch\|\lambda\|_{C[0,T];\mathbb{H}^2(\Omega)},$$
for every $n=0,1,\ldots,N_\tau.$ Since $\epsilon(t_n)=\psi(t_n)-\epsilon_{n+1,h}$ we get
\begin{multline*}
\|\epsilon(t_n)\|_{\mathbb{H}^1(\Omega)}\le C\big\{\tau\|y'\|_{L^2(0,T;\mathbb{H}^1(\Omega))}+h\|y\|_{C([0,T];\mathbb{H}^2(\Omega))}+h\|p\|_{L^2(0,T;H^1(\Omega))}\\
+h\|r\|_{H^1(\Omega)}+h\|\lambda\|_{C([0,T];\mathbb{H}^2(\Omega))}+\tau\|\lambda'\|_{L^2(0,T;\mathbb{H}^1(\Omega))} +h\|q\|_{L^2(0,T;H^1(\Omega))}+h\big\}\\
\forall n =0,1,\ldots,N_\tau.
\end{multline*}
Now, assume that $t\in (t_{n-1},t_n)$, then
$$\epsilon(t)=\lambda(t)-\lambda_{n,h}=\lambda(t)-\lambda(t_{n-1})+(\lambda(t_{n-1})-\lambda_{n,h})=\lambda(t)-\lambda(t_{n-1})+\epsilon(t_{n-1}).$$
Analogously as in the last paragraph in the proof of Lemma \ref{LM53} we have
$$\|\lambda(t)-\lambda(t_{n-1})\|_{\mathbb{H}^1(\Omega)}\le \sqrt{\tau}\|\lambda'\|_{L^2(0,T;\mathbb{H}^1(\Omega))},$$
then we get \eqref{ADE3}.
\end{proof}
\begin{remark}{\rm
\label{rm2}
According to the proof above, the constant $C$ in \eqref{ADE3} depends on $\|\lambda\|_{L^{\infty}(0,T;\mathbb{H}^1(\Omega))}$, $\|y_\sigma\|_{L^\infty(0,T;\mathbb{H}^1(\Omega))}$. However, we see that if $\|u\|_{L^2(0,T;\mathbb{L}^2(\Omega))}\le M$ then this constant depends only on $M$, not on $\lambda, y, u$.}
\end{remark}
\begin{corollary}
Assume that $\max\{\|u\|_{L^2(0,T;\mathbb{L}^2(\Omega))},\|v\|_{L^2(0,T;\mathbb{L}^2(\Omega))}\}\le M$. Let $\lambda_u\in W^{1,2}(0,T;D(A))$ be the solution of \eqref{DGE4} and $\lambda_\sigma(v)\in V_\sigma$ be the solution of the discrete equation \eqref{ADE1} corresponding to the control $v$. Then there exists a constant $C_M>0$ such that
\begin{equation}
\label{ADE9}
\|\lambda_u-\lambda_\sigma(v)\|_{L^\infty(0,T;\mathbb{H}^1(\Omega))}\le C_M \left\{h+\tau+\sqrt{\tau}+\|u-v\|_{L^2(0,T;\mathbb{L}^2(\Omega))}\right\}.
\end{equation}
\end{corollary}
\begin{proof}
From \eqref{ADE3} we have
\begin{equation}
\label{ADE9.1}
\|\lambda_u-\lambda_\sigma(v)\|_{L^\infty(0,T;\mathbb{H}^1(\Omega))}\le \|\lambda_u-\lambda_v\|_{L^\infty(0,T;\mathbb{H}^1(\Omega)}+C(h+\tau+\sqrt{\tau}),
\end{equation}
where $C$ depends on $M$. Setting $\lambda=\lambda_u-\lambda_v$, then from \eqref{DGE5} we get
\begin{multline*}
-(\lambda_t,w)+\nu a(\lambda,w) -\alpha^2a(\lambda_t,w)=c(y_v,w,\lambda_v)+c(w,y_v,\lambda_v)\\ - c(y_u,w,\lambda_u)-c(w,y_u,\lambda_u)
+\alpha_Q(y_u-y_v,w).
\end{multline*}
Taking $w=\lambda$ and using the following identities
$$c(y_v,\lambda,\lambda_v)-c(y_u,w,\lambda_u)=c(y_v-y_u,\lambda,\lambda_v),$$
$$c(\lambda,y_v,\lambda_v)-c(\lambda,y_u,\lambda_u)=c(\lambda,y_v-y_u,\lambda_v)-c(\lambda,y_u,\lambda),$$
we obtain
\begin{multline*}
-(\lambda_t,\lambda)+\nu a(\lambda,\lambda) -\alpha^2a(\lambda_t,\lambda)=c(y_v-y_u,\lambda,\lambda_v)+c(\lambda,y_v-y_u,\lambda_v)\\
-c(\lambda,y_u,\lambda)
+\alpha_Q(y_u-y_v,\lambda).
\end{multline*}
Integrating from $t$ to $T$ then integrating by parts yields
\begin{multline}
\label{ADE10}
\dfrac{1}{2}|\lambda(t)|^2-\dfrac{1}{2}|\lambda(T)|^2+\nu\int_t^T|\nabla \lambda(s)|^2ds+\dfrac{\alpha^2}{2}|\nabla \lambda(t)|^2-\dfrac{\alpha^2}{2}|\nabla \lambda(T)|^2\\
\le C\left\{\int_t^T|\nabla y_v(s)-\nabla y_u(s)||\nabla\lambda(s)||\nabla \lambda_v(s)|ds\right.\\
+\int_t^T |\nabla\lambda(s)||\nabla y_v(s)-\nabla y_u(s)||\nabla\lambda_v(s)|ds
\left.+\int_t^T |\nabla\lambda(s)|^2|\nabla y_u(s)|ds\right\}\\
+\alpha_Q\int_t^T |y_u(s)-y_v(s)||\lambda(s)|ds.
\end{multline}
Using again \eqref{DGE5} we have
$$|\lambda(T)|^2+\alpha^2|\nabla\lambda(T)|^2=\alpha_T(y_u(T)-y_v(T),\lambda(T)).$$
Therefore,
$$|\lambda(T)|^2+\alpha^2|\nabla\lambda(T)|^2\le C|y_u(T)-y_v(T)|^2,$$
where $C$ is a constant depending only on $\alpha_T$. Hence, we get from \eqref{ADE10} that
\begin{multline*}
\dfrac{1}{2}|\lambda(t)|^2+\dfrac{\alpha^2}{2}|\nabla \lambda(t)|^2\le C_M\left\{\int_0^T|\nabla y_v(s)-\nabla y_u(s)|^2ds +|y_u(T)-y_v(T)|^2\right.\\
\left.+ \int_t^T|\nabla \lambda(s)|^2ds\right\}
\end{multline*}
since $\|\lambda_v\|_{W^{1,2}(0,T;V)},\,\|y_u\|_{W^{1,2}(0,T;V)}\le C_M$. In addition, we have
$$\int_0^T\|y_v(s)-y_u(s)\|^2ds +|y_u(T)-y_v(T)|^2\le C\|y_u-y_v\|^2_{W^{1,2}(0,T;V)},$$
\begin{align*}
\|y_u-y_v\|^2_{W^{1,2}(0,T;V)}&=\|G(u)-G(v)\|^2_{W^{1,2}(0,T;V)}\\
&\le \sup_{0\le \rho\le 1}\|G'(u+\rho(v-u))\|_{W^{1,2}(0,T;V)}\|u-v\|_{L^2(0,T;\mathbb{L}^2(\Omega))}\\
&\le C_M\|u-v\|_{L^2(0,T;\mathbb{L}^2(\Omega))}.
\end{align*}
Therefore,
$$\|\lambda(t)\|^2_{\mathbb{H}^1(\Omega)}\le C_M\left\{\|u-v\|_{L^2(0,T;\mathbb{L}^2(\Omega))}^2+\int_t^T\|\lambda(s)\|^2_{\mathbb{H}^1(\Omega)}ds\right\},\;\;\forall t\in [0,T].$$
This implies that
$$\|\lambda(t)\|_{\mathbb{H}^1(\Omega)}\le C\|u-v\|_{L^2(0,T;\mathbb{L}^2(\Omega))}\;\;\forall t\in[0,T],$$
by using the Gronwall inequality. Combining this with \eqref{ADE9.1} we get \eqref{ADE9}.
\end{proof}
\subsection{Convergence of the discrete control problem and error estimates}
Since $J_\sigma$ is a continuous and coercive function on a non-empty convex closed subset of a finite-dimensional space, it is easy to see that problem $(P_\sigma)$ has at least one solution. By the similar arguments as in \cite[Sections 4.3 and 4.4]{Casas2012}, we get the following theorems.
The first theorem shows the convergence of these discrete solutions to a solution of problem $(P)$. The proof of this theorem is exactly that of Theorem 4.13 in \cite{Casas2012}.
\begin{theorem}
\label{THR4.4}
Denote by $\bar{u}_\sigma$ a global solution of problem $(P_\sigma)$. Then the sequence $\{\bar{u}_\sigma\}_\sigma$ is bounded in $L^2(0,T;\mathbb{L}^2(\Omega))$ and there exist subsequences, denoted in the same way, weakly convergent in the space $L^2(0,T;\mathbb{L}^2(\Omega))$. If $\bar{u}\in L^2(0,T;\mathbb{L}^2(\Omega))$ is one of the limit points, i.e. $u_\sigma\rightharpoonup \bar{u}$, then $\bar{u}$ is a solution of problem $(P)$. Moreover, we have
\begin{equation}
\label{CD1}
\lim_{\sigma\to 0}\|\bar{u}-\bar{u}_\sigma\|_{L^2(0,T;\mathbb{L}^2(\Omega_h))}=0\quad \text{ and } \quad \lim_{\sigma\to 0}J_\sigma(\bar{u}_\sigma)=J(\bar{u}).
\end{equation}
\end{theorem}
In general, it is not correct to claim that the sequence $\{u_\sigma\}_\sigma$ is bounded in $L^2(0,T;\mathbb{L}^2(\Omega))$, because $\bar{u}_\sigma$ is only defined in $(0,T)\times \Omega_h$. We will prove that $\{u_\sigma\}_\sigma$ is bounded in $L^2(0,T;\mathbb{L}^2(\Omega_h))$ by a constant independent of $\sigma$. Then, we take an arbitrary element $v$ in the space $L^2(0,T;\mathbb{L}^2(\Omega))$ and extend every $\bar{u}_\sigma$ to $(0,T)\times \Omega$ by setting $\bar{u}_\sigma(t,x)=v(t,x)$ for every $(t,x)\in (0,T)\times (\Omega\backslash \Omega_h)$. By \eqref{Ap1}, the sequence $\{\bar{u}_\sigma\}_\sigma$ is bounded in $L^2(0,T;\mathbb{L}^2(\Omega))$ and every weak limit point of a subsequence is a solution of $(P)$, regardless of the choice of $v$.
The next theorem, whose proof is the same that of Theorem 4.15 in \cite{Casas2012}, is important from a practical point of view because it states that every strict local minimum of problem $(P)$ can be approximated by local minima
of problems $(P_\sigma)$.
\begin{theorem}
\label{THR4.5}
If $\bar{u}$ is a strict local minimum of $(P)$ then there exists a sequence $\{\bar{u}_\sigma\}_\sigma$ of local minima of problems $(P_\sigma)$ such that \eqref{CD1} holds.
\end{theorem}
We now denote by $\bar{u}$ a locally optimal control of the problem $(P)$, and for every $\sigma$, $\bar{u}_\sigma$ denotes a local solution of $(P_\sigma)$ such that $\|\bar{u}-\bar{u}_\sigma\|_{L^2(0,T;\mathbb{L}^2(\Omega_h))}\to 0$ (see Theorems \ref{THR4.4} and \ref{THR4.5}). Each element $u\in U_\sigma$ is extended to $(0,T)\times\Omega$ by setting $u(t,x)=\bar{u}(t,x)$ for $(t,x)\in (0,T)\times (\Omega\backslash \Omega_h)$. We will also denote by $\bar{y}$ and $\bar{\lambda}$ the state and adjoint state associated to $\bar{u}$, and by $\bar{y}_\sigma$ and $\bar{\lambda}_\sigma$ the discrete state and adjoint state associated to $\bar{u}_\sigma$.
We are ready to give space-time error estimates for the discretization.
\begin{theorem}
Suppose that \eqref{DGE12} holds. Then there exists a constant $C>0$ independent of $\sigma$ such that
\begin{align}
&\|\bar{u}-\bar{u}_\sigma\|_{L^2(0,T;\mathbb{L}^2(\Omega_h))} \le C(h+\tau+\sqrt{\tau}), \label{MEST1}\\
&\|\bar{y}-\bar{y}_\sigma\|_{L^\infty(0,T;\mathbb{H}^1(\Omega))}\le C(h+\tau+\sqrt{\tau}), \label{MEST2}\\
&\|\bar{\lambda}-\bar{\lambda}_\sigma\|_{L^\infty(0,T;\mathbb{H}^1(\Omega))}\le C(h+\tau+\sqrt{\tau}). \label{MEST3}
\end{align}
\end{theorem}
\begin{proof} The estimates \eqref{MEST2} and \eqref{MEST3} are an immediate consequence of \eqref{MEST1}, \eqref{DS15.2},
and \eqref{ADE9}. We only have to prove \eqref{MEST1}. To this end, we proceed by contradiction and assume that it is false. This implies that
$$
\limsup_{\sigma \to 0} \dfrac{1}{h}\|\bar{u}-\bar{u}_\sigma\|_{L^2(0,T;\mathbb{L}^2(\Omega_h))}=+\infty;
$$
therefore, there exists a sequence of $\sigma$ such that
\begin{equation*}
\lim_{\sigma \to 0} \dfrac{1}{h}\|\bar{u}-\bar{u}_\sigma\|_{L^2(0,T;\mathbb{L}^2(\Omega_h))}=+\infty.
\end{equation*}
Now, arguing exactly as in \cite[Section 4.4]{Casas2012} and using the second-order optimality conditions, we will obtain a contradiction for this sequence.
\end{proof}
\begin{remark} {\rm
The error order in the two last estimates in the above theorem looks a bit different from those (of order $O(h)$) in \cite{Casas2012}. The reason is that we do not require the technical condition $\tau\le Ch^2$ as in \cite{Casas2012}, so the error should contain both $\tau$ and $h$. It is obvious that if $\tau\le Ch^2$ then these orders of errors are the same.}
\end{remark}
\vskip 0.5cm
\noindent{\bf Acknowledgements.} This research is funded by Vietnam National Foundation for Science and Technology Development (NAFOSTED) under grant number 101.02-2018.303.
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Eventos
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{"url":"https:\/\/gilkalai.wordpress.com\/2022\/04\/29\/joshua-hinman-proved-baranys-conjecture-on-face-numbers-of-polytopes-and-lei-xue-proved-a-lower-bound-conjecture-by-grunbaum\/","text":"## Joshua Hinman proved B\u00e1r\u00e1ny\u2019s conjecture on face numbers of polytopes, and Lei Xue proved a lower bound conjecture by\u00a0Gr\u00fcnbaum.\n\n### Joshua Hinman proved B\u00e1r\u00e1ny\u2019s conjecture.\n\nOne of my first posts on this blog was a 2008 post Five Open Problems Regarding Convex\u00a0Polytopes, now 14 years later, I can tell you about the first problem on the list to get solved.\n\nImre B\u00e1r\u00e1ny posed in the late 1990s the following question:\n\nFor a $d$-dimensional polytope $P$ and every $k$, $0 \\le k \\le d-1$,\u00a0 is it true that $f_k(P) \\ge \\min (f_0(P),f_{d-1}(P))$?\n\nNow, Joshua Hinman settled the problem! In his paper A Positive Answer to B\u00e1r\u00e1ny\u2019s Question on Face Numbers of Polytopes he actually proved even stronger linear relations. The abstract of Joshua\u2019s paper starts with the very true assertion: \u201cDespite a full characterization of the face vectors of simple and simplicial polytopes, the face numbers of general polytopes are poorly understood.\u201d He moved on to describe his new inequalities:\n\n$\\frac{f_k(P)}{f_0(P)} \\geq \\frac{1}{2}\\biggl[{\\lceil \\frac{d}{2} \\rceil \\choose k} + {\\lfloor \\frac{d}{2} \\rfloor \\choose k}\\biggr], \\qquad \\frac{f_k(P)}{f_{d-1}(P)} \\geq \\frac{1}{2}\\biggl[{\\lceil \\frac{d}{2} \\rceil \\choose d-k-1} + {\\lfloor \\frac{d}{2} \\rfloor \\choose d-k-1}\\biggr].$\n\n### Lei Xue proved Gr\u00fcnbaum\u2019s conjecture\n\nIn her 2020 paper: A Proof of Gr\u00fcnbaum\u2019s Lower Bound Conjecture for general polytopes, Lei Xue proved a lower bound conjecture of Gr\u00fcnbaum: In 1967, Gr\u00fcnbaum conjectured that any d-dimensional polytope with d+s2d vertices has at least\n\n$\\phi_k(d+s,d) = {d+1 \\choose k+1 }+{d \\choose k+1 }-{d+1-s \\choose k+1 }$\n\nk-faces. Lei Xue proved this conjecture and also characterized the cases in which equality holds.\n\nCongratulations to Lei Xue and to Joshua Hinman.\n\nThis entry was posted in Combinatorics, Convex polytopes and tagged , , , . Bookmark the permalink.\n\n### 4 Responses to Joshua Hinman proved B\u00e1r\u00e1ny\u2019s conjecture on face numbers of polytopes, and Lei Xue proved a lower bound conjecture by\u00a0Gr\u00fcnbaum.\n\n1. Can it help shed some light on your 3^d conjecture?\n\n2. kodlu says:\n\nVery nice. Are you going to blog about Park and Pham\u2019s proof of the Kahn-Kalai conjecture?\n\n\u2022 Gil Kalai says:\n\nDear kodlu, I certainly should blog about the proof! (And I am also thinking about blogging about our old 2006 program toward a proof and some related open questions.)","date":"2022-05-19 03:07:14","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 1, \"img_math\": 7, \"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\": 0.6538703441619873, \"perplexity\": 2451.237175371516}, \"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\/1652662522741.25\/warc\/CC-MAIN-20220519010618-20220519040618-00463.warc.gz\"}"}
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From Changing Tires To Tackling Household Projects, The DIY GRRL Collective Wants To Teach Chicagoans To Do It Themselves
The group is holding its first class next month with Garcia's Auto Parts. Participants will learn how to change a spare tire.
Quinn Myers
8:55 AM CDT on Apr 4, 2022
DIY GRRL founder Summer Lambert changes a tire with Miguel Garcia at Garcia's Auto Parts in Humboldt Park
Quinn Myers/Block Club Chicago
HUMBOLDT PARK — A woman-led collective is starting a series of classes designed to teach people essential life skills.
DIY GRRL is run by Summer Lambert, a Humboldt Park resident who sees the project as a way to fill in the gaps for women — and others — who want to learn how to change a tire, fix something around the house and more. Lambert hopes the classes will also serve as a place to gather with friends and neighbors.
The project's origins date back to when Lambert moved to Chicago in March 2020. She was feeling socially isolated, especially as the COVID-19 pandemic shut down local businesses and public spaces.
Lambert started frequenting local businesses along California Avenue and around Humboldt Park, and got to know business owners and residents.
"I had been trying to find my place in Chicago and find where I fit in and where I can be a part of my community. I was trying to find a way of creating a collective of sorts, a group of people who do well for the community, who give back to the community," Lambert said.
Around the same time, Lambert became a homeowner and realized she kept hiring men to do jobs around her house — tasks she wondered why she had never learned how to do herself.
"It made me really think back on the fact that when I was growing up as a young girl, you're not taught very essential skills for being independent," Lambert said.
Credit: Quinn Myers/Block Club Chicago
DIY GRRL founder Summer Lambert in Humboldt Park on April 1, 2022
Over the past few months, Lambert started to think about how the two things could come together: giving back to the community and teaching people such skills.
"I came to this realization of, 'Why don't we … know how to do these things?'" she said. "Really tapping into my community that I have here in Chicago that I've been building, I know professionals who know how to do these things that I want to do and I realized there's an opportunity here."
This month, Lambert launched DIY GRRL, which will partner with local businesses to hold classes and workshops throughout the city.
The first of those will be Garcia's Auto Parts, 1211 N. California Ave., in Humboldt Park.
Miguel Garcia and Summer Lambert pose for a photo at Garcia's Auto Parts in Humboldt Park on April 1, 2022
Lambert said she met owner Miguel Garcia, whose family has operated the auto shop since 1975, when she started bringing her car in to the shop.
Now, Lambert and Garcia are co-hosting DIY GRRL's first class together May 15. Participants will learn how to change a spare tire.
Garcia said it was something he's wanted to do for a long time, and he jumped at the chance when Lambert approached him.
"I have voiced a need for wanting to teach people the basics of owning a vehicle. Just because everyone that comes in, isn't really educated enough on their vehicle. [Lambert] said, 'Let's come up with an event where we can teach people your basic stuff' … and that's how it came about," Garcia said.
Lambert and Garcia are holding a raffle on Instagram ahead of the class. Ten winners will bring in their cars and receive free, hands-on training. Spots for spectators will also be available.
A post shared by DIY GRRL (@diygrrl)
Garcia and Lambert said they've gotten a big response so far to their first class, and not just from women.
"I was starting it to be really focused on like, by women for women. And you know, quickly we're hearing from guys and nonbinary people and trans men," Lambert said. All are welcome, she added.
Lambert's day job is in marketing, and she sees part of DIY GRRL's mission as helping the businesses she partners with expand their profile.
"This is a really great way to also highlight local businesses and give them marketing opportunities," Lambert said. "I'm reaching out to them, I want them to feel empowered, to feel like this is a really great opportunity to get their name out as a business."
Lambert is talking to a bike shop, a welder and other businesses across Chicago about holding classes later this year. She also plans to continue partnering with Garcia for more auto repair classes.
"I'm hoping that not only is it an education opportunity, but it's a really good way of just meeting up with people and getting to know them. It could be like your monthly meetup spot with your friends. It's like the new farmers market, but you're learning a skill," she said.
Summer Lambert and Miguel Garcia changing a tire at Garcia's Auto Parts ahead of their newly launched DIY GRRL class in May
DIY GRRL
Garcia's Auto Parts
|
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============
Requirements
============
Flow is being developed and tested on multiple platforms and pretty easy to set
up. Nevertheless we recommend that you go through the following list before installing
Flow, because a server with exotic *php.ini* settings or wrong file permissions can
easily spoil your day.
Server Environment
==================
Not surprisingly, you'll need a web server for running your Flow-based web
application. We recommend *Apache* (though *nginx*, *IIS* and others work too – we just
haven't really tested them). Please make sure that the
`mod_rewrite <http://httpd.apache.org/docs/current/mod/mod_rewrite.html>`_ module is
enabled.
.. tip::
To enable Flow to create symlinks on Windows Server 2008 and higher you need
to do some extra configuration. In IIS you need to configure `Authentication` for
your site configuration to use a specific user in the `Anonymous Authentication`
setting. The configured user should also be allowed to create symlinks using the
local security policy `Local Policies > User Rights Assignments > Create symbolic links`
Flow's persistence mechanism requires a `database supported by Doctrine DBAL
<http://www.doctrine-project.org/projects/dbal.html>`_. Make sure to use at least 10.2.2
for MariaDB, and 5.7.7 when using MySQL.
PHP
===
Flow was one of the first PHP projects taking advantage of namespaces and
other features introduced in PHP version 5.3. By now we started using features of
PHP 7.1, so make sure you have **PHP 7.1.0** or later available on your web server. Make
sure your PHP CLI binary is the **same version**!
The default settings and extensions of the PHP distribution should work fine
with Flow but it doesn't hurt checking if the PHP modules ``mbstring``, ``tokenizer``
and ``pdo_mysql`` are enabled, especially if you compiled PHP yourself.
.. note::
Make sure the PHP functions ``exec()``, ``shell_exec()``,
``escapeshellcmd()`` and ``escapeshellarg()`` are not disabled in you PHP
installation. They are required for the system to run.
The development context might need more than the default amount of memory.
At least during development you should raise the memory limit to about 250 MB
in your *php.ini* file.
In case you get a fatal error message saying something like ``Maximum function nesting
level of '100' reached, aborting!``, check your *php.ini* file for settings regarding
Xdebug and modify/add a line ``xdebug.max_nesting_level = 500`` (suggested value).
|
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Sherman Township is one of the nineteen townships of Huron County, Ohio, United States. As of the 2010 census the population of the township was 510.
Geography
Located on the western edge of the county, it borders the following townships:
Lyme Township - north
Ridgefield Township - northeast corner
Peru Township - east
Greenfield Township - southeast corner
Norwich Township - south
Reed Township, Seneca County - southwest
Thompson Township, Seneca County - northwest
No municipalities are located in Sherman Township.
Name and history
Sherman Township was named for Taylor Sherman, a director of the Firelands company.
It is the only Sherman Township statewide.
Government
The township is governed by a three-member board of trustees, who are elected in November of odd-numbered years to a four-year term beginning on the following January 1. Two are elected in the year after the presidential election and one is elected in the year before it. There is also an elected township fiscal officer, who serves a four-year term beginning on April 1 of the year after the election, which is held in November of the year before the presidential election. Vacancies in the fiscal officership or on the board of trustees are filled by the remaining trustees.
References
External links
County website
Townships in Huron County, Ohio
Townships in Ohio
|
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| 799
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Table of Contents
ALSO BY GRANT BUDAY
Title Page
Epigraph
Chapter One
Chapter Two
Chapter Three
Chapter Four
Chapter Five
Chapter Six
Chapter Seven
Chapter Eight
Chapter Nine
Chapter Ten
Copyright Page
ALSO BY GRANT BUDAY
_Rootbound_ , ECW Press, 2006
_A Sack of Teeth_ , Raincoast Books, 2002
_Golden Goa_ , ECW Press, 2000
_White Lung_ , Anvil Press, 1999
Though masquerading as an epic, the _Odyssey_ is the first Greek novel; and therefore wholly irresponsible where myths are concerned.
—ROBERT GRAVES
**Chapter One**
**T** HE WALLS OF TROY throb in the sun, a trick of the heat radiating off the limestone. Ten years I've studied those walls. Ten years. Sometimes they advance and sometimes they recede, at other times they waver and sway as if viewed through fire, at sunset their colour deepens, becoming richer, while in the rain they turn grey. In many places they are black from the burning pitch the Trojans pour down to drive us off. Each spring grass sprouts in the seams between the stones and the goats stand up on their hind legs to reach it. The walls are as high as five men. At the top there are oak palings battered by our catapults and charred from our burning arrows. In the early days the Trojans would stand there and wave to us as they pissed.
It's spring, the time of dragonflies. They're admirable hunters, patrolling the meadows, hovering, darting, killing. It's a dangerous season for men as well, for the jackals and wolves are on the move, and when we venture inland we wrap our ankles in leather against snakes. The days grow hot, the mosquito breeds, and the season of fever is near. Soon the meadows will dry and crack and the winds from the east will carry red dust.
I'd genuinely believed we'd be home before winter. The Trojans had other ideas, laughing at our attempts to negotiate— _keep the gold but return Helen_ —then catapulting goatskins loaded with rocks and scorpions at us. Soon that first summer ebbed into autumn, the leaves fell and the trees were bare. Under-supplied, we hunkered in our tents eating the last of our salted meat and drinking rainwater. I suggested returning in the spring. Agamemnon said no, he'd lose half the army and knew it.
He urged me to rally the men. _Explain that it's better this way. They'll be home sooner._
I dutifully tried to convince the men and they dutifully tried to believe me.
Another spring and summer passed. All too soon the leaves were on the ground again and winter was back. Our breath was cold smoke and each dawn we broke discs of ice on the water pots. By the third year resentment turned to sorrow and finally to acceptance tainted by hatred. Everyone believed—though few were bold enough to say—that it was Menelaus's fault.
Menelaus still believes Helen loves him—we're not only righting a wrong, but punishing a crime and reclaiming honour. He's a king, Agamemnon's brother, and Agamemnon is also a king, and such men expect to get their way. What I want is to go home. To see my son. To see my wife. Yet Menelaus is adamant. He wants his woman and Agamemnon wants Troy, which will make him king of kings, so we've stayed. Reluctantly, doggedly. We've dug in while Agamemnon has ranted about pride, first trying to shame us and then to inspire us. The fourth summer was a bad one for fever, and then came the bloody flux, the men groaning over their red stool. Many deserted, though where they went was a mystery. Some were so desperate they reappeared months later, starved, sick, telling tales of abduction and escape. The wheels of the seasons ground past, time turning stone to sand, young men to old, black hair grey. Six years. Seven. To acceptance and hatred add contempt. What kind of man lets another one take his woman? What kind of man leaves a foreigner, a Trojan, unwatched in his home? Menelaus is an idiot and we are paying the price.
Forget Helen and the Trojan gold, the gilded doors and bronze horses, forget the pearls, the sapphires, the silk, the opium, and the reputation. Forget it all. But Agamemnon and Menelaus don't want to forget; forgetting is an art that is beyond them, in their view akin to cowardice (you see how narrowly their minds flow). They don't approve of shrewd avoidance, in my estimation as valuable a craft as that of the metal smith or the shipwright, and one not as easily mastered.
So what to do? Years of siege had failed to break the walls of Troy. We retrieved the bow and arrows of God-like Heracles, but still failed to break the walls of Troy. And then Agamemnon committed his biggest blunder of all and took Briseis from Achilles, and all too soon Achilles was dead and our last hope shrivelled with our greatest warrior on his pyre. Our world was reduced to ash, the air stank of smoke, for days we heard nothing but the cawing of ravens. Surely some god was against us. Many gods. Every imaginable sacrifice was made, thigh bones wrapped in fat, goblets of ram's blood, owl skulls, the wings of hawks. If I have contempt for Menelaus, I hate Agamemnon. He knows what he's done and that he's to blame. So what does he do, this wise king, this great man, this leader of leaders? He decides that it's up to me, Odysseus, to break the walls of Troy —when their bright swords bend and bull-hide shields split it's always up to me. They resent this. They won't say so but they do, because they don't trust me. They say I deny the gods, that I ask too many questions, that I'm out for number one, (as if they apply their minds to higher things while I, venal and greedy, talk in circles), worst of all they fear that I'll call black white—and here's what really enrages them—that I'll convince them black _is_ white. For all that they celebrate a sleek argument they think there's too much of the eel in mine. But now our backs are to the sea. What choice do they have? Menelaus wants his woman and Agamemnon Troy.
And so the brothers pay me an official visit. I could pitch a rock from my tent to theirs, nonetheless they insist upon arriving in their tasselled chariots, sporting their lion skins, their plumes. Agamemnon has his golden staff said, by him, to have come from Zeus Cloudsplitter himself. His upper lip is freshly shaved though his lead-coloured eyes are pouched in purple. The seams caused by his brooding run deep across his brow. A leader of unshakable purpose and will, I'll give him that, but he's in decline. He's forty-seven, an old man, thick-necked and hairy, and looks nothing at all like the busts which flank the entrance to his tent, the inside of which stinks like a chicken yard.
The years have been even crueller to Menelaus. He's smaller than Agamemnon, and his shoulders, always narrow, have shrunk further. He's lost all of his lower teeth. A palsy twitches in his right hand so that he hides it behind his back. Failing eyesight makes him thrust his face forward and squint, and as for his once dazzling red hair, he now hennas it every week to hide the grey. Then there are his perpetually chapped lips. What a contrast to Paris who, though a swindler and a coward, with no more nobility than a dungheap dog, is still handsome, with a young man's clear eyes and flat gut. Ten years ago Paris arrived in Sparta wearing an indigo robe trimmed with pearls, crocodile sandals with gold clasps, his hair perfect, while all Helen had to look forward to was cotton dyed in onion skin. So off she went, taking half the treasury with her.
Menelaus and Agamemnon make their appearance just as young Sinon is plucking my ear hairs. (I ask you, why do I need hair in my ears?) I stand to greet them and we bandy the praise and usual bunk, the sort of blather I've come to despise almost as much as the brothers themselves.
"Bold Odysseus," begins Agamemnon, as if standing on a box. "Son of Laertes, grandson of Autolycus, who, with Diomedes, stole into the Trojan camp and slew the Thracians and routed their horses, I, Agamemnon, son of Atreus, grandson of Pelops, great grandson of Tantalus, call upon you once again, for we grow weary of this war. Ten years have passed. It is time for final action." And on and on. His broken nose lends timbre to his voice, as if he speaks through a pipe. A man untroubled by self-doubt, Agamemnon regards his own piss as gold and you should be honoured to have a sip. "Think of some plan, Odysseus. Succeed where main force has failed, bend your wits to this and never shall you be absent from our feast table." He grows nobler by the minute while Menelaus stands half a step behind, squinting and twitching. Agamemnon winds it all up by saying I may state my reward. "Name it, Odysseus, name it." He gestures grandly, as though the world is his to give.
But what reward can compensate for the ten years I've lost? Chopping ten years from his life? Putting a spear up his ass? A futile line of thought, one I indulge too much. I bow in acknowledgement of such a grand offer from so eminent a lord, of course it's less an offer than a demand, for you don't disappoint the unforgiving Agamemnon. He uses men like hammers, and keeps using them until they break, and then drops them. Still, I can't deny that it _would_ be a shame to stumble home empty-handed after so many years, like my father did, so I give him my thousand-mile stare and think it over, just to make him squirm, just to make him value my response, just to stake some territory. I turn and look at our beached ships parching in the sun, the eyes on the hulls overdue for repainting, the sails rotten, the woodworms relentless. Beyond them lies Tenedos and beyond that the open sea and home. "Agamemnon," I say, turning to face him at last. "King of Mycenae, Lion of the Achaeans. I'll name my reward when we are standing inside Troy, when Priam is on his knees, Paris dead, and Helen with her rightful husband, for only then will I be worthy of it."
Pleased with my response, he stands taller, though narrows his eyes as he checks my words for hidden meaning (for he is quite right that a bargain is being negotiated). "So you'll take up the task?"
"Of course."
"Good. Excellent." His hearty tone says that we are still comrades.
"The gods willing," I caution.
He inclines his head meaning he understands perfectly. "We're down to the last bulls," he adds.
I acknowledge that sacrifices are at a premium. "But can you think of a greater need?"
"None, none."
And so I send them off with praise and assurances and then call Sinon back to finish my other ear. For a time he works in silence, probing and plucking, showing me each faded auburn hair as he uproots it, every pinprick sting strangely satisfying. Hair, it sprouts all year-round, summer or winter, more relentless than grass, yet why is it that a dog's fur grows only so long and then stops? Why does my beard grow and grow while the hair on my chest stays one length?
"What will you do?" asks Sinon, interrupting these profound cogitations.
"What do you advise?" Sinon is young but has an agile and entertaining mind, and I never ignore a shrewd opinion.
He clears his throat and strikes a dignified pose. His face takes on the gravity of a king, the stern brow, the pursed lips, the elevated chin. In a chest-deep voice that is a perfect imitation of Agamemnon, he states, "Succeed, good Odysseus, and the ode masters will sing of you. For a man there can be no greater glory."
I applaud.
He bows. On a roll now, eager to perform, eager to please, he asks, "Should I do Menelaus?"
"Yes. Give me Menelaus."
Prudently glancing around in case we're being observed, he narrows his shoulders, hunches his back, takes on the pained squint of a man with weak eyes. He clenches his fists and in a cuckold's agony moans that Helen belongs to him, that she did not run away, she was kidnapped, and she loves him. " _She does. She must!_ "
Everything falls still in the afternoon heat, the dogs sleep, the ravens go silent, the flies rest, even the waves seem to grow sluggish. The day in its lull. The breeze drops and only the sun continues to burn. The ever-attentive Sinon rolls up the walls of my tent so that it doesn't get too hot. The tent is ox hide and years of sun and wind and salt have bleached it the grey of driftwood. My single bed is a sorry sight compared to the one I built for Penelope and me with its olive tree corner post. A clever bit of joinery, the pieces fitting as perfectly as we fit each other. How charmed she was, how her eyes gleamed at the idea of a live tree imparting its spirit to our nights. I sit down and ease my feet out of my sandals, roll two wads of wax until they're pliable and then press them into my ears. This taste for seclusion is a quirk that causes suspicion; the others sleep in groups, in twos and threes. I lay down and shut my eyes and begin my afternoon ritual conjuring of Penelope. First, I recall her sea-green eyes, deep and lucid and as full of surprises as the ocean itself, then I move on to her mouth, and then to her hair, thick and black, and then her voice; there's a resonance to her voice, a small richness: if oak could sing it would be in the voice of Penelope. She likes to laugh. She enjoys people and their odd ways, is intrigued by wanderers and eccentrics. That I like the taste of burnt sulphur bemuses her, as does the fact that wind unnerves me, a sailor, and that I can hold my breath for two hundred heartbeats. Once I've conjured her face and her voice, I move down to her shoulders, to the delicate valleys behind her collarbones, then to her breasts, her navel, the slope of her hip, and then glide down to that warm and hidden place where I linger as long as I can. Later, sated, our limbs twined, she tells me how life proceeds on our farm, the weight of the eggs, the quality of the goats' milk, the state of the orange trees, the days of rain versus the days of sun. And of course all about Telemachus, how big he's grown, how much he eats, the beauty of his eyes, his capacity for numbers (no trait of mine), his imagination, and the long conversations he has with the dog, Argos. Telemachus is fourteen years old now and must be well along in his lessons in the sword and the bow. It hurts not to be there teaching him these things, that he learns them from someone else, a stranger. Worse, that he knows this other man better than he knows me, and is likely growing closer to him every day, and that when I return—if I return—we'll be as formal as if we'd never met. And all of this I blame on Menelaus.
It was summer ten years ago when word reached us that Helen had run off with Paris. A tasty bit of gossip, savoury as spitted lamb. Penelope wasn't surprised. She said it was typical, if more brazen, than Helen's previous romps. We were harvesting oranges at the time.
"Poor Tyndareus," she said. "When Helen was growing up she nearly drove him insane."
She stood atop the ladder picking oranges and passing them down. Telemachus, four years old, was swatting at the wasps with a stick while the slaves worked the other trees. "Tyndareus?" I said. "What about Menelaus?"
"Menelaus? He had a choice. He should've known what he was getting involved with. You did, didn't you?" She arched an eyebrow and regarded me. A barbed question.
"Yes," I answered quickly. "It was obvious."
That wasn't quite a lie. Nearly nine years earlier I'd gone to Sparta where all the others, Menelaus, Diomedes, Palamedes, Ajax, were competing for Helen. I was twenty and an idiot, but even I soon saw that it was all theatre and that the majestic Helen would be more curse than prize.
"The spark of Zeus runs through her," I said now, unable to resist a little teasing. "Perhaps she can't help it. Blame the gods."
Penelope aimed an orange at my head. I caught it in my basket, but now Helen was in both our thoughts. Penelope frowned and plucked angrily at the fruit causing a shower of leaves. She grew up with Helen, they're cousins, and she had no patience for her and even less for the men who believed she was divine. Penelope was glad when she ran off with Paris, and hoped it would be the last we'd hear of her.
She didn't understand that when Helen looked at a man he burned. It wasn't that she was so beautiful, no, not at all. Those stories are false, nothing but rumour and delusion. Her face was hard, all planes and edges, her forehead too high, her chin too long, her brow too heavy, but it didn't matter, because she was the great and glorious Helen, and when she looked at you you felt chosen, elevated above everyone else, and honestly believed that your life—so grey until now—was about to become golden.
Everyone knew the story. Leda had been raped by Zeus who had taken the form of a swan. Leda laid an egg from which Helen was hatched, and here she was, sixteen years later, the child of the greatest god himself, destined for Olympus. Some believed all this as fact, as undeniable as the bruise on your toe when you kick a stone and yet no matter what we thought of the tale, we all believed that to lie with her was to walk unscathed through fire. So we were eager, we were curious. When I caught my first glimpse of her I saw that for all her presence she had the air of a lost child. Her eyes pleaded _Help me_ even as her posture threatened to bring down thunderbolts. You didn't doubt for a minute that she had nightmares, heard voices, burned candles for fear of the dark, wept with her head in her mother's lap. It was this combination of prowess and frailty that entranced so many of us. You wanted to lead her out to a green field and calm her as if she was a young mare.
Penelope hated all this. It made her spit. But with the kidnapping came a more pressing issue. "What are you going to tell Menelaus when he comes for your help? You know he'll come. He's probably on his way now."
I made a face that said she was talking nonsense, to which she responded by coming down off the ladder and gripping my beard with both hands. "Don't be naive. You know he is."
We'd all vowed allegiance to whomever Helen picked, a tactic to forestall civil war. Still, I argued. "It's been years. Besides, he has Agamemnon and all the others." I gestured vaguely toward the southeast, across the mainland toward Sparta, the centre of the world, the big smoke. "Why would he come all this way just for me?"
"Spare me the false modesty. You'd have to be as dead as dried fish before he'd go without you."
Still, I disagreed.
She blew air in disgust. Penelope was the least superstitious of women, but like all Achaeans she was a fatalist, and she worried that we were entwined with Helen, caught the way ivy chokes a tree, and she was right, we should have run while we had the chance, for two days later Menelaus arrived. He found me in the barn sharpening a plough. He had old Nestor with him and that chancre Palamedes; the three of them stood backlit in the doorway.
"Brave Odysseus, son of Laertes."
"Menelaus. Friend. Brother." I smiled and opened my arms as if his presence brought joy to my heart instead of dread to my gut.
"May Zeus Earth-Shaker keep your roof posts strong," said Menelaus.
"Come in," I said, "honour my humble barn." I called for wine.
Though he was doing his best to be dignified, Menelaus looked haggard; after all, he was a gull, a cuckold, and now here he was calling on others to help him get his wife back. We waded through the niceties and then he got to the point and reminded me of the oath.
The oath had been my idea, and it had returned to snare me. At the time it had seemed shrewd given my growing doubts regarding the competition for Helen. You don't get a woman like her and simply go back to the farm and live a contented life, no, you get more, a lot more, too much: envy and bitterness, and men trying to steal her, exactly what Paris did, leaving Menelaus looking like a pig on a platter.
The point was this: if I backed out I'd pay. If ever I needed help none would come, and even if that time never came, Menelaus and his big brother Agamemnon would be sure to take revenge, arranging accidents, burning my farm, maybe stealing Penelope or Telemachus. My name would be dirt and their lives a torment. So we stood there, me, Nestorthe-bore, Menelaus-the-cuckold, and Palamedes-the-haemorrhoid. The horses swivelled their ears to listen, the cows swung their heads to watch; the cats, the goats, and even the pigeons in the rafters and the mice in the hay became attentive. The sunlight poured in silt-rich and radiant, so that the air looked filled with gold dust and all you had to do was scoop it up in your palm and pour it into your pouch to be a rich man.
Menelaus talked about Trojan gold and everything else we could win. "It's about honour," he said. "Paris betrayed me. Betrayed all of us. Broke truce. Nine days I entertained him. Nine days I gave him everything he wanted, girls, boys, wine. And yet when I go off to my grandfather's funeral, what does he do, that piece of shit, that goat's ass, but steal my wife! I gave him hospitality and he robbed me!"
Nestor and Palamedes had clearly heard this rant before. They shifted impatiently, and when Menelaus finally wound down Palamedes spoke up. "Ships and men, Odysseus. We want ships and men. Plus cattle, grain, and wine."
"Wine," I said, wondering what had become of my order for a jug.
Palamedes was shark-smiling. He enjoyed seeing me squirm. He had a narrow skull, a crow's nose, eyes close-set and small, a short damp black beard that made me think of pubic hair. He knew I was stalling. There was no doubt he was shrewd, and he had a soul like pit lime. He thought he'd been through my brain like a termite through wood and seen everything inside me. Well, balls to him, I thought. I raised my arms priest-wise and began to sing, not words but yelps, and I resumed my plough filing, the noise rasping my skull bone while I howled and yipped and my visitors stared. Their shocked eyes told me that I was onto a good ruse. I kept at it, scraping at that blade until Menelaus put his fists to his ears. Then I dropped the rasp and picked up a goat and harnessed her to the plough. I did the same with a lamb then I stood back and pissed on my feet. I gave them a good soaking, but saved enough to wash the dust from my road-weary guests, the least I could do as a good host. They leapt like cats. Menelaus stared, looking around to see if there were other madmen in the barn, as if maybe all Ithaka was possessed, an island of lunatics. He backed away toward the door. Nestor was right with him, the two of them thinking no way, we don't want this crazy bastard Odysseus along. But Palamedes wasn't going for it. There's always someone who just doesn't like you and wants to see you fall. Palamedes was that man.
I wasn't done yet, though. I churned up a good froth of spit so that I foamed at the mouth. I was about to start barking, but at that moment Telemachus ran in shouting about the strange ship in the bay. Palamedes picked him up and held him like a fond uncle ready to pop a fig into his mouth, and in one smooth motion drew his knife and put it to my son's throat and gave me a querying look.
"I'll prepare my ships," I said, flat-voiced and hard.
Palamedes smiled. A cool triumph for him, one he savoured like another man's wife, while for me a humiliation that's followed me right up to the present.
"Odysseus?" asked Menelaus, frowning, bewildered at the performance he'd just witnessed.
I muttered something about too much wine at lunch and then made a show of plunging my head into a water trough.
Menelaus became brisk and practical, telling me that the fleet gathered at Aulis in thirty days.
I nodded as if I'd have it no other way and at that moment, as if on cue, Old Niko limped in with the wine and we all drank to a swift victory. So it was off to Troy for the brave Odysseus.
When Menelaus and his cronies had departed and Telemachus was put to bed, Penelope and I walked up the hill through the tart scent of citrus in the cool of the evening. I told her everything that had happened in the barn. When I reached the part about Palamedes putting his knife to Telemachus's throat, she staggered and clutched my arm so tightly her nails drew blood. "His time will come," I said, putting on a bold show of confidence I didn't feel.
The cart track led up and over the ridge to where the wind blew hard from the sea. We sat down and watched the quarter moon through the shredded clouds. The dark sea gleamed silver where the moonlight hit. The scent of grass and mint sweetened the salt breeze. I put my arm around my wife—she smelled of lavender and fear and trembled with rage, not only at Palamedes but at Helen, because she'd hoped we'd finally got out from her shadow.
"Ten weeks and I'll be back," I said. "Shall I give Helen your love?"
Penelope looked at me with those green eyes of hers that begged me not to joke. I pressed her hand. They say the Scythians believe that the hand is the house of the soul, for everything we do is via the fingers, eating, making love, killing. An interesting notion, though if I were pressed to choose an organ I would opt for the tongue.
"I hate Palamedes."
"And he hates me," I said, though I couldn't say whether this meant anything. He'd grown even more arrogant and ambitious since I'd seen him nine years ago and he'd kept close to Menelaus and Agamemnon. I could understand Menelaus not seeing through him, after all, he'd let Paris walk off with his woman. But Agamemnon? "I expect I'll see young Ajax," I said, changing the subject.
She ignored this. "Telemachus will miss you," she whispered.
Reassuring myself as much as her, I said that he had a lot to divert him. We sat a long time thinking about him. We even managed to laugh a little at all of the questions he'd been asking of late. Where do the waves come from? Can I climb a ladder and pick clouds? And there were his ongoing efforts to tame spiders and befriend beetles. Toward a father one feels ultimate loyalty; to a son ultimate obligation. Not that we were alike in all things, not at all. He loved fishing while I never had the patience to stand there holding a stick and a string. He's small for his age, (something for which he can blame both his parents), but quick and as agile as an acrobat, with Penelope's green eyes and my bow legs. When Penelope went into labour I smeared the house in pitch against evil spirits and then afterwards hung the olive branch over the door. I held him when he was born and he looked at me as though he knew me, as if he already had opinions, as if he'd been listening the entire time he'd been in his mother's belly. He evaluated everything, the oil lamps, the braids of garlic, the bundles of sage and thyme dangling from the rafters. I suspect he knew the answer to the question of all questions, which is not where we go after death but where we are before birth, except by the time he could talk he'd forgotten.
"I'll be home by the solstice," I said. "We'll break the ice on the pond together." It was a small ritual of ours to step out at dawn as the pale sun rose and go in search of ice.
Argos had followed us and now began to whine.
"He knows," she said.
"Dogs always know." We regarded the pup, scarcely half a year old, awkward and lanky and still stepping on its ears. I placed my palm cap-like over his skull and asked him not to forget me. The dog watched me with large eyes rheumy with sap, as if it understood, and maybe it did. Dogs and even chickens always knew when a storm or earthquake was coming.
Penelope began to sob so I put my arm around her and pretended to be brave. When she'd recovered she looked at me with her chin set hard as a wedge, and in a resolved voice demanded I do something for her.
"Tell me."
"Kill Palamedes."
The next morning I went for advice to my father, Laertes, and found him talking to his cat, a fat orange tom missing an ear. He called this cat Fish. He thought this clever, because the cat was always prowling around ponds and streams and tidal pools, and was surprisingly quick with its paws for a beast who spent so much time asleep. My father and mother had gone into retirement in a house of planks and stone overlooking both the orchard and the bay. When I arrived he was peeling an apple as if unwinding a ball of string, cutting the peel as thin as possible to see how long he could make it. Such are the activities that occupy old age. When he saw me he nudged the cat with his toe.
"Didn't I tell you, Fish. Eh? He'll be up. He'll be calling and wanting advice." He laughed. "Too bad it's the one gift the young are so incapable of accepting. I believe this is called a paradox. _We are too soon old and too late smart / Age and wits pass each other in the dark_."
"I'm twenty-nine. Not all that young, and maybe not utterly deaf to wisdom."
He made a face as if to say to the cat: well, well, listen to this. He continued peeling, or unwinding as it seemed, the apple. I knew enough not to push him, to let him talk when he felt ready in his roundabout way. Finally he started recalling his time with the Argonauts, and how he'd resented Jason demanding he go along because I was only four and it meant a long separation. It still hurt to think of those seven years apart from me and my mother, he said, though he also admitted that nowadays it gave him hours of pleasure to remember all he'd seen and done, that a good store of memories was more valuable than gold to an old man. "I drank my cup to the dregs," he said. "As to whether I was wise to go along on another man's quest, well . . ." He shrugged. And with that he finished the apple, measuring out the peel along the entire length of his arm from his fingertips to his shoulder. "Not bad. Not bad." He dangled the peel before the cat, who batted it once with his paw and then yawned. Laertes, taking this for a judgment, shrugged and tossed the peel to a goat and began to eat the apple, seeming to forget I was even there.
I remember how when he at last returned he was a stranger, as awkward with me as I was with him. My mother welcomed him with all the warmth due a returning hero, but there was a distance, a formality. It took months before he seemed a natural part of our lives. He was a great man, an adventurer, one of the Argonauts, a comrade of Herakles. He returned with a reputation and an endless store of tales of far off places and famous events, yet there was a remoteness in his eyes. As if he had looked at things that a man shouldn't, things he couldn't forget even though he wanted to.
When he left I missed him, and when he returned I was disappointed at this much discussed man who was so much smaller than I'd recalled, as lean as a beggar, and worst of all had hair tufting from his ears. When he embraced me I saw those hairs up close. They were long and straggly and stank of ear wax. Old men had hairy ears. My father was not only a stranger but old.
Had I been ten years younger I might have regarded sailing off to Troy as an adventure; at twenty-nine I did not. At twenty-nine I was still strong, but I could also see, like a city coming into view through the dust and distance, that I was approaching a place where I'd only be weaker and slower, where one day my arm would shake as I drew my bow, and my sight would dim as if I lived in twilight. Besides, I'd been enjoying life on Ithaka. Few things are as satisfying as harvesting your own crops, making wine, and presiding over the birth of livestock. The goats entertained me, the hens soothed me, I admired the hunting skills of the barn cats, and even came to appreciate the dull dignity of cows. Most of all I feared leaving Penelope and Telemachus, the two creatures above all others who gave purpose to my existence. What were prizes and reputation without them? And yet to remain here under a cloud of scorn would only bring them humiliation.
**Chapter Two**
**W** HEN I WAKE I think for a blessed moment that I'm back in Ithaka and that the entire war is a bad dream sent down by some meddling god. Then I see the tent, my armour, my helmet with its boar tusks, and I groan and put my hands to my face. When I eventually sit up, a wave of dizziness nearly topples me back over. I brace myself with my hands and sit a long time staring at my legs. Are those scarred limbs mine? Is it possible that I'm thirty-nine years old? Looking at the backs of my hands I can well believe I'm turning into a pine tree, my skin coarse as bark, arteries exposed roots, knuckles gnarled knots.
Sleeping in the afternoon is a bad habit. Even with a cross-breeze, the heat leaves me muddled so that I inevitably wake feeling hungover instead of refreshed, and yet more and more I feel the need to withdraw each day, for a while at least, not to court the Muse or ponder Great Things —bugger all that—but to brood. My tent is stark. I've got my blades and my bow, and a cedar trunk that holds my clothes and letters, as well as a few clever little boxes with compartments within compartments that hide rings and coins and the odd pearl, and of course I have the armour of Achilles, the one notable prize I've collected. One of Sinon's chores is to polish that armour so that it always gleams. Otherwise I lead a dull existence. The tent pole is a lance of ash; the floor an ox's hide. Gathering the strength to move, I press my hands down on my thighs and, groaning again, stand up, ride out another wave of dizziness, and step outside into the waning afternoon.
As usual, most of the men are scattered off along the beach looking under rocks and peering into the shallows seeking something to eat. We spend less and less time on strategy and training and more on scrounging, or simply lying around. Entire days I've waded the tidal pools or lain on rocks studying the antics of crabs. Other days, I walk the shore where the land has buckled, reading the exposed strata of rock and soil in all its variety, as if the very earth itself had been formed in layers like the rings of a tree. Is it possible that the earth is growing, adding a new layer each year, and that by digging down and down you might read its rings and learn its age? And does it follow that it will grow bigger and bigger until it touches the sun?
Such ideas occur when I drink poppy water and end up staring at my reflection in a pool. A disorienting experience to be thus multiplied, a me here, a me there, Odysseus in the flesh and Odysseus in the water. That this phenomenon is a trick of light does not lessen its disturbing nature. Of course, poppy and idleness are unhealthy habits that breed dark whims, such as what would occur if there were two pools, or ten, all ranged around me so that I saw a dozen Odysseuses, each acting independently, desiring to escape from their glassy imprisonment, calling for help in the voices of drowned men.
Those men who aren't foraging wander about farting and scratching and yawning, their clothes ragged with wear. The whores cuff their runts who whine for the teat while the dogs glide as silent as sharks. Hungry, everyone chews their fingernails. Soon the fires are built up, and as the sun begins to decline the shadows of our tents stretch long over the sand. The stench of the tanners is all through the camp, ox and horse hides and goatskins scraped and scalded then stretched to dry. We boil the bones for broth, the hooves for glue, and keep the sinews for bowstrings and cordage.
I totter down to the water, strip, and plunge into waves that rear like horses, white-maned with foam. Achaeans don't generally like to swim because they fear Poseidon and his harem of nymphs. The ocean is no doubt a strange thing, as vast perhaps as we can imagine, and full of even stranger creatures, many of which I've seen in the nets of fishermen. But I swim with my eyes open, for the salt water soothes them.
When I wade out of the sea, Dercynus pours a bucket of river water over me to rinse off the salt, oils me with orange flower, a gift from Agamemnon's private stock, and then kneads the muscles in my neck and shoulders, working thoroughly with his thumbs from the base of my skull on down. He has good hands, Dercynus, stronger than Sinon's. When he's done he presents me with a fresh tunic, white with blue trim, though worn so thin that it is almost as translucent as a dragonfly's wing. Not one of Dercynus's nobler chores, washing my clothes, but he never complains. Unlike the ironic Sinon, who'd go into a washerwoman's voice and make me laugh as she lamented her lot.
Dercynus has one blue eye and one brown; people don't know whether to assume him cursed or blessed. Either way, his vision is keen and he's always the first to spot the evening star. He's stockier than Sinon, square-shouldered and a stronger wrestler, (I'm still better but not for long), and as fast a runner as I was at his age. He lacks Sinon's wit and manner, but he's observant and he puts both of his eyes, the blue and the brown, to good use.
Sinon arrives with a platter of meat and a bowl of wine, play-acting like a tavern waiter, "Nuffing but the finest fer Ordysheus, wiv de complerments uv Argeememneeon." The horse meat is sinewy and the wine sour, but I'm grateful for usually it's boiled barley and seaweed. It's not uncommon for a man to cut an artery in a horse and suck a mouthful of blood. I've done it myself; a rich hot broth, though glutinous, and the taste lingers and gums up your teeth unless you rinse your mouth with lemon, or, given the scarcity of fruit, a mouthful of sea water. I always thank the horse after I've drunk its blood, it's only polite after all, and look into its eyes as I offer my appreciation, those dark orbs dim with ideas that never quite bloom. Sheep are a different beast altogether, their eyes as opaque as mud. Goats, however, are the most manlike of them all, the yellow crevices through which they spy on the world of men make me believe in satyrs. Dogs interbreed and share characteristics, why not man and goat?
"Come on, eat. Don't stand there staring." Sinon and Dercynus dig in. There's more than enough and I don't eat nearly as much as I once did. Besides, it's a pleasure to see them gorge. What beautiful young men: complexions clear, foreheads smooth, hair thick, their eyes free of veins. No need to pluck the hair from their ears. Sinon favours white tunics because he likes the contrast with his olive skin, whereas Dercynus opts for dark. Altogether an earthier fellow, Dercynus, a man of the soil, a farmer at heart, at ease with digging. He can cup soil in his hands and inhale its scent as if it is a bowl of hot stew while Sinon will worry about his nails.
"Look at them," says Dercynus, those strangely paired eyes of his round and wide, and his lips glum as he nods toward the shoreline.
We watch men scouring the beach, stepping carefully on the kelp-slick stones seeking mussels, eels, crabs, anything they can stomach. The way they bend to pry up rocks it looks as if they are sowing seeds to grow fish. Each time they straighten to ease their backs they take a moment to squint against the glare off the water—and doubtless lay down a curse on the heads of Menelaus and Helen—then bend once more to the task.
Dercynus says he hears their talk at night, that their patience is gone.
As sour as it is, I drink all my wine, suck at the meat skewer and then work my gums with the stick. How I miss hot bread and sweet onions. If I murdered Menelaus and Agamemnon, thus freeing the men to refit their ships and sail home, I'd be secretly loved, and yet at the same time publicly vilified, my name synonymous with treachery, a burden my son would bear long after I'm dead.
Dercynus draws in the dirt with his finger and observes that it will be hunger before homesickness that drives the men to desertion. He stares at his feet and frowns as if the tally of his toes is off.
"Home? I hardly remember home," says Sinon, knuckling away an invisible tear and performing the quavering voice of a maudlin waif causing both Dercynus and I to smile. "Where's me home? Where's me mum? Where's me dad? Where's me bum?"
Sinon and Dercynus-strange-eyes were press-ganged on the eve of the fleet's departure from Aulis. Servants for the kings. Agamemnon's orders. But they are my obligation as well, my wards, adopted sons, and I've taught them, though they have taught me a few lessons as well, such as patience. Ten years old when they were given to me. Children. Sinon still peed himself at night. When he or Dercynus wept it was to me they came running for solace.
I scrub my hands with sand and then contemplate the declining sun burning huge on the horizon. As it begins to set the clouds redden, and when the sun is gone the backlit sky glows like the wall of a kiln.
In the evenings there's little more to do but lie and complain, taunt each other, reminisce and listen to the old tales. Every one of us is a connoisseur of stories, recognizing a new emphasis here, a variation there— _in fact they were twins separated at birth; the father had been married before; the raven was in fact Zeus, she was the offspring of a god_—the tales handed down like figurines polished by each set of hands through which they pass.
When the tales are done there is always the fire to watch, each fuel creating a distinct flame: blue sparks erupting from salt-cured driftwood, the smooth burn and sharp heat yielded by bark so different from the rolling white smoke from damp fir branches, or sapwood that spits like fat. The flames sway like dancers, sometimes frenzied, sometimes slow, and yet always entrancing. No one among us loves fire more than Diomedes, who sees radiant cities in the quaking coals, empires erupting in burning twigs, titanic battles when sapwood screams; when a gust bends the flames flat it's a fleet of ships running downwind, pennants flying. He's the most handsome of all the captains, his face perfect in its symmetry, square forehead and jaw, straight nose, full lips, even teeth. Only the expression in his eyes betrays his incorrigible irreverence. He kills like a cat, a hunter born to the task, yet I suspect he'd just as soon juggle oranges in the market as gain glory on the battlefield. _Reputation, Odysseus, we're prisoners of reputation . . ._ I often throw questions at him: _Diomedes_ , _where does fire go when it's not burning?_ Entering into the spirit of the farce, he'll say: _The same place as the wind_. _I see, and tell me, you clever duck you, why do men have nipples? Ah, that's obvious, because we're all women in the womb before Zeus picks out the ugly ones._
We have musicians, of course. Or more accurately, men who strum the citharis, blow flutes, and sing, men who, by now, should be skilled yet still punish their instruments like peasants beating their mules. I just don't like Achaean music, something I keep to myself. I'd be wise to keep most of my thoughts to myself. I rarely speak to anyone of my Dreams, especially the ones that recur. There is one that comes down and whispers in my ear of a world without gods, a world that has no beginning or end, a world that was, is, and always will be, a world of ice and sky, rock and bone, a world without colour. In this Dream I walk the earth hearing nothing but wind. Afterwards, when I wake from this grim vision, I lay on my side as if struck down by a fist. And yet at other times I suspect that dreams are but the day's events flooding the banks of our waking hours into the fertile plains of night, an over-strung bow shooting an arrow beyond the horizon of day into the realm of sleep.
For years I've been asking everyone I meet if they've ever seen a god. Many insist they have, but it only takes a few careful questions to reveal that it was always foggy or they were drunk (ah, Dionysus, they say humorously, knowingly, winking at the god in the grape). Or they explain that it wasn't a god as such but a god in the form of an eagle or a snake. And what did this snake do that so distinguished it from others of its kind? Talk? Fly? Offer you a drink of wine? No, not really, you had to be there. The gods are beyond words. Indeed from what I can gather they most commonly manifest themselves as voices whispering in your ear. And I've heard a lot of those.
I don't doubt the gods, but I sometimes hate them. Often as a boy I sacrificed to Hermes to carry a message to my father the Argonaut asking him to return. Months I offered fruit, birds, coins, fine stones from the creek, and polished shells from the sea, the most valuable things I had, but he did not return. I gave up. I no longer had a father. And then years later a man appeared claiming to be Laertes. He looked more like a beggar than a king, and I thought that the gods were laughing at me. That's when I began to hate the gods.
I hate Menelaus and Agamemnon as well. They're right about one thing though: the time has come, in fact it's long passed. The Trojans lost Hector and we lost Achilles. Their best and ours. And yet we're no closer to resolving this stalemate. I head off up the shore to walk and think, because my brain works better when my body is occupied. I don't get far, however, before a voice calls: "Odysseus." It's Agamemnon.
"A busy world," he says as I join him. The front legs of his chair are propped on a log and the headrest is high enough that he can view the stars without straining his neck. We gaze in silence. When a shooting star marks the sky like a streak of chalk, Agamemnon grunts. "Messengers." He sounds troubled, assuming that of course the message is about him, the big man, the great king, that he's been waiting for this dispatch from the gods in council concerning our grubby bit of business down here on our briny beach. Then his right hand rises and brushes it aside as though tired of it all, and I sense that his mood is not due to Troy and Priam and the war, but something more personal: his wife. We've all heard the rumours about her and Aegisthus. No one dares joke, because we all fear the same may be happening behind our own backs.
"We'll be home soon," I reassure him, though I won't deny a small satisfaction at seeing him troubled like this, even if it means some of his frustration might be vented upon me.
"And what will I find?"
"Your people."
He snorts.
How different we are in the night. The light gone, the moon up, the lurkings of our minds rise like glowfish to feed.
"They hate me. The men hate me. You hate me."
I swallow this bone without flinching. "Love, hate. These things come and go. Hot winds and cold. Balance," I say, "the true kernel of the matter. All things find their equilibrium. After ten years we're overdue for a turn in our fortunes."
He asks me if I believe this and I lie and say yes.
"There." Another star chalks the dark. "A lot of talk tonight."
The origins of fire, the secrets of the stars, these are eminently intriguing topics—Hermes dashing about with messages—but I don't believe that every light in the sky, every glow in the sea, every snake that crosses your path is a mystery or a portent. Sometimes things are only things, a rock a rock, a skull a bit of bone. And who knows, the gods but idle ghosts.
"The men are talking," Agamemnon says.
"They always talk."
"They blame me for Achilles," he goes on, ignoring me, and yet at the same time hoping I'll prove him wrong. I can almost hear his thoughts: _Come, Odysseus, arrange some words, convince me of how popular I am._ "They think that without him we'll be defeated. That we're doomed. And it's my fault."
I make noises to the contrary, but it's true, the men do blame Agamemnon, and for good reason. He should have left Briseis to Achilles, he should have known how Achilles would react, how anyone would react. But for all Agamemnon's experience, for all his kingly wisdom, he's a clod. He was always threatened by Achilles who was younger, stronger, better looking, and had the ears of the gods. He feared him and was jealous of him, so he got rid of him. The tautest bow is also the most easily broken.
"I should cut their tongues out."
Is this for my benefit? Does he suspect me of talking, of spreading discontent? "Swift victory is the surest route to respect."
"And you know how to achieve this?" The childlike hopefulness in his tone is almost endearing.
"I'll find a way." I speak with more confidence than I feel. But Agamemnon is desperate for reassurance. He doesn't want to hear how tricky it will be, the long odds, the high risks. I stand then say goodbye and leave him to his stars and his gloom and the doubt gnawing like rats at his heart.
What a relief to walk. The camp stretches out along the coast so that we've cut off Troy's access to the sea, though they still have a corridor to the Scamander. For years our camp has been a city unto itself, if rather long and thin, between water on one side and dirt on the other, until recently lacking nothing, meat, wine, women, boys, acrobats. But the supply convoys arrive less frequently now, the usual pimps and purveyors who dog armies have drifted off because we no longer have the money to pay. The war has dragged on and bankrupted us all, so that we're raiding up and down the coast, which doesn't exactly endear us to the locals. We live in tents, cook over driftwood fires, shit in the sea, and more and more resort to eating rats and kelp. By Agamemnon's order, no permanent structures are built, lest we grow complacent and lose our fighting spirit. Lose it? Ha. It's long gone. After so many years it's tough to maintain the belief that victory's near and that we'll soon be home. Men who arrived young have withered and died, for this sort of warfare defeats you the way rot kills trees, slowly, from within, weakening the core, turning the spirit to dust. Desertions have become commonplace, men so crazed that they wade into the sea and simply start swimming west. We don't execute them; we can't afford to, and I admit that I've often imagined trying the same thing. Those are the same nights I think about my son, and about something else, which is that the only man I hate more than Menelaus, Agamemnon or Palamedes is myself for not having found an honourable way of staying home.
A half moon is pressed like a bent coin into the sky, throwing a path of light across the ocean. My left kneecap aches, thanks to a kick from a Trojan who had my bronzetipped spear in his throat. He caught me a sharp one on the kneecap—more of a spasm of the leg than a well-aimed blow—but it nearly crippled me all the same. Another injury to add to the list: the numb hands, the locked back, the torn ear, the broken toes, and sprained fingers. Some scars I'm proud of, the tusk-gash on my thigh received while boar hunting with my grandfather was my first, and it remains the one of which I am the most fond. More and more I feel like a battered old tomcat. More and more I hate both the Achaeans and the Trojans and all their gods. Enough. The company of these men has grown stale. The prostitutes who service us are fouler of mouth, grosser of figure, and tougher of hide than the men, the sex closer to relieving your bladder than rapture, much less intimacy. As for their bawling spawn of runts, they generally drown them, and if not they're fated to a life of mischief, cowering about the camp like dogs, slimy-nosed and filthy-fingered, their vocabularies even more appalling than that of their mothers.
The wise and fatherly Agamemnon—our leader, our Zeus, our Man among Men—growing concerned for our well-being has ordered us to undertake a daily regimen of callisthenics. (Masculinity being an achievement not a fact of birth.) We run and wrestle and box, and take pride—so the theory goes—in the realm of maleness; and lest our brains rot, he encourages elevated debate to keep body and mind in balance, like the spread wings of an owl. Yet how lonely the owl seems. I miss the murmur of women and the genial chaos of children who know awe at a world always new. The army bastards are nothing but a whining subspecies who rarely live three years before the bloody flux carries them off. It would be wiser to put the whores and their whelps onto a barge and ship them off, but not even Agamemnon is willing to risk depriving the noble Achaeans of such solace.
And so I take long walks. Off to my left our ships rot at anchor, overdue to be beached and have their beards scraped. For all Agamemnon's vigilance, our boats are turning to kindling. I've warned him but he worries that to set men to repairing the ships will only give them the means of escape. He's even gone so far as to order that no men go aboard in case their thoughts stray. Nonetheless, I sometimes exercise privilege and go onto mine and sleep in my old berth. Telemachus loved the ship, the eyes on the blue hull convincing him that it was not merely a clever construction of wood and rope, but a creature as alive as a dolphin, and like a dolphin breathing the air but inseparable from the sea. And who knows, maybe there's some truth to this. A ship seems a living creature breathing the wind on the glittering water. Penelope loved the ship too, though I suspect with a certain wariness, fearing that it must inevitably take me away. Against this possibility, she carved all our names on the bulkhead by my bunk, Penelope, Telemachus, Odysseus.
Beyond the sand hills and beach pines stands Troy, the new Babylon, the new Nineveh. Well, we know what happened to them. Sand and dust, home to nothing but the rat and the vulture. I plod on, footsteps filling with water, waves thumping the sand, my mouth heavy with the taste of sour wine and seared horse, food that sits like mud in my gut.
Crabs put up their claws and side-scuttle into their holes. I like crabs, they've got personality. Telemachus used to lure them with a bit of meat on a stick. He had such patience for a four-year-old, such focus. Sooner or later the crabs would edge back up and Telemachus would snag them and look at me with pride.
Bait—that's the key. Every creature can be lured. What will lure the Trojans from their hole? What do they want, besides us to go away? I try going at the problem on the oblique, for that's the best means of spotting solutions, from out of the corner of your eye. Look too directly and the answer vanishes like a mouse. Chewing my nails, I ponder. I begin thinking about my fingernails, about bird talons and dog claws, how it is that we all have them, man and beast, lizard and bird, and I wonder if there is anything to conclude in this, some grand truth other than that we all need to scratch. Penelope painted hers blue. Oh, for the delicious feel of them raking my back once again. A good long scratch is as good as sex, but the two together—ecstasy. What do the Trojans prize? Fish are lured by worms. Horses by apples. A buzzard by carrion. Achaeans love their boats. The Trojans? In their hearts they're people of the plains, with their backs to the sea and their faces to the steppe, toward Cappodoccia, Persia, India, and therefore it's obvious that they prize nothing above the horse.
A horse.
O Muse, subtle as the wind-borne scent of thyme . . . But clearly not just any horse would do. No. Perhaps many horses, a herd of Scythian mares, or else one single splendid specimen, a giant stallion of pure silver. Where would we get a silver horse? Pool our precious metals and cast one? I can hear the remarks already: give sly Odysseus our silver, our gold? Ha.
Seawater fills my footprints and each one holds a half moon. A dead-fish wind idles along the shore and the muscling waves slide over the sand. I look again at the towers of Troy. Yes, a people who liked things big, high walls, high towers, massive gates. To be honest, a little on the crude side, peasants at heart, like the northern barbarians. But there's mettle to them, they're dogged and they have more pride than a Pharaoh, but a bit blinkered and easily dazzled. So it'll have to be a big horse, the biggest they've ever seen, one they'll lust for, one so huge they'll gladly creak open those spiked gates and haul it inside because a horse so enormous can only be a Sign, and let's face it all men are fools for Signs, all of us believe an eclipse or a snake or a summer hailstorm means that the gods are near, that divine forces are at work, that the end can be seen at the beginning. Even I took heart when a heron appeared the night Diomedes and I cut up the Thracians.
I'm walking faster now, brain wheels grinding. The coast juts and scallops with points and bays, the windstunted cedars are good only for firewood or crutches, not god-like wooden horses. Big jobs demand oak, straight and tall, and the best stands are on Mount Ida. We can fell the trees and float them down the Scamander and start work, though that will take too much time and too much labour. It will mean half a year or more.
Soon I smell creek mud and then mosquitoes hit me like a fistful of hot sand. I spin away and spot our ships in the moonlight. We've lost so many men that we don't have crew enough to sail half of them home. We'll have to burn them, or—and here blessed Muse graces me again—our carpenters can unbind the planks of one and use them to build a horse, a horse the like of which no one has ever seen before, a horse not even Herakles could ride, a horse with a surprise in its belly . . . I begin to run, indifferent to my bad knee, waving my arms mad-fool-wise about my head to shoo the bugs. Hearing my laughter the dogs howl with rage for they sense that the tide of war is about to turn.
**Chapter Three**
**I** HAD, OF COURSE, anticipated debate. With Achaeans there's always debate. We thrive on it. But Ajax is even more blunt than usual.
"This is a shit plan," he says, regarding me as if I'm nothing but a wandering pedlar, a vagrant of the roads trying to beguile them with stale tricks. He finishes sucking a fish spine then makes a performance of tossing it into the fire. Belching richly, he draws his baby finger across his lips (for he's not without delicacy, our Ajax).
I could have presented an army of ten thousand strong soldiers and Ajax would find something to complain about. _Too short, too slow, no discipline. A hindrance. More trouble than they're worth . . ._ Bundles of cedar fronds smoulder against the mosquitoes. A wave crashes on the beach. Everyone awaits my reaction. I smile. I smile and incline my head as if honoured by the depth and subtlety of his analysis. "O wise one, how reassuring that I can always rely upon a considered response from you."
He's ready for this, and looks at me as if his glare alone could crush my skull. Judging by the hate in his expression I almost believe it. He spits.
And yet what a charming boy Ajax had been when we first met. A young prince with royal manners. Though he was in Sparta to compete he was ingenuous and innocent and wished the other competitors well. He was a student of bees and spent entire afternoons in the clover with his chin on his fists studying those marvellous insects, making sketches, dissecting the small dry corpses and holding them to the sunlight, convinced that any creature that ate flowers and vomited honey had a god in it. He's lost that wonder and become coarse. Worse, he's grown rigid and blinkered, and what little sense of humour he once had has died.
"A shit plan," he repeats. "The Trojans aren't idiots. They'll laugh at this toy horse of yours."
The flames reflect in the shocked eyes of everyone around the fire. I glance toward Agamemnon, the one opinion that matters, but he merely watches, waiting for the others to have their say. For all that he needs my help he's not above taking some small pleasure in seeing me squirm. Certainly Palamedes does, but even he appears surprised at the vehemence with which Ajax dismisses me.
It is young Echion who speaks out, however. Echion the Eager, Echion the Impatient, incapable of sitting still. He can scarcely contain himself. "I like it. It's good. Let's build it!" He looks around eager for others to agree.
"Shut up," says Ajax, dismissing him. Ajax has many reasons for hating me. He thinks I've betrayed him for no longer playing the fond uncle, the older brother, the way I did when we first met so many years ago. He's also convinced I think he's a blockhead, that I've decided he has cheese for a brain. And then there's the little business of my defeating him in the games for Achilles's armour, a defeat he remains suspicious about, (a suspicion fuelled by Palamedes). How could runt Odysseus—older, slower, smaller —throw the mighty Ajax? How? It wasn't all that complex, there was no drug or god involved, no Egyptian curse or Phoenician sorcery. I kept low and caught him behind the knees and used his own weight against him. Plus I did something else, something that always succeeds, at least the first time around: I praised him. Yes, praised him. The day before the match, I paid him a visit and grew nostalgic about the old days, when we first got to know each other. I spoke the truth, telling him how he, fourteen-year-old Ajax, scarcely more than a boy, had struck fear into every man competing for Helen in Sparta. I talked for some time, pouring on the honey. He was wary at first, but no man can withstand such a siege of praise, in time it will wear him down the way wind erodes stone. After such a paean how could Ajax fight me, for we never go hard against friends, we always hold back. The real challenge was for me to fight as if I hated him. It was that or be defeated, or, worse, sit out the games altogether. Then what? Have them laughing at old Odysseus Has-Been-Once-Was?
The armour of Achilles is in my tent, wrapped in a blanket so that its fire-bright gleam doesn't torment Ajax. Only when he is away from camp do I take it out and admire it: bronze chest and back plates modelled to fit Achilles's torso, a bronze helmet topped with boar tusks and a crest of horsehair, a bronze-faced shield and bronze greaves for his shins. It's too large for me and too small for Ajax.
And now he and I are fighting again. "Ajax, if Achilles had come up with the idea you'd applaud it, wouldn't you? Be honest now."
"Honest? I'm always honest," says blue-eyed Ajax. "Ask anyone. It's my weakness. Yet you. This is just another one of your schemes, the only thing you contribute." He shakes his head implying that, unlike him, I drag behind me a long chain of inane ploys and clumsy deceits, such as my idea of digging a tunnel under the walls of Troy, or of using long poles for vaulting over them. Are these any worse than offering prayers and incense to gods no one's ever seen? I've also got a long history of fairly won victories. Why are these so easily forgotten? And why is it forgotten that Achilles tried avoiding Troy by dressing as a woman? Ajax doesn't like being reminded that his hero tried escaping this whole business just as I had done. Nor does Ajax like it that I'm the one who exposed Achilles's ruse.
"A wooden horse," says Ajax. "The Trojans will set fire to it, or . . ." He struggles for words. "Pry the planks apart and stab everyone inside like crabs in a bucket." He shrugs. He turns his hands palm upward and looks to Agamemnon, Menelaus, Palamedes, old Nestor, all of whom sit like a gallery of judges. "We know Odysseus could charm the fangs out of an adder. I'm not gifted with such a slithery tongue. But I've got eyes. I see what's what. Ever since Odysseus broke into the Trojan temple and stole the Palladium our luck has gone sour. It was supposed to gain us favour. I never understood how. But who listens to me? We're being punished. This is plain. And now he comes up with this."
The response is silence, and like all silences it's loud with brooding and doubt. The men seated around the fire draw down into themselves and stare into the flames, thinking of what they could be doing, what they _should_ be doing, instead of being stuck here, far from their families and farms, growing old on this crumbling coast with an ever-diminishing prospect of going home. You only have to look to see we've had it. The firelight is both kind and cruel, hiding some scars and highlighting others. Armour keeps us alive, but can't prevent the mangled fingers, the lost ears, the crushed toes. Ajax is the only one who has all his teeth, miraculously because, to his undeniable credit, he marches foremost into every battle. His knees and arms have taken the worst, striped with slash wounds and puckered with stab marks. Agamemnon, along with his blunted nose, has lost the hearing in his right ear and has a back as rigid as a plank. Menelaus limps and has only half strength in his left hand. Old Nestor is blind in his left eye. Sinon's right collarbone was so badly broken he can scarcely heft a sword. Diomedes took one of Paris's arrows in the foot and limped for a year. Not even Palamedes has escaped injury—and no man protects himself more cleverly—he nearly bled dry from a gash across the side of the neck. None of us stands, sits, or reaches for a cup of water without a groan. Of the dead, few went quickly, lingering for days as their blood seeped through the tightest bandages. It's easy enough to avoid looking at a man as he drains away like a badly caulked barrel, but not so easy to avoid overhearing his agony. Every one of us reflected upon what Achilles said when Agamemnon took Briseis: _Fate is the same for the man who holds back and the man who fights his best._ Better a long and quiet life than one brief if glorious.
But Ajax has a point about the horse. It's a risk. A bold move, maybe a foolish one.
Uneasy with silence, Ajax starts in on me again: "The gods hate us because Odysseus denies they even exist!"
"Of course they exist," I say, quietly, sincerely (and I even believe it, for I'm not immune to my own words). "My own grandfather, Autolycus, was sired by no less than Hermes. That Hermes is sacred not only to messengers but to thieves and liars as well makes him no less a god," I add wryly, knowing perfectly well what everyone is thinking. Ajax tries to interrupt, but I raise my hand calling for patience, a moment to defend myself against such an egregious claim. "A world without the gods is a terrifying prospect . . . Of course, a world with the gods is—we must admit—also terrifying. The gods are such splendid creatures. Passionate. Fickle. Mysterious. A burning city is but a candle compared to the glory of Zeus. The beauty of Aphrodite finer than any sunrise. Hermes faster than any arrow. But perhaps this is not the point. The point is that everything exists. The air exists even though we can't see it. It pushes our ships, stirs trees, lifts the dust. We inhale it into our chests, exhale it from our nostrils, fart it from our arses. Light and dark exist. Songs stir our hearts and spur us on to victory. And yet what are songs but words, and what are words but air? And what about lies? Some lies contain truths and some truths the dark pearl of falsehood. Where is the man who has not acted honourably upon his sincere belief in a lie? Conversely, who has denied one truth in pursuit of a higher one? Where is the truth that is not a truth to one and a lie to another? Take, for instance, water—"
Ajax puts his fists to his ears. "Stop!"
The others hide their smiles.
It is Calchas, our soothsayer, who finally puts in his oar. "Spare us." He gazes around in his best effort at seeming profound. "Zeus does not tolerate disrespect." His tone made clear his meaning: I was close to blasphemy. A lesser man would have his tongue cut out.
"So now you speak for Zeus," I observe. "Tell us, what did he say the last time you two got together for a bowl of wine and a stick of meat?"
Calchas spits into the fire and grimaces as if I'm a bad taste. Sitting taller, he raises his chin as if to address higher things. "Ajax is right," he says. "Our luck has been bad. There must be a reason. For every result there is a cause. It is the thread that holds things. The fabric of the world."
"Suddenly our Calchas is a tailor." There are chuckles. Calchas grimaces in the direction of my voice. He's an ugly old man whose face looks like it's been carved from an onion. "Perhaps," I suggest, "there are many causes and effects. After all, no reasonable man will deny the variety of the world. Causes evolve into effects and effects in turn become causes. Both branch like rivers. And it only stands to reason that some causes are both more influential and indeed preferable than others."
Calchas makes a great show of yawning. It's said (by him) that as a boy a serpent licked his ears giving him the gift of prophesy. Snakes are undoubtedly among the most intriguing of creatures, but why they should have prophetic powers and be inclined to bestow them upon mortals, I'm sure I don't know. Nor can I understand the allure of putting my tongue in Calchas's ear. But to his credit, Calchas has done fairly well for himself by seeing portents in every thing and everywhere. He watches the battles from well back, and will never be seen scrubbing his own laundry. Reading signs is his way of making himself invaluable. When we were becalmed off Aulis he decided that the winds refused to carry us because Artemis was angry, and therefore Agamemnon's virgin daughter, Iphigeneia, must be sacrificed. Artemis? What did the goddess of the hunt have to do with wind? Calchas knew. Oh yes, he had it all sorted out. It was really quite simple: Artemis was angry with Agamemnon for having shot a stag in her sacred grove. To ask the location of this supposed grove was to be the meanest of hairsplitters. So Calchas said and so Agamemnon believed. He therefore called for his own daughter under the ruse that she was to marry Achilles. She arrived in a bronze-plated chariot pulled by two white stallions, with half the royal house as escort. The girl was ecstatic. Her eyes were glistening, her cheeks flushed, her breath sweet with mint, a nice looking young lady, firm of breast and round of hip, wearing a silk robe dyed in Tyrian purple. I suspect that even Achilles was tempted. Whispering sweet words, Agamemnon put his arm around her shoulders and guided her into a grotto wreathed in olive branches where he cut her throat. Bled her over a stone. The rumour that the girl was not really his daughter, but Helen's via Theseus, hardly seemed an adequate excuse for such heartless, if strategic, butchery. And all this because of old Couch Ass.
Alright, the wind did begin to blow, but winds, like words, are variable things, as fickle as men, but try telling that to sailors suddenly blessed with a favourable breeze. How Calchas's spine straightened when that breeze quickened the air. His face dropped ten years. The captains applauded his powers and the oarsmen feared to look into his eyes. Who needs a strong arm when you have the ear of the gods? And perhaps Calchas does indeed see what he says? I don't need to be told that there are mysteries beyond reason. Why is a snake not poisoned by its own venom? How do frogs and crabs breathe equally well in either water or air? Why does Dionysus inhabit the grape but not the walnut?
The hunchbacked Calchas stands now with the help of his catamite, Telkinos, an intriguer if ever there was one. Calchas is forever gliding about with his right hand perched like a bird upon the boy's shoulder while Telkinos is forever whispering in Calchas's hairy old ear, as if he liked the smell of earwax. Now Calchas adjusts his leopard skin across his sloping shoulders (as if he's ever killed anything larger than a mouse or a mosquito!) and passes his hand over the fire as if to call up Hephaestus himself. "Athena is angry," he announces. "Angry at Odysseus."
I cross my arms and wait. My fault. Always is, always has been, always will be.
"Maybe she's angry at you," suggests Diomedes poking the coals. "It was _your_ idea to take the Palladium. _You_ read the signs. Odysseus and I merely did what you can't."
When the quiet man speaks he has an attentive audience. The others look to Calchas who responds dry-voiced and trembling. "Athena has long been our ally."
"No, _Odysseus_ has long been Athena's favourite," says Diomedes.
As if cornered by dogs, Calchas becomes shrill-voiced and dire. "She carved the Palladium in honour of her sister. She demands compensation! Something valuable! A sacrifice! Either that or we'll stay here until we grow too tired to fight. Then the Trojans will drive us into the sea."
The company looks to me now, eager for my response. I leisurely toss a handful of hemp seeds onto the coals, and when they begin to smoke I inhale, once, twice, a third time. Diomedes leans in and does the same, bending the smoke into his lungs. My shoulders relax and I smile. Still, I say nothing. It never hurts to let the audience wait. True enough, the theft of the Palladium hadn't proved the success we'd all hoped, the Trojan spirit hadn't crumbled as soon as the statue was gone. I could have declined to take up the challenge in the first place, leaving it for Palamedes, but I wasn't about to give him such an opportunity. Besides, I'd regarded taking the statue as more of a rescue than a kidnapping, just like Paris snatching Helen. He did her a favour, or so he no doubt convinced himself (and I couldn't completely disagree, given Helen's alternatives in the lean and sinewy world of Sparta). So, a liberation. I liked the parallel. It had balance. It had symmetry. And I'm sure Athena thinks the same, if such a marvellous creature exists, and I rather hope she does, for how poor the world would be without her inspiration. If she hadn't wanted the image of her half-sister to leave Troy she'd have prevented it, wouldn't she?
It happened the previous year. Diomedes and I disguised ourselves. We raked our faces with thistles and clumped our hair with mud, and then we circled wide around the city, far out into the fields, waited until evening, and then staggered in calling for alms. We got ourselves a few coins, but mostly flung stones and some well-aimed kicks, but we got in. I was all set to head straight up to the temple but Diomedes took it into his head to step into a taverna and sample some Trojan beer. I tried to argue but it wasn't up for discussion. Diomedes is adamant about his whims. He lives for them, or they through him. He simply nudged me with his elbow and, affecting a grand manner, arms swinging, grin wide, strode across a plaza toward a crowd seated at benches flanking an open door. Above it hung a carved board depicting grapes and barley. Diomedes offered a jaunty wave to the good citizens. They didn't deign to notice two bums in need of a bath. Soon we were seated on a bench against a wall with a jug of beer. It was a tad bland for my taste, but Diomedes was having a fine time. He set his elbows on the plank table and, sleepy-eyed, as if he came here every evening to quench his thirst after a hard day, gazed around. The Trojans are on average taller and darker than Achaeans. They're emphatic and blunt. Even their simplest gestures, tossing down a coin, batting a fly, dismissing an irrelevant remark, are delivered with the assurance that comes from living in a great city. It's as if they think that simply by being in Troy their lives take place on a higher plane, that even grunting over their morning constipation is a more elevated activity.
"I miss city life," concluded Diomedes. "I think I could get on all right here." He raised his beaker to the barman for a refill and a serving boy limped over with a fresh jug and slopped our cups to the brim. Diomedes drank half in one go and then leaned close to my ear. "What do you think we just change sides?"
I smiled and picked a gnat from my mug. "It'd almost be worth it to see Agamemnon's face when he found out."
Diomedes clapped his thigh and let out his high howl of a laugh.
The taverna's stone walls were hung with tapestries depicting cows and lightning.
A one-armed man carrying a basket of eggs entered. In a well-rehearsed speech he announced that for a price he could stack nine eggs one atop the other. "And what do we get if you fail?" someone asked. "You get the eggs," he said, as if the answer should be obvious. He had the lean and watchful look of a man ever alert for an opportunity. I suspected he'd stolen the very eggs with which he was staging his trick. He got right down to work and began stacking the eggs. Soon a crowd had gathered. Diomedes joined them. I stood on a bench and watched from farther back. Two eggs, three, stacked end to end. The one-armed man gestured everyone away, no bumping the table, no hard breathing. Four, five, six eggs, each set just so, like brickwork. At the seventh egg someone grew suspicious and demanded to have a look at the others. The one-armed man held the three remaining eggs in his palm for all to inspect. "Fresh from the hen's arse." When the ninth egg was set on top everyone there, maybe fifty men, stared open-mouthed, as if Helen herself stood naked before them. The only sound was their breath through their beards. It was now dusk, and in the half-light of the oil lamps the men looked like troglodytes before the miracle of flame. Diomedes was the first to pay up. He tossed a coin that clattered on the table. Others followed, the coins ringing on the stone floor and wood benches. The one-armed man caught some in the air and pounced on the others like a cat on a rat. When he had every coin he briskly piled his eggs back in the basket. Someone shouted that they were hungry and wanted an egg but he was gone.
We finished out beers then followed the dirt street as it wound upward. Soon it was paved in neatly cut flags and flanked by fine villas. Feeling the beer, Diomedes enjoyed a leisurely piss against a wall. We reached the temple that sits high above the rabble with a view over the walls to the sea. "Not bad," said Diomedes, and I knew he was referring to the eggs. "The trick is salt. Did you notice how he put his hand into his pocket before placing each egg? The salt works like cement."
We hid beneath a fig tree and watched the stars.
"Salt?"
Diomedes nodded.
We listened to the sounds of Troy, harp and flute, a jug shattering, shrieks, barking. There was the syrupy scent of night blooming flowers.
When the moon had set we emerged. The Palladium is a wood statue of Pallas about my height, just over three cubits. In her right hand she grips a spear and in her left a distaff and spindle. Even in the dark it's impossible to miss her because she glows like a churned sea in the night. We approached with our eyes down whispering our apologies. Fresh roses were heaped around her. Wrapping her in a goatskin, we headed back down to the eastern gate and departed with the farm labour. We were far out in the fields when the cry arose and the trumpets sounded.
"Calchas is not altogether wrong," I say now, paying out some line. "The Trojans believe that without the Palladium they're vulnerable. So, yes, by all means, a sacrifice."
"What kind of sacrifice?" demands Agamemnon, dark-browed and suspicious, for we've nearly sacrificed ourselves dry and are down to offering up guts and hooves.
The highest sacrifice means a man and nothing less. This fact hung unspoken even as it was loud in everyone's mind.
I let Calchas have first say, but our seer, our diviner, our man intimate with the owl's mind and the eagle's heart, this man who had once condemned a young girl to the knife, suddenly becomes reticent. His mouth drops open as if to speak then claps shut, while his eyes withdraw deeper into their caves, wary of condemning someone to the slaughter stone. "I need more time," he croaks.
"You've had ten years," Agamemnon reminds him.
Calchas says nothing. Nor does anyone else. The spider of silence spins its web, for of course the question is: which man? A Trojan prisoner? Too cheap, too easy. It will have to be an Achaean, one of us. Nobody dares move much less name a name. I let the silence build. Every man there contrives to avoid the eyes of their neighbour. Who would be so selfless as to volunteer themselves? Who would be so cruel as to suggest a name?
I speak up cheerfully, as if the answer is simple. "You all know and love young Sinon." I extend my arm inviting him to step from out of the dark into the firelight. The expressions on the faces! The eyes of every man roll in terror at the fate to which I'm condemning the boy. Agamemnon watches and waits; Calchas is suspicious; Palamedes smiles his thin-lipped smirk, eager to learn what I'm up to. Sinon takes a demure pose, hands behind his back, and awaits my instructions. He's nineteen, slim, full-lipped, with long black hair and of course that smooth clean complexion of youth and the illusion of purity that comes with wearing white. What better sacrifice than a beautiful young man so excellent in every way? The horror on most every face is exquisite.
"The horse is only part of the scheme," I say, implying that only a fool and a simpleton—such as Ajax—could have imagined that it ended there, and that if they'd only have been patient I'd have already explained the rest. "We set the mood by building up the fires and putting on a show of some grand rite. Much song and lamentation. A tragic time for the noble Achaeans. The Trojans will see that something's up. They'll be curious. They'll go into council. Then Sinon slips away, into their territory, and lets himself be caught by their scouts. He begs their mercy. Pleads. A lot of boo hoo. Says he's fled because we intended to sacrifice him to Poseidon for a calm sea so that we—weary of this futile war against such an indomitable foe—can finally set sail for home."
Sinon now launches himself into a preview of his performance. He bends himself in his Protean way into old King Priam, dodders and dribbles and plays it up, earning a few smiles, even doing the accent, the way they swallow the ends of their words.
Priam: "Who are you, boy? Name your father. What do you mean by sneaking-thieving about our fair walls?"
Sinon: "King of kings, Lord of the Zeus-blessed Trojans, the foul Odysseus means to sacrifice me to Poseidon for a fair wind. Nothing less than human blood will do."
Priam: "Odysseus! A sister fucker if ever there was one."
Sinon: "To be sure, my lord, to be sure. A bum-sniffer of the lowest order."
Priam: "Give up a man for a wind? Barbarous. Come, Sinon. Forget the Achaeans. No enemy of theirs is an enemy of Troy. But tell us in plain words: what is the meaning of this wooden horse?"
Sinon: "Athena demanded reparation for the crime committed by the underhanded Odysseus. It was the only way she'd let the Achaeans depart alive. She hates them, and they knew they were beaten. They lamented around their fires each night."
Priam: "And how did Athena make her request known?"
Sinon: "Through Calchas."
Praim: "Ah, wisest of the Achaeans."
Sinon: "The owl of Minerva takes flight from his shoulders, sir. Make the horse yours. Sacrifice to it. Athena will love you."
Priam: "I have grown to suspect that the gods love no one but themselves. Perhaps we'll sacrifice you. What would you say to that?"
Sinon: "If Priam wills it, then I accept."
Sinon kneels and offers his neck to the old king's sword. Then, after a long and dramatic silence during which only the fire talks, he withdraws to a chorus of enthusiastic murmurs.
But his performance doesn't move Ajax. No, it's little more than chaff tossed at a rock wall. He doesn't like Sinon, regarding him as all too clever and mercurial. For Ajax, a man is only a man if he stands firm in the face of the enemy, keeps his mouth shut and lands his blows. He merely crosses his arms over his chest and looks away.
Nor is Palamedes impressed. "Surely, Odysseus, you don't think this _serving_ boy will convince Lord Priam—ruler of the greatest city in the world—to abandon his senses and open gates he's defended with his own sons for ten years?" He gazes at me apologetically, as if sad at having to be the one to speak so hard a truth. The crow-black gleam in his eyes betrays his joy in pointing out such a glaring flaw, for he can't bear the idea that I win fame by getting us into Troy. Better we stay here, stuck, than see Odysseus celebrated.
"Palamedes," I say, putting on the humble voice. "I'm laid bare. Yes, I'm desperate. I'll explain why. Menelaus and Agamemnon—kings, leaders, warriors before whom we all kneel—came to me requesting aid. What an honour! To be so regarded is in itself prize enough. I've rendered small service before, night actions, furtive exploits, and ever with the stalwart Diomedes to steady me." Standing up, I reach out to include all the listeners beyond the firelight and, tucking my chin to give my voice bass, clench both my fists to my chest. "Regard my scheme not as the work of a spider to be crushed, but as the web that, with patience and labour, will snare the Trojans. Help me. Offer advice. Let's work together to make this all too raw plan ripen and succeed."
Palamedes is stymied, as are Ajax and Calchas.
But my moment doesn't last. Two guards drag a hobbled man into the light. This threesome is led by Iolochus, who belongs to Palamedes. Iolochus resembles some barbarian from the northern forests. His black beard runs up over his cheekbones to his very eyes, while his hairline cuts so low across his forehead that it nearly meets his brows. The hair in his ears is long enough to braid, while his chest is a rug, and even his back is thick with fur. Sinon has often mused that Iolochus is a not man at all but a species of badger, a beast from a cave. But no one doubts that he's dangerous, that his mind is a jaw, his opinions teeth, and everyone knows to keep their distance. These new arrivals halt at the edge of the firelight. "Agamemnon. Lord," calls Iolochus a little too loudly, a bit theatrically, trying to milk his entrance, his moment at centre stage. He has a strangely nasal voice, as if something has got lodged up in his nose. "I've caught another one. Urging desertion. I heard him myself," he adds, proudly, as if hearing is an accomplishment in and of itself. And given the thickets of hair in his ears maybe it is.
The prisoner tries to argue, but one of the guards twists the rope around the man's throat and he drops to his knees, head down. For the past year Iolochus has been weeding out dissenters; this poor bugger is the second this month, more evidence of disaffection.
Agamemnon asks the man if the charge is true.
The rope is tugged as if he is a dog being taught tricks. The fellow raises his bowed head exposing his face to the firelight: Dercynus. Half strangled, his eyes—one brown and one blue—swell horribly and his mouth gapes.
"Iolochus! Set him loose." He hesitates. I draw my knife and step forward.
"Iol'," says Palamedes finally and Iolochus frees Dercynus.
"Dercynus is as loyal as Palamedes," I say.
But the badger Iolochus disagrees in a tone gilded with sarcasm. "Pardon, Lord Odysseus. I heard him."
"With all that fur clogging your ears?" This earns uneasy laughter. His hair makes him the butt of jokes, though rarely are these spoken to his face. Iolochus's eyes survey the faces around the fire as if making note of the laughers while Palamedes, feigning amusement, drinks off his wine and signals for more.
I repeat my question, " _You_ or some _other_ heard him?"
Iolochus glances once at Palamedes and then stands taller. "I."
I turn to Dercynus and ask what he has to say.
With his rope-ruined throat the best he can manage is a croak of denial.
Iolochus now claims he has witnesses, and even before I can demand that he bring them forward, three men advance from the shadows where they've been waiting, and all three echo the accusation: Dercynus urged mutiny.
"Those eyes of his say it all," says Iolochus. "They're not right. He's a betrayer. He looks in different directions at once. To Troy and to Greece."
"This is a dim argument even by your standards, Iolochus. What does all that hair say about you? That you're part rat?"
"He has witnesses," says Agamemnon. And, believing he has no recourse, he reluctantly gives the order: "His tongue."
And with no more ceremony than that, Dercynus is hauled off for amputation. "Agamemnon!" I clap my hands once, twice, a third time, as if the whole thing was theatre, a bit of farce, and that it's now time to be serious. I congratulate him on being a reasonable leader and wise judge, and yet also a man of sympathy, with a large mind and patient heart, and ask him to wait a little.
He's tired and frustrated, but he came to me to convince Achilles to return to the battle, and to me he'd come to solve the war: he owes me this much. He tells the guards to stop.
"Three witnesses?" I laugh scornfully. "You mean three paid perjurers. Give me a day, an hour, and I'll find ten men who'll vouch for Dercynus."
"Why should Iolochus lie?" asks Agamemnon.
"What's a ratter that fails to catch rats but a useless dog?"
Seeing victory about to slip away, Palamedes says, "Iolochus is not so quick to condemn as Odysseus suggests. I've been warned and I've been watching. I see the way this Dercynus hovers and whispers. I've heard his talk."
"Talk? You talk, Palamedes, almost as much as I do. Tonight we've heard Ajax and Calchas talk. We're Achaeans. We all talk. We never shut up," I add, glancing around to gather in the others, hoping to ease the atmosphere and pass this all off as an absurd lapse, a silly misunderstanding. "If we could get Priam and his generals out here around the fire we could talk them to death and never dirty our spears."
Palamedes puts on the sincere eye and speaks with an admiral façade of conviction. "But Odysseus. We enjoy hearing you talk. All your jokes and stories, your whimsies. Such an imagination, such a vocabulary, such poetry." Palamedes traces an arc with his hand and then pauses just long enough for everyone to savour the spice of his sarcasm. "Yet Dercynus . . . he talks differently. He talks treason. Besides, he'll fight as well without his tongue, maybe better."
Agamemnon looks to me with an expression that says Palamedes is right, an expression meant to remind me that Palamedes is a king, one of the grandsons of Poseidon, and his word is worth more than that of a mere soldier. He nods to the guards.
"Agamemnon. Let him go or I take my men and my ships home. You can finish your war by yourselves."
At this, Agamemnon shuts his eyes, pinches the bridge of his nose between his thumb and forefinger and exhales a long and exhausted breath.
Palamedes is quick to intervene. He inclines his head and smiles as if to say he understands the noble urges at work, but he must set me straight. "You're not Achilles, Odysseus. And justice must be done. It sets a bad precedent otherwise. There will be chaos."
I ignore him. It isn't Palamedes's decision, but Agamemnon's. Iolochus, the two guards, Dercynus, all of us, wait. The torches burn. The fire hisses. The bats tumble above us feeding on the mosquitoes while the waves shush quietly over the sand as if Poseidon is listening. Finally, with yet another enormous sigh, Agamemnon leans forward and rests one elbow on the carved arm of his cedar chair and wearily, sadly, but doggedly, regards me. "I'll kill any man who tries leaving this camp. Do you understand? King or boy, I'll kill him. If Menelaus tries to leave, I'll kill him." From the corner of my eye I see that Menelaus doesn't so much as flicker at being included in this threat, as if he expects nothing less from his elder brother.
Agamemnon is bigger than me, and in a dead lift stronger, but he's also older and slower, so I rate my chances reasonably high that I can drop him. I took down Ajax, I'll take down this bearish ungrateful old Spartan. Yet what then? I can't just sail off. Menelaus won't stand for that. He'll be on me as soon as Agamemnon is dead. And it isn't likely I'll get out of a scrap with Agamemnon uninjured. I'll be hurt, perhaps badly, meaning Menelaus will have that much more advantage. Palamedes will be there taking bets, offering odds. Ajax will wager against me, eager to see me defeated. So, it seems to be my life versus Dercynus's tongue.
Agamemnon sighs back in his chair and Dercynus is led away into the night. I remain standing, sweat seeping down my brow and burning my eyes—eyes that have never before witnessed much less been the victim of such ingratitude. When Achilles threatened to leave I talked him into staying. When Agamemnon tested the army's will by lying that he'd had a Dream from Zeus telling us to sail home, the men ran to their ships eager to be off. I went among them and convinced them to stay and fight. Me. Odysseus. When Thersites ridiculed Agamemnon and encouraged everyone to run I beat him right there in public until he wept. Then I mastered the men so that they streamed back eager to fight. And this is my reward, this is the way of the mighty Agamemnon.
After a long silence, Agamemnon pitches his wine dregs hissing into the coals. The light trembles in the mother-of-pearl inlay of his chair, and his movement stirs the scent of the orange oil his boy rubs on him. "Time's running out," he says, opting now for a soft and coaxing tone. "Frustration, fatigue, and hunger are turning us against ourselves. Exactly what Priam wants. The Trojans have their city and their wells never run dry." His eyes urge me to see it from his perspective and consider the greater picture. In the firelight his beard looks carved in cave stone and his gaze is formidable. Still, I see now, more than ever, that he is a small king, without mercy, more interested in reputation than in the lives of his people. He has demonstrated this before. But now I see what little sway I have with him. Fine. A lesson learned, one I'll never forget. _Never again will I respect you, Agamemnon. You are a king all too familiar with the blood of your own men, worse, the blood of your own daughter, for it's all over your hands._
But now he's eager to move us on by this unpleasantness. He wants to assuage me. "It must be a magnificent horse," he says smiling, trying to rekindle my enthusiasm, as if threatening me with death and cutting poor Dercynus's tongue out are quibbles not to be dwelt upon. "A horse the gods would envy," he says. "Jewels for eyes. Ivory for teeth. Beaten gold for hide. Only splendour will convince the Trojans," he adds, in case I might not know how the Trojans think. He then makes a performance of unscrewing the gold rings from each of his thumbs and drops them into his gold wine bowl. He adds the ruby that hangs from the chain around his neck. With that he passes the bowl on and everyone does likewise, until it's heaped as full as a basket of grapes at the harvest. Echion, excited at the prospect of the wooden horse, thrilled that something new is finally taking place, eager to participate, strips off every bit of metal he wears and adds it to the heap. Even Palamedes contributes a gold arm band fashioned like two twined serpents. I give the gold bracelets I took off the Trojan who kicked me in the knee. Only Ajax is blatantly reluctant, glowering as he gives his rings, but the example has been set by Agamemnon and can't be ignored. The horse will be built and it will be as splendid as the pyramids.
As the gathering disbands Menelaus appears at my side. He's been more than usually silent throughout the evening. And yet I see by his troubled eyes that he's been thinking. "It's a good plan, Odysseus. A good plan." He is adamant, his voice tense. We stand ten yards from the fire, the flame shadows slithering over the ground, a high haze veiling the moon. I want to ask why he left Paris alone with his wife and brought all this down on our heads. He reads the question in my eyes and is sorrowful. "She is everything you have ever imagined," he says.
He turns and walks away, head thrust forward, shoulders rounded under the weight of guilt.
I visit Dercynus. They've cleaned him up but he looks dreadful, pale, unconscious, breathing in spasms. Machaon says he dosed him heavily beforehand with poppy and wine. Tears nettle my eyes. "Is he going to live?" When Machaon starts in about 'the gods willing' I cut him off. "Don't give me that. Save him." I return to my tent fearing the dreams in store for me.
I don't dream as much as Penelope—who every morning had epics to relate—but my brain does its fair share of nocturnal churning. (I sometimes wonder if the world we call waking is in fact sleep for our world of dreams. When I suggested this to Penelope she looked at me as though I was a talking chicken.) I've gazed down upon armies as if a creature of the air, swum with dolphins, talked the language of ravens. For many years I've had a recurring dream about an egg. In it the yolk is the sun. And this sun hatches into a boy, only the hatching is an explosion, the shell and the gluten bursting, and the boy stands grown, a man holding a shield and gripping a spear, a crest of cock feathers atop his helmet of leather and brass, and with a shout he hurls the spear into the sky. It's a splendid throw, and the spear travels a day and a night, and pierces the sun. Only once again the sun is an egg yolk, and when the spear strikes it the yolk explodes, and the scenario begins all over.
But on this night I walk among mutes. They cling, grope, point east, point west, open their tongueless mouths which become caves leading down and down into the Underworld, where Hades waits on his throne of skulls . . .
I wake. It's long before dawn and my head is a stone. Staggering out for air I go immediately to check on Dercynus. He breathes shallowly, dried blood crusting his mouth. To watch a man sleep is to know the terror of ultimate vulnerability. I kneel and whisper into his ear. "Valour, son, valour. It's too soon to let go." I grip his hand and, to my joy he returns the pressure.
At this hour the fires have burned low and the air is cool. I return to my tent and lay on my back, taking reassurance in the weight of my forearm across my brow. Is it possible that my other son, my blood son, Telemachus, is dreaming of me right now? Is Penelope? I want to reach through to her, to both of them, plunge my arm through a waterfall and find their waiting hands. Sometimes I feel our life together continues uninterrupted like a subterranean stream, and yet all too often it seems as if we are dead to each other, as if that stream has dried up.
When my grandfather, Autolycus, died I thought I had lost my real father. For while Laertes was off with Jason and the Argonauts chasing a reputation, Autolycus showed me how to box and use the bow and the spear. He was a quirky old man, full of songs and riddles, a notorious cattle thief in his day, but he was devoted to me. He taught me the constellations, said that Egyptians worshipped cats, Indians worshipped cows, and that the northern barbarians dyed themselves blue. It was he who took me on my first pig hunt. We camped in the hills then drove the dogs down upon them before sunrise, bursting through the mist like gods from the clouds. The dogs had been muzzled all night and when we unlocked their jaw-clamps their baying racketed between the canyon walls. We cornered a boar. Huge and black with curved tusks, it faced our spears in a snarling rage. We stabbed at it from our horses and I found that I hated the beast because I feared it. At the same time I admired it the way I admired and hated a fire burning out of control. The horses tried trampling it. Their front legs pounded the air while their rear legs churned the dust. My grandfather dismounted. I did the same. Again and again we thrust our spears into the boar's neck. In its final panic it gored me, driving a tusk deep into my thigh before heaving me into the air. How clearly I remember that moment and still feel the hot stab of the tusk and how I flew, arms flailing, crying out like a hit bird.
**Chapter Four**
**O** VER THE NEXT few days I spend my time planning for the wooden horse and looking after Dercynus. I slaughter a goat and offer prayers but still he is bleeding. I dose Dercynus with poppy water and wine until he is so drunk he sings. Or tries. Without a tongue what a distorted song it is, a lament animal and otherworldly. Then I hold him as once again Machaon applies the cauterizing iron. Dercynus howls like a fate-cursed soul, and the only mercy is that between the agony and the poppy he passes out.
It's with guilty relief that I leave to meet with Epeius. He asks after the boy. I shrug and shake my head and we stand awhile in silence.
Epeius is one of those men who sees everything in the round, and can turn an image in his mind like a stone in his hand. He's also a master of siege-craft. After a suitable silence, he starts telling me that success in war can be traced to one principle: thrust, and the richest source of thrust is torsion.
"But," he cautions me, "it's also the most difficult to achieve, for it requires elasticity, and elasticity, Odysseus, elasticity is elusive, for it slackens, like a man's stomach." He grips a fold of his own belly which droops like a hag's dug. "Currently we use goat sinew, though I've had encouraging results with rope woven from horsehair. But," and here his voice grows intimate and he tilts his tall frame toward me, "I have an idea I know you'll appreciate: spider web. Yes. If it could be braided into cordage, well, you see the potential. Double maybe triple the range. Or up close, enough thrust to drive an arrow through a Trojan shield!" In demonstration, he shoots out a right jab.
I nod eagerly though I'm still not clear how this relates to the horse. When I ask him he is shocked.
"How?" His eyes darken with disappointment and he frowns, wondering how I can ask such a thing. "It all relates. If you could store torsion you could drive a wheel. Think of it, actually advance a cart without ox or horse." The strain must be showing on my face as I try envisioning what he's talking about. Frustrated, he makes as if to cuff me across the skull. "A cart, Odysseus. A chariot. A wooden horse. Anything with wheels. Think of a stone in a fire."
A stone in a fire? "All right. Yes." I'm with him so far.
"Think of the way it holds heat long after the coals have died."
I nod again. So far so good.
"If you could store thrust in the same way . . ."
"Like hate," I suggest, thinking of how it smoulders on in one's heart for days, even a lifetime.
"Hate? Yes, yes I suppose you could call it hate. Or rage."
"Latent power," I say slowly, the comprehension coming to a simmer in my brain.
"Exactly!" he shouts, gripping me by the shoulders and dancing.
We stroll arm-in-arm up and down the shore speculating upon torsion and spider webs and the heat stored in a stone, and most importantly, the construction of a colossal horse.
Dercynus dies and Machaon is resigned to punishment for failing to save him. I do nothing. What's the point? I ask if he regained consciousness or said anything? Machaon shakes his head. The corpse's face is chalk-white, the mouth twisted and the lips split. His hands, as if still prepared to fight, are clenched. We wrap him in a sheet soaked in eucalyptus.
Sinon and I bury him and then raise a modest mound set back in a discreet spot overlooking the sea. We pour blood and sing prayers and give him our best thoughts on his journey. As we look at each other over top of the mound, the fear in Sinon's eyes sparks fresh doubt in my heart. If I can't protect Dercynus, how can I expect Sinon to undertake the risk of letting himself be captured by the Trojans?
"Sinon," I begin, offering to let him withdraw from the scheme.
But he's ahead of me. He puts up his hand to block my words.
I am a cave to myself, warrened by ever-branching tunnels that lead not to epiphanies but to darkness. Gods? They're everywhere and in everything, but by that same token nowhere and in nothing. As unswayable as stones and as erratic as fire. Meaning? Meaning nothing. Riddle and paradox. Dercynus is dead and yet I sleep well. I actually wake refreshed and step out of my tent to see the sun grinding its invincible way up out of the land. On the first dawn following a funeral, no sight is crueller.
That day Epeius and I walk down the coast.
"Another grave mound," he says, meaning poor Dercynus.
The morning breeze gusts fresh off the sea. We find a tight cove with a smooth stretch of sand where, with his knife point, Epeius draws me a horse, in profile and plan and cross-section and three-quarter view, and we discuss issues of size and joinery and logistics.
"It has to be big," I caution. "With room inside for ten men. Plus water jars and a piss pot."
"And a spy hole out its eye," suggests Epeius. "You'll want to see what's going on."
"Yes. Exactly." His imagination has embraced the conceit and enlarged upon it. Talking to him always reminds me of how vast a man's mind can be, bigger than any palace, busier than any city, as full as the sea.
Then he's shaking his head as if he's made a mistake. "No, a spy hole won't be enough. It'll be too small. You'll have to use your ears. You'll have to listen. Hide stretched over a rib-work frame will be better than solid wood. And bunks," he adds, "for you could be stuck in there longer than expected."
I don't want to dwell on that possibility, but suppose it must be taken into consideration.
"Besides," he says, "lying down will allow you to wait more comfortably. You want to be rested when the moment comes."
I grow solemn and ask him what he really think of this horse idea.
He regards me as if taking my measurements, then he clears his throat and cocks his head to one side as if seeking the best words for the worst news. It's dangerous to ask for honesty, because what we really want is reassurance. He sighs and I grow wary, but at length he acknowledges that the horse has the merit of intrigue, and that intrigue is an excellent angle to have going for you. "The Trojans will have to be very wise or else as dull as cows to resist it."
"How long will it take to build?"
"You know the saying."
"I know a lot of sayings."
"Estimate your time," he says, "then double it."
I nod at such pragmatism.
"Right. First of all, the ship has to be beached then broken," he says. "That's the fun part. Allow five days. Then the more difficult business of actually building the horse: twenty days, though perhaps more. Then another five for transport by barge."
"Thirty days."
"Call it sixty."
"All right, sixty."
Epeius looks off now and narrows his eyes as if studying something in the distance. So convincing is he that I turn to look as well, but see only the pine trees and the burnt grass on the crumbling cliffs. He is nodding his head approvingly, as though he likes what he sees. He has pale grey eyes, a long nose, sparse hair and a small skull, proof that a big head does not mean a good brain. I've always liked him for he's calm and dignified and has wit. "It will be a horse as grave and beautiful as numbers," he concludes.
But which ship to break?
The question makes the rounds of the fire that night. Agamemnon looks to Calchas who calls for a cock and twists its neck. He pries open its breast with his thumbs as if splitting an orange and studies the viscera. He works his hands in its red and purple guts, sniffs his fingertips, gives them a lick and concludes that all the biremes uncaptained by death are ill-omened. This is a problem, because we can't very well take a ship out from under a Master who still lives. What to do? Good, kind, far-sighted Palamedes suggests a solution.
"Why doesn't Odysseus offer _his_ ship? It's his scheme."
A murmur travels like a tremor through the assembly.
"There's beauty in this," admits Calchas innocently, as if he himself would never have thought of it, but now that the notion has been raised, well, as reasonable men, it's only prudent to give it due consideration.
"The goddess would approve," adds Palamedes, sounding solemn, though not even he can hide the joy in his glance, a glance that looks the way piss smells. All eyes are upon me.
"Fine. Excellent. My ship it is," I say brightly, as if honoured. "And while we're at it," I say. "I've been wondering about who should be in the horse. Me, naturally—"
The irrepressible Echion strains forward. "Me, Odysseus! And me!"
"Echion. Excellent. But give me a minute. Let's work down the list."
He slumps back.
"I'm thinking first of Palamedes and Ajax."
Pride drives Ajax to immediate acceptance.
"Good. Palamedes?" I look queryingly at him, convinced it's more prudent to have him in the horse than manipulating from afar. "Such a majestic warrior must of course be included in this most glorious of ventures. Think of it: Ajax, Menelaus, Diomedes, Palamedes and me, Odysseus. How can we fail? The ode masters will sing our glory. For a thousand years men will speak of us. For eternity the gods will recall the day Troy fell. It would be unfair to exclude one such as Palamedes."
In the silence that follows, the fire burns quieter and even the waves seem to cease their pounding as if they too listen. Eyes glittering like crushed glass, Palamedes spreads his arms wide as if to embrace me. "Much honour you do me, Odysseus. I thank you. And I accept."
Cups clash in a toast. Agamemnon appears content. Even Calchas relaxes. The relief is palpable, as if a breeze has cleared a corpse stench from under our noses. Much happy chatter follows, until Palamedes interjects with an afterthought, employing the cautionary tone of a prudent elder concerned for the safety of children.
"Brave Odysseus," he says, wrinkling his humble brow in innocent reflection. "Shrewdest of us all. Master in the Arts of War. Consider this: when one stands too close to a stone you cannot see that it is a part of a wall." By way of demonstration this wise counsellor, this brother-in-arms, holds his palm against his eyes. "Nor can you see that the wall is part of a house, and that the house may harbour an enemy. If I am not mistaken, I think you and I will agree that only distance and perspective give such knowledge."
I incline my head and put on the face of one eager for his wisdom.
"Therefore I'm only wondering, Odysseus, if you're not perhaps too close to your most excellent scheme," he says. "Blades, after all, are forged to the task. No one fells an oak with a cleaver, and no one cuts onions with an axe. So with warriors. They must be deployed where most fit."
"Immense is the wisdom of Lord Palamedes. He is right to correct me. A problem is indeed best viewed from all angles by many sets of eyes. An exploit such as this is a perilous one, and wise is he who knows whether his strength lies in his arms to fight or his legs to run."
I may as well have spat in his wine bowl for the stare he gives me.
"Speed is a valuable asset," I continue, innocent of any insinuation. "In the horse a fast runner would indeed be wasted. Or a great arm with a javelin. This will be a manoeuvre favouring men—" and here I hesitate a beat so that the word _men_ should register "—favouring men who work best in close combat: sword and dagger, axe and fist." I began my little performance intending only to nettle him, but the further I go the angrier I become, because he was behind the framing of Dercynus, because he put the knife to my Telemachus's throat, because he is to blame for my wasting years here in this absurd war. So I push hard. "Your point being, Palamedes, that you're a—"
"My point being that I don't like small places and I don't like small men."
Everyone and every thing falls still. The air and the earth and the sea lay silent. Even the fire seems to cringe. And then a single blue spark pops and lands on Palamedes's bare knee. He doesn't flinch, he doesn't turn his gaze from me, he makes no move to brush away the burning ember. Perhaps he thinks it's a sign. No doubt old Couch Ass is already devising his interpretations.
At last Agamemnon slaps his thigh and announces that Palamedes would indeed be wasted in the horse. "He will stay by my side. That we may watch together for the burning arrows signalling the horse is inside Troy. And if the signal doesn't come," he adds, "I will need shrewd counsel." Is this an attempt to make amends for Dercynus? Or is he as suspicious of Palamedes as I am? Eager to move on to safer matters, Agamemnon now turns to Epeius, but Palamedes isn't finished.
"As would a man of Menelaus's stature," he observes quietly, soberly. "His very presence leading an army inspires the men and terrifies the enemy. Shut up in the horse that presence would be lost. Why keep such a flame in the dark?"
Palamedes knows Agamemnon will not abandon his brother, yet with me isolated Palamedes will happily see the horse burn. Thus he sacrifices Ajax but keeps a direct line to Agamemnon via Menelaus.
"No."
We all turn.
Menelaus. Standing. Breathing hard. "I have to reach her before Paris has a chance to take her away. He'll disappear with her. Into the East." He gestures indicating the deserts and mountains and marshes stretching away toward Persia.
I will give him that much, he has pride and he has boldness, and he wants his woman back. Even Palamedes is caught with his mouth open. Agamemnon, as if suddenly sick of the whole business, as if suddenly too old to bear a moping Menelaus, says fine, as you see fit. And so Ajax and Menelaus will join Diomedes and I in the horse.
That night I dream that suitors circle Penelope like randy cats, while others, subtle, smiling, befriend Telemachus, playing kick the pig's bladder with him, teaching him to box and wrestle and swim, the very things I should been doing. In the dream Penelope mourns me, but after ten years alone her heart begins to waver and she's ready to be swayed, for a son needs a father, and Odysseus—trapped, killed, cursed, or simply indifferent—has vanished over the sea . . .
I wake as desolate as I've been since leaving Ithaka. Tormented, I step out into the night and walk the beach. By now Telemachus has likely forgotten what I even look like. Would I recognise him?
When the midwife presented me my son wrapped in a wool blanket, I'd looked into a pair of deep green eyes that were already at work evaluating me, as if I was the one who had newly arrived, not him, as if I'd entered his world and not the other way around. And it seemed to me he was already reaching conclusions. I remember asking him what he thought and he merely gazed at me as if biding his time, as if implying that he'd wait awhile before letting me know. He wasn't disturbed in the least by my bearded face looming in upon him. I was relieved. In fact I felt honoured, the same as when a bird allows you to approach, for both are mysteries and it's a privilege to be allowed near.
Penelope worried that Helen would hear of the boy and decide to pay us a visit. Helen was Penelope's dark obsession. At times she sneered at the notion of Helen being the offspring of Zeus and Leda. At other times she did not doubt it. Penelope grew up with Helen and described flocks of swans wandering freely about the palace, fed upon the finest grain, the purest water. Anyone caught abusing them in any way, even so much as a harsh word, was whipped. Except Helen. She could get away with anything. She'd grip the biggest swan around the throat and climb onto its back demanding that it fly her up into the air. She'd pluck their feathers. Chase them. Leda said nothing. She indulged her. They all indulged her, the daughter of Zeus. Even I, in far off Ithaka, wanted to see the splendid Helen born of an egg.
Of everyone who went to Sparta only I arrived without a gift. No pearls grow on the shores of Ithaka, no rare woods on our hillsides, no gold lies in our stream beds, the craft of the weavers and potters is utilitarian at best. Besides, I didn't need a seer to predict she wouldn't be choosing an obscure prince from a small island, no matter what I might give.
There was a banquet in honour of the athletes come to impress Helen by running and jumping and wrestling like so many monkeys. I recall a marble-floored hall with pillars and iron lamp stands and caged ravens. There was so much food—roasted peacock, stuffed hens, braised dolphin, pork tongues, wolf brains, Chios wine, fig liqueur—that it was cruel, for anyone who ate and drank his fill would never make it out of bed the next morning much less be able to compete. Helen didn't make an entrance, there were no trumpets or drums or heralds making speeches—she appeared—silent, still, standing like white fire. Her hands were clasped before her as if she was there to serve, as though our comfort was paramount. She had remarkable hands, long and expressive, each finger a serpent capped with a red nail. She smiled a smile disconcertingly dark and cave-like, for it didn't show her teeth.
She wasn't beautiful. Striking, yes, beautiful no. She appeared differently to each of us. Menelaus saw the ideal woman, her features perfect in their symmetry, her manner regal, eyes calm, breasts neither too small nor too large, waist just so, ankles slim, none of your cow-hipped peasants with udders to feed a herd; a woman worthy of marble. Palamedes saw a delicate creature, small, demure, obedient, with downcast eyes, her hair woven with gold threads and blood-drop rubies. Diomedes saw a night creature, her smile as seductive as moonlight on water, and with a notorious taste for oil and the lash. Ajax whispered of a creature as majestic as an eagle; I saw a woman lean and hard and feral, her face all planes and angles. A woman in a panic.
One thing we all agreed upon were her hands. Even Penelope used to talk about "the disposition" of Helen's fingers. It was as if her hands were puppets and she was always aware of what they were up to, languid or threatening, bored or curious. She'd often find Helen alone studying her fingers, posing them like a little crowd of dolls. She'd make Penelope play shadow puppets with her. Would want to play all night and got angry when Penelope grew tired, threatening her, hitting her, twisting her arm and pulling her hair or alternately pleading with her and giving her jewelry, anything to have a playmate.
But now Helen greeted us, spreading her arms wide as if she would embrace us all. She wore no rings or bracelets or bands, and her voice was low and full of the undertones of bronze. "Welcome."
Before anyone could rise she was drawing out her own chair. Even the serving boys seemed too entranced to act. Her gaze visited each of our faces. Did her eyes linger upon me? She certainly took a moment to regard smoothfaced young Ajax who was scarcely fourteen years old.
She sipped her wine and began asking the man to her right his name and that of his father and the story of his journey. In this manner she spoke to each of us, and we did our best to charm and impress, to capture that most illusive of all prizes, another's heart. How attentive she was, how complete and utter her focus. When she spoke to you, you were at the centre of the world, the centre of _her_ world, which was the only one that mattered, and when she moved on to the next man you felt as if the sun had disappeared behind a cloud, that you'd been cast out to the dark regions of exile, and your foremost goal—your only goal—was to regain the radiance of Helen's attention. Ajax acquitted himself well.
"And would you marry me?" she asked him, her voice throaty with amusement but at the same time bearing about her brow a seriousness, as if this young man, this boy, genuinely intrigued her, and she would not be so callous as to insult him.
Gripping the table edge, Ajax stood and in a bold tone told her that he was by far the best choice, because when these old men were leaning on sticks, he would be young and vigorous.
She arched one eyebrow and had to agree that, seen in such a light, his youth was indeed an advantage, and thanking him she wished him fortune in the games.
My turn was approaching. I had a belly full of bees, for as well as having failed to bring a gift, I'd prepared no speech or story or clever reason why she should choose me before anyone else. The only thing I had to offer was exactly the same as the other men—my mortality, the fact that I would die—the one thing the daughter of Zeus would never experience. When my turn came I said, simply, teasingly, in the spirit of the evening, "In loving me you will taste the salt that gives this brief life its savour." And with that I smiled and was silent.
She said nothing for a moment, reflecting upon this, the briefest speech of the evening. Then she threw her head back and laughed so loudly she nearly blew out the torches. "But death is not unique to you, Odysseus. Every man here will die. Even the beautiful young Ajax." The way she spoke made it sound like a curse that it was in her power to declare or rescind. Could she bestow immortality on the man she chose? The question bloomed in the mind of every one of us; perhaps it had been there from the start, perhaps it was the very reason we were here at all, to lie with an offspring of the gods.
And then Helen was gone. A flame snuffed. How sudden it seemed, even though she'd thanked us again and wished us well. When she departed it was as though music had stopped, night had fallen, and the silence that remained was as desolate as dried bones on a beach.
And now, twenty years later, I could still feel that desolation right through the anger and resentment. I turned back, wishing for the hundredth time that I'd never gone to Sparta at all, though of course then I'd never have met Penelope. Irony within paradox. Something bad breeds something good, the flower in the cesspool.
Plodding back along the shore, I am shaken from these recollections of Sparta by shouting. I creep through the thornscrub and find Palamedes and Iolochus backing young Sinon against a pine tree. When I step forward Iolochus runs at me with his sword until seeing who it is and hesitates.
Palamedes is all charm. No poisoned water ever slid so smoothly over polished rock. "Odysseus. Excellent timing. Your man here—" With his blade he indicates Sinon as if directing my attention to the finer points of a slave. "He's a brave one. Attacked us both, one against two."
I don't need to interrogate Sinon to know a lie. Palamedes has it in mind to kill both of my aides. A nice project. Knock the pins out from under Odysseus. "Yes, he's a lively one. Young men. Erratic as spring winds. But they pass," I add. "No harm done."
"Oh," says Palamedes in a tone almost sprightly, "but harm has been done." He shows me a blood-stain on his robe between his hip and ribs.
"My blade's clean," argues Sinon.
"Show us," says Palamedes.
Sinon raises his sword—coated in dark blood. By his expression I could very well think he's the one who's been dirked. I'm frankly impressed. Had Sinon thought he was doing me a favour by killing Palamedes? "Palamedes," I say, making with the sincere voice, "Let me have a look at that wound."
He jumps back as if scalded. "You and yours have done enough, Odysseus. My wound, like my life, is my own." And in one swift swing he catches Sinon across the upper arm. The boy cries out and drops. Palamedes raises his blade again but I block it with my own. The reverberation of the strike runs down through my elbow. Now Iolochus manoeuvres to get at Sinon, but I drop to a crouch and slash him across the calf. I feel the tendons split; he'll never walk straight again if he walks at all. Palamedes sheaths his sword, raises his hand in a curious little gesture typical of him, half wave and half insult, maybe part incantation too, something picked up from Calchas, and withdraws into the night.
I tie Sinon's arm and hurry him back to Machaon, because Palamedes is a great one for wiping his blades in shit or venom. Iolochus we leave groaning in the dirt. It's tempting to break a few of his ribs for good measure, maybe put my heel in and snap his collarbones, and then as a finishing touch, cut out his tongue and whisper in his ear that Dercynus says hello. It's what he deserves. I've often watched Iolochus perform Nauplian folk dances, curious as to how his apelike face grows reflective and almost refined. There'll be no more dancing for him.
Machaon rubs Sinon's wound with crushed garlic, the boy protesting his innocence the entire time until I assure him I know old blood from fresh. "You'd better keep a closer eye on your weapons," I advise him. "What were you doing out there in the middle of the night?"
"Following you."
Touched by such loyalty, I mask the spasm of emotion by telling him he'd better heal quickly for he's the only one I trust to do a good job plucking my ears.
**Chapter Five**
**T** HE NEXT HIGH TIDE we lock hawsers onto bulkheads, draw up anchor stones, and twenty men drag my blue-prowed ship onto the beach, her keel shrieking against the gravel. The shipwrights are terrifying to watch. One day and they take up her decking, and in three all that remain are keel and ribs, a whale picked to the bone. I love that ship. I've often gone aboard her and sat with my back to Troy and my face to the sea, recalling the rhythm of the oars and the moan of the planks. I should be home teaching Telemachus to sail. Epeius sees my expression and steps to my side and reminds me that there are other ships in the fleet, I can take my pick. Nonetheless I call for one of the oars, stout oak, and order it kept aside. I wink at Epeius and say I'll take it home and plant it on the hill above my house and grow a new ship, for my son.
He likes that.
Men posted in the hills keep the Trojans from seeing what we're doing. I want the horse discovered only after we've burned our camp and the fleet has set sail for Tenedos. Let the horse appear as suddenly as Helen had appeared that night in Sparta. The metamorphosis from ship to horse intrigues me. I set up a tent alongside that of Epeius and remain on-site. A fifty-oared bireme will become an animal three poles high at the shoulder, the curved hull planks the hindquarters, the deck-wood the legs, the rendered feet of our dead cattle the glue, and their tails plaited into a mane. Everyone has suggestions. Carbuncle eyes, a tail of lion hair, a skin made of stitched horsehide stamped with gold, a whalebone saddle, stirrups of bronze.
Epeius, Diomedes, and I spend entire days overseeing the work. The smiths fuel their forges and hammer out bolts and plates and pins observed by statues of their crippled god Hephaestus.
"The ancients thought iron fell from the sky," I say.
"Fish have noses," says Diomedes, apropos of nothing. "But can they smell?"
"They sometimes stink."
"Watch," he says, and stands on one foot with his arms folded like wings. "A goose."
"Excellent," I say. "That'll come in handy."
Epeius and I exchange glances.
"The full moon," I say, explaining his oddness. Returning to the topic of metal, I recount the story of how, on the road to Sparta, a hermit took me for a robber and decided to finish me off before I could do any more robbering. He charged me with a sword even more ancient than he was, made of the old iron, the blade softer than gold. I side-stepped the blow, which hit the ground bending the blade sideways, so that it was as cocked as a dog's hind leg. The old warrior used his foot to straighten the bend while I drew my own sword and swatted him across the butt with the flat of my blade.
"And you became great friends?" mocks Diomedes.
"No. He tried kneeing me in the balls."
Epeius observes that if we could find a metal as light as wood but as strong as brass our infantry would be as invincible as the gods.
I ask him if he's ever seen a god.
Before he can respond Diomedes announces that he wants to kill one.
"One what?"
"One god."
"Ah." I nod politely and ask him how he proposes to do this. I love and admire Diomedes and find him endlessly intriguing even when he's being an ass.
Adopting the sober tone of a practical man, he admits it won't be easy, as if I'm the one who needs reminding that the gods are immortal. We fall into a discussion as to how best to achieve this curious goal of his. Mulling the problem, we eliminate poison and knives and spears and arrows and hammers. I suggest that perhaps he can trick one god into killing another . . .
Diomedes brightens. He waggles a finger at me meaning I've hit upon an interesting possibility. Then he falls glum. "I'm not that clever."
"You just want to cosh one on the head."
He indignantly denies this. He insists that he has a grander vision and paints the scene for us. "An arena. An audience. Flowers. Poets. A fair fight witnessed by all."
And a reputation that will last forever, I think, you who are always mocking the pursuit of fame. I don't mention this, however. Instead I inquire as to whether he has any god in particular in mind.
He looks at me as if the answer is obvious. "Ares."
The master of war . . . I begin to see his larger motivation. Kill him and kill war itself. A noble aim. A gesture I envy, a gesture I admire. As ever, I'm impressed by the man. Charmed, in fact. Still, I'm sceptical. How can a man kill a god? And if it is possible, by some trick, by some spell, the gods would take revenge, for it would be a precedent and soon they'd have nothing but insolent men on their hands. Apparently my face betrays these doubts.
Diomedes grows impatient. He says it is symbolic. "Even a wound would do. Something to drive Ares off the field." He turns to Epeius for his approval.
Epeius has remained silent all this time, bemused by our antics. He suggests, in his considered fashion, that fate has a face, and it is a grimly ironic one, and that gods are one expression of that dark wit. "Seers are another."
"Exactly," cries Diomedes. "I've always said Calchas is a joke."
I recall one so-called seer who made it across the strait to Ithaka. He had a gibbled foot, a long beard, bald head, and his front teeth were gone so his words were slush. I had Telemachus with me. He was about three at the time. The stranger said he'd tell the boy's fate, all he needed was a goat and a fire. I laughed at him. He said okay, a hen and a fire. So we built up the coals and I sent for a hen. Why not? A little diversion on a grey day. It was winter; a hard wind sloping in off the sea. The hen was scrawny. The worst of the flock. The seer was disappointed but said nothing, just pulled off the bird's head as if plucking a cork from a jug. Telemachus looked at me, blinking, not sure what had just happened, because he liked chickens and the sight of killing upset him. The seer slipped the head into his pocket. Maybe he considered it a delicacy or perhaps he'd use the beak as a whistle. I don't know. He gripped the carcass in both hands as if squeezing a wine skin and out squirted a string of blood. I half expected him to read the red scribble on the sand but he ignored it, and instead set about plucking and gutting the bird and then positioning it over the coals. He added salt from a pouch. He cut up a lemon and squeezed the juice onto the meat. A very attentive cook, this man, prodding and turning the bird with a knife so that it didn't burn. Finally he ate it, squatting there on his heels, chewing the meat, sucking the neck bones, licking his fingers. Then he belched and snugged up the salt pouch and put it away in his pocket, cleaned his hands with the last of the lemon and prepared to leave. I was smiling now, curious about the old scammer. He'd pulled one over on me, so I reminded him about reading Telemachus's fate. "And my son's future?"
"Oh, a long one," he said, "very long, but as lean as this bird."
As the horse progresses even the doubters grow enthused. Menelaus and Agamemnon wear the grins of boys; Palamedes is politely observant; Iolochus hobbles past on his crutch and spits. Among the men the horse is the sole topic of conversation. Only Ajax is indifferent. He ignores it completely despite the fact that he too will take his place inside it. Instead he begins acting erratically, forgetting to wash, talking to himself, vanishing for days at a time. I start to worry. Do I want this man in the horse with me? I ask Sinon to watch him.
Sinon usually sleeps outside my tent. At dawn the very next morning, while I'm still in bed, he positions himself just inside the flap so he can observe Ajax, whose tent is directly opposite. As he sits there, Sinon casually unwraps Achilles's armour and begins polishing it, a common enough task for him, but this morning there is something different about the way he goes about it, an element of the actor is at work as he fawns over each piece, as if it were his very own. Each time I see the armour I'm impressed. Ajax is right to covet it. Hephaestus fashioned it for Achilles at the request of Thetis. It seems almost to breathe like a bronze skin. The gift of a god to a half-god.
As if the armour calls out to him, Ajax stirs and sits up. Sinon opens the flap wider, exposing the armour to plain view. As the sun rises the armour fairly shouts with an unavoidable radiance, and it is the first thing Ajax sees. Working diligently with rag and oil, Sinon pretends to be unaware of being watched. He whistles softly, the high tone shaped by the smile curling his lips. His impersonations have always bordered upon the cruel. The very source of their brilliance is his attention to detail, keen, accurate, merciless. He delights in his art. Is more alive when in _persona_ , as he is now, pretending to be but a simple soldier absorbed in a routine task. I freely admit that I love Sinon. He is a son to me. He is the boy who was nearly killed by Palamedes and Iolochus while trying to protect me. But there is no denying that he savours the pain of others with a little too much relish. Ajax has never given him so much as a bad word. Dercynus was not so clever but he was kinder, perhaps wiser. I ask him if he has urinated long enough on Ajax's wounds?
Sinon promptly wraps up the armour. The tilt of his head expresses the sting and indignation of the unjustly rebuked. I remind him that I'd asked him to watch Ajax, not torment him.
Ajax, meanwhile, has left his tent and vanished.
Eager to redeem himself, Sinon is back by mid-afternoon reporting that Ajax has begun to spurn Palamedes. Intrigued, I set aside the wax plugs I was about to fit into my ears before settling in for a nap. I stand. I pace the tent, questioning Sinon closely. I'm inspired to make another attempt at reconciliation. In the stores taken off my ship I discovered one last pot of wildflower honey, and the next dawn, while Ajax slips from the camp on one of his wanders, I call out to him. He averts his face and hunches his shoulders as if he doesn't want to hear, as if I am a cold wind. But I mean to settle this before getting myself sealed up inside the horse with a man who hates me even more than he hates the Trojans. We walk in silence until we are beyond the camp and the stables, the sun warming the land while the sandpipers flee at our approach. I show him the honey.
He hardly glances at it, just lengthens his stride and walks faster. Reaching a rocky point we wade waist-deep into the water to round it, the sand swirling about my feet, shards of shell lodging between my sandal and sole. Holding the honey pot in one arm and balancing with the other, I suddenly wallow and grow fearful at how easy it would be for Ajax to simply turn and drown me.
"I once looked to you as a father," he announces as we regain dry land. His voice is breezy, as if referring to some fondly recalled stranger. "Sought your counsel. Confessed my fears. And you gave me riddles." He sounds bemused, almost chuckles. I run to catch up, the soft sand sucking at my sandals, my bad knee throbbing. "When I failed to solve the riddles you gave me scorn." Only now does he look at me, long hair knotted with twigs, upper lip unshaved, glance clawing me with the tethered rage of a betrayed dog. He raises his forefinger as if expecting me to interrupt. "Scorn," he repeats. "Honeyed with pleasantries. Sweet thorns whose prick you thought I was too dull to feel. No," he says, seeing my surprise. "I'm not always stumbletongued. And you, you're not as subtle as you think you are."
So, this is how he remembers those days. Is he right? My wit is dry and my jokes too often taken for insults. Even Penelope, who knows me best, sometimes grew teary at some quip or sarcasm, however innocently intended.
The first time I saw Ajax he'd just caught a moth and had it cupped in his hands as if holding water. He was standing outside the athletes' tent, so absorbed by the moth that I was able to step right up behind him undetected. He was gazing at it as if he'd never seen such a miracle. I'd only just arrived in Sparta, the new moon a sickle and torches maddening the insects. He was the first person I spoke to. "Do you remember the Death's Head Moth?" I ask him now.
Ajax slows at this and grows defensive. "You thought I was absurd. A child. A collector of bugs."
Not true. Or only partly. I was also charmed. "I was impressed, Ajax. You, fourteen years old, travelling all the way to Sparta on your own. That took guts, more mettle than some grown men ever show. But I'll tell you one thing, every man there was fearful when they saw you compete."
He shakes his head. He isn't going to be so easily flattered a second time. "Always side-smiling and smirking at me."
I admit I'd smirked and grinned along with the other athletes at the spectacle of this earnest boy so eager to win the prize. There he was, splendid and absurd, never having touched a girl much less grasped any notion of his mortality. The first day he practised with a javelin that was almost as thick as his arm. I had to wonder at his father, Telamon, permitting his son to chase such a fantasy. Was it a lesson in the school of hard knocks? A dose of the real world for a boy reared in a palace?
"You did well," I remind him now as I follow along the shore with the honey like some market tout dogging a customer.
"I lost."
"I lost, too."
"You quit."
To Ajax there is no greater humiliation. I try to argue but his mind is off on its own memories. "Do you recall telling me about the mad honey of Trebizond?" he asks.
"Of course."
He smiles the smile of a thousand-year-old man adrift upon the mornings of his youth. He angles his way up the beach to where dry grass tufts amid the rocks. "I liked that story," he says, fond now. "Rhododendron honey. With a bee god in it. I wanted to taste it. I'd have risked the madness for a taste. Didn't you ever wonder what it was like?" He halts so suddenly that I nearly bump into him. He gazes at me with the open expression he'd so often worn in those early days. He was the age then that my Telemachus is now. Would a braver man have simply refused to join this absurd siege; would a wiser man have shrugged and simply told Menelaus no, I won't go, instead of pretending to be mad? I could have risked his anger, endured the shame of breaking my vow, explained to my boy that, unlike my own father, it was more important for me to stay and see him grow up. Instead, ten years have been lost, my reputation remains uncertain, my son doesn't know me and my wife has no idea whether I am alive or dead.
"The taste of that honey has to be—" His vocabulary fails him and he shakes his head. "I thought you must be lying, but Nestor—a wise man, well-travelled, honest—he knew of it, a lot of men did. But I'll never taste it. Not now." When I ask why, he responds sarcastically. "It was a Death's Head Moth, Odysseus. I won't last long here. Hades has his eye on me. I can feel it. You think what you want, but I know what I know."
A creature of instinct, Ajax, so optimistic as a boy, so grim as a man. The most loyal of us all, believing in the honour of this war, but he needs someone or something to be loyal to, except everyone has failed him: his own father the great Telamon, me, and perhaps most of all Agamemnon, who cheated Achilles and let the war drag on.
The land rises and soft sand gives way to packed dirt. Wind-bent pines lean like a gathering of cripples to watch our passing. I break the wax seal on the honey, yet Ajax regards it as if peering into a suspect well. "Too little, too late." Not deigning to even dip his finger and taste it, he strides on, his hands clasped philosopher-wise behind his back.
I call, "I've been thinking. You'd be wasted in the horse."
This halts him with the suddenness of a stallion whose reins have been jerked. He tosses his head and laughs loudly, scornfully, as if expecting as much, as if this is all too typical of two-faced Odysseus and his sly tactics, and then he walks on, and I'm left standing there with a pot of honey already crawling with flies despite the hot wind blowing in off the glittering sea. I wander back down the slope to the beach and sit in the shade of a stone. I'd talked Achilles into staying and fighting when he'd announced he was leaving, but Ajax has walled up his ears. To him I have no honour. I look into the honey's dim gold depths as if expecting to read some omen—hoping for once to see the splendid Athena—but find nothing but bits of twig and leaf and the wing of a dragonfly.
In the days that follow, Ajax begins to drink, not just wine but wormwood, and his mind veers so that he rages at everything and everyone, stabbing the air at ghosts and phantoms, no longer bothering to take himself off in decorous exile, but going mad in plain view. Clothes piss-rank and filthy, he reels in front of my tent calling me out, telling me to put on Achilles's armour and give us all a show. "A dwarf in giant's bronze!" I try talking to him but he can only repeat that I cheated in the games for the armour, that I have the honour of a dog, and that in time men will sing songs of scorn about me. "You have more faces than a dog has hairs. Keep your prize, Odysseus. But remember—it will always be too big for you."
Agamemnon tries talking to Ajax, putting his arm around his shoulders and painting pictures of glory, speaking of prizes and home and, above all, of reputation, of the respect of his father Telamon who had been one of the Argonauts and fought alongside Herakles and Jason, but Ajax's ears remain shut. The time for prizes and listening is over. As for lesser men, Ajax cuffs them aside when they try reasoning with him; he even tells Calchas to go shove his head up a bull's ass and read its guts.
Once in Sparta I watched the young Ajax lying in a field studying bees. He enjoyed the sensation of them walking on his fingers, and they never stung him, something that pleased him immensely, something that made him proud, as if he'd won their confidence, as if he was as sweet as the flowers themselves. I can easily imagine Telemachus doing the same.
**Chapter Six**
**T** HE HORSE IS COMPLETE except for the eyes. Eyes will give the illusion of life even to a wooden animal, as if some spark of Soul has been lured to fly down and inhabit it. Lengthy discussions take place on how best to make them. Every combination of pearl and ruby, amethyst and shell, silver and gold and bronze are suggested. Epeius considers candles behind glass but is stymied by the problem of smoke. I suggest fireflies, we catch some but they refuse to glow in captivity and the philosophers among us speculate on the relationship between liberty and light. What about obsidian and silver? White marble overlaid with sapphires? The discussion continues, everyone taking part except for Ajax.
Ajax no longer talks to anyone. Like the wasps of the waning summer, he has grown sluggish and stumbling and withdrawn. We are all anxious. We all worry. The very possibility that we could soon be going home causes fear, as if regardless of all the years and the frustration, the sweat and blood and loss, it's suddenly happening too soon, but Ajax is troubled beyond all this.
One morning I rise earlier than usual and swim in the sea, floating on my back and watching the stars dim and the sky brighten. Then I walk through the camp. The dogs paw the cold ashes for bones and the ravens caw in the pines. Dogs are useful creatures, like hens and goats, yet I've always preferred cats. Dogs slobber and stink while cats have the dignity to groom themselves. They arrange their kills in a row, heads pointing in the same direction, an admirable display of symmetry that also appeals to me, though I can see the tedium of too much order. A little chaos, a little disarray, has the charm of children. And yet such thoughts make me nostalgic, and with nostalgia comes regret, an enervating indulgence to which I'm growing all too prone, another reason I admire cats, they seem impervious to self-doubt.
The farther I walk inland the heavier the dew and the mist. These are the grasslands where our horses pasture and I expect to hear the meditative sounds of grazing. Instead I hear the shriek of panicked animals, the wet chop of a blade striking meat. There. Ajax. In the dragging mist he lunges with his sword, battling not men but horses, the very last of our mounts. In his spiralling dementia the animals are Trojans. "Ajax!" He staggers on gore-slick grass and looks around, groggy, confused, as if waking, and sees his work: a slaughterhouse of heaving animals writhing and coughing blood. More painfully, he sees what he's been reduced to. He, second only to Achilles, he, the one with the greatest faith in our cause. I shout again, but he turns his blade upon himself and, arms fully extended, blade pointing inward, grips the sword with both hands. He fixes his gaze on me—a gaze that lasts half a heartbeat but bears the weight of a lifetime with all its hope and lament and accusation—and pulls death into his heart. Dropping to his knees he topples forward driving the blade through his chest and out his back. He lies face down in the grass, the protruding blade gleaming wet in the newly risen sun whose heat breathes warmly, mockingly, over my face. Is this the laughter of the sun god? Is this a lesson to be learned? First Dercynus, now Ajax. The cawing ravens flee from the trees. I run slipping in the blood, and when I reach him the flies are already at work. Ajax lies in a small cluster of blue flowers. I roll him onto his back so that he might feel the sun on his face one last time, but his Soul has already departed. My head falls back and I would cry out if I had the strength but there is nothing in me. When I open my eyes, I see that the branches above me drip with blood.
How strangely ancient Ajax looks in death, his stomach concave, skull gaunt, his fingers curled like claws, the nails so thoroughly chewed they're painful to look at. I recognize each of his scars. A man's body grows ever more distinct as it ages, bent and stunted by the life it has endured. With the help of Diomedes and Menelaus we lift the corpse onto a cypress litter while all of us, Athenians, Spartans, Cephallonians, Myceaneans, walk past with bowed heads offering our respects. Sinon too has the decency to show remorse, perhaps even genuinely feel it. Lacking women to keen, the men groan aloud like a chorus of bulls and strike their chests until the dogs begin to howl. Palamedes whispers to both Agamemnon and Menelaus and I know I'm being blamed. It's easy to kindle suspicion in an Achaean. It's a chronic flaw in our character. Like fire in a peat bog, suspicion smoulders on beneath the surface, and Palamedes fans the coals.
It's a bad omen for a man like Ajax to kill himself. It worries the men, they begin to murmur. _Where were the gods that love him? Why did one not whisper in his ear? What does this mean? We should have left long ago when we had the chance . . ._ I'm not immune to these very same questions, but anyone who knew Ajax will attest to the obsessive intensity that consumed him from within, a mortal striving vainly for immortality.
We bury Ajax on a hill looking westward over the sea. Masons polish the stones that line his tomb. His finest war axe and the sword he won from Hector accompany him. There are no more bulls to sacrifice so we lead his two best stallions to the mound, soothing their anxiety for they sense the knife even though I hide it behind my back as I whisper of barley and clover. I stand between them, my head alongside their necks, and for a time we remain that way until their breathing settles and their heartbeats slow and the flies return. Then I slit their throats. As their forelegs buckle I continue whispering of apple orchards and rich fields of sweet clover. We leave a tunnel to the open door of his crypt so that Ajax can watch the games we hold in his honour. Before we bolt the bronze hasps on the coffin and fill the tunnel, I retrieve Achilles's damned armour and place it in the tomb. Let Ajax wear it on his final journey. The gleam of the bronze will light his way. Hades will rise from his throne of skulls and welcome him.
Inescapably we discuss what we all know and fear—that the Trojans will take advantage of our loss and strike while we're grieving, exactly what we would do to them. We have to act. Calchas argues that time is needed for sacrifices, Agamemnon reminds him that there is nothing left to sacrifice.
"We move now."
Everyone looks to Epeius, who smiles and announces that the horse is ready.
It stands three poles from hoof to ear and every one of us thinks we should bring it home instead of Helen. Bronze hooves, red amethyst eyes ringed with green beryls, ivory teeth and a purple-fringed mane. The final touch is to inscribe on the horse's chest: _For their return home, the Achaeans dedicate this thank-offering to Athena_.
I climb the rope ladder hanging through the trap door in its belly. It's high enough inside to stand upright with my arms stretched overhead, smells richly of ship wood and ox-hide, and leather-hinged bunks drop from the walls. The others crowd the foot of the ladder and stare up, awaiting my response.
"Epeius," I say, "you have a god in you."
The others join me and we gaze around. It is dark even with the trap door open. Epeius points out the water jugs up front and the piss pots in the hindquarter. "It's going to be even darker at night," he warns. "You'll have to rely on your ears." He strokes the skin stretched drum-taut over the close-set wooden ribs. By way of demonstration, he shuts the trap door, dropping us into utter black but for a pale gleam from the horse's eyes. "Listen." No one moves or speaks but we can hear every word from outside, right down to the shuffling of feet in dirt. "Of course you too will have to be silent."
It soon grows stifling.
"And air?"
"Ah." I hear the grin in his voice. He demonstrates sliding panels high on the horse's back. "Impossible to spot from the ground." He encourages me to climb the ladder up the neck and inspect the eyes, each bigger than my spread hand and made with a lens of the thinnest oyster shell polished nearly translucent. "Remember," says Epeius, "if someone tears a rip in the side and holds a torch close anything metal will glint. Keep your weapons covered. No rings or buckles. And don't look at the light because the wet of your eyes will betray you."
Along with Diomedes, Menelaus, and myself, Stheneleus, Thoas, Neoptholemus, Demophon, Anticlus, and Machaon will be in the horse. It seems to me that Epeius should be there too, for he knows the horse better than anyone, but when I suggest it our engineer, usually so composed and eloquent, becomes strangely evasive. Everyone encourages him. It's his horse, after all, he designed it, and with much cajoling he finally gives in. We practise climbing in and out as quickly and quietly as possible. We spend a night in it, maintaining silence, and aside from Epeius falling ill with stomach trouble it goes well if somewhat rank. We agree that Epeius is right after all and is best utilised overseeing the evacuation with Agamemnon.
The irrepressible Echion immediately puts himself forward as a replacement. "Please, Odysseus." He has curly dark hair, wide-set eyes, and breathes through his mouth. No great thinker and a little impatient, but brave and eager. "Please."
I say all right.
That evening we burn the camp. In a frenzy of destruction we tear everything down, Agamemnon making a great show of setting his chariot ablaze. We heap tents, beds, trunks, boxes, chairs, old clothes onto the flames, we even set fire to the boats that can't be sailed home. With the camp burning and the fire reflecting in the waves, the Achaeans wade out to their ships and climb aboard and begin rowing slowly away in the dark. The ships will anchor beyond the island of Tenedos and wait for a signal, three burning arrows shot by a scout already positioned in the dunes. The honour of torching the horse when we are safely inside the walls will go to Diomedes, our connoisseur of fire.
Now all that remains of our ten-year occupation are the fire and the horse. Three men at each wheel, we roll the horse a safe distance from the flames, which fleck and glint in its jewelled trappings and bronze saddle trim. Already the Trojan sentries will be watching. Agamemnon and Nestor have lingered to bless and embrace each of us, a solemn occasion, a time to forget the small blisters. I wonder whether these two old men are relieved not to be in the horse, or fear they are missing out.
"Odysseus . . ." Agamemnon and I grip each other by the shoulders. The fire reflects in his narrowed black eyes. He's a veteran and knows better than anyone that this may be our last meeting. It's one of the few times I've known him lost for words.
"Tomorrow," I reassure him. "Troy."
"They're already watching," he warns.
"Good." If this is to be our last meeting, let him say that I went strong and confident.
Agamemnon faces his brother. I have never doubted that Agamemnon helped Menelaus win Helen in the first place; now it is up to the rest of us to get her back. They embrace in silence, there is nothing to be said. The one wants to be proud and the other approval. Finally Agamemnon turns away and walks down the beach and wades out to his ship. An old man in the night. They'll wait in a bay on Tenedos until they spot the three burning arrows. They can be back here in two hours, less if the men pull hard and the tide is with them.
One by one we climb into the belly of the horse. I draw up the rope ladder, lock down the door and drive the peg through the loop. The silence is as complete as the dark, and heavy with the brine of men.
**Chapter Seven**
**A** ND THEN it begins to rain. The initial scatter of drops patters searchingly over the horse. As the rain intensifies this dark close place grows smaller and a red-ant sweat seeps down my back. Our job now is patience, the toughest of all skills to master; the sword and the bow nothing compared to it. Careful not to shift or speak, we keep our breathing low and listen intently. Too dark to see anyone's face, I can smell fear—acrid and sour—which no amount of orange or eucalyptus can mask.
I crawl onto one of the bunks and stretch out on my back but my stomach feels exposed so I turn onto my side and draw my knees up. I can hear Diomedes occupying himself by kneading his hands, the right working the left and then the left working the right, then shifting about and applying himself to his feet. I've watched him many times and know his ritual, pressing his heels and then the insteps and the toes, probing as if in search of a gem hidden within his own flesh.
Now the rain rackets down over us like a load of pebbles. Absurdly, I look up as if to watch it fall.
For all our efforts at silence our breath shouts anxiety. Only Diomedes, in the bunk directly above mine, is calm and continues to massage himself. An Egyptian told him that if you grow relaxed enough your soul will drift off like wind through a net and that you could ride it wherever you wished. The man claimed to have flown east as far as India and south to the mountain that is the source of the Nile. He taught Diomedes that by kneading the flesh between your thumb and forefinger your headache vanishes, by probing the sole of your foot you can induce sleep, by massaging the earlobes a man's third leg stands tall. The body's tendons function like the block and tackle of a ship: adjust one line and the sail moves and the ship changes course. Perhaps it's the same with the guts, though not even the wisest Egyptian can explain the functions of the worms, tubes, and molluscs comprising a man's viscera. Certainly Machaon and his learned colleagues can't tell me how that ball of meat known as the eye actually sees. And what is light, the absence of dark or is it the other way around? Or does the one vanquish the other like two wrestlers engaged in an eternal bout, light victorious at dawn and dark victorious at dusk?
"Purpose?" Penelope once said when I was questioning everything, from our lives here on the earth to the gods on Olympus. She placed her palm upon my brow as if I was feverish. "We're all here in the middle," she said, forlorn at having to be the bearer of such news, "stuck between the millstones, grinding the seeds of time."
Penelope first saw me in the stadium. It was Helen's wish that women of all ranks be permitted as spectators, and as a result each breath of wind bore the scent of rose and lilac and orange, sweetening the air and distracting us. Helen occupied the premier seat, shielded from the sun by a fringed canopy while women fanned her with ostrich plumes. Never for a moment was she absent from our thoughts. Naked and anxious we moved through our warm-up routines, running, stretching, swinging our arms, hopelessly trying to clear our perfume-befuddled minds and focus on the task at hand.
I was about to run the first heat of the sprint. Palamedes had the lane to my right and he made a great performance of waving to the crowd as though he had already won. What a narrow skull he had; his mother must have had the hips of a boy. His beard was meticulously trimmed. We'd all cut our beards short so as not to give an opponent something to grab when it came time to wrestle. The drum was struck once, calling us to the line; the crowd grew quiet, then came the count, the drum struck once, twice, and on the third we were off. Palamedes got a good start, jumping ahead by a stride. Short, a fast start is my one advantage, and without it the long-limbed often overtake me at the finish. But I bore down, I remained relaxed and breathed evenly, and by mid-race we were shoulder-to-shoulder, and I ended up winning by five full strides, as if a breeze had aided no one else but me. Some murmured that I had a god at my back, and the odds-makers ranked me the favourite in the final. Those men who'd ignored me when I'd arrived—an obscure prince from an obscure island, a small man from a small place—now regarded me with fear and anger, and none more than Palamedes, his sneer demanding to know just who this bowlegged runt was.
At noon the day was done. As I departed the stadium Helen nodded her congratulations. Tall and so seemingly self-confident, she sat with her head tipped back, gazing down the length of her nose as if sighting along the shaft of an arrow. Menelaus trotted over and congratulated me, then added, "But I'll beat you in the final, Odysseus." His red hair was curled and his smile wet. He'd won his heat though not as impressively as I had mine. His broad hips and narrow shoulders didn't exactly inspire awe, yet he was a surprisingly strong and coordinated athlete, though above all he was dogged and confident. Besides, big brother Agamemnon was watching, and it didn't take an oracle to predict that Menelaus would die before being defeated. He clapped me on the shoulder and rejoined Palamedes who was glaring with those feral eyes.
The next day two men fell in the first heat of the long run, both from broken ribs. No surprise that these two ran in the lanes flanking Palamedes, who nonetheless placed third behind Diomedes and Menelaus. I barely noticed. I was busy imagining Helen naked, in every possible attitude, oiling herself, stretching, bathing, unpinning her hair, placing her warm hand on my cool chest as we reclined on her couch. I risked a few glances her way but she wasn't watching the games at all; maybe she was bored already, or more interested in sharing scandal with her maids. I ambled about the infield watching the heats, keeping limber, encouraging Ajax, but mostly keeping an eye on Helen.
She'd wreathed her hair in yellow petals, which made her look like a sunflower, taller and brighter than any of the women around her. Occasionally she rediscovered the games, and sat forward studying the athletes as if she actually cared about the technique of a Macedonian discus thrower, _two full turns before release or two-and-a-half?_ Then the mood would pass and she'd wilt, slumping dull-eyed and staring at her fingers. Perhaps she'd already chosen her husband on the basis of our gifts. Menelaus had given her a cape the colour of sandstone, fashioned from the mane of an Ethiopian lion, fringed with gold thread and rubies, and held at the neck by a solid gold clasp in the form of a lightning bolt. Palamedes gave her three black pearls the size of olives, Ajax gave her an elephant's tusk as tall as himself. A clown even back then, Diomedes gave her a pomegranate with a smile drawn on it in charcoal.
Throughout all of this Helen's mother, the Spartan queen, Leda, sat hollow-eyed as if her spirit were a thousand miles away, adrift. It was impossible to mistake her for she wore swan feathers on her shoulders, in her hair, down her arms, over her breasts, around her hips. No one dared raise an eyebrow much less laugh at this strange woman. Was she graced, cursed, or simply mad? Swans were everywhere, even in the stadium. All Sparta was an aviary devoted to swans nesting where they pleased, their feathers drifting in the wind, their droppings in the streets everywhere you walked.
All of this began to preoccupy me as much as images of Helen naked. As a result, I stumbled as I was warming up for the run and sprained my ankle. Moments later my heat was called, I limped to the line and finished last. No one noticed, because to everyone's delight Ajax—the boy among the men—won and advanced to the final. Already the darling of the crowd, he now owned the heart of every young girl, and likely many an old man.
Naturally he was ecstatic. He couldn't contain his happy chatter as he packed my ankle in herbs and I drank from the jug of poppy water on hand for injuries. I saw little profit in damping his joy. We'd become friends, finding an affinity in his being too young and I being too short. I worked with him on his archery and his javelin throw, while he showed me the insects he'd collected on the trip down from Salamis: moths, dragonflies, and all manner of wasp and bee. He talked mostly about his father, Telamon, who would admire and love him if he brought home Helen. "In the first battle of Troy he breached the wall ahead of everyone, even Herakles," said Ajax, as if reciting from the official history. "For this he won Hesione, Laomedon's daughter. The only way I can match that is by winning Helen."
The drug rising like a warm bath around me, I suggested that he had his work cut out for him.
He looked at me with the self-righteous solemnity of youth and ignorance. "Our children would walk unscathed through fire," he said, as if quoting.
I smiled and gave him the thumbs up.
My own father, Laertes, would say walk around the fire, or dump a bucket of water on it. He'd known Telamon, recalling him as a great warrior but also a bit of a bore, lacking wit or humour or even irony, defects far outweighing the martial virtues. My father's contribution to that battle had been to rout the Trojan cavalry by pouring oil over pigs, setting them ablaze then driving them at the horses, a spectacle I'd like to have seen. He'd strongly advised me to avoid the entire Helen business. "You're too young to listen to me," he'd said, "but all this nobility nonsense, all this glory and war, it's bollocks. Stay home, drink wine, swim in the sea. There are plenty of girls right here. Believe me, Spartans are insane. Noble, maybe. I'll give them that. But insane. Don't get mixed up with them." He shook his head and laughed at a people who, he was convinced, were the butt of some god's joke.
I lay on my pallet for the remainder of the day watching the shadows dial themselves across the tent and then dive into dusk. The tent smelled of camphor and incense against the stench of hot bodies. Only at evening, when the wind kicked up, did the air clear, but by then the mosquitoes were already out. I drew my blanket up over my head.
The next day the bad ankle messed up my footwork and I managed only one decent toss of the discus. Still, it was enough to get me to the final. When I limped out of the stadium I didn't go back to the tents with the others, but hobbled past sleeping dogs and Spartan boys wrestling in the dirt, and headed toward the edge of the city. If my ankle hadn't been so sore I might've just kept on going. The hot wind gusted ash and smoke and swan feathers. This failure business was new to me and my pride hurt as badly as my ankle. The Spartans welcomed pain as an opportunity to prove themselves, a sore tooth to bite down on. A cult of sinew that disdained the word-splitters of Athens. I just wanted the pain to go away.
Orange trees grew near the river, and the bank itself was thick with bulrushes and the inescapable swans. Some hissed and moved away, others paddled out onto the flowing water. Between the orange trees and the bulrushes was an expanse of sun-hardened mud, I limped across it and numbed my foot in the river. Rivers flow with such purpose, as if they have business that can't wait. Eventually some shouting boys arrived and began kicking an inflated pig's bladder. I hobbled back to the dormitory, drank two full beakers of poppy water and waited for oblivion. When it came I dreamed of red cloth drifting in a river, a field of ripe barley undulating in the wind, of wax slipping sweat-like down a candle, and I saw a woman walking along the shore at dawn while the incoming tide filled her footprints. I woke before sunrise and stood outside where, for the first time I could recall, I dreaded the approach of day.
Maybe it was a premonition. I was pathetic in the long jump and the day after worse in the javelin. It was a new experience for me, the rawest and most undeniable rebuke, especially when witnessed by the other athletes, the citizens in the stadium, the women, the children, the slaves, and of course Helen. Though I regained some respect by winning my round in archery, I continued retreating to the river each afternoon, disturbing the swans, and, when no one was about, wading into the water and studying my reflection. So, my eyes said, the world is more complex than you imagined, and other men hunt their desires just as hungrily as you.
The boys arrived and divided into teams to play football, but this time there was a girl with them. She played well and she played hard, as absorbed as a cat on the hunt. The next day I made a point of looking for her in the stadium and there she was, a few seats over from Helen, arms crossed under her small breasts, looking bored. I asked about her. Penelope, Helen's young cousin. It was the day of the qualifying bouts for boxing and I'd have done better to keep my attention on my opponent, a Cretan built like a hog and just as hairy, for he beat me black and blue, and it was miraculous that I came away with my teeth. While he advanced to the finals, my consolation prize was a poultice of salt and mud. I would've hidden myself if I could've found a place, but I was a stranger, I had no house to shut myself up inside, only the tent shared by forty others, so it was back to the river and the solace of the cool water and sun-heated rocks. I hoped to see Penelope though was ashamed at how I'd performed and even more at how I looked. One glimpse of my reflection in the river made me want to put a sack over my head: blackened eyes, swollen lips and ear, blood-clotted nostrils. When she appeared with the boys she spotted me right away, the stranger, the bad athlete, the lousy boxer. She gave me a rueful smile that said she knew what I was feeling. This was reassuring, this was a relief, and I began to realize what would have been obvious to an older and wiser man—my father, for instance—which was that however these games turned out my existence, such as it was, would limp on.
Penelope was so clean and lithe I would've licked the sweat off her skin. At one point she grew bored with the game and simply waded into the water and let the current carry her off, laughing at the boys who called her back to the match. She swam well and she swam strong. She turned on her back and stroked elegantly, first one arm raised as if to admire her fingertips, and then the other, content to be carried along around a bend past a floating island of swans. I too wanted her to come back. I imagined the river escorting her to the sea and out over the waves, a strange and beautiful creature that sailors might glimpse and call a mermaid and sing stories about. She would reach some island, Crete perhaps, where she would see cities and temples, and be welcomed by a young king, and this man would find himself thinking of her day and night, and so would keep her for himself and give her gifts and sweet words until she fell in love with him. I found myself gazing downriver anxious for her return, fearful that something had happened, that she was gone from my life already. And then there she was, casually strolling up the beach, letting the sun dry her, not looking at me, but aware of me watching and, I was hoping, not at all displeased by the attention.
I returned to the dormitory with my face aching not from the beating but from smiling. Eager to heap the salt of scorn onto me, the others frowned in curiosity for they saw that I was indifferent.
"Odysseus has had his brains loosened!"
"He's got birds and stars in his skull!"
One made as if to rap my head. "Odysseus! Hello! Anyone in there?"
I would've winked if my eyes weren't swollen nearly shut, so contented myself with a low bow and a jaunty twirl of the hand. Not one among them suspected that I was in love.
The state of my face frightened young Ajax. He approached me warily, as if to stand too close might injure me further.
"Odysseus . . ."
I grinned and clapped him on the shoulder and reassured him that I was fine.
"You were beaten."
I leaned toward him. "Ajax. Believe me. I have won. And let me tell you something else: you don't want Helen."
He reared back as if I'd spat on him.
I tried to explain, for I wanted him to comprehend the folly of pursuing such a creature. "Theseus couldn't cope with her. How will you? Think of it," I urged him. "You're too young. Free yourself. Give up. Quit."
He stared, bewildered. No concept was lower, no act more demeaning. "Quit? Why?"
"Because even if you win you'll lose."
Now he thought that I'd definitely had my brain rattled.
"Leave the boy alone, Odysseus," called Palamedes. "He doesn't need you pouring shit in his ears just because you took a kicking."
Ajax looked from Palamedes to me and frowned and nodded slowly as if realizing this was true, that Odysseus was sly and deceitful and a backstabber who would poison everyone else's victory to sweeten his own loss.
The others echoed the same sentiment, convinced that I only wanted to drag Ajax down.
Of course Palamedes had been busy telling Ajax that he'd never win his father's love if he listened to a malcontent such as Odysseus. I raised my hands palm upward and shrugged. "You're right. My apologies to you all. Odysseus has had his brains knocked. He's confused. He thinks up is down!" I began to canter about the tent. "Oh. Look!" I pointed to the sky. "A dead bird. And there—" Putting my hand to my ear I made as if to listen. "What a lovely song they have. Tweet, tweet."
This earned me a round of laughter, but poor Ajax was lost; why was I suddenly trying to thwart him? Miserable and confused, he joined Palamedes and Menelaus and a few others who were shaking their heads at my idiocy. Only Diomedes stood apart, frowning, bemused, knowing there was more to it.
By this point all talk was of young Ajax. The others feared to compete against him because of the humiliation of losing to one so young. He had moved on in the boxing as well as in the javelin and the long run, and here he was with his first moustache still soft on his lip, his shoulders skinny, his biceps smooth, and knees as knobby as a colt's. Yet how his manner changed with each successive victory, it was as though he grew years in mere days, holding his head higher, standing taller, but most of all his personality was changing, his innocence dissipating like a morning mist leaving only the dry glare of day.
**Chapter Eight**
**A** FEW MORNINGS LATER I was seated at the bench spooning up my gruel when a Spartan page leaned over my right shoulder and whispered that King Tyndareus wanted to talk to me. Palamedes, slit-eyed with suspicion, watched me depart. I followed the page out of the compound and through the city. For all their playing at the rustic life, the Spartan elite maintain fine houses. The approach led through an avenue lined on either side with potted orange trees, stone gods, and imperious swans. We entered a grassy courtyard equipped with dumbbells and archery targets.
Tyndareus was practising with the trident, overarm thrust, kneeling thrust, two-handed jab. He was sinewy, with the rugged complexion of a good Spartan who did his callisthenics each day and made it a principle to wear the coarsest cotton lest too much comfort make a Persian of him. His unplucked eyebrows met in a dark hedge above his nose giving him a crude demeanour. In fact he was refined, his hobbies including breeding cats and cultivating flowers.
"Odysseus!" He seemed more than happy to drop the trident. He sat down on a bench by a jug of wine and invited me to join him. He smelled of mint oil and his left forearm bore a row of three evenly-spaced puncture scars. An accident with a gardening tool or evidence that he'd put in his time on the front line? The talk began innocently enough. We spoke of Ithaka and my journey here, he remarking that in his youth he'd lived in Aetolia not so very far from my island. When the conversation lapsed he sipped his wine and then remarked, "You've stopped competing." When I argued he corrected me and said that I was merely going through the motions. Noting the state of my face and the beating I'd taken in the ring, he added, "And not very ably by the looks of you."
He was reputed to have been a fine boxer in his own day, and his blunted knuckles proved it. What could I say, I'd been thumped. I certainly wasn't going to make excuses about being diverted by lewd fantasies of Helen, so I shrugged and held his gaze so as not to appear evasive. The only thing Spartans despise more than defeat is evasiveness. Front up like a man. What did he want? To encourage me? Why should this Spartan king care about an Ithakan? What did I have to offer? He shrugged in imitation of me.
"The competition's fierce," I said, perhaps lamely.
He nodded that it was indeed. Then, as if dismissing the whole business, he drew my attention to the flowers growing in tubs of unglazed clay. The petals were a deep and fragrant red and the stems barbed with thorns. "Persian," he said. " _Dulband_ , they call them. Because the buds resemble their—" he gestured, meaning their headcloths. "The optimism of flowers," he said, as if it was a topic on which he'd been long brooding. "From out of the earth to a brief brilliance and then—" He pitched his wine dregs to the ground. "—Back to the earth." He shook his head slowly, a little sadly, as if it was all absurd and futile, like the life of a man. "Have you met my wife?"
"No."
"But you've seen the swans."
I admitted that yes, I'd seen the swans along the riverbanks and the fields and in the city and even in the stadium.
"After all these years she still sits up at night on the full moon and waits for him. Rubs herself in oils, arranges her hair, puts on her very best clothes, burns incense. She's old now and he's not interested. He likes them young and athletic—able to put up a fight." He smiled ruefully, for what else could he do? A god had raped his wife. _The god_ had raped his wife. "Apparently there was quite a tussle. He lost some feathers. She collected them. Some as long as my arm. As soft as if freshly plucked. She sewed them together with gold thread and wears them. I'm sure you've heard of her swan dance."
I confessed that the story had travelled far, Leda in wings on the full moon . . .
"And I'm sure you've seen her in the stadium," he added.
I nodded.
"For years I wondered, why Leda? What had she done? What had I done? Was it an honour or a punishment? As you so astutely observed, all we have that the gods do not is our mortality. And that's why they're so besotted by the desire to fornicate with us. Leda was a fine-looking woman in her day. But the gods are shape-shifters. No goddess need be anything less than perfection. How could a mortal woman, with her moles and moustache, her rocky teeth and pouty complexion, possibly lure one such as Zeus? One reason: because he'd taste mortality. The urgency that comes from coupling with one destined to die."
What was I supposed to say to this?
He saw that he'd made me feel awkward. Wanting to get back to the more manly world of strategy, he jutted his chin in the direction of the stadium. "How many men are competing?"
I didn't know exactly. "Perhaps forty."
"Princes and kings and first sons."
"She'll make a good marriage," I said, reassuring him.
He looked at me from the side of his eyes, as if to say I was insulting him with such bullshit. "She'll be a curse to any man. It's why Theseus gave her up. The old bull-leaper couldn't wait to get rid of her. Practically shoved her out the door when her brothers came for her. It's why you've stopped competing." He read my eyes. "Yes," he drawled, nodding shrewdly and smiling. "Whoever she marries can say goodbye forever to peace of mind. You know that. In your heart, if not in your head, and I think you're one of the new breed who know the difference. Yes . . ." He seemed to cogitate a moment upon this new breed of which I was, apparently, an example. "But typical of the young you go too far, you invent an idea and think that you can then—" He gestured like a conjurer. ". . . Inhabit your fantasy as if it was as solid as a house. That's your confusion. You have a thousand hearts and a thousand faces." He held up his forefinger. "But you only _need_ one. You should only _have_ one. All this public and private . . ." He frowned as if at unseemly habits. "A face that fits. This is the key to Sparta's greatness."
Elders. If they're so wise and understanding why do they insist on boiling the flavour out of every experience for those of us who've yet to taste them? I was twenty; he was forty. He must have seen my expression hardening because he changed tack again and asked me about the mood in the dormitory.
Ah, so I was to be an informer. I told him the obvious. "We hate each other. Every night we bicker."
"Of course." He didn't need me to tell him that. A hint of impatience weighed his voice, and I worried he was regretting having called me to him.
Offering him more, I said, "The losers will go away bitter. That bitterness will break us apart." I pointed to an old clay pot seamed with cracks. "Greece will crack. The Trojans, the Persians, anyone with a will to fight will find us fragmented and easy to shatter."
For a moment Tyndareus didn't react to this theory. Fear of scorn rushed through me like a wave of dizziness and I placed my palms on the bench to steady myself. Finally he nodded. "What do you suggest? Cancel the games?"
I spoke slowly, carefully, not sure where I was going, letting the words lead me one by one as if finding a path, step by uncertain step, through a dark wood. "A pledge," I said, eager to impress him. "Before the games end, have each competitor swear an oath to honour the man Helen selects. An oath binding him to support the victor in any crisis. For the good of us all."
Tyndareus was nodding his head in agreement even before I'd finished. He seemed renewed, unburdened. "The Oath to Tyndareus," he said, giving it a title. "Good. Very good. You're a clever young man, Odysseus. Name your reward."
I looked at the _dulbands_ , so poised and fragile, yet indomitable, each year reborn again. What was I going to say? Would my head speak or my heart, and if so, which of my heads, which of my hearts? Wasn't Helen the obvious choice? "Introduce me to Penelope."
Tyndareus raised his chin and narrowed his gaze as if to better evaluate me. Then he smiled, though I wonder now if it was in admiration or scorn.
When I returned to the tent the others stared at me like a herd of deer facing a wolf. Only the affable Diomedes seemed at ease, whistling as he trimmed his beard before a polished brass mirror. He ceased his whistling to rinse his razor and inquire whether Tyndareus and I had discussed flowers or daughters.
"Both."
"And did you talk about how to make them bloom?"
"Fertilizer. Lots of fertilizer. Nightly doses."
Diomedes barked a laugh. He shook the water from his razor, leaned toward the mirror and resumed his grooming while the others stared from behind wooden faces.
I'd wanted Tyndareus to respect me, so when he'd asked my advice I'd answered truthfully. It was shrewd advice, perhaps even wise, and benefited all Hellas. My reward was Penelope, a prize more than fair. And yet it's that very same advice that took me away from both her and my son and robbed me of ten years. Is this the work of cruel gods, blind fate, or simply chaos?
I recall a summer hailstorm. I was a boy and we were harvesting our oranges. Sudden storms weren't unknown, but this one seemed as if it had been aimed, like a fistful of stones, at our orange trees and nowhere else, for only half a mile away the ground was dry and the sky clear. Our entire crop was destroyed. My father saw it as punishment, but for what? Oracles were consulted and ludicrous responses interpreted in the convulsions of the temple virgins: his offerings had been scanty, his offerings insincere, my father had hunted in groves sacred to the gods. What I saw, even at the age of twelve, was blindly erratic nature, chaotic and uncontrollable, and men desperate for certainty.
There was a man, Petras, a citizen with land and animals and a fine reputation. He went missing on a routine journey to the mainland that should have seen him safely home in thirty days. Five years he was gone, no one knew if he had drowned or been killed or what. His wife remarried to a man called Leontes, old, litigious, notorious for his dishonesty, but he was still vital and they had a beautiful daughter together named Cleo. Petras returned. He had been shipwrecked. When he discovered that his wife had remarried and worse, had another child, he killed Leontes while his wife killed herself, and Petras, that good man, that fine citizen, that victim of Fate, had to forfeit his wealth and go into exile. His entire family, sons and brothers, were forever tainted. Cleo, the daughter born of that second union, became a whore. What wrong step had Petras taken? What god had he failed to honour?
My passion for the games now redirected to Penelope, I happily cheered for Ajax. (I'd given up trying to convince him of the folly of competing, though it was less out of love for him than sly joy at the thought of the others losing to an adolescent.) He won the javelin but managed only a draw with Menelaus in boxing. Both had to be carried from the ring. The very next day Menelaus took the discus, thanks to Diomedes falling into a laughing fit as Leda entered in her swan regalia causing him to foul on his best throw. Menelaus knew he'd got lucky there. A good athlete, Menelaus, even if he did look like a bean counter. The hard lessons of a Spartan upbringing, but there it is, men come in as many shapes and sizes as dogs. A hare lipped Thracian defeated Ajax in the wrestling. Diomedes won the sprint a half-step ahead of me, he also won the long run, and, typical, finished both races looking as composed as when he started. Truly the most enigmatic man I've ever met. The only event I won was archery, and I did it by aiming for Penelope's heart. I hit it three times.
The games ended and we stood in a row awaiting Helen's decision. I had a last minute panic—or was it desire —that she'd pick me and thwart my courting of Penelope. When she chose Menelaus I wasn't surprised, as throughout the games Agamemnon had sat next to Tyndareus. And yet at the same time I was disappointed that Helen was so easily controlled, that she hadn't asserted herself and, willful and unpredictable as the god who'd supposedly sired her, selected Ajax or Diomedes or someone else.
Diomedes laughed at her decision. Palamedes hid whatever he felt and applauded Menelaus. Poor Ajax was crushed in the way only an adolescent can be, for he'd not yet learned how to shield his heart. He hid his face in shame, then that shame turned into a rage that he directed at me, for he was convinced that I had purposely thwarted him. He accosted me after the closing ceremonies, fists clenched at his sides, tearful, trembling, betrayed.
"You sowed doubt, Odysseus. You poisoned my confidence." His eyes pleaded with me to admit what I'd done, to give him an excuse that he could carry home as evidence to his disappointed father.
"Why would I do that to you?"
But he knew what he knew. "Now I go home in disgrace," he said. "Exactly as my father predicted." What was I to say to that? I felt bad for him, so didn't laugh or walk away, but stayed there and took it on the chin. It wasn't enough. He vowed that this would stand between us forever. "Some day it must be settled, Odysseus."
I gave up and walked away saddened, but the fact was that I didn't care what he thought, not really. I'd tolerated his spittle-flecked tirade because I had something— _someone_ —more important on my mind than an indignant boy.
Penelope's first words to me were: "You're very bowlegged."
"They say it's a sign of nobility."
"They say you didn't try very hard in the Games."
I said I'd been diverted. I couldn't concentrate.
She became innocent. "Yes, all the swan feathers floating around in the air. You must have thought it was snowing."
"It wasn't the swans that diverted me." And I told her how I'd gone about winning the archery contest.
She put her fingers to her chest and checked for blood. "Good shot."
**Chapter Nine**
**T** HE RAIN CONTINUES. How I'd like to be outside, feeling the cool drops upon my face, far, far from here. My kneecap is aching again. I massage it, thinking of that Trojan with my javelin through his throat and the watery expression in his eyes, as if he was staring from the depths of a lake. I try not to trouble myself over killing. Death comes sooner or later, and if there are gods or reasons behind why and when and how, they're too subtle or shifting for me. Still, I can't help thinking how that Trojan whose blood seeped over my spear knew moments of wonder, such as the first time he felt the weight of lead in his hand or felt the confusion caused by the presence of a certain girl. Perhaps he preferred pears to apples, wept alone at night with his head under his blanket. It doesn't matter, he's in Hades now, a Shade like Ajax and Achilles and so many others, or maybe there is no Hades, no Shades, and he's just so much dust. Is it possible? Is it all merely men and their imaginings? I don't know which prospect is more frightening, gods or nothing.
Nine years passed before I met Ajax again, here, at Troy. We stood face to face, and, with Agamemnon watching, we locked arms and vowed brotherhood, all very noble, but I could see in his eyes—as he intended I should —that he still believed I'd worked against him in the games, that I'd urged Tyndareus to tell Helen to choose someone other than him. This didn't exactly bode well for the campaign. He'd become a giant, standing a full head higher than me, muscled and angry, his silver wristbands so broad they'd have slid right off my arms. But there was the task at hand of dealing with the Trojans, and for that diversion I was almost thankful. I didn't envy anyone who had to face Ajax on the field. He'd windmill his axe knocking down horses and men alike, not looking to see who was in the way, simply going whirlwind, it was your own business to avoid him. But he didn't leave opponents to suffer, not like Palamedes, who loved those moments when a man's life was bleeding out of him even while his eyes were still wide and he saw what was to come. Palamedes would cut a man's ears off and throw them to the ravens then sit the poor bastard up, hold him in his arms and direct his gaze so that he could see the birds fight over them, the very same ears with which he'd heard his mother's voice.
The snort and blow of horses stirs us from our broodings. Voices erupt as if the entire population of Troy suddenly surrounds us. Torchlight gleams through the horse's pearl-shell eyes and a spear raps the ribwork, all strangely intimate and upsetting. I find myself curling up tighter on my bunk, trying to be as small as possible. With the ebbing of the rain the mutter of voices becomes clearer.
"Burn it."
"Bring axes."
Priam, dry-voiced and hard, says wait. In my mind's eye I see the Trojans leaning their heads together in conference. No doubt the old king is thinking hard, and no doubt he's missing Hector's counsel more than ever, but it's a woman's voice we hear next, Cassandra, Priam's daughter.
"It's a trick. The Achaeans are up to something."
Priam answers with silence, a silence that smells tangibly of scorn. After all—so the story goes—Cassandra betrayed Apollo's gift of prophecy; how can she be trusted? Surely Apollo would have rescinded or corrupted this gift. Either way, she was condemned to be distrusted, her warning as futile as seed sewn on ash. She begins to argue, insisting that the only safe course of action is to burn it.
Priam calmly observes that the time has not yet come when he takes her advice in matters military, and then he orders her back into the city.
I've seen Priam often over the past decade and observed how he has aged. He's grown gaunt and bent, his shoulders brittle, his neck vulnerable, his jaw trembling, even his skull seems to have withered. As for his voice it is hard and dry and you want to give him a drink of water. He is too old for this. Instead of siege warfare he should have been enjoying his final days tending his vines, making faces at his grandchildren, and devoting his evenings to ambling along the riverside taking the cool air and reminiscing. But Hector's death broke him. His posture has become curved as a wet reed as he sits his horse. Priam, once so tall and square-shouldered with a spine like a spear. I'll always honour Achilles for having returned Hector's corpse to the old man so that he could bury him with proper ceremony. No father should outlive his son.
As Cassandra departs there is a fresh stirring among the Trojans, the slap of livery, new voices. Helen. Her delighted laughter, so perpetually amused by the workings of men. Even after all these years her laughter is unmistakable. Valuing her opinion more than that of Cassandra, Priam wastes no time in asking what she thinks.
"What I think, Lord? Perhaps the horse speaks? Let us ask it." And in a louder voice, "Tell us, horse. The Achaeans are gone and their camp burns. Are you a gift? Compensation? A trick? Speak to us."
We don't move, we scarcely breathe. I feel Menelaus's strung nerves. His thoughts ring in my own skull. He is closer to Helen than he has been in ten years, the woman he has dreamed of every night for a decade, here she is, he could speak to her without raising his voice. I reach out and find his arm and grip it firmly meaning be strong, Menelaus, be strong.
Diomedes alone knows I visited Helen only last year, the same night he and I rescued the Palladium. When we'd emerged from our hiding place he'd grinned and shaken his head when I told him my scheme, but he didn't try talking me out of it. I'd indulged him in his little side trip into the taverna, now it was my turn. We'd meet at dawn and depart with the farm labourers by the northeast gate.
To be the first Achaean to talk to Helen since she fled Sparta with Paris was a prize I'd long wanted. But there was more, I wanted to be alone with her, to look into her eyes, to see into her soul—or more accurately into my own —and satisfy certain questions I'd scarcely acknowledged even to myself. So I'd slipped away, leaving Diomedes to deal with the Palladium.
She was waiting as if it had all been pre-arranged. Her voice was as sweet as scent. "Come up, Odysseus, come up."
I climbed the wall and stepped between the potted lemon trees and saw the beaded curtain swaying over her doorway. I entered. The room smelled of orange flower and a single oil lamp burned against a brass backing giving me a glimpse of myself: bruised, ragged, filthy. My words would indeed have to be sweet.
She was in a chair in a corner. She indicated the companion seat, arching an eyebrow at my disguise.
"It's been nineteen years," she said. "Though of course I've seen you from the walls. How hot and dusty the whole business is."
The lamp burning between us blanched out her face as if I'd stared too long into the sun.
"Have I aged?" she asked. She sounded genuinely weary.
I leaned to see her better—and recoiled. My shocked reaction made her laugh. Penelope. I blinked and I squinted and looked again. Yes, Penelope, my wife. Gone were the sharp planes, the long jaw, the sharp nose, and instead I saw Penelope's heart-shaped face, the small jaw, the large eyes, the thick black hair, the full lips. But a gust rippled the lamp flame and stirred the beaded curtain. The spell departed, my head cleared. When I looked again she was Helen once more. She sat back, out of the light, and laughed deep in her throat, amused by her trick.
"Are you here to rescue me or to kidnap me?" she asked, her voice suddenly arid. "Or," she added, insinuatingly, "to seduce me?"
I tensed. I couldn't honestly say.
"Ah," she said knowingly, and chuckled. She stretched out her legs and let her hands dangle over the ends of the chair's arms. Those hands, each finger reposed like the tail of a cat. We were silent. There was only the clicking of the bead curtain in the uneasy breeze and the hiss of the wick in the oil. Now her fingers took on a new aspect, and I thought of weed in a stream. "How is Menelaus?" she asked suddenly, as if referring to a mildly entertaining acquaintance from long ago, someone we'd both found faintly absurd. Her right hand turned over with the question, palm upward, fingers spread. In that palm I saw a valley creased with roads, a land that had known battles, and I saw an oasis in the starlight, with a pool and fig trees and a tent and a bed and on the bed Helen, young and eager . . .
I put my hand to my brow and shook the images from my mind. Menelaus? I didn't want to talk about Menelaus. I put my hand on her knee; she covered it with her own, warmly, suggestively, as if she'd been awaiting me all this time.
"You know," she said, "I was very disappointed that you didn't bring me a gift all those years ago. Every other man did, but not Odysseus." She stroked the back of my hand with her forefinger. "Was the sly Odysseus the shy Odysseus? Or merely ill-mannered? And then you gave up so easily."
Here was the question that had been lodged all these years like a bit of grit under my eyelid: had I been wise in opting for Penelope, or had it been due to fear, because I knew I couldn't win, because I knew that I wouldn't be up to the mark?
Not knowing what to say, I asked her if she was content.
"Content?" She repeated the word as if bemused by the concept, as if it was a strange obsession of an absurd people. "I don't know." She looked around the room crowded with lamplight and shadows. "I've learned that there are more shades of grey than there are of black or white. I've learned that grey is an endless colour." She raised her hand and made a slicing motion. "You can divide it like an infinite apple." I watched her long hand with those precise fingers. "And you, Odysseus. Do you have regrets?" How genuinely empathetic she sounded, her with one foot on Olympus and one in Troy.
I didn't hesitate. "Yes."
This saddened her. Though to my disappointment and perhaps relief she didn't ask what exactly it was that I regretted. Perhaps it was only the strange fruit of a relentless imagination and didn't bear discussion. She inclined her head to one side and smiled ruefully. "I was hard on poor Penelope. I've been hard on a lot of people I love. Sometimes, at night," she said, "I miss all those swans."
Later, when I finally emerged from Helen's room, I found Diomedes waiting just outside. He'd followed me. He had a strange look in his eye and a brick in his hand, weighing it up and down as if gauging its potential for damage. "I thought it was Ares you wanted to kill," I said, taking him by the elbow and escorting him away. He leaned to sniff me and said I smelled of orange blossom.
**Chapter Ten**
**A** ND NOW there is a fresh stir. A new arrival. Sinon. We press our ears to the walls, but it doesn't take much to imagine Trojan sentries, one at each arm, escorting him through the crowd. Now Sinon's acting will truly be tested.
I know many of the voices from our futile parlays over the years. Sinon is deft in his responses, he puts on a convincing show of fear and sincerity as he explains how the oracles decreed that only blood sacrifice would buy us a fair wind home, and that he, Sinon, was chosen by vindictive Odysseus to bleed at the altar.
"Odysseus!" Priam mulls my name as if it can only mean trouble. "The worst and the best of the Achaeans."
In the bunk above me, Diomedes snickers.
Other voices join the questioning, foremost among them Laocoon, their seer, a blind and suspicious old man.
"How did you escape?"
"I ran."
"You ran."
"Yes."
"As simple as that? You ran."
"Yes, sir."
Someone remarks that the Achaean is wounded. When Laocoon hears this he's all the more convinced that it's a ruse. He becomes shrill. "Wounded? And a thousand Achaeans couldn't catch you?"
"I slipped away before they came for me. I knew I was the choice."
"You knew you were the choice. How?"
"The oracle. And because Odysseus hates me."
"He hates you. Why does he hate you? Stop lying. Explain yourself!"
"I refused to love him."
Priam is less interested in this than in the meaning of the horse.
"To assuage Athena," explains Sinon. "For the crime committed by Odysseus in stealing the Palladium."
A murmur runs through the crowd. Priam muses upon this. "He thought that by stealing the heart of Troy her blood would cease pumping and she would lie down and die."
"Everyone was against it," says Sinon. "But Odysseus, he convinced them. Menelaus and Agamemnon said stealing the Palladium would only make the Trojans fight harder."
"Odysseus, Odysseus!" sneers Laocoon. "I have a harder question for this spy. You, boy, why not simply return the Palladium, eh?"
"It's gone. To Mycenae."
"Why is the horse so large?" asks Priam.
It's a thought I hadn't anticipated, assuming that to the Trojans hugeness was an unquestioned virtue. Perhaps herein lies Priam's wisdom, for he isn't beyond voicing the simple question. Now is Sinon's moment, he has to come through. _The greatness of Troy merits a gift of equal greatness? The biggest city demands the biggest horse?_
"Oh noble Lord Priam," says Sinon. "The horse is large so that it won't fit inside the walls of Troy and serve as your guardian and strength. This way it placates Athena while thwarting the Trojans."
Perfect, and yet even as he delivers it I'm already worrying that Sinon will steal some of my thunder and be sung of as the man who, when the horse was at risk of being burned, when the Achaeans were on the verge of dying, when the entire ten-year campaign was about to be lost, rescued us with his wits and defeated the Trojans. The response is not merely clever, it also plants the idea of bringing the horse inside the gates in Priam's mind. How could Priam not believe Sinon? He declares that the young man is now a Trojan. A cheer goes up. The horse is a gift. The war is over.
Laocoon isn't so enthused, he insists that the Achaeans are plotting, his voice growing more shrill, more brittle, as his influence wanes.
Now Paris speaks up on his father's behalf. "They've lost Achilles and Ajax. They've burned their own camp. They're ships have sailed. They're gone. It's over."
This is too much. "Oh yes, that's good coming from you who brought all this on. She tricked you and now they're at it again."
"If you had eyes you might not think so," sneers Paris.
"At least let us slaughter a bull and read the signs," Laocoon pleads.
Priam agrees.
During all of this we hear a vast clamour of a gathering army. I have no doubt that thousands of Trojans, the entire city, is rushing to see the fantastic horse. Some stand directly under us, so close that we hear their whispering and feel their heat and urgency. They rap the horse's legs, touch and stroke them as if for a blessing. Soon we hear the bellowing of a bull. I climb the ladder up the horse's neck to its eyes.
Hundreds of torches sway like a grass fire in a wind, and there is the bull, a rope at each leg and each horn. Black-robed Laocoon is led to the bull and presented a sword. He grips it in both hands and raises it high as the bull continues to bellow and the torch light slides over its glistening hide. Laocoon swings. Brings down the blade. The bull's neck splits wet and red and spurting; the severed tendons let its great head slump. The butchers work fast, rolling the carcass onto its side and slitting the belly, releasing a slithering mass of steaming intestines. Accompanied by his sons, proteges in the arts of divining, Laocoon kneels in the blood. The onlookers press in as closely as they dare, torches an encircling wall of flame-topped palings. Sweat runs stinging into my eyes so that I have to squint, it is all a blur, a mass of swimming figures. Laocoon listens to his sons. They help him to his feet and turn him to face Priam, who sits his horse with tall dignity.
"The horse is full of snakes," declares Laocoon. "Burn it."
It is our good fortune that Priam does not always take the advice of his seers.
The horse begins to move, the oak wheels crying out despite having been pig-greased at the axle. Everyone, even the lame, rush to help, pushing or else taking a place at ropes that have been lashed to the legs. In this way we approach the city. We cling to the straps that Epeius is clever enough to have fastened to the the sides, for we are flung about as the horse dips and pitches over the terrain. I keep to the ladder and peer out the horse's eyes. The Trojans chant and dance—not the stately turns performed at the equinox—but frenzied leaps, the chant and stomp of a horse cult. Some even throw themselves under the wheels seeking some imagined glory in such a death, their exultant groans accompanying the terrifying crack of their ribs. A madness has come over them, they are possessed. We are Zeus Earth Shaker come to walk among mortals.
We advance across the marshalling ground toward the gates. The horse sways as wildly as the Trojans dance and I brace myself in the confines of the neck. It's as if I'm trapped inside the mast of a ship caught in a gale. We lurch and wobble and bounce. I've always been good at sea, but trapped in here even I begin to feel ill, for there is no horizon to look at and steady my stomach. I shut my eyes and hang on as we find each ditch and rock, every pothole and root. I'm sweating and trembling when we finally reach the stone ramp that leads to the doors where the wheels suddenly turn smoothly. We halt. Silence. There is a massive creaking as the gates of Troy crank open. Again we advance but after a few seconds we halt once more, there is discussion, the horse is too tall, the stone arch that spans the gates must be broken.
A hand grips my ankle.
"Odysseus?"
Menelaus squints up at me. In the clamour and din there's hardly need for us to keep silent any longer, we could beat drums and not be heard. I describe what I see, the people, the dogs, the waving torches, all blurred by the oyster shell. Then I see something directly in front of me, just a few feet away, on the arch above the gate: a crouching boy staring into the horse's eyes. I stare back. Then recall Epeius's warning about the reflection of our eyes and shut mine.
The next thing I hear is the bash of hammers on stone. They're breaking the arch. The stone crumbles then crashes to the flag-paved ramp. More cheers, followed by a thumping from above. I wince, then understand. Boys are leaping onto the horse to ride it into the city while women throw flowers. Again we begin to move. Once we are inside, the gates crank shut behind us. It takes an hour to haul the horse up the winding streets to the acropolis and onto the temple floor where the Palladium had stood. It is well after midnight, no victorious army enjoys a greater welcome, and for hours afterwards the people whirl and pray and drink and weep and stroke the horse. The drunker they get the louder they get. They slaughter sheep and make sacrifice. They dance and fight and sing and build fires. They cook food and they feast and I suspect that many a child will be conceived on this night of nights. I also begin to fear that they'll never go away, that the horse will never be left alone, that guardians will be posted, priests appointed, and we'll be trapped inside for days, forced in the end to make some doomed and desperate lunge. I climb down from the eyes and lie on my bunk, knees to my chest. We all try to be patient, to be calm, and not give in to twitches or whimpers. It is the longest night any of us has ever known and anxiety drives us to the chamber pots so that the stench soon grows unbearable. I mark the passage of time by counting the beats of my heart. Diomedes is the only one who retains a sense of humour, and begins whistling faintly some marketplace jig. I press my hands to my face to stifle a spasm of laughter that degenerates into a retch. And then we hear the crow of roosters. Dawn. Light filters in through the horse's eyes, the sun rises, it grows hot inside. There is a resurgence of activity around us and with it the prospect of what lies before us, an entire day in the horse, perhaps many days. This breeds panic, a panic that will lead to some desperate move. Echion is off his bunk and pacing. I catch him by the shoulders and force him to sit. A whispered argument erupts, someone—Thoas, Demophon—mutters that they've had it, they're going to open the door. There is a struggle, a brawling of mute bodies blindly wrestling, then silence but for panting. We regain self-control and without a word crawl back to our bunks. The air has become unbreathable, as humid as a steam box. Only very slowly do we sit up, realizing how quiet it is outside. I climb back up to the eyes. The Trojans are leaving, drifting away in twos and threes, or passed out, done in by drink and exhaustion. Many sing and embrace as they stagger off. What a day for Troy. The Achaeans have departed and a god has replaced them.
I draw the pin and lift the flap-hinge and pull up the door. Air. It smells of cool stone and crushed flowers. We kneel at the trap door like parched beasts at a pond. Directly below us lie drunk Trojans, men, women, clothes torn, mouths wide, most snoring, some bleeding.
Young Echion is so desperate to be first out that he begins climbing down before I've dropped the ladder. He loses his grip. I grab for his hand but he falls, fingers flailing upward, and lands on his back, the strike of skull on stone sickening to hear. Even as we watch, blood seeps out around his head. His wide eyes stare up at us. Paralysed by such an omen, we look on in horror, until I break the spell and climb down.
"Echion . . ." I kneel by him but he's dead. Letting him of all people in the horse was irresponsible, I should have known better. First Dercynus, then Ajax, now Echion; all three of them looked to me.
"Leave him," says Diomedes softly. He grips my elbow and urges me up. There is no time for memorials.
I gaze around narrow-eyed and frowning as if sizing up the situation, but in truth I am tightening my brow as much to hold back a cry of remorse as to gauge how to proceed. There are scores of bodies and the dogs prowl and sniff. Then a shout from a woman. She drops her water jug and runs shrieking. Diomedes strikes a spark to a torch and throws it spear-wise through the trap door, and within minutes the horse smokes and glows, the seams between the planks incandescent stripes. With the burning horse as a diversion, Diomedes and the others will run back down to open the gates while I guide Menelaus to Helen. Agamemnon, meanwhile, should already have our ships at the beach.
I lead the way out of the temple past villas of sandstone and cedar. Outside Helen's palace we hide between junipers and tubbed lemon trees. Now come the first shouts of fire. Off above the palace roof flames snap like shredded pennants, guards run. I climb the wall and Menelaus follows and bumps a potted fig which hits the marble in a burst of dirt. A guard sees us, Menelaus flings soil into the man's face, and while he wipes his eyes chops him across the neck leaving him sprawled in a slick of blood. The flames from the horse strain as if fighting to free themselves and take flight. All at once the entire city is shouting: voices, bells, dogs, birds. We collide with two children, a boy and a girl, rushing to see the fire. They stagger back then call out, the boy an imperious little bugger, demanding the guard, stamping his foot, not frightened but impatient, indignant, as if he is surrounded by incompetence. At that moment Helen splits the beaded curtain. She hesitates. She glances around uncertainly. Flee? Call a guard? To which side does she commit herself? Or is it merely the swaying of the beaded strings that makes her seem to vacillate? But her decision is made—the Achaeans have done it, the walls are broken, she throws herself into Menelaus's embrace.
"Mother." The boy bears a look of appalled disapproval. His resemblance to Paris is plain, the long jaw and narrow nose, the skull flat at the back, but he has Helen's brow and pale blue eyes, all planes and hard lines. Helen lets go of Menelaus and gathers both children into her arms. All three look fearfully at the strange man who has come to claim her. Now the question is what will Menelaus do? Of course he should have expected as much, ten years with Paris, children were inevitable. This made three, Hermione via Menelaus, and these two. And three husbands, Theseus, Menelaus, and Paris. A woman of some small experience.
She's grown heavier about the throat and there are creases branching from the corners of her eyes. The Trojan winters have not agreed with her, the great land to the east has taken its toll. She may be the offspring of a god but she is apparently not immune to time. Now comes the test of Menelaus's love; now comes his fiercest battle, one greater than any he'd faced on the field where victory and honour depended so directly, so simply, upon an arm and a sword. He embraces her, catches her up in his arms and spins her around and she laughs, head back, delighted, as if she too has pined for this moment, and maybe she has. Then he kneels and embraces the children. Confused but obedient, they submit in silence. I've never had great respect for Menelaus's brain, I've always regarded him as a pawn moved here and there by his big brother Agamemnon, but if he suspects Helen's loyalty he's also wise enough not to spoil their reunion, and shrewd enough not to ruin his introduction to these children to whom he will be a guardian if not a father.
During all of this I realize that I should have stabbed him, done him in while I had the chance, before he and Helen saw each other. My hand travels of its own will to my dagger. It's a good dagger, long, with a wedge-shaped blade that opens a wound that won't close. It would've been so easy to stab him, just draw the knife and plunge, it's all there in my mind's eye: dispatch Menelaus then find Helen, greet her warmly, take her by the hand, assure her that I'll guide her to Menelaus. We'd have walked together through the palace, reminiscing, maybe even sharing a laugh, then she'd halt at the sight of his corpse. I too would put on a show of shock. At that moment she'd understand everything, at that moment she'd understand she'd lost the one man who would keep her safe from the vengeance of all the Achaeans who, quite rightly, hated her for having robbed them of ten years of their lives, even better, her fall would be Penelope's revenge . . .
But it doesn't happen, it is too late, the story plays out differently.
Menelaus and Helen stand apart the better to admire each other. They grin irrepressibly and I watch Helen's eyes but can't say what she truly feels, what she hides and what she shows. As for Menelaus he looks twenty years younger, he is once again the young athlete who has just won the prize. He turns and grips my hand and my elbow in teary gratitude.
"Odysseus. Friend." His voice chokes and his eyes swim.
Friend? We've been many things to each other over the years, but friends? I return the pressure and smile warmly. "Menelaus," I say, as if his victory is my own, something he actually appears to believe. "At last."
"At last," he echoes.
I understand something that I should have grasped before: that Menelaus knew what he wanted and pursued it for ten years regardless of the reputation it earned him. Worse, I have blamed him when it is I who have been the coward. I gave up my family for his goal. If I was stronger I would have stayed home, said no, go to Troy without me, the world can say what it wants, but at the time the wise Odysseus believed that you can't sacrifice reputation and retain dignity unless you are a sage or a fool. And yet here stands Menelaus with his prize, happy.
Helen looks on radiant and imperial, the queen proud of her menfolk. Maybe it's true. More likely she's already planning ahead for the life that awaits, planning ahead and turning her heart. Even the children are convinced that something grand and god-sanctioned is taking place and so they too smile, though tentatively, uncertainly, and I anticipate trouble when they grasp what is actually in store for them, when they realize they'll never see their real father again, certainly not alive.
A new bout of shouting erupts nearby. Sandalled feet slap past on the portico. Helen takes the children by the wrists and leads us through another room and out the back to a barred gate. Menelaus slides the beam aside and edges the gate open. Clear. We step into a narrow footpath. Down to the left people pound past in the main avenue. "I know a route," Helen says. They start off but after a dozen steps halt and look back enquiringly at me.
"Go on," I say.
"Forget it," says Menelaus. "I'll give you gold. Anything. Name it."
How easy it is to be generous when you've got your prize in hand. Agamemnon also owes me a reward. What a rich man I'm going to be. My wife may have remarried, my son may have forgotten me, Ajax and Dercynus and Echion are dead, but apparently I've earned a reward, a Trojan chariot perhaps, a concubine who will always secretly hate me, another set of gleaming armour, a few trinkets from Priam's hoard, a sack of gold. Oh yes, and a reputation, a song, perhaps. Will it be difficult to find words that rhyme with Odysseus? I don't know what to say to Menelaus, so I do what I've always done best, I lie. I grin and tell him that I want one last tour of Troy, to see the sights before we depart. I even manage a stylish toss of the hand.
It takes Menelaus a moment but then he smiles. Of course, Odysseus wants a few anecdotes, for he rates no prize higher than a good story.
"Take care, Odysseus," says Helen, pronouncing my name slowly, as if it is foreign, as if we are strangers. We exchange no meaningful look, no significant glance, nothing to mark our having known each other.
When they're gone at last, I step back inside the palace and wander into Helen's bedchamber and stand there hating the sunlight that pours into the room. Gold motes roll in the shafting sun. I pass my hand through them as if to gather them up but they flee like fish while shrieks reach me muted and remote. Helen's churned bedcovers resemble swirling water, and I'm tempted to plunge in and drown. Had Tyndareus urged her to choose me? Had she defied him? Or had he told her that I'd already fallen in love with Penelope? It doesn't matter, it never mattered. My woman is home facing another day. By now there are suitors circling her, and even if Cloud Splitter himself comes down and offers me immortality and a place on Olympus I'll say no—I know that now—for I'd rather go home to Penelope and Telemachus, to the scent of fresh wood shavings in the carpenter shop, to our olive wood bed.
Calmer, I begin a leisurely perusal of Helen's room, picking up items from the tables and shelves, perfume vials, pumice stones, polished brass mirrors framed in shell-work, and all manner of rings and brooches and necklaces of strung pearls. On a stand made of antlers I find a gold ring inlaid with a silver horse that resembles the wooden one now burning in the temple. Did Helen send the idea to me in a Dream? I try the ring on all my fingers but it fits none, so I slip it into an inner pocket. The blanched stench of smoke returns me to the world. I hear pig squeals and dog barks and, bizarrely, laughter, joyous whoops of glee. Passing through the adjoining room, I step onto the portico where the guard Menelaus stabbed lies in a dried lake of blood. Dragonflies with brilliant blue bodies and two sets of wings are bogged in it. I squat for a closer look. They feed richly. Playing the god, I lift one up to free it but its wings are clotted and gummed. I pry off the dead Trojan's helmet, it's made of leather and horn and plumed with cock feathers. I also take his shield. The helmet smells rank but like the racket and the chaos it is distant, everything is distant, as though I'm watching it all from a far hill, never have I felt more aloof and alone. Looking at the sky I see that it is going to be a fine day, bright and hot, but I also see fallen oak leaves brittle on the portico, potted poppies which have burst their pods, and mint that has gone to seed. The cicadas and pine borers continue grinding away with the blind industry of insects even though summer is nearly over.
I drift around to the front of the house. Our wooden horse has burned down but other fires send up smoke and the sun shapes and contours the rolling clouds. A guard bursts at a run from the house carrying a spear in each fist; he ignores me and I him. The avenue is strewn with crockery and baskets and tipped carts. A wicker cage full of frantically flapping birds gains speed as it rolls down the hill. They cheep piteously, and without thinking I trot after them and catch the cage and hold it up. Their wings beat the air and the bars, I flip the catch and shake them free. They pour out, skimming low over the ground before veering up, a ribbon of birds assembling like a single creature into the air. The entire city smoulders, stone and brick go black and bitter, and grey fumes smart my eyes. Ur, Khajuraho, and now Troy; they rise like waves and crash on the sand and Poseidon, old kelpbeard, old fish god, yawns in his palace of coral while, high above, Zeus on his throne crosses his knees and smoothes his robe and scarcely takes any notice.
The Achaeans advance in a wave. I reach the top of the wall where Trojan bowmen shoot straight down while our archers return fire, arrows flicking past and dropping like sticks. I stroll along only mildly interested. The land slopes away to the shore where men continue streaming from our beached ships. No, the song spinners will not be kind to Priam, the king who lost the mighty city of Troy.
Ahead of me a familiar figure perches on the wall. I halt, a fist clenches in my gut, Ajax, blond hair luffing on the wind, breastplate glimmering, sword across his lap as though he might strum it like a zither. All about him hum bees. I wipe my hands over my face. Surely the smoke has addled me, but there he is, Ajax, and there they are, a swarm of honey bees, large fat ones that walk on his shoulders and his hands, rising and resettling, crawling over his armour, up his neck, across his cheek, as absorbed in their work as if Ajax is a source of nectar. At ease with this, he pays them the bemused attention one might give a young cat. I approach slowly. When I come abreast of him he tilts his head at an appraising angle to gauge me. Bees sit on his forehead, his ears, in his hair, one lights upon his very eye and he does not even blink, after a few moments it flies off.
"Odysseus," he says. His voice is no longer that of an embittered enemy or of a boy who has been rejected, but that of a traveller returned after many years in distant deserts. He notes my headgear. "So, you're a Trojan now?" he asks ironically, as though he'd not be surprised if I'd betrayed everyone at the very last moment.
I'll not quip with the dead. "The horse succeeded."
He smiles. "Small talk comes from small bones." He'll not deign to argue with the living. He raises his hand covered in a living glove of bees and turns it admiringly this way and that. He carries them close to his ear and listens, eyes closed, as though to music, to whispering voices, and seems pleased with what he hears. After a time he lowers his hand and looks out at the smoldering city. Black smoke rolls off the shops and houses. He regards it all as if reading the weather, as if admiring the very shape of the smoke itself, and then he turns the other way and gazes out over the plain toward the sea. I study him, as real as a dream. I note that he is not wearing Achilles's armour. Is he snubbing my gesture, was it too little too late? If I had the chance to do it over again would I have let him win, would it have been worth his good will? Perhaps. Either way it is one more regret to add to my ever-mounting heap. I'm close enough that I hear the hum of the bees. Some circle me but do not light, apparently I am not sweet enough. I reach to catch one but it passes through my palm with only the faintest sensation. I study my hand, frowning. Ajax turns from the view of the plain and looks at me; is that sadness or indifference in his eyes?
"What of the gods?" I ask him.
"The gods?" The concept seems remote to him. "The gods do not visit Hades."
"But Hades himself? And Persephone?"
He suppresses a smile. "Hades . . ." he says, as if he could tell me stories of the skull king and his spectral queen. At last he has both more knowledge and experience than I do, he has visited eternity.
"What do you think I can give you, Ajax, now of all times?"
"There is nothing you can give me and nothing I can have."
So the dead can lie. "You want songs."
Now he does smile. "Are you offering to sing? Come, Odysseus, sing me a song."
Looking into the eyes of the dead is like gazing at a winter sun in the hope of warmth.
As if reading my thoughts, Ajax turns his face upward and stares unblinking at the sun. He gazes at it without discomfort, as if it is but a distant candle. "You will be remembered only by dogs," he says at last, as if only now recalling the reason for his visit. Then he stands. "But we will talk again soon enough." The bees, as if trained, rise in a swarm and hover and then disperse, and with that Ajax is also gone.
I begin to run slowly along the wall, convinced that I must be dreaming, that I am like the mad who talk to the air. Indifferent to arrows and corpses and to the Achaeans now mounting their ladders, I keep on running. At a turn I find two Trojans backing a Greek against a tower. I draw my knife and advance. Seeing my helmet they let me get close and I stab the first under the ribs and the second through the throat. It is only when the second lolls dead against me that I recognize him: Sinon. Catching him as he slides to his knees I lay him gently on the stone. He does not open his eyes, he's gone already. I turn to the Greek and see that it is Palamedes. I remove my helmet.
"Odysseus." For once the sneer has left his face. His eyes are wide and innocent; a momentary lapse of suspicion.
I reach out my hand and he takes it, then I jerk him toward me and slash him across the throat. A hot spurt of blood sprays my neck then spouts again with the beat of his heart, though weaker this time. The familiar hate flares in his eyes and he almost smiles, as if even now, with his neck slit, his blood gloving my fist, he knows something I do not, that he has one more move to make, but if he does he's unable to execute it, for with my other fist I grip his hair and twist his head and whisper into his ear, "How does it feel, old friend? Tell me, how does it feel? No? Nothing to say? Well, I have: it's been worth the wait." I smile into his eyes, and with that he sinks through my arms like so much sand.
Somewhere someone shouts, Murder! Murder! I almost laugh. Murder, in a war? I walk down the steps toward the gate shaking my head at such madness, then I halt and reach into that pocket and find Helen's ring. I climb back up the steps to where Palamedes and Sinon lie in blood that already grows dull in the heat. The fighting has moved off and we are alone. Opening Sinon's mouth, I place Helen's ring under his tongue so that he will have the fare to pay Charon, as for Palamedes, he can work his passage.
I stand and look out over the city toward the sea gleaming like beaten tin. The road home. The sun is well past the midday mark on yet another trek across the sky. Its heat feels good, reassuring. Tyndareus had talked of the blind optimism of flowers, it holds for men, too, or is the heart little more than a blind mule turning a pump handle?
I leave the wall and wander through the city streets where the admirably disciplined Achaeans have been overtaken by dementia. Frenzy glazes their eyes as they rape corpses and steal anything that shines. They leap from roofs and elbow each other aside to get at women. They are crazed, they bludgeon children as if they were rabid dogs. What a proud moment. We kings. We victors. We noble warriors blessed by the gods. Only Diomedes declines to indulge, he stands forlorn, slope-shouldered and brooding, sword dangling forgotten in his hand.
"Dio . . ."
He looks at me, eyes heavy with the weight of disappointment, worse, disenchantment. "There are no gods here, Odysseus. I've looked." As if it has all been a waste, he pitches his sword down, it clatters sparking against the stone, striking blue and gold chips of fire like a hammer on an anvil. We share the same thought: Hephaestus, lame god of blacksmiths. We exchange glances. Diomedes sneers at the sky. "Hephaestus . . ." Disgusted, he stalks off saying that he'll not fight a cripple.
Outside the gate the wind gusts dirt into my eyes. I turn away but can't avoid the wet coin smell of blood. Corpses lie heaped in drifts against the walls. Sinon, Dercynus, and Echion should be here, happy, embracing, running toward our ships, and of course Ajax. It is both a wonder and a humiliation to me that despite all of this I can still take solace in a new day, in the fact that the sun is shining, that at long last the Trojan War is finished, that Palamedes is dead, Agamemnon has his city, Menelaus his woman, and soon, very soon, in a month at most, I will be home.
The author of six novels, one collection of stories, and a book on India, **Grant Buday** has travelled widely and now lives on a small island with his wife and son.
Copyright © Grant Buday, 2008
ALL RIGHTS RESERVED. No part of this publication may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying, recording, or any information storage and retrieval system, without permission in writing from the publisher.
_Library and Archives Canada Cataloguing in Publication_
Buday, Grant, 1956–
Dragonflies / Grant Buday.
eISBN : 978-1-897-23184-5
I. Title.
PS8553.U444D73 2008 C813'.54 C2008-904361-8
Edited by Daniel Wells.
PRINTED AND BOUND IN CANADA
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Robinson Cano can be a bigger asset for the Mets in 2022 than most fans think
By Adrian Cervantes
Philadelphia Phillies v New York Mets / Jim McIsaac/GettyImages
One of the biggest question marks for the New York Mets going into the 2022 season is what can they expect from second baseman Robinson Cano should he be on the Opening Day roster? While it seems more likely than not that Cano will be on the 26-man roster, many Mets fans may be forgetting the production that the 39-year old second baseman had been capable of during the 2020 season prior to his suspension.
Cano was hitting .316/.352/.544 with an OPS of .896 over 182 plate appearances. Cano has also spent some time in the Dominican Winter League this offseason, in-between making a music video cameo for Reggaeton superstar Anuel AA, and slashed a batting average of .300 over 43 plate appearances.
While the sample size is small, Cano is still proving he may have some magic left in his bat, which could provide huge for the Mets this season especially with the Designated Hitter most likely coming to the National League.
Should Robinson Cano pick up where he left off with the 2020 Mets team, he could provide a jolt of offense for a team that was sorely lacking any semblance of offense last season.
The second base position still may be a bit unsettled with Jeff McNeil rumored to be on the trade block this winter. Depending on how that situation shakes out, Cano could either slide back into his natural starting position or he may become one of the Mets' most valuable assets off the bench.
New York Mets finally retire #17 because HE is Keith Hernandez
Alan Karmin
The New York Mets Latin rich history of star players continues to provide opportunities in more ways than one
Adrian Cervantes
The New York Mets of 2022 have the same high expectations as the 1972 Mets
State of the Mets defense
The Mets had one of the best benches in all of baseball last season that carried the club through most of the season while the roster was overrun with injury after injury. Having a deep bench this season will be a key for the Mets to have sustained success. Whether or not Cano receives a starting spot, his value on this year's club could ultimately be tied to being the top bat off the bench in key situations should the Mets make some more moves this offseason.
Things are still a bit unsettled with the roster makeup going into Spring Training due to Major League Baseball's current lockout. However, it would be tough for even Billionaire Owner Steve Cohen to eat the remaining $48 million that is currently left on Cano's contract for the next two seasons.
That reasoning alone makes him a viable option for the Mets going into next season in some capacity as long as he is productive. I firmly believe new Manager Buck Showalter will also put Cano in positions to succeed early on in the season to see how he responds as well as to evaluate how much he may have left to give the organization.
Another aspect we must not forget is Cano's strong sense of leadership. I believe a big reason for the Mets midseason collapse last season was the lack of leadership in the clubhouse, which led to a wide variety of incidents, and Cano is still one of the most respected voices in the game despite the two PED suspensions.
If you need any further evidence on his potential impact, new centerfielder Starling Marte had revealed that he had signed with the Mets in large part due to his desire to play alongside Cano which speaks volumes.
There is no doubt that there will need to be a formal apology made to the current players in the clubhouse and most importantly to all of the fans this spring for his actions back in 2020. But without question, there is still the potential for a chance that Robinson Cano could make a lasting impact on the Mets as they look to compete for a potential World Series title in 2022.
As we saw last season, you can never have too many bodies ready to play, especially capable Major Leaguers, and Cano could provide the Mets that option despite the current outside noise. Picking up where he left off offensively in 2020 will make most fans forget about his previous transgressions if the team is winning, and as all of us know winning always will cure all problems.
Next. Mets icon Jose Reyes has slid head first into a successful music career. dark
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Damon Vignale
photo: John Cassini
Vancouver writer, director, producer, Damon Vignale (Little Brother of War, The Entrance, The Vitala) takes his talents for filmmaking and applies them to multi award winning TV series, Blackstone.
Blackstone is "Intense, compelling and confrontational. An un-muted exploration of First Nations' power and politics, unfolding over nine one-hour episodes. This raw, authentic drama tells the story of the fictional Blackstone First Nation, suffering disintegration by its own hand – the result of the corruption of its Chief and Council. From within the community, a new generation of leaders rise up and fight to create lasting and substantial change."
We catch up with Vignale at a local cafe where he talks about writing and producing for this new series.
DV: It's what I would call Sopranos on the rez (laughs). It's very hard-hitting. We rip from the headlines for the story lines and confront a lot of the harder issues that go on, on reservations. We lift the rug and look at the dirt underneath; we don't hide from anything. At the same time, it's very important for us that the show also presents hope, so within the arc of the season we try to steer toward something hopeful.
tV: How did you get involved with this project?
DV: I had worked with the executive producer, Ron Scott, years ago. I had been doing some writing on another show of his called Mixed Blessings, and he screened the Blackstone pilot for me. I was blown away and knew I wanted to write on this show. Fortunately he gave me the opportunity.
tV: Has there been an episode that carried a particular impact that made it challenging to write?
DV: The first season arc was around convicting a child molester. It also followed one of our lead characters, Gail (Michelle Thrush) who had a daughter who committed suicide. Gail spirals into alcoholism and the whole season is about her dealing with that. Michelle did an amazing job with that character, and that's ultimately why she won a Gemini Award for her performance.
We also go after a lot of other issues as well. The chief of the reserve is very corrupt so we tackle that always; murder cover-ups, we go after it all.
tV: What is your connection to First Nations?
DV: I think I have a great-grandmother who is First Nations but I'm actually half black so I don't really present myself as Aboriginal. Ron is Métis and very in tune with all the issues, and oversees all of that so I really look to him for direction. Outside of that, our story lines are universal. We may be focusing on a First Nations reservation but you can look in any community and find all these things. When I write I don't necessarily think I should write it a particular way because we're talking about a First Nations person. I look at the issues as being universal and I write them as telling a story that just happens to be on a reservation.
tV: How does this show compare or relate to stories you have told in the past?
DV: It's funny you should ask about my connection to the Aboriginal community. I grew up playing lacrosse, and one of the things that I was always fascinated with was the fact that that game came from the Native Americans. It was a spiritual game for them. I consider myself a spiritual person and so was always interested in that aspect of the game, so much so, that the first feature film I made, Little Brother of War, focused on that.
I'm writing on Blackstone now, and I just finished a documentary called The Exhibition, which again has Aboriginal themes. I don't know what it is. I must have some spirit guide who was an old Indian or something (laughs), but I seem to constantly find myself working with that type of material. In my work I am always interested in mythology...
(Vignale pauses briefly to greet actor John Cassini, who stars in Blackstone, as he happens into the cafe)
As a kid my grandfather used to read me stories from The Arabian Nights, so I think that's actually where it started from. Whether it be the spiritual essence around lacrosse or the Sanskrit mythology around my web series The Vitala, when I speak of the Aboriginal community I'm genuinely interested in their mythology.
tV: Can you tell us more about The Exhibition?
DV: Over the past six years I've been shooting this documentary. It's about an artist who decides to paint the 69 women on Vancouver's missing women poster, 29 of whom were found on Robert Pickton's farm, and the controversy around that artwork. It also touches on what happened to these women, who they were, and the whole botched police investigation. I have over a hundred interviews with police, advocates, and family members.
We world premiered at Hot Docs Film Festival this spring and it just played the Chicago Film Festival. I'm going to the Bergen Film Festival in Norway on friday and it's going to the United Nations Film Festival in San Francisco and it airs on Super Channel this November, so it's really making the rounds right now. It's an important film. It's very controversial because a lot of people were really opposed to the artwork, and the film tries not to say whether it is for or against it. The controversy is really only a part of it. The film ultimately tries to present an overview of who these women were, what happened to them, and how we as a society could allow it to happen. It really tries to shine a spotlight on that.
Blackstone is going into its fourth season on APTN (Aboriginal People's Television Network) and as Vignale points out, that's huge for Canada. It shoots in Alberta and around 30% of the crew and roughly 75 – 80% of the actors are from Vancouver. Check it out and support local your talent.
For more information on Vignale's work, click on Blackstone, or the exhibition
|
{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
| 7,764
|
\section{introduction}
\label{sect:intro}
The recent experimental progress on the ultracold
Fermi gases with large hyperfine spin provides an exciting
opportunity to investigate novel physical properties
\cite{Killian2010,takasu2010,wu2010, wu2006a}. In usual condensed
matter systems, large spin is not considered particularly
interesting because large values of spin suppress quantum fluctuations.
For example, in transition metal oxides, a large spin on each cation
site is usually referred as an effective spin $S$ composed of $2S$
electrons by Hund's rule. The spin exchange between two cation sites
at the leading order of the perturbation theory only involves
swapping one pair of electrons regardless of how large $S$ is. The
variation of $S_z$ is only $\pm 1$, thus increasing $S$ reduces
quantum fluctuations known as the $1/S$-effect. In contrast,
in ultracold fermion systems, the situation is dramatically
different, in which large hyperfine spin enhances quantum
fluctuations. Each atom moves as a whole object carrying a large
hyperfine spin. Exchanging cold fermions can completely flip the
entire hyperfine-spin configuration, and thus enhances quantum
fluctuations. In other words, large spin physics in solid state
systems is usually in the large $S$-limit, while in cold atom
systems it is in the large $N$ limit where $N$ is the number of fermion
components $2F+1$ \cite{wu2010}.
We follow the convention in atomic physics to use $F$ to denote
the hyperfine spin of the atom.
Ultracold fermion systems with large hyperfine spins have aroused a
great deal of theoretical interests. Early work studied the rich
structures of the Fermi liquid theory \cite{YIP1999} and the Cooper
pairing structures \cite{HO1999}. Considerable progress has been
made in the simplest large hyperfine spin systems with
$F=\frac{3}{2}$, whose possible candidate atoms are $^{132}$Cs,
$^9$Be, $^{135}$Ba, $^{137}$Ba and $^{201}$Hg. These include both
alkaline-earth-like atoms with zero electron spin due to the fully
filled electron shells, and non-alkaline-earth atoms with nonzero
electron spins \cite{wu2006a,chen2005,xu2008}. In both cases, a
generic $Sp(4)$, or, isomorphically, $SO(5)$ symmetry is proved
without fine tuning. Such a high symmetry without fine-tuning is
rare in both condensed matter and cold atom systems. It brings
hidden degeneracy in the collective modes in the Fermi liquid theory
\cite{wu2007a}, fruitful patterns of quantum
magnetism\cite{WU2003,wu2006a,WU2005a,chen2005,xu2008} and Cooper
pairing with large internal spin degrees of freedom
\cite{wu_hu_zhang2010,xu2008}. Further investigations in the
community include the study of Mott insulating states
\cite{tu2006,tu2007,zheng2011,jiang2010,Lee2010,Lewenstein2010},
Beth-ansatz solution \cite{Controzzi2006, jiang2010a}, Kondo effect
\cite{hattori2005}, and the 4-fermion quartetting superfluidity
\cite{lecheminant2005,lecheminant2008,lee2007}. Recently, $SU(N)$
models have been proposed for the alkaline-earth fermion atoms since
their interactions are insensitive to their nuclear spins. It is a
special case of the $Sp(N)$ model by further tuning interaction
parameters of spin singlet and multiplet channels to be the same
\cite{gorshkov2010,hermele2009,xu2010}. The possible ferromagnetic
states have also been studied for the $SU(6)$ symmetric system of
$^{173}$Yb \cite{cazalilla2009}. A detailed summary is presented in
a review Ref. [\onlinecite{wu2006a}] and a non-technique
introduction is published at Ref. [\onlinecite{wu2010}] by one of
the authors. In a different context of heavy fermion systems, the
effects of sympletic symmetry to quantum magnetism have also been
studied in Ref. [\onlinecite{flint2008,flint2009}].
One dimensional (1D) systems are important for the study of strong
correlation physics because of the dominant interaction effects.
Furthermore, controllable analytical and numeric methods are
available. In Ref. [\onlinecite{WU2005a}], one of the authors
performed the bosonization method to study competing phases in 1D
systems with $F=\frac{3}{2}$, including the gapless Luttinger
liquid, spin gapped Luther-Emery liquid with Cooper pairing
instability, and 4-fermion quartetting superfluid at incommensurate
fillings. At commensurate fillings with strong repulsive
interactions, a charge gap opens and the systems become
Mott-insulating. The gapless Luttinger liquid phase becomes a
gapless spin liquid phase at quarter-filling and dimerized at
half-filling, respectively \cite{wu2006a}. The Luther-Emery phase
becomes the gapped $Sp(4)$ dimer phase at quarter-filling and the
on-site singlet phase at half-filling, respectively \cite{wu2006a}.
On the other hand, the two dimensional (2D) $Sp(4)$ Heisenberg model
is still far away from clear understanding. Such a system can bring
fruitful intriguing features of quantum magnetism which do not
exhibit in usual solid state systems. For example, in the special
case of the $SU(4)$ symmetry, four particles are required to form an
$SU(4)$ singlet, thus its quantum magnetism is characterized by the
4-site correlation beyond two sites. Such a state is the analogy to
the three-quark color singlet baryon state in quantum
chromodynamics. It is also the magnetism counterpart of the
4-fermion quartetting instability with attractive interactions
\cite{wu2006a}. Recently a magnetic phase diagram in a spatially
anisotropic square lattice of the $Sp(4)$ quantum magnetism is
provided by means of large-$N$ field-theoretical
approach\cite{kolezhuk2010}. A phase transition between the
long-range Neel order state and the disordered valence bond solid
phase is discovered by the perturbative renormalization group
equations. However, the model on an isotropic square lattice is
still unexplored. In particular, quantum Monte Carlo methods for
this model suffer the notorious sign problem except in the special
case where only the singlet bond exchange exists.
In this article, we present a systematic numerical study for the
$Sp(4)$ Heisenberg model at quarter filling in both 1D systems with
large sizes and 2D systems up to $4\times 4$ by means of exact
diagonalization techniques and the density matrix renormalization
group (DMRG)\cite{white1992,white1993}. In 1D, we numerically show
that the system exhibits two competing quantum phases: a
long-range-ordered gapped dimer phase when the exchange interaction
in the bond singlet channel ($J_0$) dominates over that in the
quintet channel ($J_2$), and a gapless spin liquid phase otherwise.
The $Sp(4)$ spin correlation functions are calculated, which shows
that in the dimer phase the correlations have the 2-site
periodicity, whereas in the gapless spin liquid phase they have the
4-sites periodicity. In 2D, our numerical simulations for small
sizes indicate three different dominant correlations depending on
the values of $J_0/J_2$. We infer three competing phases: the Neel
ordering, the plaquette ordering, and another possible phase of
columnar dimer ordering, in the thermodynamic limit.
The rest of this article is organized as follows. In Sec.
\ref{sect:model}, we introduce the Hamiltonian of spin-$\frac{3}{2}$
fermions which possesses the rigorous $Sp(4)$ symmetry, and a
magnetic exchange model in the Mott-insulating state at
quarter-filling. A self-contained introduction of the $Sp(4)/SO(5)$
algebra is given. Then we separate our main discussion into two
parts: Sec. \ref{sect:1D} for 1D and Sec. \ref{sect:2D} for 2D
systems. In Sec. \ref{sect:exact}, we study the low-energy spectra
of a finite size $Sp(4)$ chain with both open and periodic boundary
conditions. In Sec. \ref{sect:DMRG}, the DMRG calculation on the
spin correlation functions are presented to identify the gapped
$Sp(4)$ dimer phase and the gapless spin-liquid phase. In the second
half part, we first analyze the $2\times 2$ cluster in Sec.
\ref{sect:foursite} and perform exact diagonalization on larger
sizes to study the low-energy spectrum behavior in Sec.
\ref{sect:2Dspectra}. Then we display the calculations of the
magnetic structure form factor in Sec. \ref{sect:mag}, the dimer
correlation in Sec. \ref{sect:dimer} and the plaquette-type
correlation in Sec. \ref{sect:plaquette}. We discuss the possible
existence of the corresponding orderings. Conclusions are made in
the last section. At the end of this paper, we present a brief and
self-contained introduction to the representation theory of Lie
group in Appendix A to C.
\section{Model Hamiltonian and the hidden $Sp(4)$ symmetry}
\label{sect:model}
\subsection{The spin-$\frac{3}{2}$ Hubbard model}
We start with the generic one-band Hubbard model
loaded with spin-$\frac{3}{2}$ fermions. By neglecting long-range
Coulomb interactions, only onsite interactions are considered in the
Hubbard model. Due to Pauli's exclusion principle, the spin
wavefunctions of two onsite fermions have to be antisymmetric. The
total spin of two onsite spin-$\frac{3}{2}$ fermions can only be
either singlet $(S_T=0)$ or quintet $(S_T=2)$. We assign an
independent interaction parameter $U_0$ (singlet) and $U_2$
(quintet), respectively, to each channel. The Hamiltonian reads \begin{eqnarray}} \def\eea{\end{eqnarray}
H&=&-t \sum_{\langle ij \rangle, \sigma} (\psi^{\dag}_{i \sigma}
\psi_{j \sigma}+h.c.)-\mu \sum_{i \sigma}\psi^{\dag}_{i \sigma}
\psi_{i \sigma} \nonumber
\\
&+&U_0 \sum_i P^{\dag}_{0}(i)P_{0}(i)
+U_2 \sum_{i,m=-2,..,2}P^{\dag}_{2m}(i)P_{2m}(i), \nonumber \\
\label{eq:hubbard} \eea where $\langle ij \rangle$ denotes the
nearest neighboring hopping; $\sigma$ represents four spin flavors
$F_z=\pm \frac{3}{2}$, $\pm \frac{1}{2}$; $P^{\dag}_{0}$ and
$P^{\dag}_{2,m}$ are the singlet and quintet pairing operators
defined through Clebsch-Gordon coefficients as \begin{eqnarray}} \def\eea{\end{eqnarray}
P^{\dag}_{0}(i)&=&\sum_{\alpha\beta}\avg{00|\frac{3}{2}\frac{3}{2}\alpha\beta}
\psi^\dagger_\alpha(i) \psi^\dagger_\beta(i),\nonumber \\
P^{\dag}_{2m}(i)&=&\sum_{\alpha\beta}\avg{2m|\frac{3}{2}
\frac{3}{2}\alpha\beta} \psi^\dagger_\alpha(i) \psi^\dagger_\beta(i).
\eea
The actual symmetry of Eq. \ref{eq:hubbard} is much larger than the
$SU(2)$ symmetry: it has a hidden and exact $Sp(4)$, or,
isomorphically, $SO(5)$ symmetry. The $Sp(4)$ algebra can be
constructed as follows. For the 4-component fermions, there exist 16
bases for the $4\times 4$ Hermitian matrices $M_{\alpha\beta}
(\alpha,\beta= \pm\frac{3}{2}, \pm\frac{1}{2})$. They serve as
matrix kernels for the bi-linear operators, {\it i.e.},
$\psi_\alpha^\dagger M_{\alpha\beta}\psi_\beta$, in the
particle-hole channel. The density and 3-component spin $F_x, F_y,
F_z$ operators {\it do not} form a complete set. The other 12
operators are built up as high rank spin tensors, including
5-component spin-quadrupoles and 7-component spin-octupoles. The
matrix kernels of the spin-quadrupole operators are defined as \begin{eqnarray}} \def\eea{\end{eqnarray}
\Gamma^1&=& \frac{1}{\sqrt 3} ( F_x F_y +F_y F_x ), \ \ \,
\Gamma^2= \frac{1}{\sqrt 3} ( F_z F_x +F_x F_z ), \nonumber \\
\Gamma^3&=& \frac{1}{\sqrt 3} ( F_z F_y +F_y F_z ), \ \ \,
\Gamma^4= ( F_z^2-\frac{5}{4} ), \nonumber \\
\Gamma^5&=& \frac{1}{\sqrt 3} ( F_x^2 -F_y^2 ), \eea which
anti-commute with each other, and thus form a basis of the
Dirac-$\Gamma$ matrices. The matrix kernels of 3 spin and 7
spin-octupole operators together are generated from the commutation
relations among the 5 $\Gamma$-matrices as \begin{eqnarray}} \def\eea{\end{eqnarray} \Gamma^{ab}=
-\frac{i}{2} [ \Gamma^a, \Gamma^b] \ \ \ (1\le a,b\le5). \eea
Consequently, these 16 bilinears can be classified as \begin{eqnarray}} \def\eea{\end{eqnarray}
n&=&\psi^\dagger_\alpha\psi_\alpha, ~
n_a=\frac{1}{2}\psi^\dagger_\alpha \Gamma^a_{\alpha\beta}
\psi_\beta, ~ L_{ab} = -\frac{1}{2}\psi^\dagger_\alpha
\Gamma^{ab}_{\alpha\beta} \psi_\beta, ~~ \label{eq:generators} \eea
where $n$ is the density operator; $n_a$'s are 5-component
spin-quadrupole operators; $L_{ab}$'s are 10-component spin and
spin-octupole operators \cite{WU2003,wu2006a}. Reversely the spin
$SU(2)$ generators $F_{x,y,z}$ can be written as
$F_+=\sqrt{3}(-L_{34}+i L_{24})-(L_{12}+iL_{25}) +i(L_{13}+iL_{35})$
and $F_z=L_{23}+2L_{15}$.
The 15 operators of $n_a$ and $L_{ab}$ together span the $SU(4)$
algebra. Among them, the 10 $L_{ab}$ operators are spin tensors with
odd ranks, and thus time-reversal (TR) odd, while the 5-component
$n_a$'s are TR even. The TR odd operators of $L_{ab}$ form a closed
sub-algebra of $Sp(4)$. The 4-component spin-$\frac{3}{2}$ fermions
form the fundamental spinor representation of the $Sp(4)$ group. In
contrast, the TR even operators of $n_a$ do not form a closed
algebra, but transform as a 5-vector under the $Sp(4)$ group. In
other words, $Sp(4)$ is isomorphic to $SO(5)$. But rigorously
speaking, the fermion spinor representations of $Sp(4)$ are not
representations of $SO(5)$. Their relation is the same as that
between $SU(2)$ and $SO(3)$. Below we will use the terms of $Sp(4)$
and $SO(5)$ interchangeably. The $SO(5)$ symmetry of Eq.
\ref{eq:hubbard} can be intuitively understood as follows. The
4-component fermions are equivalent to each other in the kinetic
energy term, which has an obvious $SU(4)$ symmetry. Interactions
break the $SU(4)$ symmetry down to $SO(5)$. The singlet and quintet
channels form the identity and $5$-dimensional vector
representations for the $SO(5)$ group, respectively, thus Eq.
\ref{eq:hubbard} is $SO(5)$ invariant without any fine-tuning.
\subsection{Magnetic exchanges at quarter-filling}
Mott-insulating states appear at commensurate fillings with strong
repulsive interactions. We focus on the magnetic exchange at quarter
filling, {\it i.e.}, one fermion per site. The Heisenberg type
exchange model has been constructed in Ref. [\onlinecite{chen2005}]
through the second-order perturbation theory. For each bond, the
exchange energies are $J_0=\frac{4t^2}{U_0}$ for the bond spin
singlet channel, $J_2=\frac{4t^2}{U_2}$ for the bond spin quintet
channel, and $J_1=J_3=0$ for the bond spin triplet and septet
channels, respectively. This exchange model can be written in terms
of bi-linear, bi-quadratic and bi-cubic Heisenberg exchange and the
Hamiltonian reads as
\begin{eqnarray}} \def\eea{\end{eqnarray}
\label{eq:heisenberg}
H_{ex}=\sum_{\langle
i,j \rangle} a (\vec{F}_i\cdot \vec{F}_j) +b (\vec{F}_i \cdot
\vec{F}_j)^2+c (\vec{F}_i \cdot \vec{F}_j)^3,
\eea
where
$a=-\frac{1}{96}(31 J_0+23 J_2)$, $b=\frac{1}{72}(5 J_0 +17J_2)$ and
$c=\frac{1}{18}(J_0+J_2)$ and $F_{x,y,z}$ are usual $4\times 4$ spin
operators. Eq. \ref{eq:heisenberg} can be simplified into a more
elegant form with the explicitly $SO(5)$ symmetry \cite{wu2006a} as
\begin{eqnarray}} \def\eea{\end{eqnarray} \label{eq:so5} H_{ex}&=&\sum_{\langle i,j \rangle} \Big\{
\sum_{1\le a < b \le 5} \frac{J_0+J_2}{4}
L_{ab}(i)L_{ab}(j)\nonumber \\
&+& \frac{3J_2-J_0}{4} \sum^5_{a=1} n_a(i)n_a(j) \Big \}. \eea In
the $SO(5)$ language, there are two diagonal operators commuting
with each other and read as \begin{eqnarray}} \def\eea{\end{eqnarray} L_{15}&=&
\frac{1}{2}(n_{\frac{3}{2}}+n_{\frac{1}{2}}-n_{-\frac{1}{2}}
-n_{-\frac{3}{2}}), \nonumber \\
L_{23}&=& \frac{1}{2}(n_{\frac{3}{2}}-n_{\frac{1}{2}}
+n_{-\frac{1}{2}}-n_{-\frac{3}{2}}). \eea Corresponding to the spin
language, each singlet-site basis state can be labeled in terms of
these two quantum numbers as $|F_z \rangle = | L_{15},
L_{23}\rangle$: $|\pm \frac{3}{2} \rangle = | \pm \frac{1}{2}, \pm
\frac{1}{2}\rangle$ and $|\pm \frac{1}{2} \rangle = |\pm
\frac{1}{2}, \mp\frac{1}{2}\rangle$. For an arbitrary many-body
state, $L^{tot}_{15}=\sum_i L_{15}(i)$ and $L^{tot}_{23}=\sum_i
L_{23}(i)$ are good quantum numbers (similar to that
$F^{tot}_{z}=\sum_i F_{z}(i)$ is conserved in $SU(2)$ cases) and can
be applied to reduce dimensions of the Hilbert space in practical
numerical calculations.
There exist two different $SU(4)$ symmetries of Eq. \ref{eq:so5} in
two special cases. At $J_0=J_2=J$, {\it i.e.}, $U_0=U_2$, it reduces
to the $SU(4)$ Heisenberg model with each site in the fundamental
representation
\begin{eqnarray}} \def\eea{\end{eqnarray}
H=\sum_{\langle i,j \rangle} \frac{J}{2} \Big\{
L_{ab}(i)L_{ab}(j)+n_a(i)n_a(j) \Big \}.
\eea
Below we denote this
symmetry as $SU(4)_A$. In this case, there is an additional good
quantum number $n_4$, \begin{eqnarray}} \def\eea{\end{eqnarray} n_4= \frac{1}{2}
(n_{\frac{3}{2}}-n_{\frac{1}{2}}-n_{-\frac{1}{2}}
+n_{-\frac{3}{2}}). \eea This $SU(4)$ model is equivalent to the
Kugel-Khomskii type model \cite{kugel1982,sutherland1975} and is
used to study the physics with interplay between orbital and spin
degree of freedom.\cite{li1998,bossche2000,bossche2001}. On the
other hand, at $J_2=0$, {\it i.e.}, $U_2\rightarrow +\infty$, Eq.
\ref{eq:so5} has another $SU(4)$ symmetry in the bipartite lattice,
which is denoted $SU(4)_B$ below. In this case, we perform the
particle-hole transformation to one sublattice but leave the other
sublattice unchanged. The particle-hole transformation is defined as
$\psi_\alpha \rightarrow R_{\alpha\beta} \psi_\beta^\dagger$ where
$R$ is the charge conjugation matrix
\begin{eqnarray}} \def\eea{\end{eqnarray}
R= \left (
\begin{array} {cc}
0 &i\sigma_2 \\
i\sigma_2 & 0 \\
\end{array}
\right ).
\label{eq:R4}
\eea
Under this operation, the fundamental representation transforms to
anti-fundamental representation whose $Sp(4)$ generators and vectors
become $L'_{ab}=L_{ab}$ and $n'_{a}=-n_{a}$.
Thus Eq. \ref{eq:so5} can be recast to
\begin{eqnarray}} \def\eea{\end{eqnarray}
H=\sum_{\langle i,j\rangle}\frac{J}{2}\big(L'_{ab}(i)L_{ab}(j)+n'_a(i)n_a(j)
\big),
\label{eq:SU4B}
\eea
which is $SU(4)$ invariant again.
These two $SU(4)$ symmetries have very different physical
properties. In the case of $SU(4)_A$, two sites are not enough to
form an $SU(4)$ singlet. It at least needs four sites to form an
$SU(4)$ singlet as
$\epsilon_{\alpha\beta\gamma\delta}\psi^\dagger_\alpha(1)\psi^\dagger_\beta(2)
\psi^\dagger_\gamma(3)\psi^\dagger_\delta(4)$, where
$\epsilon_{\alpha\beta\gamma\delta}$ is the rank-4 fully
antisymmetric tensor. Thus quantum magnetism of Eq. \ref{eq:so5} at
$J_0=J_2$ is characterized by four-site correlations. The ground
state of such a system on a 2D square lattice was conjectured to be
a plaquette $SU(4)$ singlet state without magnetic long-ranged
ordering.\cite{bossche2000,Mishra2002} On the other hand, for the
$SU(4)_B$ case, two sites can form an $SU(4)$ singlet as
$R_{\alpha\beta}\psi^\dagger_\alpha(1) \psi^\dagger_\beta(2)$. In
the 2D square lattice, a long-ranged Neel order is identified by
quantum Monte Carlo simulations\cite{harada2003} and large $N$
limit.\cite{Read1990} The square of the staggered magnetization is
numerically given as $m_s=0.091$, which is much smaller than that of
the $SU(2)$ Neel order state.
\section{Quantum magnetism in the 1D chains}
\label{sect:1D}
We start our discussion on the 1D chain. The phase diagram of the 1D
spin-$\frac{3}{2}$ Hubbard model has been studied by one of the
author using the method of bosonization \cite{WU2005a,wu2006a}. At
the commensurate quarter-filling (one particle per site) with purely
repulsive interactions $(U_0>0, U_2>0)$, the $4k_f$-Umklapp term
opens a charge gap as $K_c<\frac{1}{2}$. In this case, the physics
is captured by the exchange model of Eq. \ref{eq:so5}. It has been
found that in the regime of $J_0/J_2>1$ dimerization of spin
Perierls order is present, whereas it is a gapless spin liquid phase
at $J_0/J_2 \le 1$ (see Fig. \ref{fig:phase}) \cite{WU2005a}. In the
following, we use exact diagonalization methods and DMRG not only
identify these two competing phases but also demonstrate the ground
state profiles and 4-site periodicities in spin-spin correlations.
\begin{figure}[htb]
\centering\epsfig{file=1D_phase.eps,clip=1,width=0.6\linewidth,angle=0}
\caption{Phase diagram of the 1D chain in terms of the singlet and
quintet channel interaction $J_0$ and $J_2$. In this context,
$\theta$ is the angle defined by $\theta=\tan^{-1}(J_0/J_2)$. The
$SU(4)_A$ type ($\theta=45^{\circ}$) denoted by the dot line belongs
to the gapless spin liquid state whereas $SU(4)_B$ along $J_2=0$.
The phase boundary separating the dimerization phase and the gapless
liquid state is the $SU(4)_A$ line.} \label{fig:phase}
\end{figure}
\subsection{Exact diagonalization on low energy spectra}
\label{sect:exact}
In this subsection, we apply the exact diagonalization technique to
study the 1D $Sp(4)$ spin-$\frac{3}{2}$ chains with nearest neighbor
exchange interactions described by Eq. \ref{eq:so5}. We only
consider the case of the site number $N=4m$. For convenience, we set
$J_0=\sqrt{2}\sin \theta$ and $J_2=\sqrt{2} \cos \theta$. Regardless
of $\theta$ and sizes $N$, the ground states (GS) only exist in the
$(L^{tot}_{15},L^{tot}_{23})=(0,0)$ sector and are unique with
$C=0$, where $C$ denotes the $Sp(4)$ Casimir of the entire system
and is expressed in terms of the $Sp(4)$ generators as \begin{eqnarray}} \def\eea{\end{eqnarray} C=
\sum_{1\le a<b \le 5}\Big\{\sum_{i} L_{ab} (i) \Big \}^2. \eea In
addition to $L^{tot}_{15}$ and $L^{tot}_{23}$, the Casimir $C$ is
also a conserved quantity in the $Sp(4)$ system, analogous to the
total spin in $SU(2)$ systems. Each energy eigenstate can be labeled
by $C$ and further identified the dimension of the representation
(degeneracy). As shown in the table II in the Appendix, while $C=0$,
the state is an $Sp(4)$ singlet and unique whereas while $C > 0$ the
state is multiplet and has degeneracy which is equal to the
dimension of the associated representation.
\begin{figure}[htb]
\centering\epsfig{file=1D_OBC.eps,clip=1,width=0.43\linewidth,angle=0}
\centering\epsfig{file=1D_PBC.eps,clip=1,width=0.52\linewidth,angle=0}
\caption{The exact diagonalization on 1D chain with 12 sites for (a)
open and (b) periodic boundary conditions. The dispersion of the
ground state and low excited states, and the dimensions d of their
corresponding representations of the $Sp(4)$ group are shown. }
\label{fig:exact_8sites}
\end{figure}
In Fig. \ref{fig:exact_8sites} (a) and (b), the ground state and low
excited states with 12 sites are presented using open and periodic
boundary conditions, respectively. The GS as varying $\theta$ angles
is always an $Sp(4)$ singlet, which becomes an $SU(4)$ singlet at
$\theta=45^\circ$ ($SU(4)_A$) and $\theta=90^\circ$ ($SU(4)_B$) for
both boundary conditions. For the low energy excited states, we
first look at the regime of $45^\circ<\theta<90^\circ$, {\it i.e.},
$J_0>J_2$. With open boundary conditions (OBC), the lowest excited
states (LES) are the $Sp(4)$ 5-vector states with the quadratic
Casimir $C=4$. The next lowest excited states (NLES) are 10-fold
degenerate and belong to the 10-dimensional ($10d$) $Sp(4)$ adjoint
representation with $C=6$. The LES and NLES merge at both of the
$SU(4)_A~(\theta=45^\circ)$ and $SU(4)_B ~(\theta=90^\circ)$
points, and become 15-fold degenerate. This is the $SU(4)$ adjoint
representation with $C=8$. With periodic boundary conditions (PBC),
the 5-vector and the 10-fold states behave similarly as before.
However, a marked difference is that a new $Sp(4)$ singlet state
appears as the LES at $50^\circ<\theta<90^\circ$, which becomes
higher than the 5-vector states only very close to $45^\circ$. In
particular, it is nearly degenerate with the ground state (which is
the lowest $Sp(4)$ singlet) at $\theta=50^{\circ} \sim 60^{\circ}$.
In the regime of $0^\circ<\theta<45^\circ$, {i.e.}, $J_2>J_0$ the
excited states are many $Sp(4)$ multiplets with energies close to
each other. With OBC, the LESs form the 10d $Sp(4)$ adjoint
representation. For the PBC case, the 14-dimensional symmetric
tensor representation of $Sp(4)$ competes with the 10d adjoint one.
The appearance of two nearly degenerate $Sp(4)$ singlets at
$50^\circ<\theta<90^\circ$ with PBC and their disappearance with OBC
can be understood by the dimerization instability. The dimerization
and the spin gapped ground state was shown in the bosonization
analysis at $45^\circ<\theta<90^\circ$ \cite{WU2005a}. In the
thermodynamic limit, the ground state has double degeneracy
corresponding to two different dimer configurations, both
spontaneously breaking translational symmetry. The OBC favors only
one of the dimer configurations, but disfavors the other due to one
bond breaking. In the finite system with PBC, the two dimer
configurations tunnel between each other, which gives rise to two
nearly degenerate $Sp(4)$ singlet states. We further calculate the
gap between them, denoted by $\Delta_{ss}$, at $\theta
> 45^{\circ}$ by using exact diagonalization under PBC up to 16
sites.
\begin{figure}
\centering\epsfig{file=1D_singlet_gap.eps,clip=1,width=0.8\linewidth,angle=0}
\caption{ Exact diagonalization results on the $Sp(4)$
singlet-singlet gap with $J_0>J_2$ and periodic boundary conditions
( $\theta=60^{\circ}$ and $75^{\circ}$ with $N=8,12$ and $16$).
Finite size scaling shows the vanishing of the singlet-singlet gap
$\Delta_{ss}$.} \label{fig:gap_period}
\end{figure}
As presented in Fig. \ref{fig:gap_period}, $\Delta_{ss}$ disappears
in the finite size scaling due to the twofold degeneracy. On the
other hand, the existence of the spin gap in this parameter regime
is presented in Fig. \ref{fig:gap_open} by DMRG simulation in Sec.
\ref{sect:DMRG} below. The original Lieb-Schultz-Mattis theorem
\cite{lieb1961} was proved that for the $SU(2)$ case, the GS of
half-integer spin chains with translational and rotational
symmetries is gapless, or gapped with breaking translational
symmetry. It is interesting to observe that our results of the
$Sp(4)$ spin chain also agree with this theorem. The nature of the
GS in the parameter regime $0^\circ<\theta<45^\circ$ will be
discussed in Sec. \ref{sect:DMRG}.
\subsection{DMRG simulations on $Sp(4)$ spin chain} \label{sect:DMRG}
\label{sect:DMRG1D}
\begin{figure}[htb]
\centering\epsfig{file=1D_spin_gap.eps,clip=1,width=0.8\linewidth,angle=0}
\caption{The finite size scaling of the spin gap $\Delta_{sp}$ of
the $Sp(4)$ spin chain vs $1/N$ at various values of $\theta$.
$\theta$ is defined as $\theta=\tan^{-1}(J_0/J_2)$ and $N$ is the
system size.} \label{fig:gap_open}
\end{figure}
In this subsection, we present the DMRG calculations on the ground
state properties of the $Sp(4)$ chain up to 80 sites with OBC. We
first present the spin gap $\Delta_{sp}$ in Fig. \ref{fig:gap_open},
which is defined as the energy difference between the ground state
and the lowest $Sp(4)$ multiplet.
For chains with even number of sites, the GS is obtained with
quantum number $L^{tot}_{15}=L^{tot}_{23}=0$, and any $Sp(4)$
multiplet contains the states with quantum numbers
$(L^{tot}_{15}=\pm 1, L^{tot}_{23}=0)$ and $(L^{tot}_{15}=0,
L^{tot}_{23}=\pm 1)$. States with the same values of $(L^{tot}_{15},
L^{tot}_{23})$ may belong to different $Sp(4)$ representations,
which can be distinguished by their $Sp(4)$ Casimir. Practically, we
only need to calculate these sectors with low integer values of
$(L^{tot}_{15}, L^{tot}_{23})$ to determine the spin gaps. For the
cases of $\theta
> 45^{\circ}$, i.e., ($J_2/J_0 < 1$), $\Delta_{sp}$s saturate to
nonzero values as $1/N \to 0$, indicating the opening of spin gaps.
On the other hand, $\Delta_{sp}$'s vanish at $\theta \le 45^\circ$,
which shows that the ground state is gapless. These results agree
with the bosonization analysis \cite{WU2005a}, which shows that the
phase boundary is at $\theta=45^\circ$ with the $SU(4)_A$ symmetry,
which is also gapless. This gapless $SU(4)_A$ line was also studied
before in Ref. [\onlinecite{yamashita1998,azaria1999}].
\begin{figure}[!htb]
\centering\epsfig{file=1D_NN_corr_1.eps,clip=1,width=0.7\linewidth,angle=0}
\centering\epsfig{file=1D_NN_corr_2.eps,clip=1,width=0.73\linewidth,angle=0}
\caption{The NN correlation of $\langle L_{15}(i)L_{15}(i+1)
\rangle$ with open boundary conditions for (a) $\theta=60^{\circ}$
and (b) $\theta=30^{\circ}$, respectively. The dimer ordering is
long-ranged in (a). Note the 2-site periodicity in (a) and the
4-site periodicity in (b).} \label{fig:nncorr}
\end{figure}
To further explore the ground state profile, we calculate the
nearest neighbor (NN) correlation functions of the $Sp(4)$
generators for a chain of 80 sites. This correlation function is
similar to the bonding strength and defined as $\langle X(i)
X(i+1)\rangle$, where $X$ are $Sp(4)$ generators. We present the
result of $\langle L_{15}(i) L_{15}(i+1)\rangle$ in Fig.
\ref{fig:nncorr}, and the correlation functions of other generators
should be the same due to the $Sp(4)$ symmetry. The open boundary
induces characteristic oscillations. At $\theta=60^\circ$, {\it
i.e.}, $J_0/J_2=\sqrt{3}$, $\langle L_{15}(i) L_{15}(i+1)\rangle$
exhibits the dominant dimer pattern, which does not show noticeable
decay from the edge to the middle of the chain. This means that the
dimerization is long-range-ordered in agreement with the
bosonization analysis \cite{wu2006a}. In contrast, at
$\theta^\circ=30^\circ$, {\it i.e.}, $J_2/J_0=\sqrt 3$, $\langle
L_{15}(i) L_{15}(i+1)\rangle$ exhibits a characteristic power-law
decay with 4-site periodicity oscillations. The 4-site periodicity
is also observed at other $\theta$'s for $\theta\le 45^\circ$, same
as ones presented in the bosonization analysis.
\begin{figure}[!htb]
\centering\epsfig{file=1D_L15_dimer.eps,clip=1,width=0.4\linewidth,angle=0}
\centering\epsfig{file=1D_n4_dimer.eps,clip=1,width=0.5\linewidth,angle=0}
\caption{The finite size scaling for the dimer order parameters (a)
$D_{L_{15}}$ and (b) $D_{n_4}$ vs $1/N$ at various $\theta$'s.}
\label{fig:dimerp}
\end{figure}
We follow the definition for the dimer order parameter in Ref.
[\onlinecite{fath2001,hung2006}] as \begin{eqnarray}} \def\eea{\end{eqnarray} D_{X}=|\langle
X(\frac{N}{2}-1) X(\frac{N}{2})\rangle-\langle
X(\frac{N}{2})X(\frac{N}{2}+1)\rangle|. \nonumber \\
\label{eq:dimer} \eea As shown previously, $X$s are $Sp(4)$
generators and vectors in the $Sp(4)$ spin chain. Without loss of
generality, we choose two non-equivalent operators as $X=L_{15}$ and
$n_4$ for $Sp(4)$ generators and vectors, respectively. The open
boundary conditions provide an external field to pin down the dimer
orders. The finite size scaling of the dimer orders of the two
middle bonds is presented in Fig. \ref{fig:dimerp} (a) and (b) at
various values of $\theta$, respectively. It is evident that in the
regime of $\theta
> 45^{\circ}$ both of $D_{L_{15}}$ and $D_{n_4}$ remain finite as
$1/N \to 0$ whereas for $\theta \le 45^{\circ}$ the dimer order
parameters vanish. We conclude that the ground state is the dimer
phase for $J_0/J_2 > 1$.
\begin{figure}
\centering\epsfig{file=1D_L15_2pt_corr.eps,clip=1,width=0.9\linewidth,angle=0}
\centering\epsfig{file=1D_n4_2pt_corr.eps,clip=1,width=0.9\linewidth,angle=0}
\caption{(a) The two point correlations $\langle
L_{15}(i_0)L_{15}(i) \rangle$ at $\theta=0^{\circ}$,$15^{\circ}$,
$30^{\circ}$, $45^{\circ}$ and $60^{\circ}$. The dot line is plotted
by the fitting result using $cos(x \pi /2)/x^{1.52}$. The reference
point $i_0$($=40$) is the most middle site of the chain ($N=80$).
The inset indicates that all $S(q)$ for $\theta\le 45^{\circ}$ have
peaks located at $q=41\pi/81 \sim \pi/2$ whereas $\pi$ for
$\theta=60^{\circ}$. (b) $\langle n_{4}(i_0)n_{4}(i) \rangle$ at
$\theta=0^{\circ}$,$15^{\circ}$, $30^{\circ}$ and $60^{\circ}$ and
the fitting uses $\kappa=1.55$.}
\label{fig:corr}
\end{figure}
Next we present the two point correlation functions of $\langle X(i)
X(j) \rangle$, where $X$ is $L_{15}$ and $n_4$, in Fig.
\ref{fig:corr} (a) and (b), respectively. At $\theta>45^\circ$, say,
$\theta=60^\circ$, both correlation functions show exponential decay
due to the dimerization. In the spin liquid regime of $\theta\le
45^\circ$, {\it i.e.} $J_2\ge J_0$, however, all the correlation
functions exhibit the power-law behavior and the same $2k_f$
oscillations with the 4-site period. Their asymptotic behavior can
be written as \begin{eqnarray}} \def\eea{\end{eqnarray} \langle X(i_0) X(i) \rangle \propto \frac{\cos
\frac{\pi}{2} |i-i_0|} {|i-i_0|^\kappa}. \eea Along the $SU(4)_A$
line ($\theta=45^\circ$), the correlations of $L_{15}$ and $n_4$ are
degenerate. The power can be fitted as $\kappa\approx 1.52$, which
is in good agreement with the value of $1.5$ from bosonization
analysis and numerical studies
\cite{affleck1986,WU2005a,yamashita1998,azaria1999}. As $\theta$ is
away from $45^\circ$, the $SU(4)$ symmetry is broken. For the
correlations of $L_{15}$, the values of $\kappa$ decrease as
decreasing $\theta$, which can be fitted as $\kappa=1.41, 1.34,
1.30$ for $\theta=30^\circ, 15 ^\circ, 0^\circ$, respectively. On
the other hand, for the correlations of $n_4$, the values of
$\kappa$ can be fitted as $\kappa=1.55, 1.65, 1.60$ for
$\theta=30^\circ, 15 ^\circ, 0^\circ$, respectively. We also perform
the Fourier transforms of the correlations of $\langle
L_{15}(i_0)L_{15}(i) \rangle$, $S(q)$, and present the results in
the inset of Fig. \ref{fig:corr} (a). $S(q)$ is defined as \begin{eqnarray}} \def\eea{\end{eqnarray}
S(q)=\sum_{i,j}e^{i q(r_i-r_j)}\langle L_{15}(r_i)L_{15}(r_j)
\rangle \eea and $q=m\pi/(N+1)$, where $m=1,2 \cdots, N$ are
integers for OBC. Clearly, in the regime of $\theta \le 45^{\circ}$
all the peaks are located at $q=41\pi/81 \sim \pi/2$, indicating a
$2k_f$ charge density wave. On the other hand, $S(q)$ at
$\theta=60^{\circ}$ appears a peak at $\pi$, which denotes a $4k_f$
charge density wave and is characteristic of the dimerization phase.
\section{The $Sp(4)$ magnetism in 2D square lattice with small sizes}
\label{sect:2D}
\begin{figure} [!hbt]
\centering\epsfig{file=2D_phase.eps,clip=1,width=0.6\linewidth,angle=0}
\caption{Speculated phase diagram of the 2D $Sp(4)$ spin-$3/2$
systems at quarter filling from Ref. [\onlinecite{wu2006a}].
$\theta=\tan^{-1}(J_0/J_2)$. The
$SU(4)_A$ type drawn by the dot line is at $J_0=J_2$
($\theta=45^{\circ}$) whereas $SU(4)_B$ at $J_2=0$
($\theta=90^{\circ}$). Bold letters {\bf A}, {\bf B}, and {\bf C}
represent the plaquette, columnar dimerized and Neel order states,
respectively.}
\label{fig:2Dphasediagram}
\end{figure}
The quantum magnetism of Eq. \ref{eq:so5} in 2D is a very
challenging problem. Up to now, a systematic study is still void.
In the special case of the $SU(4)_B$ line, i.e. $J_2=0$, in the
square lattice, quantum Monte-Carlo simulations are free of the sign
problem, which shows the long-range-Neel ordering but with very
small Neel moments $n_4 =(-)^i L_{15} = (-)^i L_{23}\approx 0.05$
\cite{harada2003}. This result agrees with the previous large-$N$
analysis \cite{zhang2001}. The Goldstone manifold is
$CP(3)=U(4)/[U(1) \otimes U(3)]$ with 6 branches of spin-waves. On
the other hand, on the $SU(4)_A$ line with $J_0=J_2$, an exact
diagonalization study on the $4\times 4$ sites shows the evidence of
the four-site $SU(4)$ singlet plaquette ordering \cite{bossche2000}.
Large size simulations are too difficult to confirm this result. On
the other hand, a variational wavefunction method based on the
Majorana representation of spin operators suggests a spin-liquid
state at the $SU(4)_A$ line \cite{wang2009}. Recently, Chen {\it et
al.} \cite{chen2005} constructed an $SU(4)$ Majumdar-Ghosh model in
a two-leg spin-3/2 ladder whose ground state is solvable exhibiting
this plaquette state. An $SU(4)$ resonant plaquette model in 3D have
also been constructed \cite{xu2008,pankov2007}.
Based on these available knowledge, a speculated phase diagram was
provided in Ref. [\onlinecite{wu2006a}] as shown in Fig.
\ref{fig:2Dphasediagram}.
The Neel order state {\bf C} is expected to extend to a region with
finite $J_2$ instead of only along the $J_2=0$ line.
Furthermore, the plaquette order phase {\bf A} exists not only along the
$SU(4)_A$ line but also covers a finite range including
$\theta=45^{\circ}$.
Between {\bf A} and {\bf C}, there exists an intermediate phase {\bf B}
which renders ordered dimerizations which are two-sites spin singlets.
However, these features have not been tested due to the lack of
controllable analytic and numeric methods for 2D strongly
correlation systems.
For example, quantum Monte Carlo methods suffer notorious
sign problems at $J_2\neq 0$.
In this section, we begin with the cluster of
$2\times 2$ whose ground states can be solved analytically.
Then we perform exact diagonalization (ED) methods for the case
of $4\times 4$ sites and analyze the associated GS profiles for
different values of $\theta$.
Even though the size that we are studying is still small to draw any
conclusion for the thermodynamic limit, it provides valuable information
on the ground state properties.
\subsection{The $2 \times 2$ cluster}
\label{sect:foursite}
\begin{figure}[!htb]
\centering\epsfig{file=N4cluster.eps,clip=1,width=0.85\linewidth,angle=0}
\caption{(a) The GS wavefunctions of the $2 \times 2$ cluster at
various $\theta$. $a$ and $b$ are coefficients depending on $\theta$
and the thick bonds denote the two-site $Sp(4)$ spin singlet states.
(b) The position indices before and after the permutation
$P_{2341}$.} \label{fig:2x2cluster}
\end{figure}
We begin with the $2 \times 2$ cluster, whose ground states can be solved
analytically for all the values of $\theta$.
Such a system contains three $Sp(4)$ singlets, and the
ground states can be expanded in this singlet subspace.
These $Sp(4)$ singlets can be conveniently represented in terms
of the dimer states with the horizontal, vertical, and
cross diagonal configurations
depicted in Fig. \ref{fig:2x2cluster} (a) as
\begin{eqnarray}} \def\eea{\end{eqnarray}
|H\rangle&=&\frac{1}{4}R_{\alpha\beta} \psi^\dagger_\alpha(4)
\psi^\dagger_\beta(1)
R_{\gamma\delta}\psi^\dagger_\gamma(2) \psi^\dagger_\delta(3) |\Omega\rangle, \nonumber \\
|V\rangle&=&\frac{1}{4}R_{\alpha\beta} \psi^\dagger_\alpha(1)
\psi^\dagger_\beta(2)
R_{\gamma\delta}\psi^\dagger_\gamma(3) \psi^\dagger_\delta(4) |\Omega\rangle, \nonumber \\
|C\rangle&=&\frac{1}{4}R_{\alpha\beta} \psi^\dagger_\alpha(1)
\psi^\dagger_\beta(3) R_{\gamma\delta}\psi^\dagger_\gamma(2)
\psi^\dagger_\delta(4) |\Omega\rangle,
\eea
where $R$ is the charge
conjugation matrix define in Eq. \ref{eq:R4}.
These states are linearly independent but are not orthogonal to each
other, satisfying $\avg{H|V}=\avg{V|C}=\avg{C|H}=-\frac{1}{4}$.
Under the permutation of the four sites $P_{(2341)}$, or a rotation
at $90^{\circ}$ as shown in Fig. \ref{fig:2x2cluster} (b), they
transform as
\begin{eqnarray}} \def\eea{\end{eqnarray}
P_{2341}|H\rangle =|V\rangle, ~ P_{2341}|V\rangle
=|H\rangle, ~ P_{2341}|C\rangle =|C\rangle.
\eea
At $\theta=45^{\circ}$, i.e., the $SU(4)_A$ case, the ground state (GS) is
exactly an $SU(4)$ singlet over the sites $1$ to $4$:
\cite{li1998,chen2005}
\begin{eqnarray}} \def\eea{\end{eqnarray} \label{eq:su4singlet}
|\Psi^s_{SU(4)}\rangle=\frac{1}{\sqrt{4!}} \sum_{\mu \nu \tau \xi}
\varepsilon_{\mu \nu \tau \xi}
\psi^{\dag}_{\mu,1}\psi^{\dag}_{\nu,2}\psi^{\dag}_{\tau,3}\psi^{\dag}_{\xi,4}
|\Omega \rangle, \eea
where the indices
$\mu$,$\nu$,$\tau$,$\xi$ run over $\pm \frac{3}{2},\pm \frac{1}{2}$;
$|\Omega \rangle$ represents the vacuum state; $\varepsilon_{\mu \nu
\tau \xi}$ is a rank-four fully antisymmetric tensor.
It can also be represented as the linear combination of the dimer states as
\begin{eqnarray}} \def\eea{\end{eqnarray}
|\Psi^s_{SU(4)}\rangle=\sqrt{\frac{2}{3}} \big(|H\rangle +|V\rangle+
|C\rangle \big),
\eea
which is even under the rotation operation $P_{2341}$.
We find that in the entire range of
$0\le\theta<54^\circ$, the GS wavefunctions remain even under such
a rotation $P_{2341}$, whose wavefunctions can be represented as
\begin{eqnarray}} \def\eea{\end{eqnarray}
|\Psi\rangle= a\big(|H\rangle +|V\rangle\big)+ b|C\rangle,
\label{eq:GSWF}
\eea
where $a$ and $b$ are coefficients depending on
the values of $\theta$. In fact, the overlaps between GS
wavefunctions Eq. \ref{eq:GSWF} and the $SU(4)$ singlet state $|
\Psi_{SU(4)}\rangle$ are larger than $0.98$ at $\theta<54^\circ$.
At $\theta>54^{\circ}$, a level crossing occurs and the
GS wavefunction changes to
\begin{eqnarray}} \def\eea{\end{eqnarray}
|\Psi^s_{SU(4)}\rangle= \sqrt{\frac{2}{3}} \Big( |H\rangle
-|V\rangle \Big),
\eea
which is independent of $\theta$ and odd
odd under the rotation $P_{2341}$.
Combining the above observations, we identify that there are two
competing states in the system.
The boundary is located at $\theta=54^{\circ}$.
Next we turn to analyze large size systems.
\subsection{The Low energy spectra for the $4\times 4$ cluster}
\label{sect:2Dspectra}
In this subsection we study a larger system size of $N=4\times 4$.
Both $L^{tot}_{15}=\sum_i L_{15}(i)$ and $L^{tot}_{23}=\sum_i L_{23}(i)$
are good quantum numbers, which can used to reduce the Hilbert space.
The dimension of the Hilbert space
in the $(L^{tot}_{15},L^{tot}_{23})=(0,0)$ sector goes up to $165$ million.
On the other hand, the lowest multiplet states are located
in the sector of $(L^{tot}_{15},L^{tot}_{23})=(0,\pm 1)$ or
$(\pm 1,0)$ and the corresponding dimension is about $147$ million.
The dimensions of the subspace are too large to perform
diagonalization.
Nevertheless, by using translational symmetry, the dimension of the
Hilbert space reduces to $10$ million such that ED calculations
become doable.
The ground states are always in the sector of total momentum
$\vec K=(0,0)$, and are $Sp(4)$ singlets.
In the following, except for the specific mention in Sec.
\ref{sect:plaquette}, the systems are considered under periodic
boundary conditions.
\begin{figure}[!htb]
\centering\epsfig{file=N16spectra-a.eps,clip=1,width=0.55\linewidth,angle=0}
\centering\epsfig{file=N16spectra-b.eps,clip=1,width=0.41\linewidth,angle=0}
\caption{(a) The low-lying states for the $4\times 4$ cluster
at various values of $\theta$.
The dimensions of the corresponding $Sp(4)$ representations $d$
are marked.
The GS wavefunctions are always $Sp(4)$ singlets.
(b) The zooming-in around $\theta\approx 63^\circ$ exhibiting
various energy level crossings.
}
\label{fig:2Dspectra}
\end{figure}
The low-lying energy spectra for the $N=4 \times 4$ clusters
for $0<\theta<90^\circ$ are displayed in Fig. \ref{fig:2Dspectra}.
The ground states for all the values of $\theta$ are $Sp(4)$ singlets with
Casimir $C=0$, and that at $\theta=45^\circ$ is an $SU(4)$ singlet.
The lowest excited states are also $Sp(4)$ singlet states at
$\theta<63^\circ$.
The lowest spin multiplets appear as the 14-fold degenerate $Sp(4)$
symmetric tensor states with $C=10$.
A level crossing of the lowest excited states appears around
$\theta = 63^{\circ}$ implying that there exists competing phases nearby.
At $\theta>63^\circ$, the lowest excited states become 5-fold degenerate
$Sp(4)$ vector states with the Casimir $C=4$.
Another 10-fold degenerate states, which form the $Sp(4)$ adjoint
representation with $C=10$, appear as the next lowest excited states.
At the $SU(4)_B$ line, i.e., $\theta=90^\circ$, these two sectors
merge into the 15-fold degenerate states
forming the adjoint representation of the $SU(4)$ group whose
$SU(4)$ Casimir is $C=8$.
In Sec. \ref{sect:exact}, the appearance of the $Sp(4)$ singlet as
the lowest excited states in the small size systems implies the
dimerization in the thermodynamic limit.
This is confirmed in the large size DMRG results in Sec. \ref{sect:DMRG}.
Similarly, in the case of the $4\times 4$ cluster, the lowest excited
states are also $Sp(4)$ singlet at $\theta<63^\circ$.
This also suggests the spin disordered ground state with broken
translational symmetry in the thermodynamic limit.
Moreover, the gap between the GS and lowest
singlet excited state is very small in a narrow regime (roughly
$50^{\circ} \sim 60^{\circ}$), which implies that an intermediate
phase may exist exhibiting a different translational symmetry
breaking pattern from that with small values of $\theta$. However,
unlike the 1D case where we can justify the dimerization through
finite-size scaling of the vanishing of the $Sp(4)$ singlet-singlet
gap, it is impossible in 2D to detect the presence of the dimer
states or plaquette states from the exact diagonalization results.
Thus we will resort to other approaches to investigate the GS
profile in the following sections.
\begin{figure}[!htb]
\centering\epsfig{file=2D_dispersion.eps,clip=1,width=0.9\linewidth,angle=0}
\caption{The energy dispersion $E(\vec K)-E_0$ v.s. $\theta$
for the $4 \times 4$ cluster.
$\Gamma$, M, X are the high symmetry points for the many-body
ground state momenta, corresponding to $\vec K=(0,0)$, $(\pi,\pi)$ and
$(\pi,0)$ respectively, in the first Brillouin zone.}
\label{fig:2Ddispersion}
\end{figure}
To further clarify, in Fig. \ref{fig:2Ddispersion} we present the
spectra of lowest energy states at each crystal momentum of
$\Gamma=(0,0)$, $X=(\pi, 0)$, and $M=(\pi, \pi)$, respectively.
At $\theta \le 63^{\circ}$, the states at the $X$-point are lower than
those at $M$-point, which are $Sp(4)$ singlets with the Casimir $C=0$.
These lowest singlet excitations along $(\pi,0)$ or $(0,\pi)$
would allow the GS to shift a lattice constant along $x$ or $y$-direction,
if the gap between these singlets vanishes in the thermodynamic limit.
It would implies a four-fold degeneracy in the thermodynamic limit
breaking translational symmetry.
In comparison, as $\theta \ge 72^{\circ}$, the energy of states at
the $M$-point are lower than those at the $X$-point, which are spin
multiplet with 10-fold degeneracy and the $Sp(4)$ Casimir $C=6$.
Actually, these states are not the lowest excited states which are
5-fold degenerate located at the $\Gamma$-point. Nevertheless, their
energy splitting from the 10-fold states is very small as shown in
Fig. \ref{fig:2Dspectra}. In the thermodynamic limit, inspired by
the QMC result of the occurrence of the long-range ordering in the
$SU_B(4)$ case, we infer the long-range staggered Neel ordering of
the $Sp(4)$ spin operators $L_{ab}$ and a long-range uniform
ordering of $Sp(4)$ vector operators $n_a$. Thus we infer a phase
transition from spin disordered ground state to the Neel-like state
breaking $Sp(4)$ symmetry.
Let us make an analogy with the 2D spin-$\frac{1}{2}$ $J_1$-$J_2$ model
\cite{dagotto1989a,dagotto1989b}.
In that case, the behavior of the low-lying energy levels indicates
that the lowest excited states with nonzero momentum are triplet
while the system is a magnetic Neel ($J_2/J_1 \lesssim 0.4$) and
collinear state ($J_2/J_1 \gtrsim 0.6$), corresponding to ${\bf
K}=(\pi,\pi)$ and $(\pi,0)$, respectively.
However, there exists an intermediate phase in $0.4 < J_2/J_1 < 0.6$,
where the GS is a magnetic disordered state and the lowest excited
state with nonzero momentum, $\vec K=(\pi,0)$, is singlet.
In this region it has been conjectured that the GS is a dimerization
state or a spin-liquid (resonated-valence-bond state).
Similarly, the low-lying energy behavior in our model implies
that the GS is non-magnetic at $\theta < 63^{\circ}$.
On the other hand, at $\theta \ge 63^{\circ}$, the GS has spinful
excitations and is relevant to the Neel state.
\subsection{The magnetic structure form factor}
\label{sect:mag}
\begin{figure}[!htb]
\centering\epsfig{file=2D_mag_L.eps,clip=1,width=0.9\linewidth,angle=0}
\centering\epsfig{file=2D_mag_n.eps,clip=1,width=0.9\linewidth,angle=0}
\caption{ The magnetic structure factors for the $4
\times 4$ cluster.
(a) The structure factors $S_L(\vec q)$ for the $Sp(4)$ generator sector
The inset is the comparison between $S_L(\pi,0)$
and $S_L(\pi,\pi)$ versus $\theta$.
(b) The $Sp(4)$ vector structure factor $S_n(\vec{q})$.
} \label{fig:neel}
\end{figure}
In this subsection, we present the results of the magnetic structure
form factors for the $N=4\times 4$ cluster. Two different structure
form factors $S_L(\vec q)$ and $S_n(\vec q)$ are defined for the
$Sp(4)$ generator and vector channels, respectively, as \begin{eqnarray}} \def\eea{\end{eqnarray}
\label{eq:stucr} S_L(\vec{q})&=&\frac{1}{g_L N^2}\sum_{i,j, 1\le
a<b\le5} e^{i\vec{q} \cdot (\vec{r}_i-\vec{r}_j)}
\langle G| L_{ab}(i) L_{ab}(j)|G \rangle, \nonumber \\
S_n(\vec{q})&=&\frac{1}{g_n N^2}\sum_{i,j, a=1\sim 5} e^{i\vec{q}
\cdot (\vec{r}_i-\vec{r}_j)}\langle G| n_a(i) n_a(j) |G\rangle,
\eea where the normalization constants $g_L=10$ and $g_n=5$.
$S_L(\vec q)$ and $S_n(\vec q)$ are the analogy of the Fourier
transformation of $\langle G| \vec{S}_i \cdot \vec{S}_j | G \rangle$
in $SU(2)$ systems. If the long-range magnetic order appears, the
magnetic structure factor converges to a finite value in the
thermodynamic limit \cite{harada2003,schulz1992}.
The ED results of $S_L(\vec{q})$ for the $4 \times 4$ cluster
are presented in Fig. \ref{fig:neel} (a).
As $\theta\lesssim 60^\circ$,
$S_L(\vec q)$ distributes smoothly over all the momenta, and its
maximum is located at $\vec q =(\pi, \frac{\pi}{2})$, which is
slightly larger than other values of $\vec q$. In contrast, when
$60^\circ\lesssim\theta\le 90^\circ$, $S_L(\vec q)$ peaks at $\vec
K_M=(\pi,\pi)$.
The $Sp(4)$ vector channel structure factor $S_n(\vec q)$ is depicted in
Fig. \ref{fig:neel} (b).
At small values of $\theta$, it peaks at the M-point exhibiting
a dominate correlation at the momentum $(\pi,\pi)$.
As $\theta \gtrsim 60^\circ$, the peak changes to the $\Gamma$
point and the $M$-point becomes a minimum.
Along the $SU(4)_B$ line with $\theta=90^\circ$,
$S_n (\vec q)=S_L (\vec q + \vec K_M)$
due to the staggered definition of $Sp(4)$ vectors $n_a$
in Eq. \ref{eq:SU4B}.
This relation between $S_n(\vec{q})$ and $S_L(\vec{q}+\vec K_M)$
is consistent with the previous observation on the low-energy
spectra in Fig. \ref{fig:2Dspectra}. As $\theta \ge 60^{\circ}$
there are two nearly degenerate excited states beyond the GS, having
the total momentum of $(0,0)$ and $(\pi,\pi)$.
They correspond to the 5d vector representation with $C=4$ and the
10d tensor representation with $C=6$ in the $Sp(4)$ symmetry,
respectively.
The contributions to $S_{n}(\vec K_\Gamma)$ and $S_L(\vec K_M)$ mainly
come from the matrix
elements between the ground state and the 5d vector states,
and 10d antisymmetric tensor states, respectively.
On the other hand, in the case of $SU(4)_A$ with
$\theta=45^{\circ}$, $S_n(\vec{q})=S_L(\vec{q})$ for each $\vec{q}$.
These features highlight that the dominant Neel correlation of the
$Sp(4)$ generators $L_{ab}$'s not only exhibits along the $SU(4)_B$
line but also extends to a finite regime with non-zero values of
$J_2$. In the same parameter regime, the $Sp(4)$ vectors $n_a$'s
exhibit dominant uniform correlations. The critical value of
$\theta$ of the onset of the outstanding $S_L(\pi,\pi)$ is in good
agreement with the location of the level crossing shown in Fig.
\ref{fig:2Dspectra}, implying a transition of the GS from a non-Neel
state to a Neel type. The inset in Fig. \ref{fig:neel} (a) compares
the $S_L(\vec{q})$ behavior at $\vec{q}=(\pi,0)$ and $(\pi,\pi)$
versus $\theta$. $S_L(\pi,0)$ changes little as varying $\theta$.
Therefore, it is inferred that only the Neel-type order exists at
$\theta$ close to 90$^\circ$. The magnetic ordering at $(\pi,0)$
should not appear in the 2D $Sp(4)$ system.
Next one may raise a natural question: what is the spin pattern for
the Neel-order state as $\theta \rightarrow 90^\circ$?
According to Eq. \ref{eq:SU4B}, its classic energy can be minimized
by choosing a staggered configuration for
$\avg{G|L_{15}(i)|G}=\avg{G|L_{23}(i)|G}=(-)^i \frac{1}{2}$
and a uniform configuration of $\avg{G|n_4(i)|G}=\pm\frac{1}{2}$.
These correspond to the staggered arrangement
in the 2D lattice by using the two components of
$F_z=\pm \frac{3}{2}$, or by using the other two components of
$F_z=\pm \frac{1}{2}$.
These different classic Neel states can be connected by an
$Sp(4)$ rotation.
\subsection{The columnar dimer correlations}
\label{sect:dimer}
\begin{figure}[!htb]
\centering\epsfig{file=sus_dim_1.eps,clip=1,width=0.31\linewidth,angle=0}
\centering\epsfig{file=sus_rot.eps,clip=1,width=0.285\linewidth,angle=0}
\centering\epsfig{file=sus_dim_2.eps,clip=1,width=0.33\linewidth,angle=0}
\caption{The susceptibilities defined in Eq. \ref{eq:sus} with
respect to the perturbations $O_{dim}$ and $O_{rot}$ for the
$N=4\times 4$ cluster. (a) $\chi_{dim}(\vec Q)$ versus $\theta$ at
$\vec Q=(\pi,0)$; (b) $\chi_{rot}$ versus $\theta$; (c)
$\chi_{dim}(\vec Q)$ versus $\theta$ at $\vec Q=(\pi,\pi)$. In both
cases, a small value of $\delta=0.01$ is taken to evaluate the
susceptibilities. Both susceptibilities exhibit peaks around
$\theta\approx 60^{\circ} \sim 70^{\circ}$.} \label{fig:2Dpert}
\end{figure}
In this subsection, we discuss the possibility of the dimer-ordered state
at intermediate values of $\theta$.
We define the susceptibility to a symmetry breaking perturbation as
\begin{eqnarray}} \def\eea{\end{eqnarray}
\chi(\delta)=-\frac{2 [e(\delta)-e(0)]}{\delta^2},
\label{eq:sus}
\eea
where $e(0)$ is the GS energy per site given by the
Hamiltonian Eq. (\ref{eq:so5}) and $e(\delta)$ is
Eq. \ref{eq:so5} plus the corresponding perturbation
term $-\delta \hat{O}$\cite{santoro1999,capriotti2000}.
In the presence of long-range ordering, the corresponding susceptibility
$\chi=\lim_{\delta\to 0}\chi(\delta)$ will diverge in the thermodynamic limit.
It has been demonstrated that this approach can efficiently distinguish
dimerized and non-dimerized phases in the 1D $J_1$-$J_2$
spin-$\frac{1}{2}$ chain \cite{capriotti2000}, in which the phase
boundary $J_2/J_1 \approx 0.24$ between these two phases.
Here we employ the same method to study the dimerization correlations.
Although with small size calculations, we are unable to determine
the existence of long range order, it is still instructive
to observe the feature of $\chi$.
We have used it to test the 1D $Sp(4)$ system with the perturbation
term of $\hat{O}=\sum_i (-1)^i H_{ex}(i,i+1)$.
At $\theta = 60^{\circ}$, we found the dramatic growing behavior of
$\chi(\delta)$ upon decreasing $\delta$ and increasing the system size,
which leads to a divergent $\chi$ in the thermodynamic limit.
On the other hand, $\chi(\delta)$ at $\theta = 30^{\circ}$
has no tendency of divergence over decreasing $\delta$.
This observation is consistent with our previous analytical and numerical
studies: the 1D $Sp(4)$ system is either a gapless uniform liquid as
$\theta \le 45^{\circ}$ or a gapped dimerized state with
the breaking of translation symmetry at $\theta > 45^{\circ}$.
Next we apply this method to the 2D system with the size of $4 \times 4$,
and define two susceptibilities $\chi_{dim}(\vec Q)$ and $\chi_{rot}$
for two perturbations of $\hat O_{dim}(\vec Q)$ and $\hat O_{rot}$ as
\begin{eqnarray}} \def\eea{\end{eqnarray}
\label{eq:perturbation1}
\hat{O}_{dim} (\vec Q)&=&\sum_{i} \cos (\vec{Q} \cdot \vec{r}_i)
H_{ex}(i,i+\hat{x}), \\
\label{eq:perturbation2} \hat{O}_{rot}&=&\sum_{i}
[H_{ex}(i,i+\hat{x})-H_{ex}(i,i+\hat{y})], \eea where $H_{ex}(i,j)$
is defined as one bond of the Hamiltonian Eq. \ref{eq:so5} without
summation over $i$ and $j$. Let us set $\vec Q=(\pi, 0)$, thus
$\chi_{dim}(\pi,0)$ corresponds to the instability to the columnar
dimer configuration. Eq. \ref{eq:perturbation1} and Eq.
\ref{eq:perturbation2} break the translational symmetry along the
$x$-direction and rotational symmetry, respectively. The plaquette
ordering maintains the 4-fold rotational symmetry, thus will lead to
the divergence of $\chi_{dim}(\pi,0)$ but not $\chi_{rot}$. The ED
results for the susceptibilities with respect to the two
perturbations versus $\theta$ in Fig. \ref{fig:2Dpert} (a) and (b),
respectively. A small value of $\delta=0.01$ is taken. Both
susceptibilities exhibit a peak at $\theta$ from $60^\circ$ to
$70^\circ$, which implies a tendency to breaking both translational
and rotational symmetries. This shows that the columnar dimerization
instead of the plaquette ordering is a promising instability in this
regime in the thermodynamic limit. We have also calculated the
susceptibility of $\chi_{dim}(\vec Q)$ for $\vec Q=(\pi,\pi)$ which
corresponds to the instability to the staggered dimer configuration
in Fig. \ref{fig:2Dpert} (c). Although the magnitudes of
$\chi_{dim}(\pi,\pi)$ are smaller than $\chi_{dim}(\pi,0)$ and
$\chi_{rot}$, it suddenly raises up around $\theta=60^{\circ}$.
\subsection{The plaquette form factor}
\label{sect:plaquette}
\begin{figure}[!htb]
\centering\epsfig{file=localcasimir.eps,clip=1,width=0.75\linewidth,angle=0}
\caption{$C(\vec r)$ defined in Eq. \ref{eq:localspin} versus $\theta$.
The positions of plaquette $A$, $B$ and $C$ are defined on the right
schematics. $A$ (blue) is located at the corner whereas $C$
(black) in the most middle one.}
\label{fig:Fig5}
\end{figure}
In this subsection, we consider the plaquette type correlation. The
ground states of the $4\times 4$ cluster at $\theta\lesssim
60^\circ$ signal a different class from that in $\theta \ge
63^{\circ}$ in which the lowest excited states are spin multiplet
with $\vec K=(\pi,\pi)$. Here, the lowest excited states remain
$Sp(4)$ singlets with $\vec K=(\pi,0)$ or $(0,\pi)$. To further
elucidate the ground state profile, we define the local Casimir for
the plaquette centered at $\vec r$, \begin{eqnarray}} \def\eea{\end{eqnarray} \label{eq:localspin} C(\vec
r) =\avg{G|\sum_{1\le a<b\le 5}\big\{\sum_{i} L_{ab}(i)\big\}^2 |G},
\eea where $i$ runs over the four sites of this plaquette. The
$SU(2)$ version of Eq. \ref{eq:localspin} has been used to classify
competing dimer and plaquette orders \cite{richter1996}. If the GS
exhibits a strong plaquette pattern, for instance, indicated as
phase {\bf A} in Fig. \ref{fig:2Dphasediagram}, the magnitudes of
$C(\vec r)$ will have obvious spatial variations between
nearest-neighboring plaquette. This is analogous to the 1D
dimerization picture in Fig. \ref{fig:nncorr} (a), where the
nearest-neighboring spin-spin correlations exhibit strong and weak
alternately in magnitude. When the spins around a plaquette are
strongly bound to form an $SU(4)$ singlet, $C(\vec r)$ should be
close to zero.
Fig. \ref{fig:Fig5} depicts the behavior of $C(\vec r)$ at various
values of $\theta$ for the $4\times 4$ cluster. In order to
explicitly reflect the plaquette formation, we use {\it open}
boundary conditions rather than periodic boundary conditions. In
this case only $C_{4v}$ point group symmetry is applicable in the
ED. The $C(\vec r)$ for the corner plaquette $A$ is much smaller than
$\frac{1}{5}$ of those at the center $C$ and the middle of the edge $B$ for
small values of $\theta$. This is in sharp contrast to the 2D
spin-$1/2$ model which renders $C(A)=0.545$, $C(B)=1.015$ and
$C(C)=1.282$, which only show the difference at order of 1. The
comparison suggests the pinning-down plaquette state in the 2D
$Sp(4)$ system under the open boundary.
We observe that $C(A)$ and $C(B)$ decrease while $\theta$ goes
beyond $60^{\circ}$.
It accounts for the formation of the plaquette-type pattern weakens or
even vanishes.
Combined the above observations, it is likely that for $\theta < 60^{\circ}$
the GS has a strong plaquette-like correlation, that could be the resonate
plaquette state proposed by Bossche {\it et al.}\cite{bossche2000}
or a certain spin-liquid. It does survive not
only along the $SU(4)_A$ line but also in a finite regime.
Nevertheless, we have to emphasize that this picture cannot be
conclusively determined due to finite size effects and further
larger size calculations are needed to confirm.
\section{Conclusion}
In conclusion, we study an $Sp(4)/SO(5)$ spin Heisenberg model which
can be realized by the large spin ultra-cold fermions with
$F=\frac{3}{2}$. The $Sp(4)$ Heisenberg model describing the
magnetic exchange at the insulating state of quarter-filling is
simulated by exact diagonalization and DMRG. In 1D, our numerical
results are in agreement with previous analytic studies. There are
two competing phases: a gapped dimer phase with spin gap at $\theta>
45^{\circ}$ and a gapless spin liquid at $\theta \le 45^{\circ}$.
The phase boundary is identified as $\theta=45^{\circ}$ which
belongs to $SU(4)_A$-type symmetry. In the gapless spin liquid
phase, the static correlation functions decay algebraically with
four-site periodicity oscillations.
We also investigate the $Sp(4)$ spin model on a 2D square lattice up
to $16$ sites by means of exact diagonalization methods. Our
numerical results show three competing correlations: Neel-type,
plaquette formation and columnar spin-Peierls dimerization,
depending on $\theta$'s. Such observation can have phase behavior
analogy in comparison with the speculated phase diagram depicted in
Fig. \ref{fig:2Dphasediagram}. Due to the finite size effects,
however, we are unable to conclusively identify the existence of
these phases and the phase boundaries based on the small cluster.
More numerical studies are necessary to further explore the phase
diagram in the thermodynamic limit.
\acknowledgements H. H. H. is grateful to helpful discussions with
Stephan Rachel and computational facilities from Tunghai University.
H. H. H. also appreciates Zi Cai and Cheng-Chien Chen for fruitful
discussions and suggestions on exact diagonalization techniques. H.
H. H. and C. W. are supported by NSF under No. DMR-0804775. Y. P. W.
is supported by NSFC and 973-project of MOST China.
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 3,355
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\section{Big Macs and Correlation Coefficients}
One might think that if the correlation coefficient between two variables is high, those variables convey the same information, and thus can be used interchangably --- but this line of reasoning is erroneous. A simple example helps to illustrate. In Table~\ref{table:hamburger_correlation}, we provide two statistics for each of 22 countries: the cost of a Big Mac in local currency, and the mean hourly wage in local currency. The Pearson product-moment correlation coefficient, $\rho$, between these two statistics is 0.99. Since $\rho$ is nearly 1, one might conclude that we can use hourly wages to predict burger prices with high accuracy and one might question why anyone should waste his or her time collecting burger price information if the hourly wage rates are already known. But take a look at the column ``Real Wage''. The real wage --- the ratio of burger prices to hourly wages --- is the variable of economic interest, since it measures a worker's purchasing power. We see that real wages differ dramatically across countries. In Denmark, a worker making the mean hourly wage need only work for seven minutes to earn a Big Mac, whereas in China, a worker making the mean hourly wage must work for nearly two hours to afford a burger.
\begin{table}
\centering
{\footnotesize
\begin{tabular}{ l r r r }
Country & Burger Price & Hourly Wage & Real Wage \\
\hline
Denmark & 24.75 & 211.13 & 8.53 \\
Australia & 3.00 & 19.86 & 6.62 \\
New Zealand & 3.60 & 21.94 & 6.09 \\
Switzerland & 6.30 & 37.85 & 6.01 \\
United States & 2.54 & 14.32 & 5.64 \\
Britain/UK & 1.99 & 11.15 & 5.60 \\
Germany & 2.61 & 14.32 & 5.49 \\
Canada & 3.33 & 16.78 & 5.04 \\
Singapore & 3.30 & 15.65 & 4.74 \\
Sweden & 24.00 & 110.90 & 4.62 \\
Hong Kong & 10.70 & 44.26 & 4.14 \\
Spain & 2.37 & 8.59 & 3.62 \\
South Africa & 9.70 & 30.86 & 3.18 \\
France & 2.82 & 8.50 & 3.01 \\
Poland & 5.90 & 11.80 & 2.00 \\
Hungary & 399.00 & 704.34 & 1.77 \\
Czech Rep.& 56.00 & 85.34 & 1.52 \\
Brazil & 3.60 & 4.58 & 1.27 \\
South Korea & 3000.00 & 3134.00 & 1.04 \\
Mexico & 21.90 & 17.61 & 0.80 \\
Thailand & 55.00 & 31.69 & 0.58 \\
China & 9.90 & 5.56 & 0.56 \\
\hline
\bf{mean} & 166.01 & 207.32 & 3.72 \\
\bf{std. dev.} & 638.49 & 670.63 & 2.29 \\
\bf{std. dev./mean} & 3.85 & 3.23 & 0.62 \\
\end{tabular}
}
\caption{Hourly Wage versus Real Wage. Burger price and hourly wage are in the local currency. Burger price is the average cost of a Big Mac. The units for Real Wage are burgers per hour. Data comes from Behar's ``Who earns the most hamburgers per hour?" \cite{Behar2008burger}. The correlation coefficient between burger price and hourly wage is $\rho=0.99$.}
\label{table:hamburger_correlation}
\end{table}
In our hamburger example, it is pretty clear what is going on. The denominations of currencies vary immensely and arbitrarily. It is indeed true that differences in real wages are small relative to differences in currency denominations. But it is not true that after correcting for differences in denominations, differences in real wages are negligible. One way to think of this is that the greatest part of the variation in hourly wage comes from the relatively unimportant fact that currency is denominated differently in different countries. The standard deviation of hourly wages in nominal terms is about 300 times as large as that in real terms. Although the standard deviation of real wages across countries is tiny compared to that of nominal exchange rates, this variation is far more important for the quality of life of workers. Thus, one would be wrong to conclude from the high correlation coefficient that the real wage is constant across countries. Quite the contrary; the standard deviation of this ratio is $62\%$ of the mean.
\section{Davis's analysis}
Davis (2008) fell into a similar trap in his recent comparison of journal rankings by Eigenfactor score and by Impact Factor or Total Citations \cite{Davis2008JASIST}.
In that paper, Davis aimed to determine whether measures of ``popularity'' such as Impact Factor and total citation differ substantially from measures of "prestige" such as the journal PageRank \cite{Bollen2006js} and the Eigenfactor metrics \cite{Bergstrom2007CRL}\footnote{The same issue was the subject of a more comprehensive analysis by Bollen and colleagues in 2006 \cite{Bollen2006js}. In that paper, Bollen and colleagues compare weighted PageRank with Impact Factor and with Total Citations to explore differences between popularity and prestige. Weighted PageRank and Eigenfactor are both variants of the PageRank algorithm. See also Pinski and Narin (1976) for an early attempt at constructing prestige-based measures using citation data, and Vigna (2009) for a discussion of how Pinski and Narin's measure differs from current approaches \cite{Pinski1976InfoProcManag, Vigna2009Spectral}.}. To do so, Davis conducted a regression analysis of Eigenfactor scores on Total Citations\footnote{In his paper Davis also looked at the correlation coefficient between Eigenfactor and Impact Factor scores. This $\rho$ value is lower ($\rho=0.86$), but the point is not so much what this value is, but rather that the comparison makes little sense. Eigenfactor is a measure of total citation impact, and should (all else equal) scale with the size of the journal. Impact factor is a measure of citation impact per paper, and all else equal should be independent of journal size. If one wants to compare an Eigenfactor metric with the Impact Factor, one should use the Article Influence Score, which is a per-article measure like Impact Factor. We explore this comparison later in the paper.} for a set of 165 medical journals\footnote{Contrary to what is specified in that paper, Davis appears to have sampled from both the ``Medicine General and Internal" and ``Medicine Research and Experimental" fields, not merely the former category. In our analysis of the same subfields of medicine, we included 168 journals (of the 171 journals in this field); we eliminated 3 journals because they had an Impact Factor and/or Article Influence score of zero}. Davis reports that the correlation coefficient between 2006 Eigenfactor scores and Total Citations\footnote{Davis appears to have used citations (from year 2006) to all articles published in the journals he selected. A cleaner comparison, which would have resulted in a higher correlation, would have been to extract citations (from year 2006) to articles published in the past five years, since the Eigenfactor score takes into account only the past 5 years' citations.} is $\rho=0.9493$. Based on this result, Davis concluded that:
\begin{quote}
``At least for medical journals, it does not appear that iterative
weighting of journals based on citation counts results
in rankings that are significantly different from raw citation
counts. Or, stated another way, the concepts of popularity (as measured by total citation counts) and prestige (as measured by a weighting mechanism) appear to provide very similar information.''
\end{quote} But is Davis right? Is it really the case that if you know the number of citations, you would be wasting your time by finding the Eigenfactor score? Not at all.
First, Davis made a classic statistical error --- cautioned against by Karl Pearson in 1897 --- in comparing two measures with a common factor \cite{Pearson1897}. Second, Davis suggests that a high correlation coefficient implies that there is no significant difference between two alternative measures; this is simply false. We address these issues in turn.
\section{Journal Sizes and Spurious Correlations}
There are enormous differences in the size of academic journals, and these differences swamp the patterns that Davis was seeking in his analysis. The JCR indexes journals that range in size from tiny (\textit{Astronomy and Astrophysics Review} has published 13 articles over the previous five years) to huge (\textit{The Journal of Biological Chemistry} has published 31,045 articles over the same period) with a coefficient of variation, $c_v$, equal to 1.910. Per-article citation intensity varies less, whether measured by Article Influence or by Impact Factor (AI: range 0--27.5, coefficient of variation$=1.785$; IF: range 0--63.3, coefficient of variation$=1.548$).
We can formalize these observations by decomposing Davis' regression of Eigenfactor on
Total Citations. Davis regresses
\begin{center}
\ Log($EF_i$) \makebox[20px]{vs} Log($CT_i$),
\end{center}
\noindent where $EF_i$ is the Eigenfactor score for journal $i$ and $CT_i$ is the Total Citations received by journal $i$. We let $AI_i$ be the Article Influence for journal $i$, and $N_{i,5}$ is the total number of articles published over the last five years for journal $i$. Then by definition
\begin{eqnarray}
\log(EF_i)& = & \log(c_1 \times AI_i \times N_{i,5}) \nonumber \\
&=& \log c_1 + \log AI_i + \log N_{i,5}, \nonumber
\end{eqnarray}
\noindent where $c_1$ is a scaling constant that normalizes the Article Influence scores so that the mean article in the JCR has an Article Influence score of 1.00. Similarly, letting $IF_i$ be the Impact Factor for journal $i$,
\begin{eqnarray}
\log(CT_i)&\approx& \log(c_2 \times IF_i \times N_{i,2}) \nonumber \\
&\approx& \log(c_2\, c_3 \times IF_i \times N_{i,5}) \nonumber \\
&=&\log c_2 \,c_3 + \log IF_i + \log N_{i,5} \nonumber
\end{eqnarray}
\bigskip
\noindent where $c_2$ and $c_3$ are additional scaling constants. The scaling constant, $c_2$, accounts for the fact that Davis compared citations for \textit{all} years and not just citations for 2 years. The scaling constant $c_3$ relates the number of articles published in two years to the number of articles published in five years (and thus is approximately 5/2). As a result, Davis is effectively calculating a regression between
\[
\log(\mbox{Article Influence}) + \textbf{log(Total Articles)}\]
and
\[ \log (\mbox{Impact Factor}) +\textbf{log(Total Articles)}.
\]
Having the ``log(Total Articles)'' term on both sides of the regression --- especially given that it varies more than the other two terms --- obscures the relation between the variables that one would actually wish to observe when trying to evaluate the difference between ``popularity'' and ``prestige''.
This pitfall is famous in the history in mathematical statistics. In 1897, two years after pioneering statistician Karl Pearson developed the product-moment correlation coefficient, he presented a paper to the Royal Society in which he noted that fellow biometrician W. F. R. Weldon had made precisely this mistake in the analysis of body dimensions of crustaceans \cite{Pearson1897, Weldon1892}. Explaining this error, Pearson wrote
\begin{quote}``If the ratio of two absolute measurements on the same or
different organs be taken it is convenient to term this ratio an index.
If $u = f_1(x, y)$ and $v = f_2(z, y)$ be two functions of the three variables
$x$, $y$, $z$, and these variables be selected at random so that there exists
no correlation between $x$,$y$, $y$,$z$, or $z$,$x$, there will still be found to
exist correlation between $u$ and $v$. Thus a real danger arises when a
statistical biologist attributes the correlation between two functions,
like $u$ and $v$ to organic relationship.''
\end{quote}
It was to describe this danger that Pearson coined the term {\em spurious correlation} \cite{Pearson1897, aldrich1995correlations}. He imagined a set of bones assembled at random. Based on correlations between measurements that share a common factor, a biologist could easily make the mistake of concluding that the bones were properly assembled into their original skeletons:
\begin{quote}``For example, a quantity of bones are taken from an
ossuarium, and are put together in groups, which are asserted to be
those of individual skeletons. To test this a biologist takes the
triplet femur, tibia, humerus, and seeks the correlation between the
indices femur/humerus and tibia/humerus. He might reasonably
conclude that this correlation marked organic relationship, and
believe that the bones had really been put together substantially in
their individual grouping. As a matter of fact, since the coefficients
of variation for femur, tibia, and humerus are approximately equal,
there would be, as we shall see later, a correlation of about 0.4 to
0.5 between these indices had the bones been sorted absolutely at
random. I term this a spurious organic correlation, or simply a
spurious correlation. I understand by this phrase the amount of
correlation which would still exist between the indices, were the
absolute lengths on which they depend distributed at random.''
\end{quote}
The reason for this correlation will be that some of the random femur and tibia pairs will be combined with a large humerus; in this case both the femur/humerus and tibia/humerus ratio will tend to be smaller than average. Other femur and tibia pairs will be combined with a small humerus; in this case both the femur/humerus and tibia/humerus ratio will tend to be larger than average. Correlation coefficients of the two ratios give the illusion that tibia and femur length covary, even when they in fact do not. For his part, Weldon was forced to concede that nearly 50\% of the correlation he had observed in body measurements was actually due to this effect.
Just over a decade later, another important figure in the development of mathematical statics, G. U. Yule, noted that when absolute values share a common factor, they are just as susceptible to this problem as are "indices" or ratios \cite{Yule1910RoyalStat}:
\begin{quote}``Suppose we combine at random two indices $z_1$ and $z_2$, e.g. two death-rates, and also combine at random with each pair a denominator or population $x_3$. The correlations between $z_1$, $z_2$, and $x_3$ will then be zero within the limits of sampling. But now suppose we work out the total deaths $x_1=z_1 x_3$ and $x_2=z_2 x_3$; the correlation $r_{12}$ between $x_1$ and $x_2$ will not be zero, but positive.''
\end{quote}
This is precisely the form of spurious correlation that arises in Davis's analysis. Per-article popularity as measured by Impact Factor takes the role of $z_1$ in Yule's example, and per-article prestige as measured by Article Influence score takes the role of $z_2$. Total Articles takes the role of Yule's $x_3$. Even if Impact Factor and Article Influence were entirely uncorrelated, Davis still would have observed a high correlation coefficient in his regression of Eigenfactor and Total Citations ($\sim \rho=0.6$ for all journals), because both share number of articles as a common factor. What Davis discovered is not that popularity and prestige are the same thing; he discovered that big journals are big and small journals are small. Because of this wide variation in journal size, one would also observe a high correlation coefficient between pages and total cites, though very few would argue that the former is an adequate surrogate for the latter\footnote{We collected page and citation information for 149 Economics journals in 2006. The correlation coefficient between total pages and total citations is $\rho=0.615$.}.
To avoid this problem, we might want to look at the correlation between popularity \textit{per article} and prestige \textit{per article}. That is, we need to look at the comparison
\begin{center}
Log(Article Influence) \makebox[20px]{vs.} Log(Impact Factor).
\end{center}
Since its inception in January 2007, Eigenfactor.org has provided exactly this information at {\tt http://www.eigenfactor.org/correlation/}, for the entire JCR dataset and also for each individual field of scholarship as defined by the JCR\footnote{Falagas et. al (2008) presented a similar comparison of Impact Factor and the SJR indicator (a per-article measure of prestige) \cite{Falagas2008FASEB}. Waltman and van Eck look at a correlations among a number of bibliometric measures; their discussion of differences between Impact Factor and Article Influence is noteworthy \cite{WaltmanAndvanEck10}.}. Figure~\ref{fig:hist} is a histogram of the correlation coefficients between Impact Factor and Article Influence scores for all 231 categories in the 2006 JCR. The mean for all fields was 0.853 with a standard deviation of 0.099. The field with the lowest correlation coefficient is Communication ($\rho=0.478$). Marine Engineering has the highest correlation ($\rho=0.986$). The sample of medical journals that Davis selected, with $\rho=0.954$, ranks in the 90th percentile when compared to all 231 fields. Correlation coefficients within fields typically exceed the correlation coefficient for all journals together. For all $7,611$ journals considered together, $\rho=0.818$. This value is lower than the mean of individual-field correlation coefficients, which is $\rho=0.853$.
\begin{figure}
\begin{center}
\includegraphics[scale=0.75]{hist_fig03.pdf}
\end{center}
\caption{Histogram of correlation coefficients between Impact Factor and Article Influence scores. This includes all 231 categories in the 2006 Science and Social Science JCR. The mean of all fields is 0.853 (intra-field mean) and the standard deviation is 0.099. The correlation for all journals considered together is $0.818$. The correlation for the field of Medicine as studied by Davis is 0.954. The correlation coefficients for all fields can be found at {\tt {http:/www.eigenfactor.org/correlation/.}}}
\label{fig:hist}
\end{figure}\section{Correlation and significant differences}
To evaluate Davis's claim that Eigenfactor score and Total Citations are telling us the same thing, we can focus on the \textit{ratio} of Eigenfactor score to Total Citations (EF/TC). (When we look at the ratio, the common factor "Total Articles" divides out.) Notice that a journal's EF/TC ratio is a measure of ``bang per cite received" -- that is, how much Eigenfactor boost does this journal receive, on average, when it is cited. In the hamburger example, the corresponding notion is ``burgers per hour," the real wage or purchasing power of an hour's work. Does a high correlation between Total Citations and Eigenfactor score mean that the bang per cite received is about constant? If it is, there really would be no point to looking at Eigenfactor scores instead of Total Citations. So let's see what happens.
\begin{figure}
\begin{center}
\includegraphics[scale=0.75]{EF_ratioB.pdf}
\end{center}
\caption{Ratio of Eigenfactor score to Total Citations. Data are normalized by the median ratio of the data set. The dashed line indicates a ratio of one. The journals are ordered from those with the highest ratio to the lowest. The inset shows only the 168 medical journals from Davis's analysis.}
\label{fig:EF_ratio}
\end{figure}
Figure~\ref{fig:EF_ratio} shows the ratio of Eigenfactor score to Total Citations for every journal in the JCR, and the insert shows just the medical journals. The standard deviation of this ratio is $1.1 \times 10^{-5}$ and the mean is $1.56 \times 10^{-5}$. The standard deviation, in this case, is $71\%$ of the mean. This is even more variable than the Big Mac case! Moreover, there are nearly 1000 journals with twice the mean ``bang per cite".
The thing to notice in both the Big Mac and the journal example is that if you are interested in the ratio of $A$ to $B$ and if $A=ax$ and $B=bx$ for some $x$ with a very high variance relative to that of $a$ and of $b$, you will get a very high $\rho$ value when you regress $B$ on $A$. However, if what really interests you is the ratio $A/B$, you will note that the $x$'s cancel and $A/B=ax/bx=a/b$. Thus, the variance of $x$ has literally nothing to tell you about the variance of the ratio $a/b$. You don't learn about whether $a/b$ is nearly constant or highly variable from looking at the correlation of $B$ on $A$.
If, as Davis claims, Eigenfactor scores do not differ significantly from Total Citation counts, the ratio EF/TC should be constant across different groups of journals. To evaluate this claim, we look at the EF/TC ratios of social journals with those of science journals, with groupings determined by whether a journal is listed in the Social Science JCR or the Science JCR. (Journals listed in both are omitted from the analysis). The mean EF/TC ratio for science journals is $1.42 \times 10^{-5}$, whereas the mean for social science journals is $2.12 \times 10^{-5}$. A Mann-Whitney U test shows that this difference is highly significant, at the $p<10^{-167}$ level.
These differences are not only statistically significant, but also economically relevant. The $49\%$ difference in mean EF/TC ratios indicates that a librarian who uses Total Citations to measure journal value will underestimate the value of social science journals by $49\%$ relative to a librarian who uses Eigenfactor scores to measure value.
There are also significant differences within the sample of journals that Davis considered. Based on the difference between science and social science ratios described above, one might expect medical journals more closely associated with the social sciences, such as those in public health, to have higher-than-average EF/TC ratios. Seven of the publications in Davis's sample of medical journals are cross-listed in the JCR category of public, environmental, and occupational health. Indeed, this group of journals has a $29\%$ higher EF/TC ratio than do the rest of the journals in Davis's sample, again statistically significant (Mann-Whitney U test, $p<.01$).
Note that there is nothing special about this particular comparison between sciences and social sciences; one could test any number of alternative hypotheses and would find significant differences between EF/TC ratios for many other comparisons as well.
\section{The value of visualization}
So, if correlation coefficients are misleading, what is the alternative? First, we argue for a deeper examination of the data. Figure~\ref{fig:b_graphEF} is an example of this strategy\footnote{\textbf{Figure~\ref{fig:b_graphEF} caption:} Journal ranking comparisons by Total Citations and Eigenfactor score. The journals listed are the top $50\%$ from the field of Medicine that Davis analyzed. Journals in the left column are ranked by Total Citations for all years. Journals in the right column are ranked by Eigenfactor score. The lines connecting the journals indicate whether the journal moved up (green), down (red) or stayed the same (black) relative to their ranking by Total Citations. Journal names in black can also be journals that do not exist in both columns.}. Listing the journals in this way, one is able to quickly see the ordinal differences that exist between this highly correlated data. This type of graphical display illustrates the interesting stories that can be lost behind a summary statistic such as the Spearman correlation.
\begin{figure}
\begin{center}
\includegraphics{b_graph_EF.pdf}
\end{center}
\caption{See footnote in text for caption.}
\label{fig:b_graphEF}
\end{figure}
Figure~\ref{fig:b_graphEF} illustrates the ordinal ranks of the top 50\% of the medical journals used in Davis's study. In the left column, the journals in this subfield of medicine are ranked by the total number of citations. In the right column, the journals are ordered by Eigenfactor score. The lines connecting the journals indicate whether the journal moved up (green), down (red) or stayed the same (black) relative to their ranking by Total Citations. The figure highlights the differences between the metrics. For example, \textit{Aviation Space and Environmental Medicine} drops 30 places while \textit{PLoS Medicine} raises 31 places. Davis claims in his paper that the ordering of journals does not change drastically. Figure~\ref{fig:b_graphEF} suggests otherwise.
Figure~\ref{fig:b_graph} compares the ordinal ranking by Impact Factor and Article Influence for 84 journals --- the top-ranked half --- from Davis's study\footnote{\textbf{Figure~\ref{fig:b_graph} caption:} Comparing Impact Factor and Article Influence. The journals shown are from the same field that Davis analyzed (because of limited space, only the top 84 journals are shown). For these 84 journals, the correlation coefficient between IF and AI is $\rho=0.955$. The relative rankings by Impact Factor and Article Influence are listed in the left and right column, respectively. The third column lists the Article Influence scores. The journal names in green indicate those that fare better when ranked by Article Influence; the journal names in red fare better when ranked by Impact Factor. The names in black are journals that exhibit no change or exist outside the range of the journals shown.}. Changes in ranking are even more dramatic when we look at the lower-ranked 84 journals. The correlation coefficient between Impact Factor and Article Influence for these 84 journals is $\rho=0.955$. Despite this high correlation, the figure highlights the fact that the two metrics yield substantially different ordinal rankings.
\begin{figure}
\begin{center}
\includegraphics[scale=1]{b_graph02.pdf}
\end{center}
\caption{See footnote in text for caption.}
\label{fig:b_graph}
\end{figure}
Figure~\ref{fig:b_graph} reveals that the top few journals change in rank less than those further down the hierarchy. For example, going from Impact Factor to Article Influence, the journals in the top ten change in rank by only 1 or 2 positions. By contrast, there are many larger changes further on in the rankings\footnote{Bollen (2006) observed a similar pattern in a series of scatterplots contrasting PageRank and Impact Factor values for all journals \cite{Bollen2006js}. In these scatterplots the rankings of top-tier journals differ relatively little whereas more variation is found in the middle and bottom portions of the hierarchy.}. For example, as we go from Impact Factor to Article Influence, the {Journal of General Inernal Medicine} rises 18 spots to number 19 while \textit{Pain Medicine} drops 35 spots to end up at number 80. These are just two of the many major shifts (in a field with a correlation of 0.955!). These changes in relative ranking would certainly not go unnoticed by editors or publishers.
Furthermore, while ordinal changes are interesting, cardinal changes are often more important. Figure~\ref{fig:c_graphEF} shows the top ten journals from Figure~\ref{fig:b_graphEF} --- those with the least ordinal change from one metric to another --- now in their cardinal positions. Even those journals that do not change ordinal rank from one metric to another may be valued very differently under the two different metrics. For example, {\em Nature Medicine} is the \#2 journal regardless of whether one uses Impact Factor or Article Influence. But under Impact Factor, it has barely half the prestige of the first-place {\em New England Journal of Medicine}, whereas by Article Influence it makes up a good deal of that ground.
\begin{figure}
\begin{center}
\includegraphics[scale=1]{Cardinal_AI03.pdf}
\end{center}
\caption{Cardinal differences between Impact Factor and Article Influence score. The top ten journals by Impact Factor are shown in the left column. The scores are scaled vertically, reflecting their cardinal positions. The smallest Impact Factor score is on the bottom, and the highest Impact Factor score is on the top. The right column shows the same journals scaled by Article Influence.}
\label{fig:c_graphEF}
\end{figure}
\section{Conclusion}
Correlation coefficients can be useful statistical tools. They can help us identify some kinds of statistically significant relationships between pairs of variables, and they can tell us about the sign (positive or negative) of these relationships. One must use considerably greater caution, however, when drawing conclusions from the magnitude of correlation coefficients --- all the more so in the presence of spurious correlates and in the absence of a formal hypothesis-testing framework. In particular, we have illustrated that just because two metrics have a high correlation --- 0.8 or 0.9 or even higher ---- we cannot safely conclude that they convey the same information, or that one has little additional information to tell us beyond what we learn from the other.
Comparative studies of alternative measures can be very useful in choosing an appropriate bibliometric toolkit. We close with a few suggestions for how one might better conduct these sorts of analyses. First, be wary of what correlation coefficients say about the relationship of two metrics \cite{Tukey1954JStat,Anscombe1973AmStat}. High correlation does \textit{not} necessarily mean that two variables provide the same information any more than a low correlation means that two variables are unrelated. Purchasing power varies wildly despite the high correlation between wage and hamburger price in our Big Mac example. At the other end of the spectrum, in the chaotic region of the logistic map, successive iterates have an immediate algebraic relationship yet a correlation of zero.
Second, appropriate data visualization can bring out facets of the data that are obscured by summary statistics. Different forms of data graphics can be better suited for certain tasks; for example the comparison plots such as those in Figure~\ref{fig:b_graph} better highlight the differences between bibliometric measures than do standard scatter plots.
Finally, simple observations can be at least as powerful as rote statistical calculations in understanding the nature of our data. For example, the median of the burgers/hour in the top third of the countries is about five times the median of the burgers/hour in the bottom third. This says a great deal about the differences in purchasing power across countries. The median ``bang per cite received" in the top third of journals is almost 2.4 times of the median in the bottom third. This says a great deal about the difference in how journals are valued under the Eigenfactor metrics, and helps us understand why the Eigenfactor metrics offer a substantially different view of journal prestige than that which we get from straight citation counts.
\section{Acknowledgements}
We would like to thank Ben Althouse for assistance with figures 3, 5, and 6, Cosma Shalizi for helpful discussions, Johan Bollen for extensive feedback on the manuscript, and an anonymous reviewer for provocative commentary. This research was supported in part by NSF grant SBE-0915005 to CTB.
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in Batista, Maia & Farr, Revta Biol. , Lisb. 1(3-4): 287 (1958)
#### Original name
Asterinema glabratae Bat. & H. Maia
### Remarks
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\begin{document}
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\begin{flushright}\footnotesize
\texttt{CALT-68-2695}\\
\vspace{2.1cm}
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\begin{center}
{\Large\textbf{\mathversion{bold} Efficient precision quantization in AdS/CFT}\par}
\vspace{2.1cm}
\textrm{Nikolay Gromov$^{\alpha}$, Sakura Sch\"afer-Nameki$^{\beta}$ and Pedro Vieira$^{\gamma}$}
\vspace{1.2cm}
\textit{$^{\alpha}$ Service de Physique Th\'eorique,
CNRS-URA 2306 C.E.A.-Saclay, F-91191 Gif-sur-Yvette, France;
Laboratoire de Physique Th\'eorique de
l'Ecole Normale Sup\'erieure et l'Universit\'e Paris-VI,
Paris, 75231, France;
St.Petersburg INP, Gatchina, 188 300, St.Petersburg, Russia } \\
\texttt{nikgromov@gmail.com}
\vspace{3mm}
\textit{$^{\beta}$ California Institute of Technology\\
1200 E California Blvd., Pasadena, CA 91125, USA } \\
\texttt{ss299@theory.caltech.edu}
\vspace{3mm}
\textit{$^{\gamma}$ Laboratoire de Physique Th\'eorique
de l'Ecole Normale Sup\'erieure et l'Universit\'e Paris-VI, Paris,
75231, France; Departamento de F\'\i sica e Centro de F\'\i sica do
Porto Faculdade de Ci\^encias da Universidade do Porto Rua do Campo
Alegre, 687, \,4169-007 Porto, Portugal} \\
\texttt{pedrogvieira@gmail.com}
\vspace{3mm}
\par\vspace{1cm}
\textbf{Abstract}\vspace{5mm}
\end{center}
\noindent
Understanding finite-size effects is one of the key open questions in solving planar AdS/CFT.
In this paper we discuss these effects in the $AdS_5 \times S^5$ string theory at one-loop in the world-sheet coupling.
First we provide a very general, efficient way to compute the fluctuation frequencies, which allows to determine the energy shift for very general multi-cut solutions. Then we apply this to two-cut solutions, in particular the giant magnon and determine the finite-size corrections at subleading order. The latter are then compared to the finite-size corrections from L\"uscher-Klassen-Melzer formulas and found to be in perfect agreement.
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\tableofcontents
\newpage
\section{Introduction and Summary}
Semi-classical quantizing around generic classical configurations is a challenging problem in field theory.
In two-dimensional integrable field theories this situation is ameliorated, but it remains a difficult problem
to quantize the theory around an arbitrary classical motion if we simply try to expand the action around the classical solution at stake.
In general the quadractic Lagrangian will not be time independent, one needs to find the stability angles and the explicit determination of the fluctuation energy spectrum becomes computationally involved.
On the other hand, in general, classically integrable theories admit a finite gap description.
In this construction each classical motion is mapped to a Riemann surface and semi-classical quantization amounts to
pinching this surface by adding extra singularities to the algebraic curve.
The superstring in $AdS_5\times S^5$ falls precisely into this class of theories:
Sharpening our understanding of the quantum spectrum of the superstring in $AdS_5 \times S^5$ is of crucial importance.
However, despite much progress in semi-classical quantization of classical string configurations in $AdS_5 \times S^5$,
it has remained a daunting problem to quantize around a generic classical string solution.
Applying the conventional methods of semi-classical quantization becomes particularly challenging for so-called multi-cut solutions, which in terms of the finite-gap description correspond to higher-genus curves.
The classical $AdS_5 \times S^5$ world-sheet theory however precisely admits a finite-gap description in terms on an algebraic curve
\cite{Kazakov:2004qf, Kazakov:2004nh, Beisert:2004ag, SchaferNameki:2004ik, Beisert:2005bm}. Each classical string motion maps to a Riemann surface, and semi-classical quantization can be performed by pinching this curve \cite{Gromov:2007aq}. This approach has been
successfully applied to various string configurations, and were shown to reproduce
the standard world-sheet results of \cite{Frolov:2002av, Frolov:2003tu, Park:2005ji}.
In this paper we propose a very general, efficient quantization method,
which is applicable to a very large class of classical string configurations. We will derive this within the framework of the algebraic curve.
The key idea that it is based on is the concept of off-shell fluctuation energies, which we advocate in the main text,
and allows one to find the full spectrum around a vast set of classical solutions from the knowledge of
\begin{equation}
\text{one $S^3$ and one $AdS_3$ fluctuation frequency ("frequency basis").}
\end{equation}
Compared to standard semi-classical quantization of string solutions one is not required to compute the fluctuations in all bosonic and fermionic fields. In particular it is interesting to note that the fermionic excitations can be constructed from these building blocks alone. Furthermore, this method is applicable to multi-cut solutions, which are from the conventional point of view, difficult to quantize. The concept of quasi-energy for the $\mathfrak{su}(2)$ sector, which is related to that of off-shell frequency, was introduced earlier in \cite{Vicedo:2008jy}. We will show however, that this is not only an abstract concept but can be put to practical use.
We will demonstrate the efficiency of our quantization method by computing the semi-classical spectrum
around the (dyonic) giant-magnon solution \cite{Hofman:2006xt, Dorey:2006dq, Chen:2006gea}. This describes a classical string
moving in $S^3$ with angular momenta $J$ and $Q$ and world-sheet momentum $p$.
When $J\to \infty$ this solution becomes the fundamental excitations of the two dimensional field theory
defined in the infinite volume and its dispersion relation reads
\begin{equation}
\epsilon_\infty(p)=\sqrt{Q^2 + {\lambda\over \pi^2} \sin^2 {p\over 2}} \,.
\end{equation}
When $J$ is large but not infinite this expression receives exponential corrections \cite{Arutyunov:2006gs,Astolfi:2007uz,Minahan:2008re} which can be physically traced back to the existence of wrapping interactions \cite{Luscher:1985dn,Klassen:1990ub,Ambjorn:2005wa,Janik:2007wt,Hatsuda:2008gd,Hatsuda:2008na,Gromov:2008ie,Heller:2008at,Bajnok:2008bm}.
The giant magnon solution is paradigmatic for the efficiency of this approach. For example,
in infinite volume $\epsilon_\infty(p)$ has no constant term in the large $\lambda$ expansion.
This means that the one-loop shift should vanish.
Showing this fact from a direct world-sheet field theory computation is a rather involved computation \cite{Papathanasiou:2007gd}, whereas
from the point of view of the algebraic curve point this result is obtained in a trivial way \cite{Chen:2007vs,Gromov:2008ie}.
\begin{figure}
\begin{center}
\includegraphics*[width =12cm]{CurveSheets.eps} \caption{\small Algebraic curve for classical superstrings on $AdS_5 \times S^5$. The macroscopic green cuts corresponds to a classical configuration. The wavy lines depict the several physical fluctuations. From left to right we have four bosonic $S^5$ fluctuations , four $AdS_5$ and eight fermionic fluctuations respectively. Any physical configuration has to cross the dashed line. We depict only the physical $|x|>1$ region. \label{CurveSheets}
}
\end{center}
\end{figure}
For the quantization of the finite volume dyonic giant magnon a direct world-sheet approach would most certainly be incomparably more involved than the one based on the classical algebraic curve which we carry out in this paper.
Needless to say the method we propose here and illustrate with the giant magnon solution can be applied to other integrable theories and to very different classes of classical solutions.
The plan of the paper is as follows:
we will begin with a lightning review of the algebraic curve and its semi-classical quantization. In section \ref{sec:QTools} we prove our efficient quantization method and provide a closed formula for the one-loop energy shift in terms of our "frequency basis".
In section \ref{sec:GeneralTwoCut} this approach is exemplified for the generic two-cut $\mathfrak{su}(2)$ solution and in section
\ref{sec:GMTwoCut} we compute the energy shift to the giant magnon and extract the subleading correction. Finally in section \ref{sec:Luscher} we compute these corrections from the L\"uscher-Klassen-Melzer formulas and show their agreement with our semi-classical quantization method.
Throughout all the paper we use
\begin{equation}
g=\frac{\sqrt{\lambda}}{4\pi} \,,\qquad
\mathcal{E}=\frac{\Delta}{\sqrt{\lambda}} \,,\qquad
{\cal J}=\frac{J}{\sqrt{\lambda}} \,,\qquad
\mathcal{Q}=\frac{Q}{\sqrt{\lambda}}\,.
\end{equation}
\section{Quantizing the algebraic curve}
\label{sec:SemiClassics}
\subsection{Classical algebraic curve}
In \cite{Kazakov:2004qf} a beautiful map between classical superstring motion in $AdS_5\times S^5$ and Riemann surfaces was presented. The idea is that using the Bena-Polchinski-Roiban \cite{Bena:2003wd} flat connection $A(x)$ -- where $x$ is an arbitrary complex number, the so-called spectral parameter -- we can diagonalize the monodromy matrix
\begin{equation}
\Omega(x)={\rm Pexp\,}\oint_\gamma A(x) \label{mono} \,,
\end{equation}
where $\gamma$ is any path starting and ending at some point $(\sigma,\tau)$ and wrapping the worldsheet cylinder once, to obtain a set of (eight) eigenvalues
\begin{equation}
\{e^{i\hat p_1},e^{i\hat p_2},e^{i\hat p_3},e^{i\hat p_4}|e^{i\t p_1},e^{i\t p_2},e^{i\t p_3},e^{i\t p_4} \} \,,
\end{equation}
which, due to flatness of the current, are $\gamma$ independent. As they depend on the arbitrary complex number $x$ they give rise to conserved charges by Taylor expansion around any point in the $x$-plane.
Since they are obtained from the diagonalization of an (almost) regular matrix $\Omega(x)$ they are obtained by solving a characteristic equation and thus define an (eight-sheeted) algebraic curve. The properties of this curve \cite{Kazakov:2004qf} follow from those of the flat connection $A(x)$ and are summarized
in appendix A.
The quasimomenta $p_i(x)$, being the log of the eigenvalues of $\Omega(x)$, do not define a Riemann surface. Rather, when evaluated on the algebraic curve for $e^{i p_i(x)}$ they might jump by an integer multiple of $2\pi$ as one crosses one of the square root cuts of the algebraic curve, i.e.
\begin{equation}
p_i^+ (x)- p_j^-(x)=2\pi n_{ij} \,\, , \,\, x\in \mathcal{C}_{n}^{ij} \label{AlgCur} \,,
\end{equation}
where $p^{\pm}_i(x)$ is the value of the quasimomentum above/bolow the cut.
Moreover to each cut we can associate a filling fraction given by integrating the quasimomenta around the cut
\begin{equation}\label{FillFrac}
S_{ij}=\pm\,\frac{\sqrt{\lambda}}{8\pi^2i}\oint_{\mathcal{C}_{ij}} \(1-\frac{1}{x^2}\) p_i(x) dx \,.
\end{equation}
Thus, each cut of the algebraic curve is characterized by a discrete label $(i,j)$, corresponding to the two sheets being united, an integer $n$, the multiple of $2\pi$ mentioned above, and a real filling fraction. These three quantities are the analogues of the polarization, mode number and amplitude of the flat space Fourier decomposition of a given classical solution. The (sixteen) superstring physical polarizations correspond to the pairing of sheets
\begin{equation}\label{Polarisations}
\begin{aligned}
S^5 :& \quad (\t 1,\t 3)\,,(\t 1,\t 4)\,,(\t 2,\t 3)\,,(\t 2,\t 4) \cr
AdS_5 :& \quad (\hat 1,\hat 3)\,,(\hat 1,\hat 4)\,,(\hat 2,\hat 3)\,,(\hat 2,\hat 4) \cr
\text{Fermions}
:& \quad (\t 1,\hat 3)\,,(\t 1,\hat 4)\,,(\t 2,\hat 3)\,,(\t 2,\hat 4)\cr
& \quad (\hat 1,\t 3)\,,(\hat 1,\t 4)\,,(\hat 2,\t 3)\,,(\hat 2,\t 4) \,.
\end{aligned}
\end{equation}
These physical polarizations are determined by the constraint that the lines connecting the sheets $(ij)$ have to cross the yellow line in figure \ref{CurveSheets}.
A simple rule of thumb is that they always connect sheets with index $1$ or $2$ with $3$ or $4$.
The classical energy of the string is obtained from the asymptotics (\ref{AsympInfty})
\begin{equation}
E = {\sqrt{\lambda} \over 4 \pi} \lim_{x\to\infty} x \left(\hat{p}_1 (x) + \hat{p}_2 (x)\right) \,.
\end{equation}
\subsection{Quantization}
Semi-classical quantisation of the algebraic curve proceeds by adding small number of fluctuations on top of the classical configuration \cite{Gromov:2007aq}. This treatment is equivalent to the semi-classical computation of quadratic fluctuations in the sigma-model \cite{Frolov:2002av, Frolov:2003tu, Park:2005ji}, however we will show that the algebraic curve approach is far more efficient.
We consider fluctuations around the classical curve for each polarization $(i, j)$ and mode number $n$. Adding a fluctuation amounts to shifting the quasimomenta as
$p_k (x) \rightarrow p_k (x) + \delta^{ij}_n p_k (x)$ where $\delta^{ij}_n p_k(x)$ is constrained by precise analytical properties as listed in Appendix A.2. In particular, the quasimomenta $\delta^{ij}_n p_i$ and $\delta^{ij}_n p_j$ which are the quasimomenta connected by the fluctuation at stake must behave as
\begin{equation}\label{PolePosition}
\delta^{ij}_n p_i (x) \simeq \pm \frac{\alpha(x_n^{ij})}{x-x_n^{ij}}
\end{equation}
close to the pole position $x_n^{ij}$ which is determined by
\begin{equation}
p_i (x_n^{ij}) - p_j (x_n^{ij}) = 2 \pi n_{ij} \,. \label{PolePosition}
\end{equation}
The physical poles correspond to solutions of this equation with $|x_n^{ij}|>1$. The precise choice of signs above as well as $\alpha(y)$ is given in Appendix A.2. Having found $\delta^{ij}_n p_k$ we read off the fluctuation energy with mode number $n$ and polarization $(i,j)$ from the large $x$ asymptotics
\begin{equation}
\Omega_n^{ij} =-2\,\delta_{i,\h1}+ {\sqrt{\lambda} \over 2 \pi} \lim_{x\to\infty} x \, \delta_n^{ij}\hat{p}_1 (x) \,. \label{largex}
\end{equation}
In the next section we will explain that in fact we do not need to compute separately each of the sixteen physical fluctuations corresponding to the various string polarizations (\ref{Polarisations}) but that it suffices to compute two of them, at least for a huge number of interesting solutions. In particular we shall see that the fermionic fluctuations can be obtained from the $S^3$ and $AdS_3$ fluctuation energies.
\subsection{Quantizers toolkit}
\label{sec:QTools}
Notice that the dependence on $n$ of the shift in the quasimomenta $\delta_n^{ij} p_k$ only appears through $x_n^{ij}$ as determined in (\ref{PolePosition}). In other words the shift in the quasimomenta is actually a function of the position of the pole, i.e.
\begin{equation}
\delta_n^{ij} p_k(x)=\left. \delta^{ij} p_k(x;y) \right|_{y=x_n^{ij}} \,.
\end{equation}
Moreover the \textit{off-shell} quantity $ \delta^{ij} p_k(x;y)$ is a well defined function of $y$. It is determined by the same asymptotics as for the \textit{on-shell} shift of quasimomenta $\delta_n^{ij} p_k(x)$ except that the position of the pole is left unfixed. An obvious consequence of what we just observed is that the fluctuation energies read off from (\ref{largex}) are, by construction, of the form
\begin{equation}
\Omega_n^{ij}=\left.\Omega^{ij}(y)\right|_{y=x_n^{ij}} \,,
\end{equation}
where the function $\Omega^{ij}(y)$ is independent of the mode number $n$. We call $\Omega^{ij}(y)$ the \textit{off-shell} fluctuation energies. The off-shell frequency is related for the particular case of the $SU(2)$ principal chiral model to the quasi-energy introduced in \cite{Vicedo:2008jy}.
Given an on-shell fluctuation energy $\Omega_n^{ij}$ as a function of the mode number $n$, we can always reconstruct the off-shell frequencies by first computing the quasimomenta $p_i(x)$ for the underlying classical solution and then we simply replace $n$ using (\ref{PolePosition}), that is
\begin{equation}
\Omega^{ij}(y) =\left. \Omega_n^{ij} \right|_{n\to \frac{p_i(y)-p_j(y)}{2\pi}} \,.
\end{equation}
In Appendix C this is exemplified for a simple $S^3$ circular string.
We will now explain how, using the inversion symmetry (\ref{Auto}), we can relate the several off-shell fluctuation energies. In this way we will find a powerful reduction algorithm for the computation of the fluctuation energies and thus the one loop energy shift
\begin{equation}\label{OneLoopE}
\delta \Delta^{1-loop}=\frac{1}{2} \sum_{ij,n}(-1)^{F_{ij}} \Omega_n^{ij} \,,
\end{equation}
around a generic classical solution.
\subsubsection{Frequencies from inversion symmetry}
\begin{figure}
\begin{center}
\includegraphics*[width =15cm]{exchange.ps} \caption{\small As we analytically continue a fluctuation energy $\Omega^{\t2 \t3}(y)$ from a point $|y|>1$ to the interior of the unit circle we see that its mirror image becomes physical. \label{Exchange}
}
\end{center}
\end{figure}
An important property of the quasi-momenta, which follows from the $\mathbb{Z}_4$-grading of the $\mathfrak{psu}(2,2|4)$ superalgebra, is the inversion symmetry (\ref{Auto}) under $x\rightarrow 1/x$, which exchanges the quasi-momenta $p_{\t1,\t4} \leftrightarrow p_{\t2,\t 3}$ and likewise for the $AdS$ hatted quasi-momenta. Thereby, a pole connecting the sheets $(\t2, \t3)$ at position $y$, always comes with an image pole at position $1/y$ connecting the sheets $(\t1,\t4)$.
We can obtain a physical frequency $\Omega^{\t 1\t 4}(y)$, by analytically continuing the off-shell frequency $\Omega^{\t 2 \t 3}(y)$, inside the unit circle. This is because when we cross the unit-circle, the physical pole for $(\t2\t3)$ becomes unphysical, thereby rendering its image, which lies now outside the unit-circle, a physical pole for $(\t1\t4)$ as depicted in figure \ref{Exchange}.
Let us consider in detail how this works for the $AdS$ fluctuations. As we will now demonstrate
\begin{equation}\label{OmegaMapAdS}
\Omega^{\h1\h4}(y)=-\Omega^{\h2 \h3}(1/y)-2 \,.
\end{equation}
Thus, suppose we know $\Omega^{\hat 2 \hat 3}(y)$. We know that this fluctuation energy appears in the asymptotics of the shifted quasimomenta $\delta^{\hat 2 \hat 3} p_k(x;y)$ defined by the analytic properties listed in Appendix A.2. Consider now $ - \delta^{\hat 2 \hat 3} p_k(x;1/y)$. From the analytic properties of $ \delta^{\hat 2 \hat 3} p_k(x;y)$ we conclude that
\begin{itemize}
\item
Close to $x=y$ we have
\begin{equation}
- \delta^{\hat 2 \hat 3} p_{\hat 1}(x;1/y) \simeq \frac{\alpha(y)}{x-y} \,,\qquad
- \delta^{\hat 2 \hat 3} p_{\hat 4}(x;1/y) \simeq - \frac{\alpha(y)}{x-y} \,.
\end{equation}
\item
The poles at $x=\pm 1$ for these functions $- \delta^{\hat 2 \hat 3} p_k(x;1/y)$ are also synchronized as in equation (\ref{DeltapVirConst}).
\item
Close to the branch points of the original solution these functions exhibit inverse square root singularities.
\end{itemize}
These are precisely the required properties for $\delta^{\hat 1 \hat 4} p_k(x;y)$ as listed in Appendix A.2! Thus
\begin{equation}
\delta^{\hat 1 \hat 4} p_k(x;y) =- \delta^{\hat 2 \hat 3} p_{ k}(x;1/y) \label{ident} \,.
\end{equation}
From the large $x$ asymptotics we have
\beqa
- {\sqrt{\lambda} \over 4 \pi} \lim_{x\to\infty} x\, \delta^{\h2\h3}\hat{p}_{\hat 1} (x;1/y) &=&- \frac{\Omega^{\h2\h3}(1/y)}{2}\,,
\eeqa
while by definition $\Omega^{\hat 1 \hat 4}(y)$ can be read off from
\beqa
{\sqrt{\lambda} \over 4 \pi} \lim_{x\to\infty} x\, \delta^{\h1\h4}\hat{p}_{\hat 1} (x;y) &=& \frac{\Omega^{\h1\h4}(y)}{2}+1 \,.
\eeqa
From the identification
(\ref{ident})
we thus conclude (\ref{OmegaMapAdS}).
Similarly we can proceed for the $S^5$ frequencies and relate $\Omega^{\t2\t3}(y)$ with $\Omega^{\t1 \t4}(y)$.
It is clear that $\Omega^{\t1\t4}(y) = - \Omega^{\t2 \t3}(1/y)$ +constant, and to find this constant we can either repeat the analysis we just did applied to the sphere fluctuations or we can be smarter and fix it from $\Omega^{\t1\t4}(\infty)=0$. This must of course hold -- the energy shift when we add an extra root at infinity is obviously zero, in other words, roots at infinity are zero modes.
Thus, the relation we find is similar to (\ref{OmegaMapAdS}), except that the constant term differs:
\begin{equation}\label{InversionOmega}
\Omega^{\t1\t4}(y) = - \Omega^{\t2 \t3}(1/y) +\Omega^{\t2 \t3}(0)\,.
\end{equation}
Obviously for the purpose of computing the one-loop shift these constants are irrelevant as they will cancel in the sum.
So far we have obtained the frequencies $(14)$ from $(23)$. In the next subsection we will show how to derive all remaining frequencies. For a very large class of classical solutions we will be able to extract all fluctuation energies, including the fermionic ones, from the knowledge of a single $S^3$ and a single $AdS_3$ fluctuation energy.
\subsubsection{Basis of fluctuation energies}
For simplicity let us consider only symmetric classical configurations that have pairwise symmetric quasi-momenta
\begin{equation}\label{SymSol}
p_{\h1, \hat 2 ,\t 1, \t2 } = - p_{\h4, \h3, \t 4 , \t3} \,,
\end{equation}
as depicted in figure \ref{CurveSheets}.
This is in particular the case for all rank one solutions, i.e. $\mathfrak{su}(2)$ and $\mathfrak{sl}(2)$.
\begin{figure}
\begin{center}
\includegraphics*[width =16cm]{OmegaLinearCombi.eps} \caption{\small
Depiction of equation (\ref{LinearCombiExample}). On top: we see that for symmetric configurations we can obtain the off-sheel fluctuation frequency $\Omega^{\h2\t2}=\Omega^{\t3,\h3}$ from the knowledge of the two $S^5$ and $AdS_5$ frequencies. On bottom: With this unphysical fluctuation at hand we can compute the fermionic fluctuation frequency $\Omega^{\h2\t3}=\Omega^{\t2\t3}+\Omega^{\h2\t2}$ in terms of the two bosonic fluctuations. \label{FigureLinearCombi}
}
\end{center}
\end{figure}
Consider e.g. the fermionic frequency $\Omega^{\h2 \t3}(y)$. This energy can be thought of as a linear combination of the physical fluctuation $\Omega^{\t 2 \t 3}(y)$ and an unphysical fluctuation $\Omega^{\h2 \t2}(y)$, which in particular does not appear in the table (\ref{Polarisations}) of physical, momentum-carrying polarisations
\begin{equation}\label{FermLinearCombi}
\Omega^{\h2 \t3}(y) = \Omega^{\t 2 \t 3}(y)+ \Omega^{\h2 \t2}(y) \,.
\end{equation}
Since we are considering symmetric configurations, this unphysical fluctuation energy is identical to $\Omega^{\t3\h3}(y)$, i.e.
\begin{equation}
\Omega^{\h2 \t2}(y) = \Omega^{\t3\h3}(y) \,.
\end{equation}
As in (\ref{FermLinearCombi}), these unphysical fluctuations can be linearly combined in terms of physical fluctuations
\begin{equation}
\Omega^{\h2 \h3}(y) =\Omega^{\h2 \t2}(y) + \Omega^{\t2 \t3}(y)+ \Omega^{\t3 \h3}(y) \,.
\end{equation}
Combining all these relations we obtain
\begin{equation}\label{LinearCombiExample}
\Omega^{\h2 \t3}(y) = {1\over 2} \left( \Omega^{\t 2 \t 3}(y) + \Omega^{\h2 \h3}(y) \right) \,,
\end{equation}
as depicted in figure \ref{FigureLinearCombi}.
Proceeding in a similar fashion we can derive all frequencies as linear combinations of $\Omega^{\t 2 \t 3}(y)$ and
$\Omega^{\h2 \h3}(y)$. Table (\ref{FreqTable}) summarizes all these relations.
\subsubsection{Final result}
The physical frequencies are labeled by the eight bosonic and eight fermionic polarizations (\ref{Polarisations}), so we can label them by
\begin{equation}
\Omega^{ij} \,,\qquad \hbox{where}\quad i= (\h1, \h2, \t1, \t 2) \qquad j = (\h3 ,\h4, \t3 ,\t4) \,.
\end{equation}
To construct the complete set of off-shell frequencies for a symmetric solution (\ref{SymSol})
in terms of the two fundamental $S^3$ and $AdS_3$ ones $\Omega^{\t2 \t3}(y)$ and $\Omega^{\h2 \h3}(y)$ and their images under $y\rightarrow 1/y$, we first construct by inversion
\begin{equation}
\begin{aligned}\label{FreqTable}
\Omega^{\t1\t4} (y) &= - \Omega^{\t2 \t3}(1/y) + \Omega^{\t2 \t3}(0) \cr
\Omega^{\h1\h4} (y) & = - \Omega^{\h2 \h3}(1/y) -2 \,.
\end{aligned}
\end{equation}
The remaining frequencies are then obtained by linear combination of these four fluctuation frequencies. In this way we obtain the following concise form for all off-shell frequencies
\begin{equation}\label{OmegaFinal}
\Omega^{ij} (y) = {1\over 2} \left(\Omega^{ii'} (y) + \Omega^{j'j} (y) \right) \,,
\end{equation}
where
\begin{equation}
(\h1, \h2, \t1, \t 2,\h3 ,\h4, \t3 ,\t4)' = (\h4, \h3, \t 4 , \t 3 ,\hat 2, \hat 1 , \t 2, \t 1) \,.
\end{equation}
This generalizes (\ref{LinearCombiExample}), and we have made explicit these linear combinations in appendix B, (\ref{FreqTable}).
In the complete one-loop energy shift (\ref{OneLoopE}) the constant terms in (\ref{FreqTable}) will drop out and thus do not need to be computed. This is particularly obvious, when performing the graded sum over $\Omega^{ij} (x_n^{ij})$ with the explicit frequencies in (\ref{FreqTable}).
For the general case of not symmetric solutions, we can repeat the above analysis, however the minimal set of required off-shell fluctuation frequencies will generically be larger than two. It would be interesting to analyse this further.
In the rest of this paper we will consider only $\mathfrak{su}(2)$ solutions which means that only $\t p_2$ (and $\t p_3$) will be connected by square root cuts (outside the unit circle). For these solutions it is clear that
\begin{equation}
\t p_2=-\t p_3 \,,\qquad \t p_1=-\t p_4\qquad \text{and} \qquad \hat p_1=\hat p_2 =-\hat p_3=-\hat p_4 \,,
\end{equation}
so that we will generically have 6 different frequencies, namely:
\begin{enumerate}
\item{One internal fluctuation corresponding to a pole shared by $\t p_2$ and $\t p_3$ which we denote by
\begin{equation}
\Omega_S(y)= \Omega^{\t2 \t3}(y)
\end{equation}}
\item{Another $S^3$ fluctuation connecting $\t p_1$ and $\t p_4$
\begin{equation}
\Omega_{\bar S}(y)= \Omega^{\t1 \t4}(y)
\end{equation}}
\item{Two fluctuations which live in $S^5$ but are orthogonal to the ones in $S^3$,
\begin{equation}
\Omega_{S_{\perp}}(y)= \Omega^{\t1 \t3}(y)= \Omega^{\t1 \t4}(y)
\end{equation} }
\item{Four $AdS_5$ fluctuations
\begin{equation}
\Omega_{A}(y)= \Omega^{\h1 \h3 }(y)= \Omega^{\h1 \h4 }(y)=\Omega^{\h2 \h3 }(y)=\Omega^{\h2 \h4 }(y)
\end{equation} }
\item{Four fermionic excitations which end on either $p_{\t 2}$ or $p_{\t 3}$ (which are the sheets where there are cuts outside the unit circle)
\begin{equation}
\Omega_{F}(y)= \Omega^{\h1 \t3 }(y)= \Omega^{\h2 \t3 }(y)=\Omega^{\t2 \h3 }(y)=\Omega^{\t2 \h4 }(y)
\end{equation} }
\item{Four fermionic poles which end on either $p_{\t 1}$ or $p_{\t 4}$ (which are the sheets where there are cuts inside the unit circle)
\begin{equation}
\Omega_{\bar F}(y)= \Omega^{\h1 \t4 }(y)= \Omega^{\h2 \t4 }(y)=\Omega^{\t1 \h3 }(y)=\Omega^{\t1 \h4 }(y) \,.
\end{equation} }
\end{enumerate}
These fluctuations are depicted in figure \ref{CurveSheets} from left to right.
\section{General $\mathfrak{su}(2)$ two-cut solution}
\label{sec:GeneralTwoCut}
In this section we explain how to compute the fluctuation energies around a general $2$-cut $\mathfrak{su}(2)$ solution\footnote{General two-cut solutions for the $\mathfrak{su}(2)$ Heisenberg magnet were discussed in \cite{Bargheer:2008kj}.} with branch points $a,\bar{a},b,\bar{b}$. We will find out that the fluctuation energies can be obtained
from the surprisingly simple expressions
\begin{equation}\label{OmegaTwo}
\begin{aligned}
\Omega_A(y) &= {2 \over y^2-1} \left( 1+ y {f(1) - f(-1) \over f(1) + f(-1)} \right)
\cr
\Omega_S(y) &= {4\over f(1) + f(-1) } \left( {f(y) \over y^2 -1 } -1 \right)
\,,
\end{aligned}
\end{equation}
with the remaining fluctuation energies obtained through table (\ref{FreqTable}). Here
\footnote{The proper definition of $f(y)$ is
\begin{equation}
f(y)=(2x-a-\bar a)(2x-b-\bar b)\sqrt{\frac{(x-a)(x-\bar a)}{(2x-a-\bar a)^2}}\sqrt{\frac{(x-b)(x-\bar b)}{(2x-b-\bar b)^2}}
\end{equation}.
}
\beqa
f(y)\equiv \sqrt{(y-a)(y-\bar a)(y-b)(y-\bar b)} \,.
\eeqa
Note that this is a very simple elegant expression for the off-shell fluctuation energies. All the intricate structure that appears for the on-shell frequencies is hidden in the equation for the pole positions $x_n^{ij}$ (\ref{PolePosition}).
Let us first review the construction of the quasi-momenta for a two-cut $\mathfrak{su}(2)$ solution. The AdS-quasi-momenta have no cuts, and therefore are rational functions with at most simple poles at $x= \pm1$ and large $x$ asymptotics given by
\begin{equation}
p_{\hat 1, \hat 2} = - p_{\hat 3, \hat 4} = {2 \pi \mathcal{E} \over x} + O\left({1\over x^2}\right) \,.
\end{equation}
This determines the AdS quasi-momenta uniquely to be
\begin{equation}
p_{\hat 1, \hat 2} = - p_{\hat 3, \hat 4} = {2 \pi \mathcal{E} x \over x^2 -1} \,.
\end{equation}
The derivatives of the sphere quasi-momenta are
\begin{equation}\label{Pprime}
p'_{\t2} = - p'_{\t3} = - { \pi \over f(x)}
\left(
{\mathcal{E} f(1) \over (x-1)^2}
+ {\mathcal{E} f'(1) \over x-1} +
{\mathcal{E} f(-1) \over (x+1)^2} +
{\mathcal{E} f'(-1) \over x+1} +2 (\mathcal{J}_1 - \mathcal{J}_2 )
\right) \,.
\end{equation}
The remaining sphere quasi-momenta $\t p_1=-\t p_4$ are obtained by the inversion $x\rightarrow 1/x$ as in (\ref{Auto}).
The first four terms inside the parethesis ensure that the poles of the
quasi-momenta at $x = \pm 1$ are synchronized with the corresponding poles of the AdS quasi-momenta (\ref{VirConst}). Note that $p'_{\tilde{i}}$ is required to have a double pole at $x=\pm1$, with vanishing residue. The function $1/f(x)$ is needed for the correct inverse square root behaviour close to the branch-points.
The constant terms in the parenthesis are engineered to ensure the correct large $x$ asymptotics (\ref{AsympInfty}).
The moduli of the algebraic curve fix the A and B cycle integrals, and thereby the branch-points. More precisely the moduli are hyperelliptic functions of the branch-points.
Finally to get the quasi-momenta we would have to integrate the meromorphic differential $p' dx$. These last steps will again yield the quasi-momenta as hyperelliptic functions of $x$ and of the branch-points.
In certain instances there can be considerable simplifications due to a degenerate choice of moduli for the curve. This is for example the case for the well-studied symmetric two-cut $\mathfrak{sl}(2)$ solution. Also in the case of the giant magnon, where
the two cuts are very close, $a \sim b$ and $\bar{a} \sim \bar{b}$, \cite{Minahan:2006bd,Vicedo:2007rp}, we will see that this leads to considerable computational efficiency.
In terms of these unfixed branch-points, however the expression for the derivative of the quasimomenta (\ref{Pprime}) is quite simple as are the expressions for the fluctuation energies anticipated above (\ref{OmegaTwo}).
To discuss the fluctuation frequencies we now perturb the quasi-momenta and fix $\delta p$ by the required asymptotics (\ref{DeltaPAsymp}). We consider only the $(\h2, \h3)$ and $(\t2, \t3)$ fluctuations with $N_{\h2, \h3} = N_{\t2 \t3} =1$, located at $x=z$ and $x=y$ respectively.
The shift in quasi-momenta are
\begin{equation}
\begin{aligned}\label{PShift}
\delta p_{\h2} (x; y, z)
& = {\alpha (z) \over x-z} +
{\delta \alpha_- \over x-1} + { \delta \alpha_+ \over x+1} \cr
\delta p_{\t2} (x; y, z)
&= {1 \over f(x)} \left(
- { f(y) \,\alpha (y) \over x -y} +
{\delta \alpha_- f(1) \over x-1} + { \delta \alpha_+ f(-1) \over x+1} - {4 \pi \over \sqrt{\lambda }} \,x + A
\right)\,,
\end{aligned}
\end{equation}
where the asymptotics at large $x$ for $\delta p_{\h2}$, $\delta p_{\t2}$, and also $\delta p_{\h1}$, $\delta p_{\t1}$ obtained by inversion symmetry (\ref{Auto}) fix the constants $\delta \alpha_{\pm}$, $A$ and $\delta \Delta$. We provide the details in Appendix D. The result is
\begin{equation}
\delta \Delta =
\Omega_S(y) + \Omega_A(z) \,,
\end{equation}
with the notation of (\ref{OmegaTwo}). The remaining constants are summarized in appendix D.
Now that we have found the two off-shell frequencies $\Omega_S$ and $\Omega_A$ we can apply our method from section \ref{sec:QTools} and construct the remaining frequencies as in table (\ref{FreqTable}). In this way we obtain the complete set of fluctuation energies around a generic two cut solution. As an application we will consider in the next section the Giant Magnon solution which corresponds to a particular (singular) limit of the general treatment we considered so far.
Notice also that our simple treatment can be used trivially generalized for $K\ge 3$ cuts.
\section{GM as a two-cut solution}\label{sec:GMTwoCut}
The Giant Magnon solution is a degenerate case of the $2$-cut solution studied in the previous section where the branch points of the algebraic curve are pairwise close.
We will use the explicit formulas (\ref{OmegaTwo}) to compute the frequencies for the giant magnon solution.
In the next subsection we will summarize all the results and then provide the derivations in the subsequent parts.
\subsection{Results}
From the analysis in the last section we have learned that in order to compute the one-loop energy shift (\ref{OneLoopE}), we need the following ingredients:
\begin{itemize}
\item the two off-shell $S^3$ and $AdS_3$ fluctuation energies $\Omega_S (y)$ and $\Omega_A(y)$
\item the various quasi-momenta, which are required to determine the position of the physical poles as a function of $n$ (\ref{PolePosition}).
\end{itemize}
Parametrize the branch-points as
\begin{equation}\label{BranchPts}
a = X_+ + {\delta \over 2} \,,\qquad
b = X_+ - {\delta \over 2} \,,
\end{equation}
and $\bar{a}$ and $\bar{b}$ are complex conjugate to these branch-points, where we denote $X_- = (X_+)^\ast $.
We will always work up to second order in $\delta$.
Away from the branch-points the two-cuts become indistinguishable, $a \simeq b \simeq X_+$ etc., and the quasi-momenta can be obtained from (\ref{Pprime}) as
\begin{equation}
p'_{\t2} (x) = {d \over dx} \left({ 2 \pi \mathcal{E} x \over x^2-1} +{2 \pi (\mathcal{E } - \mathcal{J} + \mathcal{Q}) \over X_+ - X_- } \log\frac{x-X_+}{x-X_-} \right) \,,
\end{equation}
where we replaced $\mathcal{J}_1 \rightarrow \mathcal{J}$ and $\mathcal{J}_2 \rightarrow \mathcal{Q}$.
The expression inside the paranthesis is obviously $p_{\t2}(x)$, and the log-cut is the condensate of two cuts with consecutive mode-numbers \cite{Vicedo:2007rp}. The discontinuity by crossing the log-cut is given by $\pi (n+1) - \pi n$ and therefore we can fix the prefactor of the log to be $1/i$, that is to leading order we find
\begin{equation}
\mathcal{E } - \mathcal{J} + \mathcal{Q} = {1\over 2 \pi i} (X_+ - X_-) + O(\delta^2) \,,
\end{equation}
and therefore
\begin{equation}
p'_{\t2} (x) \simeq p_{far}' (x)
\equiv {d \over dx} \left({ 2 \pi \mathcal{E} x \over x^2-1} +{1\over i} \log\frac{x-X_+}{x-X_-} \right) \,,\qquad
|x- X_+|, |x-X_-| \gg \delta
\,.
\end{equation}
The quasi-momentum itself is given by
\begin{equation}
p_{far}(x)=\frac{\Delta}{2g}\frac{x}{x^2-1}+\frac{1}{i}\log\frac{x-X_+}{x-X_-}+\tau \label{PnotPrime} \,,
\end{equation}
where the twist $\tau$ is required to account for the not periodic boundary conditions for the giant magnon and is given by \cite{Gromov:2008ie}
\begin{equation}\label{TauDef}
\tau=-p/2=\frac{i}{2}\log\frac{X_+}{X_-} \,.
\end{equation}
Also, far from the branch-points, $\t p_1(x)=\t p_2(0)+\tau-\t p_2(1/x)$.
Close to the branch-points $a$ and $b$ are given in (\ref{BranchPts}), and the quasi-momentum (\ref{Pprime}) becomes
\begin{equation}\label{LogP}
p_{\t2}' (x) \simeq p_{close}' (x) \equiv
{1\over \sqrt{(x-X_+ -{\delta\over 2}) (X_+ -{\delta\over 2} -x )}} \,,\qquad
|x-X_+| \ll 1\,,
\end{equation}
where we again used the leading order expression for the energy. Note that up to an overall constant this is obvious, as this is the only function that has the correct branch-cut. Imposing further the same asymptotics for the overlap region $\delta \ll x- X_+ \ll1 $ as $p_{\t 2}$ in (\ref{LogP}) fixes the overall factor. Alternatively we could fix this constant by imposing $p(b)-p(a)=
\int_a^b p'dx=\pi$ which is precisely what we used above to find the prefactor of the log.
As we will explain below, the classical energy, total filling fraction and momenta of this solution, obtained by integrating the quasi-momenta with suitable measures around the two cuts, will be given by
\beqa
\Delta-J &=&{g\over i} \left(X_+ - {1\over X_+} - {\delta^2 \over 8 (X_+)^3} \right) + c.c. \cr
Q &=& {g\over i} \left(X_+ + {1\over X_+} + {\delta^2 \over 8 (X_+)^3} \right)+ c.c. \label{ChargesExpanded}
\\
P &=& {1\over i } \left(\log X_+ - {\delta^2 \over 16 (X_+)^2}\right) + c.c. \nonumber \,.
\eeqa
Finally, $\delta$ is fixed by imposing the B-cycle condition $\int_\infty^a p' = \pi n$, which yields\footnote{The twist $\tau$ is fixed as in the appendix of \cite{Gromov:2008ie}.}
\begin{equation}\label{DeltaSquare}
\delta^2 = 16 (X_+ -X_-)^2 \exp\left( - 2 i \tau - i {4 \pi \Delta \over \sqrt{\lambda}}
{X_+ \over (X_+)^2 -1}\right)\,.
\end{equation}
These relations allow to parametrize the branch-points $X^{\pm}$ in terms of $\mathcal{Q}$ and $P$, from which then the classical energy $\mathcal{E}$ can easily be computed.
We have determined the off-shell frequencies in the previous section. To obtain the on-shell frequencies $\omega_n$ we compute the positions of
the poles $x_n^{ij}$ from (\ref{PolePosition}) and evaluate them at $x_n$.
There are two case we have to consider. Mainly $x_n$ are situated relatively far from the
branch points of the two cuts and we can expand off-shell frequencies
\begin{equation}
\begin{aligned}
\label{ExpandedOmegas}
\Omega_A(y) &= \Omega^{(0)} (y) -\left( {y \over y^2 -1 } {X_+ (X_-^2 - 1) \over \, 2\, (X_+^2 -1)(X_+ X_- + 1)^2 }\, \delta^2 + c.c.\right) \cr
\Omega_S (y) &=
\Omega_A (y)-\(\frac{1}{y-X_+}\frac{X_+-X_-}{4(X_+^2-1)(X_-X_++1)}\delta^2+c.c.\)\,.
\end{aligned}
\end{equation}
The first term is the leading order frequency, as determined in \cite{Gromov:2008ie}, which is
\begin{equation}
\Omega^{(0)}(y) = {2\over y^2-1} \left(1- y \, {X_+ + X_- \over X_+ X_- +1} \right)\,.
\end{equation}
The remaining frequencies are of course determined as in (\ref{OmegaFinal}).
However there are fluctuations corresponding to the
variations of the filling fractions of the two cuts.
These are situated right at the branch points. To compute their
contributions to the 1-loop energy shift we have to expand
$\delta E^{\rm BP}\equiv \frac{1}{2}\Omega^{\t 2\t3}(a)+\frac{1}{2}\Omega^{\t 2\t3}(b)$.
That leads to
\begin{equation} \label{OmBP}
\delta E^{\rm BP}\simeq \Omega^{(0)}(X_+)+\(\frac{1-X_-X_+}{4(X_-X_++1)^2(X_+^2-1)}\delta^2+c.c.\) \,.
\end{equation}
We will assume that the fluctuations are
situated along the real axis, except the fluctuations
at the branch points, which we will treat separately.
Now, we have the off-shell fluctuation frequencies, the classical energy as well as the quasi-momenta, and therefore the position of the physical poles, and thus we have all ingredients assembled to compute the one-loop energy shift.
In the next subsection we will derive all the above expressions and subsequently, we will sum up the fluctuation energies to obtain the one-loop energy shift for a generic $Q$-magnon solution.
For the simple giant magnon $\mathcal{Q}\ll 1 $ and the one-loop energy shift organizes as a series in these two exponential \cite{Gromov:2008ie}
\begin{equation}
\delta \Delta^{1-loop} = \sum_{n, m} a_{n,m}(P, \mathcal{J}) \, \Big( e^{- 2 \pi \mathcal{J}} \Big)^n
\Big(e^{- {2 \pi \mathcal{J}\over \sin{p\over 2}}} \Big)^m \,.
\end{equation}
In \cite{Gromov:2008ie} we determined the complete set of $a_{n, 0}$ coefficients (see also \cite{Janik:2007wt}), which correct the one-loop shift of the giant magnon in finite volume, by properly summing the leading frequencies as opposed to approximating them by an integral over their momenta. In this paper we determine $a_{1, 1}$, which is the leading correction to the one-loop shift due to the fine-structure of the condensate cut.
Combining the methods in the present paper and in \cite{Gromov:2008ie}, it should be straight forward to compute $a_{1, n}$.
\subsection{Derivations}
We now provide the details for the results in the last subsection. First let us consider $\delta^2$.
It is determined by fixing the B-cycle integral. We will compute this integral using different approximations to the quasi-momentum, depending on how far the integration point is from the branch-point
\begin{equation}
\pi n=\int_\infty^a p_{\t 2}' = \int_{\infty}^c p'_{far} + \int_{c}^a p'_{close} \,,
\end{equation}
where $c= X_+ + \epsilon$ is an arbitrary point in the overlapping region $\delta \ll |x-X_+|\ll 1$, i.e. $\delta \ll \epsilon \ll1$. We depicted the integration region in figure \ref{picasso}.
Evaluating the integrals yields
\begin{equation}
\pi n \simeq \left[ { 2 \pi \mathcal{E} X_+ \over (X_+)^2-1} +{1\over i} \log\frac{\epsilon}{X_+-X_-} + \tau \right]
+ \left[{1\over i}\log {\delta \over 4 \epsilon} \right] \,.
\end{equation}
Here $\tau$ is the value of the quasi-momenta at infinity. As required the dependence on $\epsilon$ cancels and we obtain $\delta$ as function of $X^\pm$ in (\ref{DeltaSquare}).
\begin{figure}
\begin{center}
\includegraphics*[width =14cm]{picasso.eps} \caption{\small
Integration regions. \label{picasso}
}
\end{center}
\end{figure}
Next we derive the expressions for the charges from the general relations
\begin{equation}
\begin{aligned}
\Delta-J=& { g \over 2 \pi i} \oint dx\, p'(x) \left(x - {1\over x}\right) \cr
Q =& { g \over 2 \pi i} \oint dx\, p'(x)\left(x + {1\over x}\right) \cr
P =& { 1 \over 2 \pi i} \oint dx\, p'(x) \log x \,.
\end{aligned}
\end{equation}
For this purpose we write
\begin{equation}
\oint dx \, p_{\t2}' (x) f (x) =
\oint dx \, p_{far}'(x) f(x) + \oint dx \left( p_{\t2}'(x) - p_{far}'(x)\right)f(x) \,.
\end{equation}
The first term obviously yields the leading order charges (\ref{ChargesExpanded}). The second term can be evaluated by deforming the contour to the region where the integrand is singular, i.e. $x \sim X_+$, where in particular $p_{\t2}$ can be approximated by $p_{close}$
\begin{equation}
\oint dx \left( p_{\t2}'(x) - p_{far}'(x)\right)f(x)
\simeq \oint \left({1\over \sqrt{(x-X_+- {\delta\over 2} )(X_+ - {\delta\over 2} -x) }} - {i\over x- X_+} \right) f(x) + c.c. \,,
\end{equation}
where the contour integral encircles all the poles of the integrand. The integrals can be easily computed and yield (\ref{ChargesExpanded}).
\section{Finite-size correction to the GM} \label{sec5}
The one-loop energy shift is obtained by the weighted sum over all fluctuation frequencies
\begin{equation}
\delta\Delta_{1-loop}=\frac{1}{2}\sum_{n,ij}(-1)^{F_{ij}} \Omega_{ij}\(x_n^{ij}\) \,.
\end{equation}
To deal with this sum we first split this sum into the fluctuation energies corresponding to a variation of the filling fractions of the two cuts, $\delta E^{\rm BP}$ and the remaining fluctuations.
To sum the latter we transform the sum over $n$ into an integral with $\cot \pi n$ and then we pass from the $n$ to the $x$ plane using the map (\ref{PolePosition}). Actually as we will explain later there is an additional third contribution coming from fluctuations which got trapped between the two cuts when they collapsed into the log cut.
This contribution, denoted by $\delta E^{UP}$ is considered in section \ref{sec:UP}. Thus we have
\begin{equation}
\delta\Delta_{1-loop}=\frac{1}{2}\sum_{ij}(-1)^F\ccw\oint_{{\cal C}_{\mathbb R}}\Omega^{ij}(y)\cot_{ij}\frac{dy}{2\pi i }+\delta E^{\rm BP}+\delta E^{\rm UP}\;,
\end{equation}
where
\begin{equation}\label{Cotij}
\cot_{ij}\equiv \partial_y\log\sin\(\frac{p_i-p_j}{2}\)\;,
\end{equation}
and the contour $\cal C_{\mathbb R}$ encircles all the fluctuations on the real axis.
Our goal will be to deform this contour to the unit circle, where the argument of
the $\cot$ has a large imaginary component
everywhere and the integral can be computed by standard saddle point method.
However, when deforming the contour we will obtain several poles from $\cot_{ij}$ located close to the points $x=X_+,X_-$ and $x=1/X_+,1/X_-$. The contribution from these poles is computed in the next section and is denoted by $\delta E^{\rm PL}$. We find therefore
\begin{equation}
\delta\Delta_{1-loop}=\delta E^{\rm INT}+\delta E^{\rm PL}+\delta E^{\rm BP}+\delta E^{\rm UP}\;,
\end{equation}
where
\begin{equation}
\delta E^{\rm INT}=\cw\oint_{{\cal C}_{\mathbb U}}\(\frac{1}{2}\sum_{ij}(-1)^F\Omega^{ij}(x)\cot_{ij}\)\frac{dx}{2\pi i } \,. \label{unitc}
\end{equation}
Notice that since we already dealt with the zero mode contribution $\omega_{BP}$ separately we can (and will) use the far away quasi-momenta (\ref{PnotPrime}) in the rest of the paper. In the following four sections we will consider each of these four contributions in detail.
The splitting of the one-loop shift into a unit circle contribution plus the rest was proposed in \cite{Gromov:2007cd} where the unit circle contribution was analyzed and related with the Hernandez-Lopez phase \cite{Hernandez:2006tk}. In \cite{Gromov:2007ky} the remaining contribution was considered around general non-singular classical curves and matched with the usual finite size corrections, known as anomalies, appearing in the Beisert-Staudacher equations \cite{Beisert:2005fw}.
\subsection{Extracting poles}
We now determine the positions of the poles mentioned above.
Consider first the polarization $(\t2,\t3)$. We have
\begin{equation}
\exp(-i\t p_2+i\t p_3)=\exp\(-i\frac{x\Delta}{g(x^2-1)}-2i\tau\)\frac{(x-X_-)^2}{(x-X_+)^2}\,,
\end{equation}
so we will have an obvious pole from (\ref{Cotij}) at $x=X^+$ but we will also have some less trivial poles
if the denominator in (\ref{Cotij}) vanishes, i.e. for $\exp(-i\t p_2+i\t p_3)=1$,
\begin{equation}
\exp\(-i\frac{x\Delta}{g(x^2-1)}-2i\tau\)\frac{(x-X_-)^2}{(x-X_+)^2}=1\,.
\end{equation}
The first factor is exponentially small. When $x\sim X_+$
the exponent is of order $\delta^2$ as one can see from (\ref{DeltaSquare}).
However we can compensate that if the second factor diverges.
To be able to compensate the exponential supression we will
require that $x-X_+\sim \delta$. What one finds is the poles at
$x-X_+=\epsilon^\pm_1$, where
\begin{equation}
\begin{aligned}
\epsilon^\pm_1
&= \pm\frac{\delta}{4}+ \frac{\delta^2}{16} \(\frac{1}{X_+-X_-}+i\frac{\Delta}{2g}\frac{X_+^2+1}{(X_+^2-1)^2}\)\cr
&\pm\frac{\delta^3}{64}
\(\frac{1}{(X_--X_+)^2}-\frac{3\Delta^2(X_+^2+1)^2}{8g^2(X_+^2-1)^4}
+\frac{i\Delta}{2g}\frac{2 X_+^4+X_- X_+^3-3X_+^2+3X_-X_+-3}{(X_+-X_-)(X_+^2-1)^3}\)
+{\cal O}\(\delta^4\)
\end{aligned}
\end{equation}
Proceeding in the same way for the different polarizations we would find the position of all existing poles. We have summarized all poles, and whether they are physical or unphysical (around $X_+$ or $1/X_+$, respectively) in table \ref{tab:1}. In Appendix E we listed the explicit values of the small deviations $\epsilon_j$.
\begin{table}[h]
\begin{center}
\begin{tabular}{@{}cll@{}}
\toprule
\bf Polarization & \bf Poles around $X_+$ & \bf Poles around $1/X_+$ \\ \midrule
$A\times 4$ & & \\ \midrule
$F\times 4$ & $x-X_+=0,\epsilon_{3}$ & \\ \midrule
$\bar F\times 4$ & & $1/x-X_+=0,\epsilon_{3}$ \\ \midrule
$S$ & $x-X_+=\epsilon^-_{1},0,\epsilon^+_{1}$ & \\ \midrule
$\bar S$ & & $1/x-X_+=\epsilon^-_{1},0,\epsilon^+_{1}$ \\ \midrule
$S_\perp\times 2$ & $x-X_+=0,\epsilon_{2}$ & $1/x-X_+=0,\epsilon_{2}$ \\ \bottomrule
\end{tabular}
\end{center}
\caption{Poles of different $\cot_{ij}$ in the upper half plane close to the logarithm branch points }\label{tab:1}
\end{table}
In summary, the contribution to the contour integral from these singularities is
\begin{equation}\label{Epl}
\delta E^{\rm PL}=\(
\frac{e^{i\tau}}{(X_-X_++1)(X_+^2-1)}+
\frac{2-X_+(X_-+X_+)}{(X_-X_++1)(X_+^2-1)^2}
+\frac{i\Delta}{4g}\frac{(X_--X_+)(X_+^2+1)}{(X_-X_++1)(X_+^2-1)^3}
\)\frac{\delta^2}{4}+c.c. \,.
\end{equation}
which for small $Q$ values becomes
\begin{equation}\label{PlQ}
\delta E^{\rm PL}\simeq
8 e^{-\frac{J}{2g\sin\frac{p}{2}}-2} \sin^2\frac{p}{2} \,.
\end{equation}
\subsection{Unphysical fluctuations}
\label{sec:UP}
Consider a general finite gap solution.
Let us assume first that all filling fractions
are sufficiently small. By other words we are dealing with a slightly deformed BMN
curve. Then we know that the equation
\begin{equation}
p_i(x_n^{ij})-p_j(x_n^{ij})=2\pi n\label{xijn}
\end{equation}
for a physical pair $(ij)$
always has a solution\footnote{In fact one should add twists to ensure this statement.}.
When we gradually start increasing the filling fractions, the cuts become bigger and at some point
a cut could collide with some $x_n$.
After this point we will not be able to find solutions to (\ref{xijn}) for some values of $n$.
This however does not imply any non-analyticity of the fluctuation
energies $\Omega^{ij}(x^{ij}_n)$
as a function of the filling fractions and we can analytically continue the fluctuation energies
below this point. What happens is that the fluctuation $x_n$ passes through a cut and afterwards is connecting two different sheets. This will generically yield unphysical fluctuations. We have depited this process in figure \ref{UnFl}.
Indeed for each missing solution of $\eq{xijn}$ one could find the corresponding unphysical fluctuation.
We conclude that we also have to consider all possible solutions of $\eq{xijn}$ for unphysical pairs $(ij)$.
In the calculation above we have taken into account only physical fluctuations. However there are
$2+4$ unphysical fluctuations $(\t1,\t2),(\t3,\t4)$ and $(\hat 1\t2),(\h2\t2),(\h3\t3),(\h4\t3)$, which by the above reasoning we also need to take into account. We denote these fluctuations by $S_u$ and $F_u$
\beqa
\Omega_{S_u}(x)&=&\frac{\Omega_{\bar S}(x)-\Omega_S(x)}{2}+c\\
\Omega_{F_u}(x)&=&\frac{\Omega_A(x)-\Omega_S(x)}{2}+c\,,
\eeqa
where the specific values of the constant $c$ and of the position of these fluctuations $x^{S_u}$ and $x^{F_u}$ are collected in Appendix E.
\begin{figure}[ht]
\begin{center}
\includegraphics*[width =12cm]{UnphFl.eps}
\caption{When increasing the filling fraction of a cut, the fluctuation could pass through the cut and reappear uniting different sheets. The physical fluctuation $\t2\t3$ could become the unphysical one $\t1\t2$.\label{UnFl}
}
\end{center}
\end{figure}
Combining this together with the branch-point contribution (\ref{OmBP}) one obtains
\begin{equation}\label{Eun}
\delta E^{\rm UP}+\delta E^{\rm BP}=\delta E^{BP}+\frac{2\Omega_{S_u}(0)-4\Omega_{F_u}(x^{F_u})}{2}=
\frac{1-\sqrt{X_-/X_+}}{4(X_-X_++1)(X_+^2-1)}\delta^2+c.c. \,,
\end{equation}
where in particular the leading order term correctly cancels!
In the $\mathcal{Q}\to 0$ limit we obtain for this combined contribution
\begin{equation}
\delta E^{\rm UP}+\delta E^{\rm BP}\simeq-8 e^{-\frac{J}{2g\sin\frac{p}{2}}-2}\sin^2\frac{p}{2} \,.
\end{equation}
This contribution precisely cancels the contribution of $\delta E^{\text PL}$ in (\ref{PlQ}). Thus, for the simple giant-magnon solution the only contribution is given by the integral over the unit circle (\ref{unitc})
\subsection{On the unit circle}
In the previous two section we took into account the extra poles in the complex $x$ plane, the branch-point fluctuations and the unphysical excitations. For a general Dyonic magnon these contributions are given by (\ref{Epl}) added to (\ref{Eun}) while for a simple giant-magnon this sum vanishes.
In this section we consider the remaining contribution given by the integral (\ref{unitc}) over the unit circle. There are three contributions into which this integral is naturally split.
On the upper/lower half of the unit circle we have
\begin{equation}
\cot\(\frac{p_i-p_j}{2}\)=\pm i\(1+2 e^{\mp i (p_i-p_j)}+\dots \) \label{cote} \,,
\end{equation}
while the fluctuation energies are given by
\begin{equation}
\Omega_{ij}(y)=\Omega^{(0)}(y)+\delta \Omega_{ij}(y) \,.\label{flucte}
\end{equation}
Thus the we can pick the leading term in (\ref{cote}) times the leading term in (\ref{flucte}) to get
\begin{equation}
\delta E^{\rm INT,(0)}=\cw\oint_{{\cal C}_{\mathbb U}^+} \frac{dy}{2 i}(-1)^{F_{ij}} \partial_y\Omega^{(0)}(y) \frac{p'_i-p'_j}{2\pi} \,,
\end{equation}
where the integral goes over the upper half of the unit circle from $x=-1$ to $x=+1$. Since $\sum_{i=1}^4 \t p_i-\hat p_i=0$ this contribution vanishes and therefore the one-loop shift around the infinite volume giant magnon is zero.
We are therefore left with the exponentially suppressed contributions.
The second contribution comes from picking the subleading term in (\ref{flucte}) and the leading value in (\ref{cote}). This gives
\begin{equation}
\begin{aligned}\label{Eint}
\delta E^{{\rm INT},(1)}
&\simeq 2\cw\oint_{{\cal C}_{\mathbb U}^+}\frac{h(x)-h(1/x)/x^2+g(x)}{(X_+^2-1)(X_+X_-+1)}\frac{dx}{2\pi i}+c.c.\cr
&=\frac{i\delta^2}{4\pi}\[\frac{X_+-X_-}{(X_+X_-+1)(X_+^2-1)^2}+\frac{(X_-^2-1)({\rm arccoth}\,X_+-{\rm arccoth}\,X_-)}{(X_+^2-1)(X_+^2X_-^2-1)}\]+c.c.\,,
\end{aligned}
\end{equation}
where
\begin{equation}
\begin{aligned}
h(x)&=\frac{\delta^2}{16}\[\frac{X_--X_+}{(x-X_+)^2}+\frac{X_--2X_++X_-X_+^2}{X_+(X_-X_+-1)}\(\frac{1}{x-X_+}-\frac{1}{x-X_-}\)\]\cr
g(x)&=\frac{\delta^2}{8}\frac{(X_+-X_-)^2}{(x X_+-1)(x X_--1)X_+} \,.
\end{aligned}
\end{equation}
Expanding this result in the $\mathcal{Q}\to 0$ limit we obtain
\begin{equation}\label{Eintone}
\delta E^{{\rm INT}, (1)}\simeq
16 e^{-\frac{J}{2g\sin\frac{p}{2}}-2}
\(
\frac{g\sin^3\frac{p}{2}}{Q}-
\frac{\sin\frac{p}{2}}{\pi}
\) \,.
\end{equation}
Notice that this contribution is singular in the $\mathcal{Q}\rightarrow 0$ limit. This singularity will cancel however with the third contribution we will now analyze.
Finally we have the contribution coming from picking the leading term in (\ref{flucte}) multiplied by the subleading term in (\ref{cote}). This was the contribution analyzed in \cite{Gromov:2008ie} and \cite{Janik:2007wt}. This gives
\beqa
&&\delta E^{{\rm INT}, (2)}=\cw\oint_{U^+} \frac{dx}{2\pi i} \partial_x \Omega_0 \(e^{-i\tau} \frac{x-X_-}{x-X_+}+e^{-i\tau} \frac{x-1/X_+}{x-1/X_-}-2\)^2 e^{-\frac{ix\Delta}{g(x^2-1)}} \,,
\eeqa
which in the small $\mathcal{Q}$ limit is divergent and becomes
\begin{equation}\label{Eexp}
\begin{aligned}
\delta E^{{\rm INT}, (2)}
\simeq \,&V.P. \ \cw \oint_{U^+} \frac{dx}{2\pi i} \partial_x \Omega_0 \(
2\frac{xX_+-1}{x-X_+}-2
\)^2 e^{-ix\frac{J+4g\sin\frac{p}{2}}{g(x^2-1)}}\\
&+e^{-\frac{J}{2g\sin\frac p2}-2}\(-\frac{16g\sin^3\frac p2}{Q}+\frac{4iJ\cos\frac{p}{2}}{g}-8i\sin\frac{p}{2}+8i\sin p\)\,,
\end{aligned}
\end{equation}
where $V.P.$ stands for the principal value of the integral.
Finally, we can combine (\ref{Eintone}) and (\ref{Eexp}) to obtain the final result
\begin{equation}\label{GMFinal}
\begin{aligned}
\delta \Delta_{1-loop}&\simeq
V.P.\cw \oint_{U^+} \frac{dx}{2\pi i} \partial_x \Omega_0 \( 2\frac{xX_+-1}{x-X_+}-2
\)^2 e^{-ix\frac{J+4g\sin\frac{p}{2}}{g(x^2-1)}}\cr
&\qquad +e^{-\frac{J}{2g\sin\frac p2}-2}\(-\frac{16\sin\frac p2}{\pi}+\frac{4iJ\cos\frac{p}{2}}{g}-8i\sin\frac{p}{2}+8i\sin p\) \,.
\end{aligned}
\end{equation}
We will show in the next section that this is in precise agreement with the
$F$ and $\mu$ terms of the L\"uscher-Klassen-Melzer formulas!
Note that the expression above is real by construction and the divergence at $\mathcal{Q}=0$ has cancelled among the various contributions.
\subsection{Combined energy shift for a generic Dyonic magnon}
Notice that we are by no means obliged to take the simple magnon magnon and our previous formulas are absolutely general and also yield the finite size $1$-loop shift around a generic Dyonic magnon. Combining all the contributions computed in the previous sections we get
\beqa\label{Combi}
\delta \Delta_{1-loop}&=&
\cw\oint_{U^+} \frac{dx}{2\pi i} \partial_x \Omega_0 \(e^{-i\tau} \frac{x-X_-}{x-X_+}+e^{-i\tau} \frac{x-1/X_+}{x-1/X_-}-2\)^2 e^{-\frac{ix\Delta}{g(x^2-1)}} \\
&+&\( \frac{\delta^2}{4(X_-X_++1)(X_+^2-1)^2}\[ 1-X_-X_++i\frac{X_+-X_-}{\pi} \nonumber
-i\,\frac{\Delta}{4g}\frac{(X_+^2+1)(X_+-X_-)}{X_+^2-1}\right. \right. \\ \nonumber
&+&\left.\left.i\frac{(X_-^2-1)(X_+^2-1)}{2\pi(X_-X_+-1)}\log\(\frac{(X_++1)(X_--1)}{(X_+-1)(X_-+1)}\)
\]+c.c.\)\,.
\eeqa
\section{L\"uscher-Klassen-Melzer formulas}
\label{sec:Luscher}
Finally we compute the finite-size correction (\ref{GMFinal}) using the L\"uscher-Klassen-Melzer formulas \cite{Luscher:1985dn, Luscher:1986pf, Klassen:1990ub, Ambjorn:2005wa, Janik:2007wt, Gromov:2008ie, Heller:2008at}.
There are two contributions, the $F$- and the $\mu$-term
\beqa
&&\delta\epsilon^F_a=-V.P.\int_{\mathbb R}\frac{dq}{2\pi}\(1-\frac{\epsilon'(p)}{\epsilon'(q^*(q))}\)
e^{-iq^*(q)L}\sum_b(-1)^{F_b}S_{ba}^{ba}(q^*(q),p)
\\
&&\delta \epsilon^{\mu}_a =
- i \left(1 - {\epsilon'(p) \over \epsilon'(\tilde{q}^*)} \right)
e^{- i\tilde{q}^* L} \hbox{Res}_{q = \tilde{q}} \left(\sum_b (-1)^{F_b} S_{ba}^{ba} (q_*(q) , p) \right) \,,
\eeqa
which describe the corrections to the dispersion relation of a single magnon with momentum $p$ due to virtual particles running in the loop, and bound state formation, respectively.
We have used the notation for the on-shell momentum
\begin{equation}
q^2 + \epsilon(q_*)^2 = 0 \,,
\end{equation}
and $\tilde{q}$ denotes the Euclidean energy of the bound state.
Inserting the all-loop AdS/CFT S-matrix \cite{Beisert:2005tm, Beisert:2006ib, Beisert:2006ez, Arutyunov:2006yd}, one can expand to arbitrary order and obtain the leading-volume correction.
Through a trivial change of variables, the F-term can be written as \cite{Gromov:2008ie}
\begin{equation}\label{Fterm}
\delta \epsilon^F =
V.P. \cw\oint_{\mathbb{U}^+} \frac{dx}{2\pi i}\,\partial_x\Omega_0(x) \, e^{-4\pi\frac{iJ}{\sqrt{\lambda}}\frac{x}{x^2-1}}
e^{-4\pi \frac{i(\Delta-J)}{\sqrt{\lambda}}\frac{x}{x^2-1}}
\(2\, \frac{x-X_-}{x-X_+}\sqrt\frac{X_+}{X_-}-2\)^2 \,,
\end{equation}
where
\begin{equation}
\Delta = J + {\sqrt{\lambda} \over \pi} \sin{p\over 2} \,.
\end{equation}
In the limit of $\mathcal{Q}\rightarrow 0$, $X_+ \sim 1/X_-$ and thus the F-term agrees precisely with the first line in (\ref{GMFinal})!
For the $\mu$-term we have to evaluate the residue at the bound states as done in \cite{Janik:2007wt}, to subleading order. Since the computation is exactly as done in this paper we omit the details. There are
three contributions, which arise from poles up to the one-loop dressing factor, the effect from the one-loop dressing factor and the higher-loop contributions, respectively. In summary we obtain
\begin{equation}
\delta \epsilon^\mu=e^{-\frac{2\pi J}{\sqrt{\lambda}\sin\frac{p}{2}}} \delta_1 \delta_2 \delta_3 \,,
\end{equation}
where
\beqa
\delta_{1}&=&-4 g \sin^3\frac{p}{2}+i \({J \over g} \cos\frac{p}{2}-2\sin\frac{p}{2}+\sin p\) +\mathcal{O}\(\frac 1g\) \\
\delta_{2}&=&\frac{1}{2}+\frac{1}{g}\(\frac{1}{2\pi\sin^2 \frac
p2}-\frac{i\cos\frac p2}{4\sin^2 \frac p2}\)+\mathcal{O}\(\frac 1{g^2}\)\\
\delta_{3}&=&\frac{8}{e^2} +\mathcal{O}\(\frac 1{g^2}\) \,,
\eeqa
so that the $\mu$-term up to this order is
\begin{equation}
\delta\epsilon_\mu=
-e^{-\frac{2 \pi J}{\sqrt{\lambda} \sin\frac p2}-2}\(16 g \sin^3\frac{p}{2}+\frac{16}{\pi}\sin\frac{p}{2} -4i\({\cal J}\cos\frac{p}{2}-2\sin\frac{p}{2}+2\sin p\)\)
+ \mathcal{O}\left({1\over g}\right)\,.
\end{equation}
The leading $\mathcal{O}(g)$ contribution to the $\mu$-term is precisely the one in \cite{Janik:2007wt}. The subleading terms are in complete agreement with the corrections appearing in the second line of our result (\ref{GMFinal}).
Thus we have successfully demonstrated the agreement of our result (\ref{GMFinal}) with the L\"uscher-Klassen-Melzer approach of computing finite-size effects. As we emphasized already in the introduction, we strongly believe that this efficient quantization method that we developed in this paper will be useful in systematic studies of the finite-size effects for strings in $AdS_5 \times S^5$. In particular it would be interesting to reproduce from a L\"uscher like approach the full result (\ref{Combi}) for the finite size corrections to the Dyonic magnon.
\subsection*{Acknowledgments}
We thank V.~Kazakov for discussions. NG was partially supported by RSGSS-1124.2003.2, by RFBR grant 08-02-00287 and ANR grant INT-AdS/CFT (contract ANR36ADSCSTZ).
The work of SSN is supported by John A. McCone Postdoctoral Fellowship. SSN would like to thank the IPMU and the University of Tokyo for hospitality during some stages of this work.
PV is funded by the Funda\c{c}\~ao para a Ci\^encia e Tecnologia fellowship {SFRH/BD/17959/2004/0WA9}. PV would like to thank KITP for warm hospitality. PV would like to thank California Institute of Technology where part of this work was done for warm hospitality.
\newpage
\setcounter{section}{0}
\setcounter{subsection}{0}
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 4,115
|
Q: how to upload images in flutter hello i wonder to upload images in flutter
i try to use http.MultipartRequest
like this
request.fields["name"] = "$RegisterName";
request.fields["description"] = "$RegisterDescription";
request.fields["caution"] = "$RegisterCaution";
request.fields["price"] = "$RegisterPrice";
request.fields["price_prop"] = "$RegisterPriceProp";
request.fields["user.id"] = "1";
request.fields["lend"] = "$RegisterCategory";
request.fields["category"] = "Digital";
request.fields["place_option"] = "true";
var multipartFile = http.MultipartFile.fromBytes(
'file',
(await rootBundle.load('assets/images/main_1.jpg')).buffer.asUint8List(),
filename: 'test01.jpg',
contentType: MediaType('image', 'jpg'),
);
request.files.add(multipartFile);
var response = await request.send();
if (response.statusCode == 200) print('Upload');
}
but this code is not working
if i use this code, upload only another data
upload things
then json type is this
json type image
i want upload images files ...:(
A: i use this to send picture with formData
var head = Api().bearerHeader; ////just bearerToken
var request = http.MultipartRequest(
'POST',
Uri.parse(
'https://c.....'));
request.files
.add(await http.MultipartFile.fromPath('TITLEOFFORMDATA', imageFile.path));
request.headers.addAll(head);
http.StreamedResponse response = await request.send();
if (response.statusCode == 200) {
String varo = await response.stream.bytesToString();
}
A: This is how you can send image to your server with MultipartRequest with http package
try {
final uri = Uri.parse(your_url);
final request = http.MultipartRequest('POST', uri);
final multipartFile =
await http.MultipartFile.fromPath('Image', 'your_path_of_image'); // Image is the parameter name
request.files.add(multipartFile);
request.fields['userId_if_required'] = value;
final response = await request.send();
if (response.statusCode == 200) {
print('success');
} else {
print('Something went wrong');
}
} catch (e) {
print('Something went wrong');
}
A: How to upload your image to a Django rest API server
this will work for sure, let me know if you have any issues.
Please be sure to add the necessary packages to your pubspec.yaml file
image_picker
http
if there is some I missed please ask me or add it and add as a reply
import 'dart:convert';
import 'package:http/http.dart' as http;
import 'dart:io';
import 'package:get/get.dart';
import 'package:image_picker/image_picker.dart';
final _picker = ImagePicker();
File? _image;
// use this to send your image
Future<void>uploadImage(filePath) async {
// your token if needed
try{
var headers = {
'Authorization':
'Bearer ' + "token",
};
// your endpoint and request method
var request = http.MultipartRequest(
'POST',
Uri.parse("https://api.imgur.com/3/image"));
request.fields
.addAll({'yourFieldNameKey1': 'yourFieldNameValue1', 'yourFieldNameKey2': 'yourFieldNameValue2'});
request.files.add(await http.MultipartFile.fromPath(
'yourPictureKey', filePath));
request.headers.addAll(headers);
http.StreamedResponse response = await request.send();
if (response.statusCode == 200) {
print(await response.stream.bytesToString());
} else {
print(response.reasonPhrase);
}
}catch(e){
print(e);
}
}
// Use this to pick your image
Future<void> _openImagePicker() async {
try {
var pickedImage = await _picker.pickImage(source: ImageSource.gallery);
if (pickedImage != null) {
setState(() {
_image = File(pickedImage.path);
});
uploadImage(pickedImage.path);
}
} catch (e) {
//print(e);
}
}
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 2,818
|
{"url":"https:\/\/hackage.haskell.org\/package\/chimera-0.3.2.0\/docs\/Data-Chimera-WheelMapping.html","text":"chimera-0.3.2.0: Lazy infinite streams with O(1) indexing and applications for memoization\n\nData.Chimera.WheelMapping\n\nDescription\n\nHelpers for mapping to rough numbers and back. This has various applications in number theory.\n\nExample\n\nLet isPrime be an expensive predicate, which checks whether its argument is a prime number. We can memoize it as usual:\n\nisPrimeCache1 :: UChimera Bool\nisPrimeCache1 = tabulate isPrime\n\nisPrime1 :: Word -> Bool\nisPrime1 = index isPrimeCache1\n\nBut one may argue that since the only even prime number is 2, it is quite wasteful to cache isPrime for even arguments. So we can save half the space by memoizing it for odd numbers only:\n\nisPrimeCache2 :: UChimera Bool\nisPrimeCache2 = tabulate (isPrime . (\\n -> 2 * n + 1))\n\nisPrime2 :: Word -> Bool\nisPrime2 n\n| n == 2 = True\n| even n = False\n| otherwise = index isPrimeCache2 ((n - 1) quot 2)\n\nHere \\n -> 2 * n + 1 maps n to the (n+1)-th odd number, and \\n -> (n - 1) quot 2 takes it back. These functions are available below as fromWheel2 and toWheel2.\n\nOdd numbers are the simplest example of numbers, lacking small prime factors (so called rough numbers). Removing numbers, having small prime factors, is sometimes called wheel sieving.\n\nOne can go further and exclude not only even numbers, but also integers, divisible by 3. To do this we need a function which maps n to the (n+1)-th number coprime with 2 and 3 (thus, with 6) and its inverse: namely, fromWheel6 and toWheel6. Then write\n\nisPrimeCache6 :: UChimera Bool\nisPrimeCache6 = tabulate (isPrime . fromWheel6)\n\nisPrime6 :: Word -> Bool\nisPrime6 n\n| n elem [2, 3] = True\n| n gcd 6 \/= 1 = False\n| otherwise = index isPrimeCache6 (toWheel6 n)\n\nThus, the wheel of 6 saves more space, improving memory locality.\n\n(If you need to reduce memory consumption even further, consider using Bit wrapper, which provides an instance of unboxed vector, packing one boolean per bit instead of one boolean per byte for Bool)\n\nSynopsis\n\n# Documentation\n\nfromWheel2 n is the (n+1)-th positive odd number. Sequence A005408.\n\nmap fromWheel2 [0..] == [ n | n <- [0..], n gcd 2 == 1 ]\n>>> map fromWheel2 [0..9]\n[1,3,5,7,9,11,13,15,17,19]\n\n\nLeft inverse for fromWheel2. Monotonically non-decreasing function.\n\ntoWheel2 . fromWheel2 == id\n\nfromWheel6 n is the (n+1)-th positive number, not divisible by 2 or 3. Sequence A007310.\n\nmap fromWheel6 [0..] == [ n | n <- [0..], n gcd 6 == 1 ]\n>>> map fromWheel6 [0..9]\n[1,5,7,11,13,17,19,23,25,29]\n\n\nLeft inverse for fromWheel6. Monotonically non-decreasing function.\n\ntoWheel6 . fromWheel6 == id\n\nfromWheel30 n is the (n+1)-th positive number, not divisible by 2, 3 or 5. Sequence A007775.\n\nmap fromWheel30 [0..] == [ n | n <- [0..], n gcd 30 == 1 ]\n>>> map fromWheel30 [0..9]\n[1,7,11,13,17,19,23,29,31,37]\n\n\nLeft inverse for fromWheel30. Monotonically non-decreasing function.\n\ntoWheel30 . fromWheel30 == id\n\nfromWheel210 n is the (n+1)-th positive number, not divisible by 2, 3, 5 or 7. Sequence A008364.\n\nmap fromWheel210 [0..] == [ n | n <- [0..], n gcd 210 == 1 ]\n>>> map fromWheel210 [0..9]\n[1,11,13,17,19,23,29,31,37,41]\n\n\nLeft inverse for fromWheel210. Monotonically non-decreasing function.\n\ntoWheel210 . fromWheel210 == id","date":"2021-12-02 00:41:42","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\": 0.5207101106643677, \"perplexity\": 7462.098206779332}, \"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-2021-49\/segments\/1637964361064.58\/warc\/CC-MAIN-20211201234046-20211202024046-00337.warc.gz\"}"}
| null | null |
package com.google.api.ads.dfp.jaxws.v201308;
import javax.xml.bind.annotation.XmlEnum;
import javax.xml.bind.annotation.XmlType;
/**
* <p>Java class for RateCardStatus.
*
* <p>The following schema fragment specifies the expected content contained within this class.
* <p>
* <pre>
* <simpleType name="RateCardStatus">
* <restriction base="{http://www.w3.org/2001/XMLSchema}string">
* <enumeration value="ACTIVE"/>
* <enumeration value="INACTIVE"/>
* <enumeration value="UNKNOWN"/>
* </restriction>
* </simpleType>
* </pre>
*
*/
@XmlType(name = "RateCardStatus")
@XmlEnum
public enum RateCardStatus {
/**
*
* The rate card has been activated.
*
*
*/
ACTIVE,
/**
*
* The rate card has not been activated.
*
*
*/
INACTIVE,
/**
*
* The value returned if the actual value is not exposed by the requested API
* version.
*
*
*/
UNKNOWN;
public String value() {
return name();
}
public static RateCardStatus fromValue(String v) {
return valueOf(v);
}
}
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 8,541
|
Schedule Read this! Rideshare About Food & Friends Speakers Music Directions Facebook RSVP #LifeStock
Evelyn's Crackers Dawn Woodward & Edmund Rek
Evelyn's Crackers exclusively uses older grain varieties grown and milled by Ontario farmers committed to nutrition and long-term agricultural diversity.
Building a community one cracker at a time.
Dawn Woodward (The Baker):
Inspired by her mother, Dawn began cooking at a very young age. Following a degree in Philosophy from Bard University, she continued her interests in professional cooking and baking to positions as a Pastry Chef in New Orleans and Head Baker at artisan bakeries in New York and Toronto. Extensive travels expanded her repertoire by exposure to such cuisines as: Eastern Mediterranean, Thai, Chinese, Cambodian, Georgian, and Middle Eastern. Committed to using heritage grain varieties grown close by, Dawn sources from Ontario farmers and millers and is constantly experimenting in ways to incorporate them into the crackers. In addition to leading all production at Evelyn's Crackers, she is a speaker an educator at the annual Kneading Conference in Maine/Seattle, can be found at the Evergreen Brickworks Farmers Market and offers her expertise as a professional consultant.
Edmund Rek (The Chef):
While growing up in a diplomatic family, Edmund experienced a wide array of foods and cultures. Drawn to the face-paced world of professional kitchens, he began his career cooking in small family style Italian restaurants. After culinary school in he established his cooking career in competitive 4-star luxury hotels and celebrity award-winning restaurants in Washington DC and as a sous chef for Iron Chef Morimoto in Philadelphia. After taking yearly food sabbaticals to California, New York City, Europe and Asia he opened a farm-to-table restaurant advocating organic and local food. After moving to Toronto he jumped at the chance to participate in the local farmers markets and has been instrumental in building Evelyn's Crackers brand spearheading sales and marketing. He can be found in production, or on delivery and most Saturdays mornings at Wychwood Farmers Market. In addition Edmund offers small business consulting services and mentorships to food entrepreneurs.
Pancake Making & Whole Grains
Workshops & Demonstrations
powered by foxpress.design
|
{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
| 7,812
|
package org.umlg.sqlg.sql.parse;
import com.google.common.base.Preconditions;
import org.apache.commons.collections4.set.ListOrderedSet;
import org.apache.commons.lang3.tuple.Pair;
import org.apache.tinkerpop.gremlin.structure.Direction;
import org.umlg.sqlg.structure.PropertyType;
import org.umlg.sqlg.structure.SchemaTable;
import org.umlg.sqlg.structure.SqlgGraph;
import org.umlg.sqlg.structure.topology.Topology;
import java.util.*;
import java.util.stream.Collectors;
/**
* List of column, managing serialization to SQL
*
* @author jpmoresmau
* @author pieter
*/
public class ColumnList {
/**
* Column -> alias
*/
private final LinkedHashMap<Column, String> columns = new LinkedHashMap<>();
/**
* Alias -> Column
*/
private final LinkedHashMap<String, Column> aliases = new LinkedHashMap<>();
/**
* Indicates that the query is for a {@link org.apache.tinkerpop.gremlin.process.traversal.step.filter.DropStep}
* In this case only the first column will be returned.
*/
private final boolean drop;
/**
* the graph to have access to the SQL dialect
*/
private final SqlgGraph sqlgGraph;
/**
* A map of all the properties and their types.
*/
private final Map<String, Map<String, PropertyType>> filteredAllTables;
private ListOrderedSet<String> identifiers;
/**
* build a new empty column list
*
* @param graph
* @param drop
* @param filteredAllTables
*/
public ColumnList(SqlgGraph graph, boolean drop, ListOrderedSet<String> identifiers, Map<String, Map<String, PropertyType>> filteredAllTables) {
super();
this.sqlgGraph = graph;
this.drop = drop;
this.filteredAllTables = filteredAllTables;
this.identifiers = identifiers;
}
/**
* add a new column
*
* @param schema
* @param table
* @param column
* @param stepDepth
* @param alias
*/
private Column add(String schema, String table, String column, int stepDepth, String alias) {
Column c = new Column(schema, table, column, this.filteredAllTables.get(schema + "." + table).get(column), stepDepth);
this.columns.put(c, alias);
this.aliases.put(alias, c);
return c;
}
/**
* add a new column
*
* @param schema The column's schema
* @param table The column's table
* @param column The column
* @param stepDepth The column's step depth.
* @param alias The column's alias.
* @param foreignKeyParts The foreign key column broken up into its parts. schema, table and for user supplied identifiers the property name.
*/
private void addForeignKey(String schema, String table, String column, int stepDepth, String alias, String[] foreignKeyParts) {
Column c = add(schema, table, column, stepDepth, alias);
c.isForeignKey = true;
if (foreignKeyParts.length == 3) {
Map<String, PropertyType> properties = this.filteredAllTables.get(foreignKeyParts[0] + "." + Topology.VERTEX_PREFIX + foreignKeyParts[1]);
if (foreignKeyParts[2].endsWith(Topology.IN_VERTEX_COLUMN_END)) {
c.propertyType = properties.get(foreignKeyParts[2].substring(0, foreignKeyParts[2].length() - Topology.IN_VERTEX_COLUMN_END.length()));
c.foreignKeyDirection = Direction.IN;
c.foreignSchemaTable = SchemaTable.of(foreignKeyParts[0], foreignKeyParts[1]);
c.foreignKeyProperty = foreignKeyParts[2];
} else {
c.propertyType = properties.get(foreignKeyParts[2].substring(0, foreignKeyParts[2].length() - Topology.OUT_VERTEX_COLUMN_END.length()));
c.foreignKeyDirection = Direction.OUT;
c.foreignSchemaTable = SchemaTable.of(foreignKeyParts[0], foreignKeyParts[1]);
c.foreignKeyProperty = foreignKeyParts[2];
}
} else {
c.propertyType = PropertyType.LONG;
c.foreignKeyDirection = (column.endsWith(Topology.IN_VERTEX_COLUMN_END) ? Direction.IN : Direction.OUT);
c.foreignSchemaTable = SchemaTable.of(foreignKeyParts[0], foreignKeyParts[1].substring(0, foreignKeyParts[1].length() - Topology.IN_VERTEX_COLUMN_END.length()));
c.foreignKeyProperty = null;
}
}
/**
* add a new column
*
* @param stt
* @param column
* @param alias
*/
public void add(SchemaTableTree stt, String column, String alias) {
add(stt.getSchemaTable(), column, stt.getStepDepth(), alias);
}
/**
* add a new column
*
* @param st
* @param column
* @param stepDepth
* @param alias
*/
public void add(SchemaTable st, String column, int stepDepth, String alias) {
add(st.getSchema(), st.getTable(), column, stepDepth, alias);
}
public void addForeignKey(SchemaTableTree stt, String column, String alias) {
String[] foreignKeyParts = column.split("\\.");
Preconditions.checkState(foreignKeyParts.length == 2 || foreignKeyParts.length == 3, "Edge table foreign must be schema.table__I\\O or schema.table.property__I\\O. Found %s", column);
addForeignKey(stt.getSchemaTable().getSchema(), stt.getSchemaTable().getTable(), column, stt.getStepDepth(), alias, foreignKeyParts);
}
/**
* get an alias if the column is already in the list
*
* @param schema
* @param table
* @param column
* @return
*/
private String getAlias(String schema, String table, String column, int stepDepth) {
//PropertyType is not part of equals or hashCode so not needed for the lookup.
Column c = new Column(schema, table, column, null, stepDepth);
return columns.get(c);
}
/**
* get an alias if the column is already in the list
*
* @param stt
* @param column
* @return
*/
public String getAlias(SchemaTableTree stt, String column) {
return getAlias(stt.getSchemaTable(), column, stt.getStepDepth());
}
/**
* get an alias if the column is already in the list
*
* @param st
* @param column
* @param stepDepth
* @return
*/
public String getAlias(SchemaTable st, String column, int stepDepth) {
return getAlias(st.getSchema(), st.getTable(), column, stepDepth);
}
@Override
public String toString() {
String sep = "";
StringBuilder sb = new StringBuilder();
int count = 1;
for (Map.Entry<Column, String> columnEntry : this.columns.entrySet()) {
Column c = columnEntry.getKey();
String alias = columnEntry.getValue();
sb.append(sep);
sep = ",\n\t";
c.toString(sb);
sb.append(" AS ");
sb.append(this.sqlgGraph.getSqlDialect().maybeWrapInQoutes(alias));
if (this.drop && (this.identifiers.isEmpty() || count++ == this.identifiers.size())) {
break;
}
}
return sb.toString();
}
public Pair<String, PropertyType> getPropertyType(String alias) {
Column column = this.aliases.get(alias);
if (column != null) {
return Pair.of(column.column, column.propertyType);
} else {
return null;
}
}
public String toString(String prefix) {
StringBuilder sb = new StringBuilder();
int i = 1;
List<String> fromAliases = this.aliases.keySet().stream().filter(
(alias) -> !alias.endsWith(Topology.IN_VERTEX_COLUMN_END) && !alias.endsWith(Topology.OUT_VERTEX_COLUMN_END))
.collect(Collectors.toList());
for (String alias : fromAliases) {
sb.append(prefix);
sb.append(".");
sb.append(this.sqlgGraph.getSqlDialect().maybeWrapInQoutes(alias));
if (i++ < fromAliases.size()) {
sb.append(", ");
}
}
return sb.toString();
}
public Map<SchemaTable, List<Column>> getInForeignKeys(int stepDepth, SchemaTable schemaTable) {
return getForeignKeys(stepDepth, schemaTable, Direction.IN);
}
public Map<SchemaTable, List<Column>> getOutForeignKeys(int stepDepth, SchemaTable schemaTable) {
return getForeignKeys(stepDepth, schemaTable, Direction.OUT);
}
private Map<SchemaTable, List<Column>> getForeignKeys(int stepDepth, SchemaTable schemaTable, Direction direction) {
Map<SchemaTable, List<Column>> result = new HashMap<>();
for (Column column : this.columns.keySet()) {
if (column.isForeignKey && column.foreignKeyDirection == direction && column.isFor(stepDepth, schemaTable)) {
List<Column> columns = result.computeIfAbsent(column.getForeignSchemaTable(), (k) -> new ArrayList<>());
columns.add(column);
}
}
return result;
}
public LinkedHashMap<Column, String> getFor(int stepDepth, SchemaTable schemaTable) {
LinkedHashMap<Column, String> result = new LinkedHashMap<>();
for (Column column : this.columns.keySet()) {
if (column.isFor(stepDepth, schemaTable)) {
result.put(column, this.columns.get(column));
}
}
return result;
}
public void indexColumns(int startColumnIndex) {
int i = startColumnIndex;
for (Column column : columns.keySet()) {
column.columnIndex = i++;
}
i++;
}
public int indexColumnsExcludeForeignKey(int startColumnIndex) {
int i = startColumnIndex;
for (String alias : this.aliases.keySet()) {
if (!alias.endsWith(Topology.IN_VERTEX_COLUMN_END) && !alias.endsWith(Topology.OUT_VERTEX_COLUMN_END)) {
this.aliases.get(alias).columnIndex = i++;
}
}
return i++;
}
/**
* simple column, fully qualified: schema+table+column
*
* @author jpmoresmau
*/
public class Column {
private final String schema;
private final String table;
private final String column;
private final int stepDepth;
private PropertyType propertyType;
private final boolean ID;
private int columnIndex = -1;
//Foreign key properties
private boolean isForeignKey;
private Direction foreignKeyDirection;
private SchemaTable foreignSchemaTable;
//Only set for user identifier primary keys
private String foreignKeyProperty;
Column(String schema, String table, String column, PropertyType propertyType, int stepDepth) {
super();
this.schema = schema;
this.table = table;
this.column = column;
this.propertyType = propertyType;
this.stepDepth = stepDepth;
this.ID = this.column.equals(Topology.ID);
}
@Override
public int hashCode() {
final int prime = 31;
int result = 1;
result = prime * result + getOuterType().hashCode();
result = prime * result + ((column == null) ? 0 : column.hashCode());
result = prime * result + ((schema == null) ? 0 : schema.hashCode());
result = prime * result + ((table == null) ? 0 : table.hashCode());
result = prime * result + stepDepth;
return result;
}
@Override
public boolean equals(Object obj) {
if (this == obj)
return true;
if (obj == null)
return false;
if (getClass() != obj.getClass())
return false;
Column other = (Column) obj;
if (!getOuterType().equals(other.getOuterType()))
return false;
if (column == null) {
if (other.column != null)
return false;
} else if (!column.equals(other.column))
return false;
if (schema == null) {
if (other.schema != null)
return false;
} else if (!schema.equals(other.schema))
return false;
if (table == null) {
if (other.table != null)
return false;
} else if (!table.equals(other.table))
return false;
return this.stepDepth == other.stepDepth;
}
private ColumnList getOuterType() {
return ColumnList.this;
}
public String getSchema() {
return schema;
}
public String getTable() {
return table;
}
public String getColumn() {
return column;
}
public int getStepDepth() {
return stepDepth;
}
public PropertyType getPropertyType() {
return propertyType;
}
public boolean isID() {
return ID;
}
public int getColumnIndex() {
return columnIndex;
}
public boolean isForeignKey() {
return isForeignKey;
}
public Direction getForeignKeyDirection() {
return foreignKeyDirection;
}
public SchemaTable getForeignSchemaTable() {
return foreignSchemaTable;
}
@SuppressWarnings("BooleanMethodIsAlwaysInverted")
boolean isForeignKeyProperty() {
return foreignKeyProperty != null;
}
@Override
public String toString() {
StringBuilder sb = new StringBuilder();
toString(sb);
return sb.toString();
}
/**
* to string using provided builder
*
* @param sb
*/
void toString(StringBuilder sb) {
sb.append(sqlgGraph.getSqlDialect().maybeWrapInQoutes(schema));
sb.append(".");
sb.append(sqlgGraph.getSqlDialect().maybeWrapInQoutes(table));
sb.append(".");
sb.append(sqlgGraph.getSqlDialect().maybeWrapInQoutes(column));
}
boolean isFor(int stepDepth, SchemaTable schemaTable) {
return this.stepDepth == stepDepth && this.schema.equals(schemaTable.getSchema()) && this.table.equals(schemaTable.getTable());
}
public boolean isForeignKey(int stepDepth, SchemaTable schemaTable) {
return this.stepDepth == stepDepth && this.schema.equals(schemaTable.getSchema()) && this.table.equals(schemaTable.getTable());
}
}
}
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 5,990
|
{"url":"https:\/\/www.jobilize.com\/online\/course\/11-8-homework-the-chi-square-distribution-by-openstax?qcr=www.quizover.com","text":"# 11.8 Homework\n\n Page 1 \/ 4\nThis module provides homework on Chi-Square Distribution as a part of Collaborative Statistics collection (col10522) by Barbara Illowsky and Susan Dean.\n\u2022 Explain why the \u201cgoodness of fit\u201d test and the \u201ctest for independence\u201d are generally right tailed tests.\n\u2022 If you did a left-tailed test, what would you be testing?\n\n## Word problems\n\nFor each word problem, use a solution sheet to solve the hypothesis test problem. Go to The Table of Contents 14. Appendix for the chi-square solution sheet. Round expected frequency to two decimal places.\n\nA 6-sided die is rolled 120 times. Fill in the expected frequency column. Then, conduct a hypothesis test to determine if the die is fair. The data below are the result of the 120 rolls.\n\nFace Value Frequency Expected Frequency\n1 15\n2 29\n3 16\n4 15\n5 30\n6 15\n\nThe marital status distribution of the U.S. male population, age 15 and older, is as shown below. ( Source: U.S. Census Bureau, Current Population Reports )\n\nMarital Status Percent Expected Frequency\nnever married 31.3\nmarried 56.1\nwidowed 2.5\ndivorced\/separated 10.1\n\nSuppose that a random sample of 400 U.S. young adult males, 18 \u2013 24 years old, yielded the following frequency distribution. We are interested in whether this age group of males fits the distribution of the U.S. adult population. Calculate the frequency one would expect when surveying 400 people. Fill in the above table, rounding to two decimal places.\n\nMarital Status Frequency\nnever married 140\nmarried 238\nwidowed 2\ndivorced\/separated 20\n\u2022 The data fits the distribution\n\u2022 The data does not fit the distribution\n\u2022 3\n\u2022 19.27\n\u2022 0.0002\n\u2022 Decision: Reject Null; Conclusion: Data does not fit the distribution.\n\nThe next two questions refer to the following information . The columns in the chart below contain the Race\/Ethnicity of U.S. Public Schools for a recent year, the percentages for the Advanced Placement Examinee Population for that class and the Overall Student Population. ( Source: http:\/\/www.collegeboard.com ). Suppose the right column contains the result of a survey of 1000 local students from that year who took an AP Exam.\n\nRace\/Ethnicity AP Examinee Population Overall Student Population Survey Frequency\nAsian, Asian American or Pacific Islander 10.2% 5.4% 113\nBlack or African American 8.2% 14.5% 94\nHispanic or Latino 15.5% 15.9% 136\nAmerican Indian or Alaska Native 0.6% 1.2% 10\nWhite 59.4% 61.6% 604\nNot reported\/other 6.1% 1.4% 43\n\nPerform a goodness-of-fit test to determine whether the local results follow the distribution of the U. S. Overall Student Population based on ethnicity.\n\nPerform a goodness-of-fit test to determine whether the local results follow the distribution of U. S. AP Examinee Population, based on ethnicity.\n\n\u2022 5\n\u2022 13.4\n\u2022 0.0199\n\u2022 Decision: Reject null when $a=0\\text{.}\\text{05}$ ; Conclusion: Local data do not fit the AP Examinee Distribution. Decision: Do not reject null when $a=0\\text{.}\\text{01}$ ; Conclusion: There is insufficient evidence to conclude that Local data do not fit the AP Examinee Distribution.\n\nThe City of South Lake Tahoe, CA, has an Asian population of 1419 people, out of a total population of 23,609 ( Source: U.S. Census Bureau ). Suppose that a survey of 1419 self-reported Asians in Manhattan, NY, area yielded the data in the table below. Conduct a goodness of fit test to determine if the self-reported sub-groups of Asians in the Manhattan area fit that of the Lake Tahoe area.\n\nRace Lake Tahoe Frequency Manhattan Frequency\nAsian Indian 131 174\nChinese 118 557\nFilipino 1045 518\nJapanese 80 54\nKorean 12 29\nVietnamese 9 21\nOther 24 66\n\nwhere we get a research paper on Nano chemistry....?\nwhat are the products of Nano chemistry?\nThere are lots of products of nano chemistry... Like nano coatings.....carbon fiber.. And lots of others..\nlearn\nEven nanotechnology is pretty much all about chemistry... Its the chemistry on quantum or atomic level\nlearn\nda\nno nanotechnology is also a part of physics and maths it requires angle formulas and some pressure regarding concepts\nBhagvanji\nhey\nGiriraj\nPreparation and Applications of Nanomaterial for Drug Delivery\nrevolt\nda\nApplication of nanotechnology in medicine\nwhat is variations in raman spectra for nanomaterials\nI only see partial conversation and what's the question here!\nwhat about nanotechnology for water purification\nplease someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.\nDamian\nyes that's correct\nProfessor\nI think\nProfessor\nNasa has use it in the 60's, copper as water purification in the moon travel.\nAlexandre\nnanocopper obvius\nAlexandre\nwhat is the stm\nis there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?\nRafiq\nindustrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong\nDamian\nHow we are making nano material?\nwhat is a peer\nWhat is meant by 'nano scale'?\nWhat is STMs full form?\nLITNING\nscanning tunneling microscope\nSahil\nhow nano science is used for hydrophobicity\nSantosh\nDo u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq\nRafiq\nwhat is differents between GO and RGO?\nMahi\nwhat is simplest way to understand the applications of nano robots used to detect the cancer affected cell of human body.? How this robot is carried to required site of body cell.? what will be the carrier material and how can be detected that correct delivery of drug is done Rafiq\nRafiq\nif virus is killing to make ARTIFICIAL DNA OF GRAPHENE FOR KILLED THE VIRUS .THIS IS OUR ASSUMPTION\nAnam\nanalytical skills graphene is prepared to kill any type viruses .\nAnam\nAny one who tell me about Preparation and application of Nanomaterial for drug Delivery\nHafiz\nwhat is Nano technology ?\nwrite examples of Nano molecule?\nBob\nThe nanotechnology is as new science, to scale nanometric\nbrayan\nnanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale\nDamian\nIs there any normative that regulates the use of silver nanoparticles?\nwhat king of growth are you checking .?\nRenato\nWhat fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?\nwhy we need to study biomolecules, molecular biology in nanotechnology?\n?\nKyle\nyes I'm doing my masters in nanotechnology, we are being studying all these domains as well..\nwhy?\nwhat school?\nKyle\nbiomolecules are e building blocks of every organics and inorganic materials.\nJoe\nhow did you get the value of 2000N.What calculations are needed to arrive at it\nPrivacy Information Security Software Version 1.1a\nGood\nGot questions? Join the online conversation and get instant answers!","date":"2020-10-20 03:01:24","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 2, \"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\": 0.5253660082817078, \"perplexity\": 4334.753129085037}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2020-45\/segments\/1603107869785.9\/warc\/CC-MAIN-20201020021700-20201020051700-00537.warc.gz\"}"}
| null | null |
package com.taobao.tddl.rule.virtualnode;
import java.util.HashSet;
import java.util.Map;
import java.util.Set;
import java.util.concurrent.ConcurrentHashMap;
import com.taobao.tddl.common.model.lifecycle.AbstractLifecycle;
import com.taobao.tddl.rule.enumerator.handler.CloseIntervalFieldsEnumeratorHandler;
import com.taobao.tddl.rule.enumerator.handler.IntegerPartDiscontinousRangeEnumerator;
import com.taobao.tddl.rule.model.sqljep.Comparative;
/**
* @author <a href="junyu@taobao.com">junyu</a>
* @version 1.0
* @since 1.6
* @date 2011-8-20 02:58:34
*/
public class WrappedLogic extends AbstractLifecycle {
private static final String SLOT_PIECE_SPLIT = ",";
private static final String RANGE_SUFFIX_SPLIT = "-";
private CloseIntervalFieldsEnumeratorHandler enumerator = new IntegerPartDiscontinousRangeEnumerator();
protected String valuePrefix; // 无getter/setter
protected String valueSuffix; // 无getter/setter
protected int valueAlignLen = 0; // 无getter/setter
protected String tableSlotKeyFormat = null;
public void setTableSlotKeyFormat(String tableSlotKeyFormat) {
if (tableSlotKeyFormat == null) {
return;
}
this.tableSlotKeyFormat = tableSlotKeyFormat;
int index0 = tableSlotKeyFormat.indexOf('{');
if (index0 == -1) {
this.valuePrefix = tableSlotKeyFormat;
return;
}
int index1 = tableSlotKeyFormat.indexOf('}', index0);
if (index1 == -1) {
this.valuePrefix = tableSlotKeyFormat;
return;
}
this.valuePrefix = tableSlotKeyFormat.substring(0, index0);
this.valueSuffix = tableSlotKeyFormat.substring(index1 + 1);
this.valueAlignLen = index1 - index0 - 1;// {0000}中0的个数
}
protected String wrapValue(String value) {
StringBuilder sb = new StringBuilder();
if (valuePrefix != null) {
sb.append(valuePrefix);
}
if (valueAlignLen > 1) {
int k = valueAlignLen - value.length();
for (int i = 0; i < k; i++) {
sb.append("0");
}
}
sb.append(value);
if (valueSuffix != null) {
sb.append(valueSuffix);
}
return sb.toString();
}
/**
* <pre>
* 参数oriMap的value格式为 <b>0,1,2-6</b> 0,1表示2个槽,'-'表示一个范围
*
* 此函数中将范围枚举成一个个槽,并将槽变为key,原本的key变为value
*
* example 1:key为 1 value为 1,2,3-6
*
* 返回结果为 1->1,2->1,3->1,4->1,5->1,6->1
*
* example 2:key为db_group_1 value为1,2 db_group_2 value为3,4-6
* 返回结果为 1->db_group_1,2->db_group_1
* 3->db_group_2,4->db_group_2,5->db_group_3,2->db_group_2
*
* <b>
* 暂时不支持任何形式的value格式化.即_0000,0001之类的字符串,只接受
* 数学形式上的integer,long
*
* 后续改进
* </b>
* </pre>
*
* @param tableMap
* @return
*/
protected Map<String, String> extraReverseMap(Map<String, String> oriMap) {
ConcurrentHashMap<String, String> slotMap = new ConcurrentHashMap<String, String>();
for (Map.Entry<String, String> entry : oriMap.entrySet()) {
String[] pieces = entry.getValue().trim().split(SLOT_PIECE_SPLIT);
for (String piece : pieces) {
String[] range = piece.trim().split(RANGE_SUFFIX_SPLIT);
if (range.length == 2) {
Comparative start = new Comparative(Comparative.GreaterThanOrEqual, Integer.valueOf(range[0]));
Comparative end = new Comparative(Comparative.LessThanOrEqual, Integer.valueOf(range[1]));
int cumulativeTimes = Integer.valueOf(range[1]) - Integer.valueOf(range[0]);
Set<Object> result = new HashSet<Object>();
enumerator.mergeFeildOfDefinitionInCloseInterval(start, end, result, cumulativeTimes, 1);
for (Object v : result) {
slotMap.put(String.valueOf(v), entry.getKey());
}
} else if (range.length == 1) {
slotMap.put(piece, entry.getKey());
} else {
throw new IllegalArgumentException("slot config error,slot piece:" + piece);
}
}
}
return slotMap;
}
}
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 1,446
|
Q: JFrame freezes after clicking a button I have a JFrame that has a JTextField and a JButton. I am trying to get the count of elements from a user's input in the JTextField. However, after pressing the button, the JFrame freezes. Here is my code:
private void bubbleSortButtonActionPerformed(java.awt.event.ActionEvent evt) {
// read the user's input from text field and store it in to the elements string
String elements = inputField.getText();
// initialize a scanner for the elements
Scanner input = new Scanner(elements);
// initialize a counter variable for counting the number of elements input by the user
int n = 0;
// increment the counter variable as long as it could read a next token
while (input.hasNext())
n++;
}
I've already tried to search for solutions, but nothing answered my problem. What's wrong with my code?
A: Change:
while (input.hasNext())
n++;
..to something like..
while (input.hasNext()) {
input.getNext();
n++;
}
Otherwise the conditional will be true forever.
Sourced via comment.
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 6,925
|
{"url":"https:\/\/stats.stackexchange.com\/questions\/141309\/does-rejection-of-null-hypothesis-in-multiple-regression-entail-causation","text":"# Does rejection of null hypothesis in multiple regression entail causation?\n\nWe make a model of the following form:\n\n$$y = \\beta_0 + \\beta_1x_1 + \\beta_2x_2 + \\epsilon$$\n\nand with $n=1,000$, $\\hat\\beta_1$ has a p-value <0.001.\n\nIf our data and data collection meets assumptions of multiple linear regression, we can say:\n\n\u2022 If the population shows NO relationship between $x_1$ and $y$, then properly random samples of $n=1,000$ will show this degree of fit (or relationship) in <0.001 of the samples.\n\nWhat then are the logically possible relationships between $x_1$ and $y$?\n\nI see the following but want to know if there are others:\n\n\u2022 If $\\beta_1 = 0$:\n1. $x_1$ and $y$ are linearly independent and uncorrelated\n2. $x_1$ and $y$ are two independent variables \"rendered dependent\" by observations on their effect, $x_2$\n\u2022 If $\\beta_1 \\ne 0$:\n1. $x_1$ is linear cause of $y$ (more umbrellas cause wetter sidewalks)\n2. $y$ is linear cause of $x_1$ (wetter sidewalks cause more umbrellas)\n3. $Z$ is linear cause of $x_1$ AND $y$ (more rain, GDP,... causes both)\n4. []\n5. $Cor(x_1,y) \\ne 0$ and linearly dependent but no causal chain, parent, or child\n\nAre the final three choices mutually exclusive and exhaustive if we are correct that $\\beta_1 \\ne 0$? Illustrative examples would help.\n\n*Note, when I say \"cause,\" I do not mean the direct, immediate cause, but instead that there is a shared causal chain from $l,m,n, ... \\to x_1 \\to p,q,r,... \\to y$. Also, $\\beta_1 \\ne 0$. $\\hat\\beta_1$ is irrelevant.\n\nSuggested additions that I dispute (but open to change):\n\n1. $x_1$ and $y$ are (jointly) linear cause(s) of $x_2$ (less heat and more water cause more time to boil water)\n\u2022 if this is not really a case of (3), hence new example, then it seems we incorrectly concluded that $\\beta_1 \\ne 0$ with two independent variables ($x_1$ and $y$) such as heat and water quantity. (I think \"independent variables\" $\\to \\beta_1 = 0 \\to$ (10), not (4))\n\u2022 also violates multicollinearity assumption needed for $\\hat{\\beta_1} = \\beta_1$ ($x_1$ and $x_2$ should be linearly independent and uncorrelated).\n2. $x_1$ and $y$ have no linear causal relationships at all (margarine consumption\/capita $\\to$ divorces in Maine\/capita(k))\n\u2022 then it seems we incorrectly concluded that $\\beta_1 \\ne 0$. ($\\beta_1 = 0 \\to$ (10), not (5))\n\u2022 time-series autocorrelation (of values and\/or errors) may pose a problem??\n\u2022 what is the population from which these values represent a random sample?\n\u2022 if this is the \"population\", $N=10$, how do we talk about it?\n\u2022 $Z \\to x_1,y$ is true but unbelievable with current knowledge\n\u2022 No. Some readings that may help you understand how to weave causal inference into your statistical models: Greenland, S., Pearl, J., and Robins, J. M. (1999). Causal diagrams for epidemiologic research. Epidemiology, 10(1):37\u201348. and Maldonado, G. and Greenland, S. (2002). Estimating causal effects. International Journal of Epidemiology, 31(2):422\u2013438. Mar 11 '15 at 17:23\n\u2022 For more depth, see also: Pearl, J. (2000). Causality: Models, Reasoning, and Inference. Cambridge University Press. And the forthcoming Hern\u00e1n, M. A. and Robins, J. M. (2015). Causal Inference. Chapman & Hall\/CRC. Mar 11 '15 at 17:25\n\u2022 Note also that $\\beta_1= 0$ is perfectly compatible with the idea that $x_1$ causes $x_2$, which causes $y$. Mar 11 '15 at 18:10\n\u2022 @Scortchi: or just $x_1$ causes $y$, but not linearly. Mar 11 '15 at 18:23\n\u2022 \"If we correctly reject the null hypothesis...\": Note that one would not be justified in saying that in the population B1 is 99.9% likely to be nonzero. One could say that if the null were true, 99.9% of sample B1's would be closer to zero than the observed one. Mar 12 '15 at 0:07\n\nIt sounds like you're asking if two variables $x_1, y$ are dependent, which causal relationships between them might account for the dependence.\n\nYou pointed out three. A fourth is the following where neither is the cause of the other, nor do they have a common cause:\n\n$x_1 \\rightarrow x_2 \\leftarrow y$.\n\nOr in general something like this. For example,\n\n$x_1 \\rightarrow a \\leftarrow b \\rightarrow c \\leftarrow y$ where $a \\rightarrow x_2 \\leftarrow c$.\n\nYour choices are not mutually exclusive because there could also be a common cause $z$.\n\n\u2022 Thanks! But I don't understand your arrows or your fourth possibility. Could you give an illustrative example of an $x_1$ and $y$ that are dependent and meet all assumptions as in original question that clearly is your (4) (or clearly does not fall into 1-3)?\n\u2013\u00a0jtd\nMar 11 '15 at 20:19\n\u2022 @jtd: What don't you understand? If $x_1$ and $y$ are the causes of $x_2$, they will appear dependent given that $x_2$ is observed. Mar 11 '15 at 20:33\n\u2022 (I should say, \"may\" appear dependent. They could nevertheless remain independent.) Mar 11 '15 at 21:39\n\u2022 @jtd: Just rearrange the terms in your equation so that $x_2$ is a function of $y$ and $x_1$. Why is anything violated? Anyway, the linear relationship doesn't tell you anything about the actual direction of causality \u2014 just that things are dependent. Mar 11 '15 at 23:12\n\u2022 @jtd but $\\beta_1$ won't be zero because $x_1$ and $y$ won't be independent given observation of $x_2$. A pair of common causes can be rendered dependent by observation of their effect. Mar 12 '15 at 16:01","date":"2021-09-27 08:12:25","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 1, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.7304359674453735, \"perplexity\": 1089.1312383740694}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2021-39\/segments\/1631780058373.45\/warc\/CC-MAIN-20210927060117-20210927090117-00068.warc.gz\"}"}
| null | null |
This book is more oriented to children who have recently lost their hearing and their parents, however, I would recommend for all readers.
The story follows Cece a small 4 year old rabbit how after coming ill with meningitis loses her hearing. The story follows her feelings as to dealing with being deaf and how she comes to view her first difference and then superpower.
Hearing loss can be very challenging especially for children, however, as this book shows, there's nothing wrong with being deaf and vocalizing your feeling goes a long way to make children come to terms with their conditions.
|
{
"redpajama_set_name": "RedPajamaC4"
}
| 4,011
|
\section{Introduction}
\label{intro}
Object classification and detection are fundamental tasks in computer vision. Previous mainstream object detectors based on hand-designed features (HOG\cite{dalal2005histograms}) and classifiers like linear discriminative analysis (LDA) \cite{welling2005fisher} or Support Vector Machines~\cite{boser1992training,lsvm-pami} required training of limited parameters, and were thus sustainable with small datasets. More recent Deep Convolutional Neural Network (DCNN)-based object detectors\cite{RCNN,renNIPS15fasterrcnn} utilize more powerful DCNN features and yield a significant performance boost, both for image classification\cite{he2015deep,vgg,alexnet,GoogleLenet} and object detection\cite{RCNN,renNIPS15fasterrcnn}. Nevertheless, deep convolutional neural networks need lots of labeled images to train their millions of parameters. Collecting these images and annotating the objects is cumbersome and expensive. Even the most popular detection datasets provide a limited number of labeled categories, e.g., 20 classes in PASCAL VOC\cite{pascal} and 200 in ImageNet\cite{ImageNet}. Hence, a question arises: is it possible to avoid the frustrating collection and annotation process and still train an effective object classifier or detector?
To achieve this goal, researchers have proposed several recent models for object model training. \cite{LSDA} introduces a transfer learning method that gains an object detector by transferring learned object knowledge from a classifier. \cite{siva2011weakly,song2014learning} propose to train an object localization and detection model with image-level labels. However, these methods still count on per-image manual supervision. In contrast, the methods in the previous literature that assume no per-image labeling data can be categorized into two groups: 1) methods that utilize on-line search results or an existing unlabeled dataset \cite{chen2015webly,divvala2014learning,zhou2015conceptlearner}; 2) methods that render domain-specific synthetic images \cite{peng2015learning,BMVC}. For example, \cite{chen2015webly,divvala2014learning} propose to learn a visual representation and object detectors from on-line images. \cite{zhou2015conceptlearner} leverages a concept learner to discover visual knowledge from weakly labeled images (weak labels can be in the form of keywords or short description). On the other hand, \cite{peng2015learning,BMVC} proposed to generate synthetic training images from 3D CAD models. Such synthetic data is shown to be useful for augmenting small amounts of real labeled data to improve object detection, as bounding boxes can be obtained ``for free'' and objects can be rendered in arbitrarily many viewpoints.
\begin{figure}[t]
\centering
\includegraphics[width=\textwidth]{2.png}
\vspace{-1cm}
\caption{\textbf{Two-Stream Texture and Shape Model:}
We propose a framework to combine texture cues and shape information for object recognition with minimal supervision. The Texture-CNN stream is trained on images collected from on-line image search engines and the Shape-CNN stream is separately trained on images generated from domain-specific 3D CAD models. To combine the two streams, an average of the two last layers' activations is computed to create the Fusion-CNN object classifier and detector. In the test phrase, the model will forward an image patch through the two networks simultaneously and compute average fusion of the activations from the last layers. (Best viewed in color.) }
\label{fig:overview}
\vspace{-0.73cm}
\end{figure}
While these approaches can work effectively in some cases, there are still many challenges that need to be addressed:
\begin{itemize}
\item \textbf{Lack of bounding boxes:} Unsupervised machine learning algorithms tackle learning where no labeled training data is provided. This setting leads to a great challenge for object detection because the performance of a detection system depends heavily on differentiated positive and negative training examples labeled with tight bounding boxes. Without such annotations, it is difficult for a model to learn the extent and shape of objects, i.e. which parts of the image correspond to the target object and which are background.
\item \textbf{Missing shape or texture cues:} Prior literature uses either web search~\cite{chen2015webly,divvala2014learning} or synthetic images~\cite{peng2015learning,su2015render,BMVC}, but rarely combines the intrinsic cues like object shape and characteristic appearance patterns, or ``texture'', which are critical for recognition systems. Some rigid objects can be easily recognized by its shape, such as aeroplanes and sofa; other objects can be easily recognized by their unique texture, such as leopards and bees.
\item \textbf{Domain Shift:} Images from different sources have different statistics for background, texture, intensity and even illumination \cite{saenko2010adapting}, which consequently results in domain shift problems. Unlike images taken in the wild, most photographs returned by image search engines lack diversity in viewpoint, background and shape, for image search engines follow a high-precision low-recall regime. In addition, synthetic images used in current work~\cite{peng2015learning} are far from photorealism in terms of intensity contrast, background, and object texture.
\end{itemize}
In this paper, we address these shortcomings by proposing a two-stream DCNN architecture (See Figure \ref{fig:overview}) that decomposes the input into texture and shape feature streams. The texture stream learns realistic texture cues from images downloaded from the Internet. Web images (which contain noise from backgrounds and unrelated results) are collected by searching for the names of categories in image search engines, i.e. Google Image Search. We prune the noise data and use the cleaned data to train the texture-based stream for classification or detection. The shape stream is trained exclusively on shape information from 2D images rendered from CAD models annotated with category labels.
Note that images from web search also contain shape information, but due to the lack of bounding box annotations, it is not accurate enough for localization models. Synthetic images can also be generated from 3D CAD models with added texture mapping, but the result is non-photorealistic and lacks real-world variety. Therefore, synthetic images rendered by 3D CAD models can be viewed as primarily shape-oriented and the web-search images can be viewed as primarily texture-oriented. The outputs of the two streams are combined through averaging the two top layers' activations. Our method requires no tedious manual bounding box annotation of object instances and no per-image category labeling and can generate training data for almost any novel category. Table \ref{tab_compare} shows a comparison of the amount of supervision with other methods. The only supervision in our work comes from labeling the CAD models as positive examples vs. ``outliers'' while downloading them, and choosing proper textures for each category while generating the synthetic images.
We evaluate our model for object classification and detection on the standard PASCAL VOC 2007 dataset, and show that the texture and shape cues can reciprocally compensate for each other during recognition. In addition, our detector outperforms existing webly supervised detectors~\cite{divvala2014learning}, the approach based on synthetic images only~\cite{peng2015learning}, and the weakly supervised learning approach of~\cite{siva2011weakly}, despite using less manual supervision.
\begin{table*}[t]
\scriptsize
\noindent\begin{tabular}{ |c | c|c| }
\hline
Type &VOC Training Data Used&Supervision\\
\hline
CAD Supervision, Ours, \cite{peng2015learning} & NO & Labeling 3D CAD models and their texture\\
\hline
Webly Supervision \cite{chen2015webly,divvala2014learning}& NO & Semantic Labeling \\
\hline
Selected Supervision \cite{zhou2015conceptlearner}& NO & Per-image Labeling With Text \\
\hline
Weak Supervision \cite{siva2011weakly,song2014learning}& YES & Per-image Labeling \\
\hline
Full Supervision \cite{RCNN}& YES & Per-instance Labeling\\
\hline
\end{tabular}
\vspace{0.1cm}
\caption{\textbf{Comparison of supervision between our method and others.} The supervision in our work only comes from labeling the CAD models and their texture when generating the synthetic images. The second column indicates whether in-domain data (VOC \emph{train+val} set) is used in the methods. The amount of supervision used in each method increases from top to bottom.}
\label{tab_compare}
\vspace{-1cm}
\end{table*}
To summarize, this work contributes to the computer vision community in the following three aspects:
\begin{itemize}
\item we propose and implement a recognition framework that decomposes images into their shape and texture cues;
\item we show that combining these cues improves classification and detection performance while using minimal supervision;
\item we present a unified schema for learning from both web images and synthetic CAD images.
\end{itemize}
\section{Related Work}
\label{related}
\noindent \textbf{Webly Supervised Learning.} The explosion of visual data on the Internet provides important sources of data for vision research. However, cleaning and annotating these data is costly and inefficient. Researchers have striven to design methods that learn visual representations and semantic concepts directly from the unlabeled data. Because the detection task requires stronger supervision than classification, most previous research work \cite{bergamo2010exploiting,fergus2010learning,li2010optimol,schroff2011harvesting} involving web images only tackles the object classification task. Some recent work \cite{chen2013neil,divvala2014learning} aims at discovering common sense knowledge or capturing intra-concept variance. In the work of \cite{chen2015webly,divvala2014learning}, webly supervised object detectors are trained from image search results. We follow a similar approach as in~\cite{chen2015webly} to train our texture model from web search data, but also add shape information using a CAD-based CNN.
\vspace{0.1in}
\noindent \textbf{Utilization of CAD Models.} CAD models had been used by researchers since the early stages of computer vision. Recent work involving 3D CAD models focuses on pose prediction \cite{liebelt2010multi,stark2010back,su2015render,sun2009multi}. Other recent work applied CAD models to 2D object detection \cite{peng2015learning,sun2014virtual} by rendering synthetic 2D images from 3D CAD models and using them to augment the training data. The main drawback of these methods is that the rendered images are low-quality and lack real texture, which significantly hurts their performance. In contrast, we propose a two-stream architecture that adds texture information to the CAD-based shape channel. \cite{peng2015learning} explored several ways to simulate real images, by adding real-image background and textures onto the 3D models, but this requires additional human supervision to select appropriate background and texture images for each category. In this work, we propose more effective ways to simulate real data with less supervision.
\vspace{0.1in}
\noindent \textbf{Two-stream Learning.} The basic aim of two-stream learning is to model two-factor variations. \cite{tenenbaum2000separating} proposed a bilinear model to separate ``style" and ``content" of an image. In \cite{lin2015bilinear}, a two-stream architecture was proposed for fine-grained visual recognition, and the classifier is expressed as a product of two low-rank matrices. \cite{fragkiadaki2015learning,simonyan2014two} utilized two-stream architectures to model the temporal interactions and aspect of features. We propose a CNN-based two-stream architecture that learns intrinsic properties of objects from disparate data sources, with one stream learning to extract texture cues from real images and the other stream learning to extract shape information from CAD models.
\vspace{-0.2cm}
\section{Approach}
\label{appro}
Our ultimate goal is to learn a good object classifier and object detector from the massive amount of visual data available via web search and from synthetic data. As illustrated in Figure \ref{fig:overview}, we introduce a two-stream learning architecture to extract the texture cues and shape information simultaneously. Each stream consists of three parts: the data acquisition component, the DCNN model and the object classifier or detector. The intuition is to utilize texture-oriented images from the web to train the texture stream and correspondingly, use shape-oriented images rendered from 3D CAD models to train the shape stream.
\subsection{DCNN-based Two-Stream Model}
The history of two-stream learning can be traced back to over a decade ago when a ``bilinear model'' was proposed by \cite{tenenbaum2000separating} to separate the ``style'' and ``content'' of an image. More recent use~\cite{lin2015bilinear,fragkiadaki2015learning,simonyan2014two} of two-stream learning is based on a similar philosophy: employ different modalities to model different intrinsic visual properties, e.g. spatial features and temporal interactions. Inspired by this idea, we propose a two-stream learning architecture, with one stream modeling real image texture cues and the other modeling 3D shape information. We demonstrate that the texture and shape cues can reciprocally compensate for each other's errors through late fusion.
For fair comparison with other baselines, within each stream, we use the eight-layer ``AlexNet'' architecture proposed by \cite{alexnet}. It contains five convolutional layers, followed by two fully connected layers ($fc6$, $fc7$). After $fc7$, another fully connected layer ($fc8$) is applied to calculate the final class predictions. The network adopts ``dropout'' regularization to avoid overfitting and non-saturating neurons ($ReLU$ layers) to increases the nonlinear properties of the decision function and to speed up the training process. The network is trained by stochastic gradient descent and takes raw RGB image patches of size 227x227.
The last layer (\emph{fc8}) in each stream is represented by a softmax decision function. To combine the learned texture cues and shape information, we fuse the streams to render the final prediction as follows:
\begin{equation}
P(I=j|\mathbf{x}) = \frac{e^{\mathbf{x}^{T}\mathbf{w}_{j}^{t}}}{2 \sum_{i=1}^{N} e^{\mathbf{x}^{T}\mathbf{w}_{i}^{t}}}+\frac{e^{\mathbf{x}^{T}\mathbf{w}_{j}^{s}}}{2 \sum_{i=1}^{N} e^{\mathbf{x}^{T}\mathbf{w}_{i}^{s}}}
\label{equ_fusion}
\end{equation}
where $P(I=j|\mathbf{x})$ denotes the probability that image $I$ belongs to category $j$ given feature vector $\mathbf{x}$ (\emph{fc7} feature in this case); $\mathbf{x}^{T}$, $N$, $\mathbf{w}_{i}^{t}$, $\mathbf{w}_{i}^{s}$ are the transpose of $\mathbf{x}$, the number of total categories, weight vector for category $i$ in Texture CNN, weight vector for category $i$ in Shape CNN, respectively.
The final probability $P(I=j|\mathbf{x})$ is used as the score for Two-Stream classifier and detector.
\subsection{Texture CNN Stream}
\label{sub:texture}
Previous work \cite{peng2015learning,viscnn} has shown that discriminative texture information is crucial for object classification and object detection systems. The challenge is how to obtain large scale accurate texture data with the least effort and how to prune the noisy images from unrelated search results. Previous approaches~\cite{chen2015webly,chen2013neil,divvala2014learning} have tried various search engines to form the texture bank, while other research work~\cite{fan2010harvesting,xia2014well} attempt to clean the data.
\noindent \textbf{Noise Data Pruning.} We assume the distribution of features of the higher CNN layers follows a multivariate normal distribution, thus we can fit the data from each class to the domain-specific Gaussian distribution as follows:
\begin{equation}
f_{\mathbf{x}} (x_{1},x_{2} ... x_{k}) =
\frac{1}{\sqrt{(2\pi)^{k}|\sum|}}*exp(-\frac{1}{2}(\mathbf{x}-u)^T \sum{}{} ^{-1} (\mathbf{x}-u))
\end{equation}
where $\mathbf{x}$ is an k-dimensional feature vector and $\sum$ is the covariance matrix.
To remove outliers, for each category $j$ ($j\in [1,N]$, $N$ is the category number), we start from the downloaded image set $S_{j}$ with noise data and an empty set $T_{j}$. For each image $i$, we perform outlier removal by
\begin{equation}
T_{j}=\left\{\begin{matrix}
T_{j}\cup S_{j}(i), & P(S(i)=j|u^{j},\sum{}{}^{j}) \geqslant \varepsilon^{j} \\
T_{j},& P(S(i)=j|u^{j},\sum{}{}^{j}) < \varepsilon^{j}
\end{matrix}\right.
\label{equ:gus}
\end{equation}
where $P(S(i)=j|u,\sum)$, $\varepsilon^{j}, u^{j}, \sum{}{}^{j}$ are the probability that image $i$ belongs to category j, the pruning threshold for category $j$, the mean and covariance matrix of domain specific Gaussian distribution, respectively. We then use \{$T_{1}$, $T_{2}$, ..., $T_{N}$\} to train the Texture CNN.
\subsection{Shape CNN Model}
\label{appsub:shape}
Crowdsourced 3D models are easily accessible online and can be used to render unlimited images with different backgrounds, textures, and poses~\cite{su2015render,peng2015learning}. The widely used 3D Warehouse\footnote{https://3dwarehouse.sketchup.com}, Stanford Shapenet\footnote{http://shapenet.cs.stanford.edu/} provide numerous 3D CAD models for research use. Previous work has shown the great potential of synthetic images rendered from 3D CAD models for object detection~\cite{peng2015learning} and pose estimation~\cite{su2015render}.
The flexibility and rigidity give 3D CAD models the unique merit for monitoring image properties such as background, texture and object pose, with constant shape information. However, the drawbacks of synthetic data are also obvious:
\begin{enumerate}
\item \textbf{Lack of realism:} Sythetic images generally lack realistic intensity information which explicitly reflect fundamental visual cues such as texture, illumination, background.
\item \textbf{Statistic Mismatch:} The statistics (eg. edge gradient) of synthetic image are different from realistic images. Thus the discriminatory information preserved by the DCNN trained on synthetic images may lose its effect on realistic images.
\end{enumerate}
\begin{wrapfigure}{r}{0.5\textwidth}
\vspace{-1cm}
\begin{center}
\includegraphics[width=0.48\textwidth]{edge}
\end{center}
\vspace{-0.7cm}
\caption{\small{\textbf{Illustration of Edge Gradient}. Synthetic images rendered with white backgrounds tend have higher contrast edges around the outline of the object than natural images taken in the wild. Note that this figure is an illustration, not representing real pixel intensities.}}
\vspace{-0.7cm}
\label{fig:simulation}
\end{wrapfigure}
\noindent\textbf{Simulating Real-Image Statistics:} One flaw of synthetic images is that the instance is inconsistent with the background.
Thus, even rendered with very sophisticated parameters (pose variation, illumination, etc.), the statistics mismatch in intensity level still remains. Figure \ref{fig:simulation} illustrates the difference of edge gradients between synthetic images rendered with white backgrounds and real images.
After analysing the synthetic data, we find the objects in the synthetic images tend to have higher contrast edges compared to real images taken in the wild. Adding more realistic backgrounds~\cite{peng2015learning} is a good way to decrease the contrast, but may obscure the object if the background is not chosen carefully. Instead, in this work, for each image $I$, we process $I$ by:
\begin{equation}
I' = \psi_{\mathbf{G}}(I) + \xi_{G}
\end{equation}
where $\psi_{\mathbf{G}}(.)$ is a smoothing function based on a Gaussian filter and $\xi_{G}$ is a Gaussian noise generator. $\psi_{\mathbf{G}}(.)$ is used to mitigate the sharp edge contrast and $\xi_{G}$ to increase the intensity variations.
\vspace{-0.2cm}
\section{Experiments}
\label{exper}
In this section, we describe our experimental settings in detail. We start from downloading texture images via web search and rendering shape data from 3D CAD models. We evaluate our Two-Stream CNN classifier and detector on the standard benchmark PASCAL VOC 2007 \cite{pascal} dataset.
\vspace{-0.3cm}
\subsection{Data Acquisition}
\label{exp:data}
\noindent \textbf{Texture Data} As illustrated in Figure \ref{fig:overview}, we leverage a text-based image search engine (Google image search engine in our experiments) to collect the image data. Most of the images returned by Google contain a single object centered in the picture. This is good news for an algorithm attempting to learn the main features of a certain category. However, the drawback is the returned data is noisy and highly biased. For example, the top results returned by searching ``aeroplane'' may contain many toy aeroplane and paper plane images. To make matters worse, some returned images contain no ``aeroplane'', but objects from other categories.
We use the name of each object category as the query for Google image search engine to collect the training images. After removing unreadable images, there are about 900 images for each category and 18212 images in total.
\begin{figure}[t]
\centering
\includegraphics[width=\textwidth]{purify_1.png}
\vspace{-0.2cm}
\caption{\textbf{Illustration Of Noisy Data Pruning}. We fit the downloaded data to a multivariate normal distribution and remove the outliers if their probability is less than a learned threshold. (Best viewed in color) }
\vspace{-0.5cm}
\label{fig:purify}
\end{figure}
With millions of parameters, the CNN model easily overfits to small dataset. Therefore, data augmentation is valuable. Since most of the images are object-centered, we crop 40 patches by randomly locating the top-left corner $(x_{1}, y_{1})$ and bottom-right corner $(x_{2}, y_{2})$ by the following constraint:
\begin{equation}
\left\{\begin{matrix}
x_{1} \in[\frac{W}{20},\frac{3W}{20}],& y_{1} \in[\frac{H}{20},\frac{3H}{20}]\\
&\\
x_{2}\in[\frac{17W}{20},\frac{19W}{20}], & y_{2} \in[\frac{17H}{20},\frac{19H}{20}]
\end{matrix}\right.
\end{equation}
$(W, H)$ are the width and height of original image. The constraint ensures that 49\%-81\% of the center area of the image is reserved. This image subsampling process leaves us about 0.7 million images to train the Texture CNN.
We further utilize the approach illustrated in Section~\ref{sub:texture} to remove outliers from the downloaded data. For each category $j$ ($j\in[1,N]$, $N$ is the class number), we denote all the image patches after image subsampling as $S_{j}$. We adopt a DCNN architecture, known as ``AlexNet" to extract $fc7$ feature for each patch $i \in S_{j}$ to form the $fc7$ feature set $F_{j}$. We fit the $F_{j}$ to a multivariate normal distribution $\mathcal{N}(u, \sum)$ and compute the probability of each image path $i$. Through the fitting process we can find domain-specific variables $u^{j}$ and $\sum{}{}^j$. The threshold $\varepsilon^{j}$ in Equation \ref{equ:gus} is set so that the probabilities of 80\% of patches from $S_{j}$ are larger than it. Figure \ref{fig:purify} shows some samples which have been pruned out from the keyword search for ``aeroplane''.
\vspace{0.3cm}
\noindent \textbf{Shape Data} 3D CAD models of thousands of categories are available online. We utilize the 3D CAD models provided by \cite{peng2015learning} to generate our training images. These 3D CAD models were downloaded from 3D Warehouse by querying the name of the target categories. However, these models contain many ``outliers'' (eg. tire CAD models while searching car CAD models). To solve this, we manually selected the positive examples and delete the ``outliers''. To be consistent with our target dataset, we only adopt 547 3D CAD models for the 20 categories in PASCAL 2007, ranging from ``aeroplane'' to ``tv-monitor''.
\begin{figure}[t]
\centering
\includegraphics[width=\textwidth]{synthetic.png}
\vspace{-0.4cm}
\caption{\textbf{Synthetic Data Rendering Process}. For each 3D CAD model, we first align the model to the front view and set rotation parameters $(X, Y, Z)$. The 3D CAD model is then rotated by the chosen parameters. To mitigate the intensity contrast around object edges, we add a background and texture to the final synthetic image.}
\vspace{-0.5cm}
\label{fig:syn}
\end{figure}
AutoDesk 3ds MAX\footnote{http://www.autodesk.com/store/products/3ds-max} is adopted to generate the synthetic images, with the entire generation process completed automatically by a 3ds Max Script. The rendering process is almost the same as \cite{peng2015learning} (we refer the reader to this work for more details), except that our approach generates possible poses by exhaustively selecting $(X,Y,Z)$ rotation parameters, where $X, Y$ and $Z$ are the degree that the 3D CAD model needs to rotate around X-axis, Y-axis, Z-axis. As shown in figure \ref{fig:syn}, for each 3D CAD model, we first align the model to front view and set rotation parameters $(X, Y, Z)$. The 3D CAD model is then rotated by the chosen parameters.
In our experiments, we only increment one variable from $(X,Y,Z)$ by 2 degrees at one time, constrained by $X,Y\in [-10, 10]$ and $Z\in [70,110]\cup [250,290]$ to cover possible intra-category variations. In total, we generate 833,140 synthetic training images to train our Shape CNN model.
To mitigate the intensity contrast around object edges, we add the mean image of ImageNet\cite{ImageNet} as the background to the final synthetic images. In our ablation study, we try either texture-mapping objects with real image textures as in~\cite{peng2015learning}, or using uniform gray (UG) texture.
\begin{figure}[t]
\centering
\includegraphics[width=0.8\textwidth]{edge_gradient.png}
\caption{\textbf{Illustration of different edge gradients.} The edge gradients of synthetic images with white background differ drastically from images taken in the wild. In the third image, we replace the white background with a mean image computed from ImageNet\cite{ImageNet} images and apply $\psi_{G}$(.) to smooth the edge contrast. We further add some Gaussian noise $\xi_{G}$ to the image to increase background variation.}
\label{fig:edge}
\end{figure}
As addressed in Section~\ref{appsub:shape}, one shortcoming of synthetic data is the statistic mismatch, especially for the intensity contrast around the object edges. Figure \ref{fig:edge} shows the difference in edge gradient between synthetic images with white background and images taken in the wild.
Recent work on DCNN visualization \cite{zeiler2014visualizing} has shown that parameters in lower layers are more sensitive to edges. Thus matching the statistics of synthetic images to those of real images is a good way to decrease the domain shift. We apply a Gaussian filter based smoothing function $\psi_{G}$(.) for every synthetic image, and then add Gaussian noise $\xi_{G}$ to the smoothed images. In our experiment, we use $\mathcal{N}$(0, 1) for $\psi_{G}$(.) and $\mathcal{N}$(0, 0.01) for $\xi_{G}$.
The third image in Figure \ref{fig:edge} illustrates how the edge gradients become more similar to real image gradients after $\psi_{G}$(.) and $\xi_{G}$ are applied.
\subsection{Classification Results}
We evaluate our approach on standard benchmark PASCAL VOC 2007\cite{pascal}. PASCAL VOC dataset is originally collected for five challenges: classification, detection, segmentation, action classification and person layout. PASCAL VOC 2007 has 20 categories ranging from people, animals, plant to man-made rigid objects, and contains 5011 training/validation images and 4952 testing images. In our experiments, we only use testing images for evaluation.
Among the 4952 testing images, 14976 objects are annotated with tight bounding boxes. In our classification experiments, we crop 14976 patches (one patch for one object) with the help of these bounding boxes to generate test set.
The DCNN in each stream is initialized with the parameters pre-trained on ImageNet\cite{ImageNet}. The last output layer is changed from a 1000-way classifier to a 20-way classifier and is randomly initialized with $\mathcal{N}(0, 0.01)$. The two DCNNs are trained with same settings, with the base learning rate to be 0.001, momentum to be 0.9 and weight decay to be 0.0005. Two dropout layers are adopted after $fc6,fc7$ with the dropout ratio to be 0.5.
The results in Table \ref{tab_cls} show when adding texture cues to Shape-CNN with Equation \ref{equ_fusion}, the performance rises from 28.3\%, 29.9\%, 31.7\% to 38.1\%, 38.7\%, 39.3\%, respectively. On the other side, adding shape cues to Texture-CNN can boost the performance from 36.7\% to 39.3\%, with $\psi_{G}$(.) and $\xi_{G}$ applied. The results also demonstrate that simulating the real statistics can benefit the classification results.
\begin{table*}[t]
\scriptsize
\noindent\begin{tabular}{ |p{2.0cm} | p{0.37cm} p{0.37cm} p{0.37cm} p{0.37cm} p{0.37cm} p{0.37cm} p{0.37cm} p{0.37cm} p{0.37cm} p{0.37cm} p{0.37cm} p{0.37cm} p{0.37cm} p{0.37cm} p{0.37cm} p{0.37cm} p{0.37cm} p{0.37cm} p{0.37cm} p{0.37cm} |p{0.6cm}| }
\hline
Method & aer & bik & bir & boa & bot & bus & car & cat & chr & cow & tbl & dog & hrs & mbk & prs & plt & shp & sof & trn & tv & All\\
\hline
T-CNN& \textbf{61}& \textbf{76}& \textbf{36}& 39& 12& 44& \textbf{53}& 68& 12& 52& \textbf{16}& \textbf{46}& \textbf{58}& 56& 19& \textbf{86}& \textbf{63}& 22& \textbf{76}& \textbf{89}&36.7\\
\hline
\hline
S-CNN& 37& 24& 18& 50& \textbf{60}& \textbf{82}& 36& 54& 16& 78& 2& 10& 3& 56& 19& 31& 6& 26& 7& 77&28.3\\
S-CNN-G& 31& 20& 21& 58& 44& 80& 44& 42& 20& 73& 3& 9& 3& 66& 21& 42& 8& 34& 6& 80&29.9\\
S-CNN-GN& 32& 20& 20& 51& 53& 80& 47& 47& \textbf{25}& 68& 2& 12& 5& \textbf{69}& 21& 53& 8& \textbf{37}& 3& 85&31.7\\
\hline
F-CNN& 52& 66& 30& 50& 42& 74& 45& \textbf{70}& 16& \textbf{84}& 13& 33& 41& 59& 24& 71& 53& 27& 50& \textbf{89}&\textbf{38.1}\\
F-CNN-G& 56& 65& 31& \textbf{59}& 27& 76& 48& 63& 18& 77& 14& 35& 46& 64& \textbf{25}& 71& 55& 30& 51& \textbf{89}&\textbf{38.7}\\
F-CNN-GN& 56& 65& 31& 48& 32& 74& 51& 65& 21& 68& 13& 35& 47& 67& \textbf{25}& 77& 54& 31& 46& 90&\textbf{39.3}\\
\hline
\hline
S-CNN-UG&30&25&16&51&20&80&40&11&13&49&1&5&16&76&27&64&1&24&35&46&27.2\\
S-CNN-G-UG&25&16&15&43&20&86&43&8&15&57&1&3&5&81&21&53&1&30&14&38&27.5\\
S-CNN-GN-UG&35&23&13&55&33&82&37&9&22&34&0&4&3&78&29&56&0&29&19&39&29.4\\
\hline
F-CNN-UG&56&68&30&52&16&70&50&61&14&57&12&38&49&70&28&83&55&23&68&81&39.3\\
F-CNN-G-UG&56&59&29&48&16&79&50&62&15&61&11&35&46&78&26&80&54&26&56&81&38.1\\
F-CNN-GN-UG&62&65&28&55&20&74&49&61&19&50&13&38&48&75&32&81&56&26&62&84&41.4\\
\hline
\end{tabular}
\vspace{0.2cm}
\caption{\textbf{Classification Results}. Prefix ``S-'', ``T-'' and ``F-'' denote ``Shape'', ``Texture'', ``Fusion'', respectively. Suffix ``-G'', ``-N'', ``-UG'' indicate synthetic data is smoothed with $\psi_{G}$(.), is colored with Gaussian noise $\xi_{G}$, and is generated with uniform gray texture. The results show Fusion-CNN model outperforms Shape-CNN model and Texture-CNN model and simulating real statistics benefit object classification system with minimal supervision. Note the last column is not mean accuracy, but accuracy over all test set.}
\label{tab_cls}
\vspace{-0.7cm}
\end{table*}
To better analyze how texture cues and shape information compensate for each other, we plot the confusion matrix (with X-axis representing ground truth labels and Y-axis representing predictions) for Texture-CNN, Shape-CNN, Fusion-CNN in Figure \ref{fig:confusion} (Networks with ``-G'' or ``-GN" have similar results). For the convenience of comparison, we re-rank the order of categories and plot per-category accuracy in Figure \ref{fig:confusion}. There are some interesting findings:
\begin{itemize}
\item The top-right confusion matrix (for Shape-CNN) in Figure \ref{fig:confusion} shows that CNN trained on synthetic images mistakes most of the ``train'' images for ``bus'', and mistakes ``horse'' and ``sheep'' images for ``cow'', which is not very surprising because they share similar shape visual information.
\item From two confusion matrices on the top, we can see that Texture-CNN trained on web images tends to mistake other images for ``plant'' and ``TV'', while Shape-CNN is keen on categories like ``cow'', ``bus'', ``bottle'' etc.
\item The last sub-figure in Figure \ref{fig:confusion} is per-category accuracy. We re-rank the order of categories for inspection convenience. Shape-CNN (green line) tends to perform well for the categories presented on the left and Texture-CNN (blue line) is more likely to get a high performance for the categories presented on the right. Taking a closer look at the categories, we find that Shape-CNN will work well for shape-oriented or rigid categories, e.g. bus, bottle, motorbike, boat, etc. Inversely, Texture-CNN is more likely to obtain higher performance on texture-oriented categories, such as cat, sheep, plant, horse, bird etc. The performance of Fusion-CNN is mostly between Shape-CNN and Texture-CNN and never gets a very poor result, which is why it can work better than CNNs based on single cues. We also tried performing a max fusion over the two streams, but the performance improvement is not comparable with average fusion.
\end{itemize}
\begin{figure}[t]
\centering
\includegraphics[width=0.4\textwidth]{texture.png}
\includegraphics[width=0.4\textwidth]{shape.png}
\includegraphics[width=0.4\textwidth]{fusion.png}
\includegraphics[width=0.4\textwidth]{plot_accuracy.png}
\caption{\textbf{Confusion matrix and classification results}. The confusion matrix has been normalized by the number of total images per category. From up to bottom, left to right, the four figures are: confusion matrix of texture CNN, confusion matrix of shape CNN, confusion matrix of fusion CNN, classification accuracy for each category. (Best viewed in color!) }
\label{fig:confusion}
\vspace{-0.4cm}
\end{figure}
We also try removing the texture on the object from the synthetic image by replacing it with uniform gray (UG) pixels. As shown in Table \ref{tab_cls}, this achieves similar results, indicating that synthetically adding real texture to CAD models may not be important for this classification task.
\subsection{Object Detection Results}
In our detection experiments, we find ``S-CNN'' (``F-CNN''), ``S-CNN-G'' (``F-CNN-G'') and ``S-CNN-GN'' (``F-CNN-GN'') get comparable results, thus we only report result for ``S-CNN'' (``F-CNN'') in this section.
For detection, we followed the standard evaluation schema provided by \cite{pascal}: a prediction bounding box P is considered to be a valid detection if and only if the area of overlap $IoU$ exceeds 0.5. The $IoU$ is denoted with the following formula: $IoU$ = $\frac{area(B_{p}\cap B_{gt})}{area(B_{p} \cup B_{gt})}$, where $B_{p} \cap B_{gt}$ denotes the intersection of the predicted bounding box and the ground truth bounding box and $B_{p} \cup B_{gt}$ their union.
\vspace{0.2cm}
\noindent \textbf{Region Proposal} An excellent region proposal method contributes to the performance of both supervised and unsupervised learning. In our experiments, we adopt EdgeBox\cite{zitnick2014edge} to generate region proposals. EdgeBox is a efficient region proposal algorithm which generates bounding box proposals from edge maps obtained by contour detector. The bounding boxes are scored by the number of enclosed contours inside the boxes.
\vspace{0.2cm}
Like in the classification task, for each region proposal, we pass it to Shape-CNN and Texture-CNN simultaneously, and fuse the last layers' activations. Similar to \cite{chen2015webly}, we randomly crop patches from YFCC\cite{thomee2015new} as the negative samples. Further, we follow the schema of R-CNN\cite{RCNN} to compute mAP. We compare our method to following baselines.
\begin{itemize}
\item \textbf{VCNN(ICCV'15)\cite{peng2015learning}} In this work, the authors propose to render domain-specific synthetic images from 3D CAD models and train an R-CNN\cite{RCNN} based object detector. Some results may involve minor supervision, e.g. selecting background and texture. We compare to their W-RR model (white background, real texture) where the amount of supervision is almost the same as in this work.
\item \textbf{LEVAN(CVPR'14)\cite{divvala2014learning}} LEVAN uses items in Google N-grams as queris to collect training images from Internet. They propose a fully-automated approach to organize the visual knowledge about a concept and further apply their model to detection task on PASCAL VOC 2007.
\item \textbf{Webly Supervised Object Detection(ICCV'15)\cite{chen2015webly}} The webly supervised learning approach collects images from Google and Flickr by searching for the name of a certain category and utilizes Examplar-LDA~\cite{hariharan2012discriminative} and agglomerative clustering~\cite{chen2014enriching} to generate the potential ``ground truth" bounding box. For fair comparison, we only compared to their results of images downloaded from Google.
\item \textbf{Concept Learner(CVPR'15)\cite{zhou2015conceptlearner}} Concept learner is designed to discover thousands of visual concepts automatically from webly labeled images. It first trains a concept learner on the SBU dataset and selects the learned concept detectors to compute the average precision.
\end{itemize}
Results listed in Table \ref{tab_det} demonstrate that combining real-image texture information with Shape-CNN will boost the mAP from 15.0 to 19.7, a relative 31.3\% increase! Inversely, adding shape information to Texture-CNN boost the mAP from 18.1 to 19.7, which shows texture cues and shape information can compensate for each other in detection task. Despite the minimal amount of required supervision, our Fusion-CNN also obtains higher performance than a purely 3D CAD model based method like \cite{peng2015learning}, a webly supervised approach like \cite{divvala2014learning} and a weakly supervised method where in-domain training data from PASCAL VOC 2007 is available \cite{divvala2014learning}. The results show that Fusion-CNN outperforms DCNN based on single visual cues and other methods where similar or higher levels of supervision are adopted.
As an ablation study, we perform the same experiments on synthetic images generated without texture (``S-CNN-UG'', ``F-CNN-UG'' in table \ref{tab_det}). The results reveal that, unlike classification, adding some texture into the synthetic images helps to boost performance for the detection task.
\begin{table*}[t]
\scriptsize
\noindent\begin{tabular}{ p{2.4cm} |p{0.35cm} p{0.35cm} p{0.35cm} p{0.35cm} p{0.35cm} p{0.35cm} p{0.35cm} p{0.35cm} p{0.35cm} p{0.35cm} p{0.35cm} p{0.35cm} p{0.35cm} p{0.35cm} p{0.35cm} p{0.35cm} p{0.35cm} p{0.35cm} p{0.35cm} p{0.35cm} |p{0.6cm} }
\hline
Method & aer & bik & bir & boa & bot & bus & car & cat & chr & cow & tbl & dog & hrs & mbk & prs & plt & shp & sof & trn & tv & mAP\\
\hline
T-CNN & 22& 20& 19& 18& 9& 35& 28& \textbf{21}& 9& 13&\textbf{4}& 16&\textbf{29}& 31& 6& \textbf{11}& 15& 11& \textbf{25}& 19& 18.1\\
S-CNN & 20& 18& 18& 15& 9& 29& 23& 4& 9& 16& 0& 13& 19& 26& 13& 9& 14& 12& 4& 27& 15.0\\
F-CNN & 29& 23& \textbf{19}& \textbf{22}& 9& \textbf{41}& 29& 17& 9& \textbf{21}& 1& \textbf{20}& 23& \textbf{33}& 9& 9& \textbf{17}& \textbf{13}& 16& 27&\textbf{19.7}\\
\hline
S-CNN-UG & 22&17&13&12&9&23&26&2&2&13&0&6&11&29&5&9&2&10&1&17&11.4\\
F-CNN-UG &25&18&15&18&10&38&30&14&3&18&1&17&20&31&8&10&12&12&11&19&17\\
\hline
\hline
VCNN \cite{peng2015learning} & \textbf{36}& \textbf{23}& 17& 15& \textbf{12}& 25& \textbf{35}& 21& \textbf{11}& 16& 0.1& 16& 16& 29& \textbf{13}& 9& 4& 10& 0.6& \textbf{29}& 17\\
\hline
\hline
Levan, Webly\cite{divvala2014learning}& 14& 36& 13& 10& 9& 35& 36& 8& 10& 18& 7& 13& 31& 28& 6& 2& 19& 10& 24& 16& 17.2\\
Chen's Webly\cite{chen2015webly}& 35& 39& 18& 15& 8& 31& 39& 20& 16& 13& 15& 4& 21& 34& 9& 17& 15& 23& 28& 19& 20.9\\
\hline
\hline
Zhou's, Selected\cite{zhou2015conceptlearner}& 30& 34& 17& 13& 6& 44& 27& 23& 7& 16& 10& 21& 25& 36& 8& 9& 22& 17& 31& 18& 20.5\\
\hline
\hline
Siva's, Weakly\cite{siva2011weakly}& 13& 44& 3& 3& 0& 31& 44& 7& 0& 9& 10& 2& 29& 38& 5& 0& 0& 4& 34& 0& 13.9\\
Song's, Weakly\cite{song2014learning}& 8& 42& 20& 9& 10& 36& 39& 34& 1& 21& 10& 28& 29& 39& 9& 19& 21& 17& 36& 7& 22.7\\
\hline
\hline
RCNN, Full\cite{RCNN}& 58& 58& 39& 32& 24& 51& 59& 51& 20& 51& 41& 46& 52& 56& 43& 23& 48& 35& 51& 57& 44.7\\
\hline
\end{tabular}
\vspace{0.2cm}
\caption{\textbf{Illustration Of Detection Results}. Methods are categorized by their supervision type. ``Webly'', ``Selected'', ``Weakly'', ``Full'' represent webly supervision, selected supervision, weak supervision, full supervision, respectively. The definitions of these supervisions are the same as in \cite{zhou2015conceptlearner}. The supervision in our work only comes from labeling the CAD models and choosing proper texture for the CAD models when generating synthetic images. The supervision used in \cite{peng2015learning} is almost the same as in our approach, except that they also labeled pose for the CAD models. The results demonstrate that our Fusion-CNN model outperforms methods based on single visual cues and other methods with similar or higher required supervision effort.}
\label{tab_det}
\vspace{-0.6cm}
\end{table*}
\vspace{-0.4cm}
\section{Conclusion}
\label{con}
In this work, we proposed and implemented a novel minimally-supervised learning framework that decomposes images into their shape and texture and further demonstrated that texture cues and shape information can reciprocally compensate for each other. Furthermore, a unified learning schema, including pruning noise web data and simulating statistics of real images is introduced, both for web image based learning and synthetic image based learning. Finally, our classification and detection experiments on VOC 2007 show that our Fusion-CNN with minimal supervision outperforms DCNNs based on single cues (only shape, only texture) and previous methods that require similar or more supervision effort. We believe our model is valuable for scaling recognition to many visual object categories and can be generalized to other generic tasks such as pose detection, robotic grasping and object manipulation.
\vspace{0.2cm}
\noindent {\bf Acknowledgement}. This research was supported by NSF award IIS-1451244 and a generous donation from the NVIDIA corporation.
\section{Appendix}
\noindent {\bf A. Detection Result Diagnosis}\\
We further use the diagnosis tools provided in~\cite{hoiem-analysis} to analyze our detectors, Table~\ref{tab_diag} and Figue~\ref{fig_topfp} highlight some of the interesting observations.
Table \ref{tab_diag} illustrates the diagnosis results. From top to bottom, each row represents diagnosis for ``\textbf{animals}'' (including ``birds'', ``cat'', ``cow'', ``dog'', ``horse'', ``person'', ``sheep''), ``\textbf{vehicles}'' (including ``aeroplane'', ``bike'',``boat'', ``bus'', ``car'', ``motorbike'', ``train'') and ``\textbf{furniture}'' (including ``chair'', ``table'', ``sofa'').
The diagnosis tools \cite{hoiem-analysis} will categorize the false positive samples into four categories: \textbf{Localization error} (\textbf{Loc}), \textbf{Confusion with similar objects} (\textbf{Sim}), \textbf{Confusion with dissimilar objects} (\textbf{Oth}), \textbf{Confusion with background} (\textbf{BG}). (We refer the reader to \cite{hoiem-analysis} for more details). After analyzing the distribution of these four type of errors, we list some interesting observations:
\begin{itemize}
\item Localization error (\textbf{Loc}, blue area in the figures) is the majority of the false positives for \textbf{T-CNN}, mainly because training images (downloaded from Google image search engine) for Texture-CNN do not provide ground truth bounding boxes. Other issues like multiple instances in one bounding box also cause localization error.
\item For \textbf{S-CNN}, confusion with similar objects (\textbf{Sim}, red area in the figures) accounts for a large portion of the false positives, especially for ``animals''. The synthetic data used to train Shape-CNN lacks discriminative texture cues. For example, the rendered sheep images, horse images and cow images share the same visual representations in shape.
\item Compared to ``anmimals'', ``vehiches'' will have less confusion with similar objects (\textbf{Sim}) errors and confusion with dissimilar objects (\textbf{Oth}) errors; on the contrary, ``furniture'' will have more.
\item The distribution of false positives for \textbf{F-CNN} is a compromise of \textbf{T-CNN} and \textbf{S-CNN}.
\end{itemize}
\begin{table}
\begin{center}
\begin{tabular}{ |c|c|c| }
\hline
\textbf{T-CNN}&\textbf{S-CNN}&\textbf{F-CNN}\\
\hline
\includegraphics[width=0.31\textwidth]{tx_animals.png} &
\includegraphics[width=0.31\textwidth]{shape_animals.png} &
\includegraphics[width=0.31\textwidth]{fusion_animals.png} \\
\hline
\includegraphics[width=0.31\textwidth]{tx_vehicles.png} &
\includegraphics[width=0.31\textwidth]{shape_vehicles.png} &
\includegraphics[width=0.31\textwidth]{fusion_vehicles.png} \\
\hline
\includegraphics[width=0.31\textwidth]{tx_funiture.png} &
\includegraphics[width=0.31\textwidth]{shape_funiture.png} &
\includegraphics[width=0.31\textwidth]{fusion_funiture.png} \\
\hline
\end{tabular}
\end{center}
\caption{\textbf{Illustration of diagnosis.} We visualize the failure model of our models. From up to bottom, each row shows the diagnosis for ``animals'', ``vehicles'' and ``furniture''. The colored area indicates the percentage of false positives of each failure type. (Best viewed in color!)}
\label{tab_diag}
\end{table}
\noindent {\bf B. Examples Of Top False Positives}\\
In figure \ref{fig_topfp}, we show the top four false positives for category ``aeroplane'', ``bottle'', ``bus'', ``cat'', ``cow'', ``dog'' and ``train''. These figures demonstrate that localizing the objects precisely is still a great challenge for object detector with little supervision.
\vspace{2cm}
\begin{figure}
\centering
\includegraphics[width=0.24\textwidth]{aero_1.png}
\includegraphics[width=0.24\textwidth]{aero_2.png}
\includegraphics[width=0.24\textwidth]{aero_4.png}
\includegraphics[width=0.24\textwidth]{aero_5.png}
\includegraphics[width=0.24\textwidth]{bot_1.png}
\includegraphics[width=0.24\textwidth]{bot_2.png}
\includegraphics[width=0.24\textwidth]{bot_3.png}
\includegraphics[width=0.24\textwidth]{bo_4.png}
\includegraphics[width=0.24\textwidth]{bus_1.png}
\includegraphics[width=0.24\textwidth]{bus_2.png}
\includegraphics[width=0.24\textwidth]{bus_3.png}
\includegraphics[width=0.24\textwidth]{bus_4.png}
\includegraphics[width=0.24\textwidth]{cat_1.png}
\includegraphics[width=0.24\textwidth]{cat_2.png}
\includegraphics[width=0.24\textwidth]{cat_3.png}
\includegraphics[width=0.24\textwidth]{cat_4.png}
\includegraphics[width=0.24\textwidth]{cow_1.png}
\includegraphics[width=0.24\textwidth]{cow_2.png}
\includegraphics[width=0.24\textwidth]{cow_3.png}
\includegraphics[width=0.24\textwidth]{cow_4.png}
\includegraphics[width=0.24\textwidth]{dog_1.png}
\includegraphics[width=0.24\textwidth]{dog_2.png}
\includegraphics[width=0.24\textwidth]{dog_3.png}
\includegraphics[width=0.24\textwidth]{dog_4.png}
\includegraphics[width=0.24\textwidth]{train_1.png}
\includegraphics[width=0.24\textwidth]{train_2.png}
\includegraphics[width=0.24\textwidth]{train_3.png}
\includegraphics[width=0.24\textwidth]{train_4.png}
\caption{\textbf{Examples of top false positives.} We show the top four false positives for \textbf{F-CNN} ``aeroplane'', ``bottle'', ``bus'', ``cat'', ``cow'', ``dog'' and ``train'' detectors. The text on the bottom of each image shows the type of error, the amount of overlap (``ov'') with a ground truth object, and the fraction of true positives that are ranked after the given false positive (``1-r'', for 1-recall) }
\label{fig_topfp}
\end{figure}
\bibliographystyle{splncs}
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
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{"url":"https:\/\/www.groundai.com\/project\/topological-quasiparticles-and-the-holographic-bulk-edge-relation-in-21d-string-net-models\/","text":"Topological quasiparticles and the holographicbulk-edge relation in 2+1D string-net models\n###### pacs:\n71.10.-w, 02.20.Uw, 03.65.Fd\n\nString-net models allow us to systematically construct and classify 2+1D topologically ordered states which can have gapped boundaries. We can use a simple ideal string-net wavefunction, which is described by a set of F-matrices [or more precisely, a unitary fusion category (UFC)], to study all the universal properties of such a topological order. In this paper, we describe a finite computational method \u2013 Q-algebra approach, that allows us to compute the non-Abelian statistics of the topological excitations [or more precisely, the unitary modular tensor category (UMTC)], from the string-net wavefunction (or the UFC). We discuss several examples, including the topological phases described by twisted gauge theory (i.e., twisted quantum double ). Our result can also be viewed from an angle of holographic bulk-boundary relation. The 1+1D anomalous topological orders, that can appear as edges of 2+1D topological states, are classified by UFCs which describe the fusion of quasiparticles in 1+1D. The 1+1D anomalous edge topological order uniquely determines the 2+1D bulk topological order (which are classified by UMTC). Our method allows us to compute this bulk topological order (i.e., the UMTC) from the anomalous edge topological order (i.e., the UFC).\n\n## I Introduction\n\nA major problem of physics is to classify phases and phase transitions of matter. The problem was once thought to be completely solved by Landau\u2019s theory of symmetry breakingLandau (1937), where the phases can be classified by their symmetries. However, the discovery of fractional quantum Hall (FQH) effectTsui\u00a0et\u00a0al. (1982) indicated that Landau\u2019s theory is incomplete. There are different FQH phases with the same symmetry, and the symmetry breaking theory failed to distinguish those phases. FQH states are considered to possess new topological ordersWen (1989); Wen\u00a0and\u00a0Niu (1990); Wen (1990) beyond the symmetry breaking theory.\n\nWe know that all the symmetry breaking phases are labeled by two groups , where is the symmetry group of the Hamiltonian and is the symmetry group of the ground state. This fact motivates us to search for the complete \u201clabel\u201d of topological order.\n\nHere, the \u201clabel\u201d that labels a topological order corresponds to a set of universal properties that can fully determine the phase and distinguish it from other phases. Such universal properties should always remain the same as long as there is no phase transition. In particular, they are invariant under any small local perturbations. Such universal properties are called topological invariants in mathematics.\n\nIn 2+1D, it seems that anyonic quasiparticle statistics, or the modular data matrices, are the universal properties. The set of universal properties that describes quasiparticle statistics is also referred to as unitary modular tensor category (UMTC). matrices (i.e., UMTC) can fully determine the topological phases, up to a bosonic FQH state.Wen (1990); Keski-Vakkuri\u00a0and\u00a0Wen (1993); Wen (2012); Zhang\u00a0and\u00a0Vishwanath (2013); Cincio\u00a0and\u00a0Vidal (2013) In Section II we will introduce topological quasiparticle excitations and their statistics, i.e. fusion and braiding data, in 2+1D topological phases and on 1+1D gapped edges.\n\nSince the universal properties do not depend on the local details of the system, it is possible to calculate them from a simple renormalization fixed-point model. In this paper we will concentrate on a class of 2+1D fixed-point lattice model, the Levin-Wen string-net modelLevin\u00a0and\u00a0Wen (2005). As a fixed-point model, the building blocks of Levin-Wen models are effective degrees of freedom with the form of string-nets. The fixed-point string-net wavefunction is completely determined by important data \u2013 the F-matrices. The F-matrices are also referred to as unitary fusion category (UFC).\n\nTherefore, a central question for string-net models is how to calculate the matrices from F-matrices (or how to calculate the UMTC from the UFC). In Ref. Levin\u00a0and\u00a0Wen, 2005 the matrices can be calculated by searching for string operators. String operators are determined by a set of non-linear algebraic equations involving the F-matrices. However, this algorithm is not an efficient one. The equations determining string operators have infinite many solutions and there is no general method to pick up the irreducible solutions. In this sense it is even not guarantied that one can find all the (irreducible) string operators.\n\nIn this paper we try to fix this weak point. Motivated by the work of Kitaev and Kong\u00a0Kitaev\u00a0and\u00a0Kong (2012); Kong (2012), we introduce the Q-algebra approach to compute quasiparticle statistics. The idea using Q-algebra modules to classify quasiparticles is analog to using group representations to classify particles. It is well known that in a system with certain symmetry the energy eigenspaces, including excited states of particles, form representations of the symmetry group. String-net models are fixed-point models thus renormalization can be viewed as generalized \u201csymmetry\u201d. Moreover we show that renormalization in string-net models can be exactly described by evaluation linear maps. This allows us to introduce the Q-algebra, which describes the renormalization of quasiparticle states. Quasiparticles are identified as the invariant subspaces under the action of the Q-algebra, i.e., Q-algebra modules.\n\nRoughly speaking, the Q-algebra is the \u201crenormalization group\u201d of quasiparticles in string-net models, a linearized, weakened version of a group. The notions of algebra modules and group representations are almost equivalent. Modules over the group algebra are in one to one correspondence with group representations up to similarity transformations. The only difference is that \u201cmodule\u201d emphasizes on the subspace of states that is invariant under the action of the group or algebra, while \u201crepresentation\u201d emphasizes on how the group or algebra acts on the \u201cmodule\u201d.\n\nThe specific algorithm to compute the Q-algebra modules is also analog to that to compute the group representations. For a group, firstly, we write the multiplication rules. Secondly, we take the multiplication rules as the \u201ccanonical representation\u201d. Thirdly, we try to simultaneously block-diagonalize the canonical representation. Finally, the irreducible blocks correspond to irreducible representations, or simple modules over the group algebra. The canonical representation of a group contains all types of irreducible representations of that group. This is also true for the Q-algebra. The multiplication rules of the Q-algebra are fully determined by the F-matrices (i.e., the UFC, see (49) and (86)). Therefore, following this block-diagonalization process we have a finite algorithm to calculate the quasiparticle statistics from F-matrices. We are guarantied to find all types of quasiparticles by block-diagonalizing the canonical representation of the Q-algebra. Simultaneous block-diagonalization is a straightforward algorithm, however, it is not a quite efficient way to decompose the Q-algebra. The algorithm used in this paper is an alternative one, idempotent decomposition.\n\nThe notions of algebra, module and idempotent play an important role in our discussion and algorithm. On the other hand, we think it a necessary step to proceed from \u201cgroups and group representations\u201d to \u201calgebras and modules\u201d, since we are trying to extend our understanding from \u201csymmetry breaking phases\u201d to \u201ctopologically ordered phases\u201d. We provide a brief introduction in Appendix A to these mathematical notions in case the reader is not familiar with them.\n\nAnother weak point of the original version of Levin-Wen model in Ref Levin\u00a0and\u00a0Wen, 2005 is that the F-matrices are assumed to be symmetric under certain index permutation. More precisely, the F-matrices have 10 indices which can be associated to a tetrahedron, 6 indices to the edges and 4 indices to the vertices. If we reflect or rotate the tetrahedron the indices get permuted and the F-matrices are assumed to remain the same. In this paper we find that such tetrahedral symmetry can be dropped thus the string-net model is generalized.\n\nIn Section III we will first drop the tetrahedron-reflectional symmetry of the F-matrices but keep the tetrahedron-rotational symmetry and reformulate the string-net model. We keep the tetrahedron-rotational symmetry because in this case the relation between string operators and Q-algebra modules is clear. We give the formula to compute quasiparticle statistics, the matrices from Q-algebra modules by comparing them to string operators.\n\nNext, in Section IV we will drop the tetrahedron-rotational symmetry assumption, and generalize string-net models to arbitrary gauge. In arbitrary gauge the string operators are not naturally defined, but we can still obtain the formula of quasiparticle statistics by requiring the formula to be gauge invariant and reduce to the special case if we choose the tetrahedron-rotation-symmetric gauge.\n\nFinally, in Section V we briefly discuss the boundary theoryKitaev\u00a0and\u00a0Kong (2012) of generalized string-net models which shows the holographic bulk-edge relation. In 2+1D there are many different kinds of topological orders, classified by the non-Abelian statistics of the quasiparticles plus the chiral central charge of the edge state. Mathematically, the non-Abelian statistics, or the fusion and braiding data of quasiparticles form a UMTC. On the other hand, in 1+1D, there is only trivial topological order.Verstraete\u00a0et\u00a0al. (2005); Chen\u00a0et\u00a0al. (2011) However, if we consider anomalous topological orders that only appear on the edge of 2+1D gapped states, we will have nontrivial anomalous 1+1D topological orders. In these anomalous 1+1D topological orders, the fusion of quasiparticles is also described by a set of F-matrices. Mathematically, the F-matrices give rise to a UFC, and anomalous 1+1D topological orders are classified by UFCs. The F-matrices we use to determine a string-net ground state wavefunction turn out to be the same F-matrices describing the fusion of quasiparticles on one of the edges of the string-net model.Kitaev\u00a0and\u00a0Kong (2012); Kong\u00a0and\u00a0Wen (2014) Thus, our algorithm calculating the bulk quasiparticle statistics (UMTC) from the F-matrices (UFC) can also be understood as calculating the bulk topological order (UMTC) from the anomalous boundary topological order (UFC). Since the same bulk topological order may have different gapped boundaries, it is a natural consistency question: Do these different gapped boundaries lead to the same bulk? The answer is \u201cyes\u201d.Kitaev\u00a0and\u00a0Kong (2012) Mathematically, we give an algorithm to compute the Drinfeld center functor that maps a UFC (that describes a 1+1D anomalous topological order) to a UMTC (that describes a 2+1D topological order with zero chiral central charge).M\u00fcger (2003) Different gapped boundaries of a 2+1D topological phase are described by different UFCs, but they share the same Drinfeld center UMTC. In Appendix E we discuss the twisted string-net model in detail to illustrate this holographic relation.\n\n## Ii Quasiparticle excitations\n\n### ii.1 Local quasiparticle excitations and topological quasiparticle excitations\n\nTopologically ordered states in 2+1D are characterized by their unusual particle-like excitations which may carry fractional\/non-Abelian statistics. To understand and to classify particle-like excitations in topologically ordered states, it is important to understand the notions of local quasiparticle excitations and topological quasiparticle excitations.\n\nFirst we define the notion of \u201cparticle-like\u201d excitations. Consider a gapped system with translation symmetry. The ground state has a uniform energy density. If we have a state with an excitation, we can measure the energy distribution of the state over the space. If for some local area, the energy density is higher than ground state, while for the rest area the energy density is the same as ground state, one may say there is a \u201cparticle-like\u201d excitation, or a quasiparticle, in this area (see Figure 1).\n\nAmong all the quasiparticle excitations, some can be created or annihilated by local operators, such as a spin flip. This kind of particle-like excitation is called local quasiparticle. However, in topologically ordered systems, there are also quasiparticles that cannot be created or annihilated by any finite number of local operators (in the infinite system size limit). In other words, the higher local energy density cannot be created or removed by any local operators in that area. Such quasiparticles are called topological quasiparticles.\n\nFrom the notions of local quasiparticles and topological quasiparticles, we can further introduce the notion topological quasiparticle type, or simply, quasiparticle type. We say that local quasiparticles are of the trivial type, while topological quasiparticles are of nontrivial types. Two topological quasiparticles are of the same type if and only if they differ by local quasiparticles. In other words, we can turn one topological quasiparticle into the other one of the same type by applying some local operators.\n\n### ii.2 Simple type and composite type\n\nTo understand the notion of simple type and composite type, let us discuss another way to define quasiparticles:\nConsider a gapped local Hamiltonian qubit system defined by a local Hamiltonian in dimensional space without boundary. A collection of quasiparticle excitations labeled by and located at can be produced as gapped ground states of where is non-zero only near \u2019s. By choosing different we can create all kinds of quasiparticles. We will use to label the type of the quasiparticle at .\n\nThe gapped ground states of may have a degeneracy which depends on the quasiparticle types and the topology of the space . The degeneracy is not exact, but becomes exact in the large space and large particle separation limit. We will use to denote the space of the degenerate ground states.\n\nIf the Hamiltonian is not gapped, we will say (i.e.,\u00a0 has zero dimension). If is gapped, but if also creates quasiparticles away from \u2019s (indicated by the bump in the energy density away from \u2019s), we will also say . (In this case quasiparticles at \u2019s do not fuse to trivial quasiparticles.) So, if , only creates quasiparticles at \u2019s.\n\nIf the degeneracy cannot not be lifted by any small local perturbation near , then the particle type at is said to be simple. Otherwise, the particle type at is said to be composite. The degeneracy for simple particle types is a universal property (i.e.,\u00a0a topological invariant) of the topologically ordered state.\n\n### ii.3 Fusion of Quasiparticles\n\nWhen is composite, the space of the degenerate ground states has a direct sum decomposition:\n\n V(Md;\u03be1,\u03be2,\u03be3,\u22ef) =V(Md;\u03b61,\u03be2,\u03be3,\u22ef)\u2295V(Md;\u03c71,\u03be2,\u03be3,\u22ef) \u2295V(Md;\u03c81,\u03be2,\u03be3,\u22ef)\u2295\u22ef (1)\n\nwhere , , , etc. are simple types. To see the above result, we note that when is composite the ground state degeneracy can be split by adding some small perturbations near . After splitting, the original degenerate ground states become groups of degenerate states, each group of degenerate states span the space or etc. which correspond to simple quasiparticle types at . We denote the composite type as\n\n \u03be1=\u03b61\u2295\u03c71\u2295\u03c81\u2295\u22ef. (2)\n\nWhen we fuse two simple types of topological particles and together, it may become a topological particle of a composite type:\n\n \u03be\u2297\u03b6=\u03b7=\u03c71\u2295\u03c72\u2295\u22ef, (3)\n\nwhere are simple types and is a composite type. In this paper, we will use an integer tensor to describe the quasiparticle fusion, where label simple types. When , the fusion of and does not contain . When , the fusion of and contain one : . When , the fusion of and contain two \u2019s: . This way, we can denote that fusion of simple types as\n\n \u03be\u2297\u03b6=\u2295\u03c7N\u03c7\u03be\u03b6\u03c7. (4)\n\nIn physics, the quasiparticle types always refer to simple types. The fusion rules is a universal property of the topologically ordered state. The degeneracy is determined completely by the fusion rules .\n\nLet us then consider the fusion of 3 simple quasiparticles . We may first fuse , and then with , . We may also first fuse and then with , . This requires that . If we further consider the degenerate states , it is not hard to see fusion in different orders means splitting the space as different direct sums of subspaces. Thus, fusion in different orders differ by basis changes of . The F-matrices are nothing but the data to describe such basis changes.\n\nFor 1+1D anomalous topological orders (gapped edges of 2+1D topological orders), the quasiparticles can only fuse but not braiding. So, the fusion rules and the F-matrices are enough to describe 1+1D anomalous topological orders. Later, we will see fusion rules and F-matrices are also used to determine a string-net wavefunction, which may seem confusing. However, as we have mentioned, this is a natural result of the holographic bulk-edge relation. Intuitively, one may even view the string-net graphs in 2D space as the 1+1D space-time trajectory of the edge quasiparticles.\n\nFor 2+1D topological orders, the quasiparticles can also braid. We also need data to describe the braiding of the quasiparticles in addition to the fusion rules and the F-matrices, as introduced in the next two subsections.\n\n### ii.4 Quasiparticle intrinsic spin\n\nIf we twist the quasiparticle at by rotating at by 360 (note that at has no rotational symmetry), all the degenerate ground states in will acquire the same geometric phase provided that the quasiparticle type is a simple type. We will call the intrinsic spin (or simply spin) of the simple type , which is a universal property of the topologically ordered state.\n\n### ii.5 Quasiparticle mutual statistics\n\nIf we move the quasiparticle at around the quasiparticle at , we will generate a non-Abelian geometric phase \u2013 a unitary transformation acting on the degenerate ground states in . Such a unitary transformation not only depends on the types and , but also depends on the quasiparticles at other places. So, here we will consider three quasiparticles of simple types , , on a 2D sphere . The ground state degenerate space is . For some choices of , , , , which is the dimension of . Now, we move the quasiparticle around the quasiparticle . All the degenerate ground states in will acquire the same geometric phase . This is because, in , the quasiparticles and fuse into , the anti-quasiparticle of . Moving quasiparticle around the quasiparticle plus rotating and respectively by 360 is like rotating by 360. So, moving quasiparticle around the quasiparticle generates a phase . We see that the quasiparticle mutual statistics is determined by the quasiparticle spin and the quasiparticle fusion rules . For this reason, we call the set of data quasiparticle statistics.\n\nIt is an equivalent way to describe quasiparticle statistics by matrices. The matrix is a diagonal matrix. The diagonal elements are the quasiparticle spins\n\n T\u03be\u03b6=T\u03be\u03b4\u03be\u03b6=ei\u03b8\u03be\u03b4\u03be\u03b6. (5)\n\nThe matrix can be determined from the quasiparticle spin and quasiparticle fusion rules [see Eq. (223) in Ref. Kitaev, 2006] :\n\n S\u03be\u03b6=1DZ(C)\u2211\u03c7N\u03c7\u03be\u03b6\u2217ei\u03b8\u03c7ei\u03b8\u03beei\u03b8\u03b6d\u03c7, (6)\n\nwhere is the largest eigenvalue of the matrix , whose elements are .\n\nOn the other hand matrix determines the fusion rules via the Verlinde formula[see (60) in Section III.5]. So, and fully determine the quasiparticle statistics , and the quasiparticle statistics fully determines and .\n\nWe want to emphasize that the fusion rules and F-matrices of bulk quasiparticles and edge quasiparticles are different. In this paper we use only the F-matrices of edge quasiparticles, which is also the F-matrices describing the bulk string-net wavefunctions. Although our Q-algebra module algorithm can be used to compute the F-matrices of bulk quasiparticles, we did not explain in detail how to do this, because calculating the matrices is enough to distinguish and classify 2+1D topological orders with gapped boundaries.\n\n## Iii String-net models with tetrahedron-rotational symmetry\n\nThe string-net condensation was suggested by Levin and Wen as a mechanism for topological phases.Levin\u00a0and\u00a0Wen (2005) We give a brief review here.\n\nThe basic idea of Levin and Wen\u2019s construction was to find an ideal fixed-point ground state wave function for topological phases. Such an ideal wave function can be fully determined by a finite amount of data. The idea is not to directly describe the wave function, but to describe some local constraints that the wave function must satisfy. These local constraints can be viewed as a scheme of ground state renormalization.\n\nLet us focus on lattice models. We put the lattice on a sphere so that there is no nontrivial boundary conditions. Since renormalization will change the lattice, we will consider a class of ground states on arbitrary lattices on the sphere. One way to obtain \u201carbitrary lattices\u201d is to triangulate the sphere in arbitrary ways. There may be physical degrees of freedom on the faces, edges, as well as vertices of the triangles. Any two triangulations can be related by adding, removing vertices and flipping edges. The ideal ground state must renormalize coherently when re-triangulating.\n\nThe string-net picture is dual to the triangulation picture. As an intuitive example, one can consider the strings as electric flux lines through the edges of the triangles. Like the triangulation picture, there are some basic local transformations of the string-nets, which we call evaluations. Physically, evaluations are related to the so-called local unitary transformationsChen\u00a0et\u00a0al. (2010), and states related by local unitary transformations belong to the same phase. If we evaluate the whole string-net on the sphere, or in other words, we renormalize the whole string-net so that no degrees of freedom are left, we should obtain just a number. We require that this number remains the same no matter how we evaluate the whole string-net. This gives rise to the desired local constraints of the ideal ground state wave function. We now demonstrate in detail the formulation of the string-net model with the tetrahedron-rotational symmetry.\n\n### iii.1 String-net\n\nA string-net is a 2-dimensional directed trivalent graph. The vertices and edges (strings) are labeled by some physical degrees of freedom. By convention, we use for string labels and for vertex labels. We assume that the string and vertex label sets are finite.\n\nA fully labeled string-net corresponds to a basis vector of the Hilbert space. If a string-net is not labeled, it stands for the ground state subspace in the total Hilbert space spanned by the basis string-nets with all possible labellings. A partially labeled string-net corresponds to the projection of the ground state subspace to the subspace of the total Hilbert space where states on the labeled edges\/vertices are given by the fixed labels. This way, we have a graph representation of the ground state subspace, which will help us to actually compute the ground state subspace.\n\nThere is an involution of the string label set, satisfying , corresponding to reversing the string direction\n\n \\vbox\\includegraphics[]{i}i\u00a0=\u00a0\\vbox\\includegraphics[]{ir}i\u2217\u00a0=\u00a0\\vbox\\includegraphics[]{i}i\u2217\u2217. (7)\n\nWhen an edge is vacant, or not occupied by any string, we say it is a trivial string. The trivial string is labeled by 0 and . Trivial strings are usually omitted or drawn as dashed lines\n\n vacuum=\u00a0\\vbox\\includegraphics[]{0.pdf}0\u00a0=\u00a0\\vbox% \\includegraphics[]{0.pdf}0\u2217. (8)\n\nIn addition we assume that trivial strings are totally invisible, i.e.,\u00a0can be arbitrarily added, removed and deformed without affecting the ideal ground state wave function. To understand this point, suppose we have a unlabeled string-net on a graph. It corresponds to a subspace of the total Hilbert space on the graph. Now, we add a trivial string to the string-net which give us a partially labeled string-net on a new graph (with an extra string carrying the label ). Such a partially labeled string-net on a new graph corresponds to subspace of the total Hilbert space on the new graph. The two subspaces and are very different belonging to different total Hilbert spaces. The statement that trivial strings are totally invisible implies that the two subspaces are isomorphic to each other . In other words, there exists a local linear map from to , such that the map is unitary when restricted on . Such a map is called an evaluation, which will be discussed in more detail below.\n\n### iii.2 Evaluation and F-move\n\nA string-net graph represents a subspace, which corresponds to the ground state subspace on that graph. When we do wavefunction renormalization, we change the graph on which the string-net is defined. However, the ground state subspace represented by the string-net, in some sense, is not changed since the string-net represents a fixed-point wavefunction under renormalization. To understand such a fixed-point property of the string-net wavefunction, we need to compare ground state subspaces on different graphs. This leads to the notion of evaluation.\n\nWe do not directly specify the ground state subspace represented by a string-net. Rather, we specify several evaluations (i.e. several local linear maps). Those evaluations will totally fix the ground state subspace of the string-net for every graph.\n\nConsider two graphs with total Hilbert space and . Assume that the two graphs differ only in a local area and dim dim. An evaluation is a local linear map from to . Here \u201clocal\u201d means that the map is identity on the overlapping part of the two graphs. Note that the evaluation maps a Hilbert space of higher dimension to a Hilbert space of lower dimension. It reduces the degrees of freedom and represents a wave function renormalization.\n\nAlthough evaluation depends on the two graphs with and , since the graphs before and after evaluation are normally shown in the equations, we will simply use to denote evaluations. We will point out the two graphs only if it is necessary.\n\nLet us list the evaluations that totally fix the ground state subspace. For a single vertex, we have the following evaluation\n\n ev\\vbox\\includegraphics[]{vertex}=\u03b4ijk,\u03b1\\vbox\\includegraphics[]{vertex}, (9)\n\nwhere\n\n \u03b4ijk,\u03b1=0\u00a0or\u00a01, (10) \u03b4ijk,\u03b1=\u03b4kij,\u03b1=\u03b4k\u2217j\u2217i\u2217,\u03b1, (11) \u03b4ij0,\u03b1=\u03b4ij\u2217\u03b40\u03b1, (12) \u2211mNijmNm\u2217kl=\u2211nNinlNn\u2217jk. (13)\n\nWe note that the above evaluation does not change the graph and thus . The evaluation is a projection operator in whose action on the basis of is given by (9).\n\nThe vertex with is called a stable vertex. is the dimension of the stable vertex subspace, called fusion rules. To determine the order of the labels, one should first use (7) to make the three strings going inwards, then read the string labels anticlockwise. If one thinks of strings as electric flux lines, enforces the total flux to be zero for the ground state.\n\nThe next few evaluations are for 2-edge plaquettes, -graphs, and closed loops:\n\n ev\u00a0\\vbox\\includegraphics[]{2-plaquette}=\u0398ijkOi\u03b4ijk,\u03b1\u03b4\u03b1\u03b2\u03b4il\u00a0\\vbox\\includegraphics[]{i}i, (14) ev\u00a0\\includegraphics[]{theta}=\u0398ijk\u03b4ijk,\u03b1\u03b4\u03b1\u03b2, (15) ev\u00a0\\vbox\\includegraphics[]{di}=Oi, (16)\n\nwhere\n\n \u0398ijk=\u0398kij=\u0398k\u2217j\u2217i\u2217, (17) \u0398ii\u22170=\u0398i\u2217i0=Oi=Oi\u2217, (18) O0=ev(vacuum)=1. (19)\n\nwhere is called the quantum dimension of the type string. When is self-dual , the phase factor corresponds to the Frobenius-Schur indicator. Otherwise can be adjusted to 1 by gauge transformations. is because for any closed string-net on the sphere, the half loop on the right can be moved to the left across the other side of the sphere. Those evaluations change the graph. They are described by how every basis vector of is mapped to a vector in .\n\nThe last evaluation is called F-move. It changes the graph. In fact, the F-move is the most basic graph changing operation acting on local areas with two stable vertices. It is given by\n\n ev\\vbox\\includegraphics[]{fl}=\u2211n\u03bb\u03c1Fijm,\u03b1\u03b2kln,\u03bb\u03c1\\vbox\\includegraphics[]{fr}. (20)\n\nIt is equivalent to flipping edges in the triangulation picture. The rank 10 tensor are called F-matrices. are considered as column indices and as row indices. is zero if any of the four vertices is unstable. Otherwise, is a unitary matrix.\n\nNote that the evaluations can be done recursively. When two graphs within and are connected by different sequences of evaluations, the induced maps from to by different sequences must be the same. Firstly the F-matrices must satisfy the well known pentagon equations\n\n \u2211n\u03c4\u03bb\u03b7Fijq,\u03b1\u03b2kln,\u03b7\u03bbFn\u2217jk,\u03bb\u03b3rsp,\u03c4\u03bcFlin,\u03b7\u03c4pst,\u03c1\u03bd=\u2211\u03c3Flq\u2217k,\u03b2\u03b3rst,\u03c1\u03c3Fijq,\u03b1\u03c3rt\u2217p,\u03bd\u03bc. (21)\n\nWe also assume the tetrahedron-rotational symmetry. The tetrahedron-rotational symmetry is actually the symmetry of the evaluation, not of the graphs. For example, if one rotates the graphs in (15)by , the result of the evaluation should be and the tetrahedron-rotational symmetry requires that . In general, with tetrahedron-rotational symmetry, doing evaluation is \u201crotation-invariant\u201d. When the evaluation of tetrahedron graphs, and simpler graphs such as -graphs or closed loops, is rotation-invariant, the evaluation of all graphs is rotation-invariant. Therefore, we call it tetrahedron-rotational symmetry.\n\nThe tetrahedron-rotational symmetry puts the following constraints on the F-matrices. Firstly, it is necessary that the trivial string is totally invisible. So, if in (20) we set the label to 0, the corresponding F-matrix elements should be 1 when the labels match and 0 otherwise, i.e.,\n\n ev\u00a0\\vbox\\includegraphics[]{fl0}=\u2211n\u03bb\u03c1Fijm,\u03b1\u03b20ln,\u03bb\u03c1\\vbox\\includegraphics[% ]{fr0}, (22) Fijm,\u03b1\u03b20ln,\u03bb\u03c1=\u03b4ijl,\u03b1\u03b4\u03b1\u03bb\u03b4ml\u03b4nj\u03b4\u03b20\u03b4\u03c10. (23)\n\nSecondly consider the tetrahedron graphs. After one step of F-move, the tetrahedron graphs have only 2-edge plaquettes. Thus, the amplitude can be expressed by , and , i.e.,\n\n ev =Fijm,\u03b1\u03b2kln,\u03bb\u03c1\u0398nli\u0398n\u2217jkOn, (24) ev =Fklm\u2217,\u03b2\u03b1ijn\u2217,\u03c1\u03bb\u0398nli\u0398n\u2217jkOn, (25) ev =Fjmi,\u03b1\u03bbl\u2217n\u2217k,\u03c1\u03b2\u0398m\u2217kl\u0398n\u2217jkOk, (26) ev =Fk\u2217j\u2217n,\u03c1\u03bbi\u2217l\u2217m,\u03b2\u03b1\u0398mij\u0398m\u2217klOm, (27)\n\nwhere the F-move is performed in the boxed area. These four results must be the same. Thus, we got another constraint on the F-matrices.\n\n Fijm,\u03b1\u03b2kln,\u03bb\u03c1 =Fklm\u2217,\u03b2\u03b1ijn\u2217,\u03c1\u03bb=Fjmi,\u03b1\u03bbl\u2217n\u2217k,\u03c1\u03b2On\u0398m\u2217klOk\u0398nli =Fk\u2217j\u2217n,\u03c1\u03bbi\u2217l\u2217m,\u03b2\u03b1On\u0398mij\u0398m\u2217klOm\u0398nli\u0398n\u2217jk. (28)\n\nNote that (28) is different from that in Ref. Levin\u00a0and\u00a0Wen, 2005 because we do not allow reflection of the tetrahedron. This is necessary to include cases of fusion rules like , for example the finite group model with a non-Abelian group [see Section III.6.4]. It turns out that the conditions above are sufficient for evaluation of any string-net graph to be rotation-invariant.\n\nWith these consistency conditions, given any two string-net graphs with total Hilbert spaces and , , there is a unique evaluation map from to , given by the compositions of simple evaluations listed above. Thus, evaluation depends on only the graphs before and after, or and , not on the way we change the graphs. As we mentioned before, usually it is not even necessary to explicitly point out and , since they are automatically shown in the equations and graphs.\n\nWe want to emphasize that the fusion rules (9-13), the F-move (20), and the pentagon equation (21) are the most fundamental ones. The rest of the equations (14-19)(23)(28) are either normalization conventions, gauge choices, or conditions of the tetrahedron-rotational symmetry. With the tetrahedron-rotational symmetry, are encoded in F-matrices. In (28) set some indices to 0, and we have\n\n Fii\u22170,00ii\u22170,00 =1Oi, (29) Fijk,\u03b1\u03b2j\u2217i\u22170,00 =\u0398ijkOiOj\u03b4ijk,\u03b1\u03b4\u03b1\u03b2, (30) Fjj\u22170,00i\u2217ik,\u03b1\u03b2 =Ok\u0398ijk\u03b4ijk,\u03b1\u03b4\u03b1\u03b2. (31)\n\nMoreover, in (21) set to 0 and one can get\n\n \u2211n\u03bb\u03c1Fijm,\u03b1\u03b2kln,\u03bb\u03c1Fjkn\u2217,\u03c1\u03bblim\u2032,\u03b1\u2032\u03b2\u2032=\u03b4mm\u2032\u03b4\u03b1\u03b1\u2032\u03b4\u03b2\u03b2\u2032. (32)\n\nThus, satisfies\n\n \u2211kNijkOk=\u2211k\u03b1\u03b2Fijk,\u03b1\u03b2j\u2217i\u22170,00Fjj\u22170,00i\u2217ik,\u03b1\u03b2OiOj=OiOj. (33)\n\nThis implies that is an eigenvalue of the matrix , whose entries are , and the corresponding eigenvector is .\n\n### iii.3 Fixed-point Hamiltonian\n\nDoes the evaluation defined above really describe the renormalization of some physical ground states? What is the corresponding Hamiltonian? A sufficient condition for the string-nets to be physical ground states is that the F-move is unitary, or that the F-matrices are unitary\n\n \u2211n\u03bb\u03c1Fijm,\u03b1\u03b2kln,\u03bb\u03c1(Fijm\u2032,\u03b1\u2032\u03b2\u2032kln,\u03bb\u03c1)\u2217=\u03b4mm\u2032\u03b4\u03b1\u03b1\u2032\u03b4\u03b2\u03b2\u2032. (34)\n\nThis requires a special choice of . From (34)(32) we know\n\n Fjkn\u2217,\u03c1\u03bblim,\u03b1\u03b2=(Fijm,\u03b1\u03b2kln,\u03bb\u03c1)\u2217, (35)\n\nwhich implies that , are real numbers, or , and , i.e.,\u00a0if\n\n \u2223\u2223\u0398ijk\u2223\u22232=OiOjOk=didjdk>0. (36)\n\nMoreover, (33)(36) together imply that\n\n \u2211kNijkdk=didj. (37)\n\nHence has to be the largest eigenvalue (Perron-Frobenius eigenvalue) of the matrix and the corresponding eigenvector is .\n\nTo find the corresponding Hamiltonian, note that\n\n evev\u2020\u00a0\\vbox% \\includegraphics[]{i}i=ev\u2211jkl\u03b1\u03b2\u0398\u2217ijkOi\u03b4ijk,\u03b1\u03b4\u03b1\u03b2\u03b4il\u00a0\\vbox\\includegraphics[]{2-plaquette} =\u2211jkNijk|\u0398ijk|2O2i\u00a0\\vbox\\includegraphics[]{i}i=\u2211kO2k\u00a0\\vbox% \\includegraphics[]{i}i=D2C\u00a0\\vbox\\includegraphics[]% {i}i, (38)\n\nwhere is the total quantum dimension.\n\nFor a local area with plaquettes, consider the evaluation that removes all the plaquettes and results in a tree graph, as sketched in Figure 2. Since F-move does not change the number of plaquettes, we can first use F-move to deform the local area and make all the plaquettes 2-edge plaquettes. Thus, we have\n\n evev\u2020D2KC=1. (39)\n\nConsider\n\n P=ev\u2020evD2KC, (40)\n\nwhich means that first use to remove all the plaquettes in the local area, and then use to recreate the plaquettes and go back to the original graph. It is easy to see that . Thus, is a Hermitian projection. Like evaluation, can also act on any local area of the string-net. We can take the Hamiltonian as the sum of local projections acting on every vertex and plaquette\n\n H=\u2211verticesplaquettes(1\u2212P), (41)\n\nwhich is the fixed-point Hamiltonian.\n\nWe see that is exactly the projection onto the ground state subspace. acting on a single vertex projects onto the stable vertex; acting on a plaquette is equivalent to the operator.Levin\u00a0and\u00a0Wen (2005); Kitaev\u00a0and\u00a0Kong (2012); Kong (2012); Lan (2012) The operator is more general because there may be \u201cnonlocal\u201d plaquettes, for example when the string-net is put on a torus, in which case evaluation cannot be performed. But in this paper we will not consider such \u201cnonlocal\u201d plaquettes. Evaluation is enough for our purpose.\n\nIf we evaluate the whole string-net, the evaluated tree-graph string-net represents the ground state. For a fixed lattice on the sphere with plaquettes, the evaluated tree graph is just the void graph, or the vacuum. Therefore, the normalized ground state is\n\n |\u03c8\u27e9ground=ev\u2020DKC|vacuum\u27e9. (42)\n\nGenerically the ground state subspace is\n\n### iii.4 Cylinder ground states, quasiparticle excitations and Q-algebra\n\nNow we have defined the string-net models with tetrahedron-rotational symmetry. We continue to study the quasiparticles excitations.\n\nLet us first discuss the generic properties of quasiparticle excitations from a different point of view. By definition, a quasiparticle is a local area with higher energy density, labeled by , surrounded by the ground state area (see Figure 3). We want to point out that, a topological quasiparticle is scale invariant. If we zoom out, put the area and ground state area together, and view the larger area as a single quasiparticle area , then should be the same type as . Moreover, if we are considering a fixed-point model such as the string-net model, the excited states of the quasiparticle won\u2019t even change no matter how much surrounding ground state area is included. Intuitively, we may view this renormalization process as \u201cgluing\u201d a cylinder ground state to the quasiparticle area. \u201cGluing a cylinder ground state\u201d is then an element of the \u201crenormalization group\u201d that acts on (renormalizes) the quasiparticle states. Thus, quasiparticle states form \u201crepresentations\u201d of the \u201crenormalization group\u201d. Of course \u201crenormalization group\u201d is not a group at all, but the idea to identify quasiparticles as \u201crepresentations\u201d still works. We develop this idea rigorously in the following. We will define the \u201cgluing\u201d operation, introduce the algebra induced by gluing cylinder ground states and show that quasiparticles are representations of, or modules over this algebra. This algebra is nothing but the \u201crenormalization group\u201d.\n\nSince any local operators acting inside the area will not change the quasiparticle type, we do not quite care about the degrees of freedom inside the area, Instead, the entanglement between the ground state area and the area is much more important, and should capture all the information about the quasiparticle types and statistics. Since we are considering systems with local Hamiltonians, the entanglement should be only in the neighborhood of the boundary between the ground state area and the area.\n\nTo make things clear, we would first forget about the entanglement and study the properties of ground states on a cylinder with the open boundary condition. Here open boundary condition means that setting all boundary Hamiltonian terms to zero thus strings on the boundary are free to be in any state. Later we will put the entanglement back by \u201cgluing\u201d boundaries and adding back the Hamiltonian terms near the \u201cglued\u201d boundaries.\n\nOn a cylinder with the open boundary condition, the ground states form a subspace of the total Hilbert space. should be scale invariant, i.e.,\u00a0not depend on the size of the cylinder. We want to show that, the fixed-point cylinder ground states in allows a cut-and-glue operation.\n\nGiven a cylinder, we can cut it into two cylinders with a loop, as in Figure 4. The states in the two cylinders are entangled with each other; but again, the entanglement is only near the cutting loop. If we ignore the entanglement for the moment, in other words, imposing open boundary conditions for both cylinders, by scale invariance, the ground state subspaces on the two cylinders should be both . Next, we add back the entanglement (this can be done, e.g., by applying proper local projections in the neighborhood of the cutting loop), which is like \u201cgluing\u201d the two cylinders along the cutting loop, and we should obtain the ground states on the bigger cylinder before cutting, but still states in . Therefore, gluing two cylinders by adding the entanglement back gives a map\n\n Vcyl \u2297Vcyl glue\u27f6Vcyl h1 \u2297h2 \u27fch1h2 (43)\n\nIt is a natural physical requirement that such gluing is associative, . Thus, it can be viewed as a multiplication. Now, the cylinder ground state subspace is equipped with a multiplication, the gluing map. Mathematically, forms an algebra [see Appendix A].\n\nWe can also enlarge a cylinder by gluing another cylinder onto it. Note that when two cylinders are cut from a larger one as in Figure 4, there is a natural way to put them back together, however, when we arbitrarily pick two cylinders, simply putting them together may not work. To glue, or enforce entanglements between two cylinders, we need to first put them in such a way that there is an overlapping area between their glued boundaries (see Figure 5). In this overlapping area, we identify degrees of freedom from one cylinder with those from the other cylinder; this way we \u201cconnect and match\u201d the boundaries. Next, we apply proper local projections in the neighborhood of the overlapping area, such that the two cylinders are well glued. But, the ground state subspace remains the same, i.e.,\u00a0\u201cmultiplying\u201d by is still ,\n\n VcylVcyl =Vcyl. (44)\n\nNow, we put back the quasiparticle . Since the entanglement between and the ground state area is restricted in the neighborhood of the boundary, it can be viewed as imposing some nontrivial boundary conditions on the cylinder. Equivalently, we may say that the quasiparticle picks a subspace of . should also be scale invariant. If we enlarge the area by gluing a cylinder onto it, in other words, multiply by , remains the same, Mathematically, is a module over the algebra . In this way, the quasiparticle is identified with the module over the algebra . A reducible module corresponds to a composite type of quasiparticle, and an irreducible module corresponds to a simple type of quasiparticle (see section II.2).\n\nAs for string-net models, recall that ground state subspaces can be represented by evaluated tree graphs. The actual ground state subspace can always be obtained by applying to the space of evaluated tree graphs. Thus, we can find out by examining the possible tree graphs on a cylinder. A typical tree graph on a cylinder is like Figure 6. Assuming that there are legs on the outer boundary and legs on the inner boundary, we denote the space of these graphs by . As evaluated graphs, all the vertices in the graphs in must be stable. In principle can take any integer numbers. But note that if , we can add trivial legs on the outer boundary, and can be viewed as a subspace of . Similarly for . Therefore, we know the largest space is .\n\nWe find that the gluing of cylinder ground states can be captured by the spaces . The gluing is nothing but adding back the entanglement. For string-net model the proper local projections are just . But before doing evaluation we have to \u201cconnect and match\u201d the boundaries. i.e.,\u00a0make sure the strings are well connected. Note that and acting inside each cylinder do not affect the boundary legs. can be glued onto from the outer side only if . We need to first connect the legs on the inner boundary of with those on the outer boundary of and make their labels match each other\u2019s; broken strings are not allowed inside a ground state area. This defines a map , where w.c. means restriction to the subspace in which the strings are well connected. Thus, is the desired gluing if there are plaquettes in . Recall that evaluation can be performed in any sequence. We know the following diagram\n\n (45)\n\ncommutes. Thus, gluing with to obtain ground states in can be done by first considering the evaluation of the tree graphs, and then applying to get the actual ground states.\n\nHowever, it is impossible to deal with an infinite-dimensional algebra . We want to reduce it to an algebra of finite dimension. Again our idea is to do renormalization. When we glue the cylinder ground states, we renormalize along the radial direction. Now, we renormalize along the tangential direction, or reduce the number of boundary legs, to reduce the dimension of the algebra.\n\nMore rigorously, our goal is to study the quasiparticles, which correspond to modules over , rather than the algebra itself. So, if we can find some algebra such that its modules are the \u201csame\u201d as those over (here \u201csame\u201d means that the categories of modules are equivalent), this algebra can also be used to study the quasiparticles. Mathematically, two algebras are called Morita equivalentMorita (1958); Kong (2012) if they have the \u201csame\u201d modules. Thus, we want to find finite dimensional algebras that are Morita equivalent to .\n\nNote that with the multiplication forms an algebra. From (45) we also know that and are isomorphic algebras (the isomorphisms are just and ). It turns out that all the algebras are Morita equivalent for [see Section VI]. Thus, we know and have the \u201csame\u201d modules. We choose the algebra to study the quasiparticles of string-net models for has the lowest dimension among the algebras . Now, we reduced the infinite-dimensional algebra to the finite-dimensional . Since a graph in is like a letter Q, and describes the physics of quasiparticles, we name it the Q-algebra, denoted by\n\n Q=V11=\\includegraphics[]{q-space}\u00a0. (46)\n\nThe subtlety of Morita equivalence will be discussed further in Section VI.\n\nIn detail, the natural basis of is\n\n Qi,\u03bc\u03bdrsj=\\vbox\\includegraphics[]{q-basis}\u00a0. (47)\n\nThe notation looks like a tensor. But denotes a basis vector rather than a number. On one hand, represents a cylinder ground state ; on the other hand, when glued onto other cylinder ground states, can be viewed as a linear operator . Both of and are incomplete and misleading. That is why we choose the simple notation ; just keep in mind that it stands for a vector\/operator. As an evaluated graph, the two vertices are stable, . Thus, the dimension of the Q-algebra is\n\n dimQ=\u2211rsijNrj\u2217iNsij\u2217=\u2211rsTr(NrNs). (48)\n\nIn terms of the natural basis, the multiplication is\n\n Qi,\u03bc\u03bdrsjQk,\u03c3\u03c4s\u2032tl=evp\u00a0(Qi,\u03bc\u03bdrsj\u2297Qk,\u03c3\u03c4s\u2032tl) (49) =\u03b4ss\u2032\u2211mn\u03bb\u03c1Qm,\u03bb\u03c1rtn\u2211\u03b1\u03b2\u03b3Fij\u2217s,\u03bd\u03c3k\u2217ln\u2217,\u03b1\u03b2Fr\u2217i\u2217j,\u03bc\u03b2k\u2217nm\u2217,\u03bb\u03b3Ftkl\u2217,\u03c4\u03b1in\u2217m,\u03c1\u03b3\u0398kim\u2217Om.\n\nWe know that the identity is\n\n 1=\u2211rQ0,00rrr=\u2211r\\vbox\\includegraphics[]{qr-id% }. (50)\n\nWe can study the quasiparticles by decomposing the Q-algebra. The simple quasiparticle types correspond to simple -modules. The number of quasiparticle types is just the number of different simple -modules. As of the Morita equivalence of algebras, we also want to mention that the centers [see Appendix A] of Morita equivalent algebras are isomorphic. Thus, the center is an invariant. We argue that is exactly the ground state subspace on a torus and is the torus ground state degeneracy, also the number of quasiparticle types.\n\nWe give a more detailed discussion on the Q-algebra in Appendix B.\n\nAssume that we have obtained the module over the Q-algebra, or the invariant subspace , that corresponds to the quasiparticle . Since , it is possible to choose the basis vectors of from respectively. Such a basis vector can be labeled by , namely,\n\n e\u03ber\u03c4=\\vbox\\includegraphics[]{m-basis}\u2208Q0,00rrrM\u03be. (51)\n\nThen we can calculate the representation matrix of with respect to this basis\n\n Qi,\u03bc\u03bdrsje\u03bet\u03c3 =\u2211q\u03c4Mi,\u03bc\u03bd\u03be,rsj,q\u03c4t\u03c3e\u03beq\u03c4 =evp(\\vbox\\includegraphics[]{mrl}) =\u03b4st\u2211\u03c4Mi,\u03bc\u03bd\u03be,rsj,\u03c4\u03c3\\vbox\\includegraphics[]{m-basis} =\u03b4st\u2211\u03c4Mi,\u03bc\u03bd\u03be,rsj,\u03c4\u03c3e\u03ber\u03c4, (52)\n\nwhere is still the map that connects legs and matches labels. We know that the representation matrix of is , which is a block matrix. Since is an idempotent, . Later we will see that the representation matrices are closely related to the string operators, and can be used to calculate the quasiparticle statistics.\n\n### iii.5 String operators and quasiparticle statistics\n\nThe string operatorLevin\u00a0and\u00a0Wen (2005) is yet another way to study the quasiparticles. A string operator creates a pair of quasiparticles at its ends (see Figure 7). It is also the hopping operator of the quasiparticles, i.e.,\u00a0a quasiparticle can be moved around with the corresponding string operator. First recall the matrix representations of string operators. For consistency we still label the string operator with ,\n\n \\vbox\\includegraphics[]{sol}=\u2211rsj\u03bc\u03bd\u03c4\u03c3\u03a9i,\u03bc\u03bd\u03be,rsj,\u03c4\u03c3\\vbox\\includegraphics[]{sor}, (53)\n\nwhere is zero when either vertex is unstable.\n\nFor a longer string operator, one can apply (53) piece by piece, and contract the or labels at the connections. In particular, , since means simply extend the string operator. We define , which means the number of type strings the string operator","date":"2021-01-27 17:40:10","metadata":"{\"extraction_info\": {\"found_math\": false, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.8428720235824585, \"perplexity\": 604.4438160176047}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2021-04\/segments\/1610704828358.86\/warc\/CC-MAIN-20210127152334-20210127182334-00615.warc.gz\"}"}
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{"url":"https:\/\/socratic.org\/questions\/how-do-you-simplify-5-12-2-3","text":"# How do you simplify 5\/12 + 2\/3?\n\nJun 3, 2016\n\n$\\frac{5}{12} + \\frac{2}{3} = 1 \\frac{1}{12}$\n\n#### Explanation:\n\nTo simplify $\\frac{5}{12} + \\frac{2}{3}$, we should first modify the fractions so that they have common denominator.\n\nAs $3$ is a factor of $12$, the GCD of denominators $12$ and $3$ is $12$ and\n\n$\\frac{5}{12} + \\frac{2}{3} = \\frac{5}{12} + \\frac{2 \\times 4}{3 \\times 4}$\n\n= $\\frac{5}{12} + \\frac{8}{12} = \\frac{13}{12} = \\frac{12 + 1}{12}$\n\n= $\\frac{12}{12} + \\frac{1}{12} = 1 \\frac{1}{12}$","date":"2022-08-20 02:38:29","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 10, \"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\": 0.9816418886184692, \"perplexity\": 824.9931129304782}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2022-33\/segments\/1659882573876.92\/warc\/CC-MAIN-20220820012448-20220820042448-00680.warc.gz\"}"}
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Q: Remove line from line graph in d3.js I have a button that displays a line graph using d3.js. But I want to remove the line from the graph on clicking the same button. I have created a toggle button, but how do i remove the line from the graph ? I have the following function that plots the graph. svg.selectAll("path").remove() is removing the axis and but not the line.
function plotGraph(file) {
var color = d3.scale.category10();
var svg = d3.select('#mySvg');
svg.selectAll("path").remove();
var line = d3.svg.line().interpolate("basis").x(function(d) {
return x(d.date);
}).y(function(d) {
return y(d.mvalue);
});
d3.csv(file,function(error, data) {
color.domain(d3.keys(data[0]).filter(function(key) {
return key !== "date";
}));
data = data.map(function(d) {
return {
mvalue : +d.mvalue,
date : parseDate(d.date)
};
});
x.domain(d3.extent(data, function(d) {
return d.date;
}));
y.domain([ 0, 100 ]);
svg.append("path").datum(data).attr("class", "line").attr("d",line);
});
}
A: You can store the line in a variable and then use that as a handle to remove it later.
In d3, the .append operator returns the appended child, so to do this, all you need to do is this:
var myLine;
function(appendLine){
...
myLine = svg.append("path").datum(data)...
...
}
function(removeLine){
myLine.remove()
}
Use appendLine when you want to create the line and removeLine to remove it. With this method, you can have a variable for each line you want to control, or else use variable scoping to not have to worry about it. It depends on what the rest of your code looks like.
Alternately, if you have a line with an ID that you want to remove, d3.select('#myId').remove() should work.
A: You have a few options to select a specific element you want to remove. If that element is identified by a class, you can do
d3.select("path.line").remove();
If you want to remove all lines on the graph, you should use
d3.selectAll("path.line").remove();
If, as in your example, there are several of these elements, you can assign an ID to them and use that to remove it.
svg.append("path")
// ...
.attr("id", "id");
// ...
d3.select("#id").remove();
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"redpajama_set_name": "RedPajamaStackExchange"
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https://www.phonescoop.com/articles/article.php?a=21173
OnePlus Pushing Android 9 Pie to the OnePlus 6
Article Comments 7
Sep 21, 2018, 10:46 AM by Eric M. Zeman @zeman_e
OnePlus today began delivering Android 9 Pie to its flagship phone, the OnePlus 6. The company had been testing a beta version of the new operating system from Google for several weeks. Android 9 Pie carries with it a brand new user interface, adaptive battery support, and revised gesture navigation. It adds a reworked Do Not Disturb mode, as well as a new Gaming Mode. The update also improves text and call notifications, and allows people to adjust the accent color. Last, the update includes the September security patch from Google. Android 9 Pie is free for owners of the OnePlus 6 to download and install. The company is rolling it out over the next few days.
Hands On with the OnePlus 6
The OnePlus 6 is here to convince you that no one needs a $700, $800, $900, or $1,000 smartphone. Heck, you don't even need a $600 smartphone.
Review: OnePlus 6
The OnePlus 6 is the company's latest attempt to convince you that ultra-pricey flagships are unnecessary; why spend $800 to $1000 on a phone when you can get one that's nearly as good for just over $500? The 6 is an attractive metal-and-glass device that has the latest design from OnePlus, the latest specs from Qualcomm and others, and the latest Android software from Google.
OnePlus Offers Open Beta of Android 9 Pie to the OnePlus 6
OnePlus has made the first Open Beta build of OxygenOS based on Android 9 Pie available to the OnePlus 6. OnePlus had previously been testing the beta with a closed group.
OnePlus 6 Comes In New Red Finish
OnePlus today announced a new red color option for its OnePlus 6 phone. This red special edition also features a new six-layer glass back to enhance the color, including an "anti-reflective layer to create a sense of depth" and a translucent orange layer.
OnePlus 6 Receiving OxygenOS Update with Selfie Portraits
OnePlus has begun pushing OxygenOS 5.1.6 to its OnePlus 6 phone. The update contains a few new features, including Portrait Mode for the front camera, which enables bokeh-style selfies.
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{
"redpajama_set_name": "RedPajamaCommonCrawl"
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Q: Generating unique fixed length strings I need to generate unique non-sequential IDs of specified length. I've looked up some implementations of Feistel ciphers but I don't understand why exactly specific numbers are chosen and how do they affect output. Basically, I need a function int pseudo_encrypt(int seed, int max) that produces unique result less than max for each seed less than max.
UPD: It turns out it's called 'format-preserving encryption'. I've tried AES CTR mode but the problem is the 'length' is specified in bytes. So if max is greater than 255 the cipher will produce a result between 0 and 65535. How to deal with this problem?
A: Using a cipher is correct. Since a cipher is reversible, each seed number will encrypt to a unique cyphered value and can be decrypted back to the original value.
To generate your required sequence of unique numbers, you encrypt seeds 0, 1, 2, 3, ... for as many unique numbers as you require.
Your major problem is likely the value of max you need. For max = (2^64) - 1 then use DES in ECB mode with a fixed key. For max = (2^128) - 1 use AES in ECB mode with a fixed key. Different keys give a different mapping from plaintext to cyphertext, so you need to keep using the same key to avoid duplicates.
If your max is a different size, then you will have to do more work. Basically you need a block cipher with the same block size as your value of max, measured in bits. You can construct your own Feistel cipher with the appropriate block size, or else you can use Hasty Pudding cipher, which allows any reasonable block size you want.
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Q: null handling in ORACLE Stored Procedure is like below.
CREATE OR REPLACE PROCEDURE FIRSTPROC (ID1 IN VARHAR2, ID2 OUT NUMBER )
AS
BEGIN
SELECT id2val INTO ID2
FROM
WHERE id1val = ID1;
END
If i am passing
if i pass valid value then i am getting the output.
ID1 := 5;
ID1 as NULL VALUE I AM GETTING run time ERROR in the SELECT statement. Please note my table contains NULL values also. How to handle this condition.
ID1 := null;
calling the procedure
Please let me know what changes i need to make?
A: NULL is not equal to NULL. So when you are passing in NULL, and doing a query using only an equality condition, you can't get a match. So I assume you are getting a no-data-found exception (P.S. instead of saying you are getting "an error" it can be helpful to say specifically what error you are getting).
You have to handle NULL explicitly.
SELECT id2val INTO id2
FROM whatever
WHERE ( id1val = id1 OR (id1val IS NULL AND id1 IS NULL) )
Another possibility is to use a condition like NVL( id1val, -1 ) = NVL( id1, -1 ), where -1 is some value that will never actually occur in your data. But this is dependent on choosing a good sentinel value, and can impact the effectiveness of indexes on the query.
Note that if your table contains multiple rows with NULL values for ID1VAL, this query will then raise too-many-rows.
A: NULL is not equals to NULL, but you can use the nvl function to fix your equality like this
WHERE nvl(id1val, 'na')=nvl(id1, 'na')
A: Comparing whether a NULL value is equal to another value (including another NULL) will always return false. Instead you can compare if the two values are equal or if the first value IS NULL and the second value IS NULL.
Amending my previous answer to your last question:
CREATE OR REPLACE PROCEDURE FIRSTPROC (
ID1 IN table_name.id1val%TYPE,
ID2 OUT table_name.id2val%TYPE
)
AS
BEGIN
SELECT id2val
INTO ID2
FROM table_name
WHERE ( id1val = ID1 OR ( ID1 IS NULL AND id1val IS NULL ) );
EXCEPTION
WHEN NO_DATA_FOUND THEN
ID2 := NULL;
WHEN TOO_MANY_ROWS THEN
ID2 := NULL;
END;
/
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\section{Introduction}~An important aim of globalization is ensuring that, alongside capital, labor moves freely across national borders~\cite{Grubel, Docquier, Hufbauer, Dani16}. In the EU, therefore, a country lacking labor force should welcome laborers from another EU country with excess labor. This economic argument is somewhat negligent of how such a welcoming policy may affect public opinion, expressed in democratic societies via popular vote. Especially volatile situations may arise if either the native majority or the migrant minority sense that their national or religious identity is being threatened. In this context, an unprecedented inflow of immigrants into the EU during the recent migrant crisis seems to offer a fresh insight into the relationship between popular vote and immigration and, by extension, into the factors that directly affect the success of further globalization.
Although a large body of literature is dedicated to the analysis of how migration affects the global economy~\cite{Grubel, Docquier, Hufbauer, Chiswick, Borjas, Borjas1, Borjas2, Swank, Gibson} and right-wing (RW) populism~\cite{Betz, Knight, Fysh, Voerman, Chapin, Smith, Bonikowski}, much less is known about the limitations of globalization~\cite{Dani00, Dani16}, especially how large-scale migrations sway popular vote and what the economic consequences may be. Borjas reported slow integration of immigrants into the US, taking four generations to catch up with the earnings of natives instead of one or two as commonly believed~\cite{Borjas3}. In a slightly broader context, particularly interesting is the relation of immigration to globalization. Rodrik's argument~\cite{Dani00, Dani16} states that globalization, democracy, and national sovereignty are mutually irreconcilable, leading to a conclusion that democracy is compatible with national sovereignty only if globalization is restricted.
The recent migrant crisis in the EU was motivated by political turmoil and armed confrontations rather than globalization itself, yet as an unintended consequence some of the central tenets of globalization have been put to test. In particular, the increase of immigrants in the total population was paralleled with a surge in voters who support RW populist parties. The growing number of RW voters across the EU suggests that tolerance towards immigrants is conditional~\cite{BP16} because parts of the general public with little regards for RW populism at first, demonstrably turned into RW populist supporters. Understanding this "change of heart" is of utmost importance for the success of globalization. If the immigration inflow exceeds the speed of integration for a prolonged period of time~\cite{BP16}, the RW populist movement may democratically prevail and eventually lead to a demise of globalization. Even if one is not concerned with globalization, humanity needs to better prepare for the potential massive displacements of the global population expected due to global climate change.
Herein we find that during the last three years, in the EU countries hit by the recent migrant crisis, the speed of immigrant inflow substantially exceeds the speed of their integration. We further find a significant relationship between the percentage of RW voters and the fraction of immigrants, on the one hand, as well as the total immigration inflow into the entire EU, on the other hand. By contrast, injuries and casualties in violent incidents and the integration rate proved to be poor predictors of the percentage of RW voters. Seeing globalization as a tolerant mode of democracy, wherein cooperation between nations supplements national interest, an important question arises naturally: Can we predict under what circumstances the RW populist movement receives enough support to overthrow this tolerant mode of democracy, potentially leading to drastic political and economic upheaval? Moreover, because globalization makes many countries experience similar problems, should we expect a cascade (i.e., a domino) effect, whereby the rise of a populist movement in one country triggers similar movements in other countries?
\section{Results}~RW populism is often characterized by intolerance which, in turn, is among the most widespread social phenomena, commonly responsible for conflicts and segregation \cite{Schelling, Axelrod97, McPherson, Antal, Marvel, Moreno, Parravano}. Together with the associated phenomenon of radicalization, intolerance is the main cause of violence and terrorism \cite{Lim, Dancygier1, Dancygier2, Galam, Sampson}. Here, we use the fraction of RW populist voters in a given country as a proxy for RW populism. For each of the countries most affected by the recent migrant crisis, we estimate the percentage of immigrants from September 2013 to June 2016, starting from the official value in 2013 and complementing it with the number of monthly recorded visa applicants \cite{inflow}. For the same time period, we also collect the available election polls and election results \cite{DATApolls}.
The situation in the EU in June 2016 is summarized in Fig.~\ref{f1}(a) that reveals a rising trend in the percentage of RW populist supporters in response to the increasing percentage of immigrants in the general population. From the cumulative exponential function that fits the data well, we find that as the percentage of immigrants approaches approximately 22\%, the percentage of RW populist voters exceeds 50\%, a threshold needed in democratic societies for any party, and thus RW parties too, to take over the government. Considerable scatter in the data furthermore suggests that there may be visible differences in (in)tolerance between the countries---the larger the percentage of voters in favor of RW populism compared to the percentage of immigrants, the smaller the tolerance. Taking Austria as an example, in Fig.~\ref{f1}(b) we find that the 50\% threshold would be reached even if the percentage of immigrants was below 20\%.
\begin{figure}[t]
\centering \includegraphics[scale=0.80]{Fig01.pdf}
\caption{\textbf{Immigration affects the support for right-wing populism I.} (a) Among the EU countries involved in the recent migrant crisis, support for RW populism is generally higher in those countries that accepted a larger number of immigrants relative to the country's population size. Shown is June 2016. Seeing democracy as the majority rule principle, we presume that RW populism becomes a dominant political option when the percentage of RW voters exceeds 50\%. Judging based on a cumulative exponential function that fits the data reasonably well, RW populism in the examined EU countries may take over if the percentage of immigrants in the total population approaches 22\%. (b) Similar as in the other EU countries, Austrian data reveal that the increase in the percentage of immigrants is accompanied with an increase in the percentage of RW populist voters. Here too a cumulative exponential function fits the data well. This function predicts the rise of RW populism in Austria when the percentage of immigrants is slightly below the 20\% mark.}
\label{f1}
\end{figure}
The fraction of immigrants in the general population is not the only factor affecting the sentiment of voters. In Fig.~\ref{f2}, using the data for Austria and Germany over the past three years (2013-2016), we demonstrate that the percentage of RW populist supporters also depends on the inflow of immigrants into Europe. Illustrative is the Austrian example, where in 2013 parliamentary election the far-right party won 20.5\% of the popular vote, roughly reflecting the sentiment predicted from the percentage of immigrants living in Austria at the time. However, due to a high inflow of immigrants that in the second half of 2015 reached unprecedented proportions~\cite{inflow}, the local Vienna election saw the percentage of RW voter suddenly jump to 33\%. This sudden change in popular vote is reminiscent of phase transitions (i.e., tipping or critical points)---well documented in social sciences~\cite{PRL, Mas}---whereby the closer a country to a tipping point, the more abruptly voters turn their back to moderate parties and start voting for more extreme alternatives. A qualitatively similar phenomenon is seen in the case of Germany in Fig.~\ref{f2}(b)-(c).
\begin{figure}[t]
\centering \includegraphics[scale=0.80]{Fig02.pdf}
\caption{\textbf{Immigration affects the support for right-wing populism II.} (a) An unprecedented inflow of immigrants into Austria coincided with a steady increase in the fraction of RW populist voters.
A solitary black dot represents the results of Austrian presidential election in May 2016 in which an RW populist candidate secured almost 50\% of votes. This election shows that even after the record immigrant inflow at the end of 2015 had subsided, a decreasing trend in the number of immigrants that enter Austria did not automatically translate into lower support for the RW populist political option, i.e., RW populism seems to be more than just a craze. (b) In Germany, the increasing inflow of immigrants (monthly data~\cite{inflow}) rather clearly coincided with the increasing support for an RW populist party. (c) A significant regression emerges when the German case is presented as a scatter plot between the inflow of immigrants and the percentage of far-right voters.}
\label{f2}
\end{figure}
\begin{figure}[t]
\centering \includegraphics[scale=0.80]{Fig03.pdf}
\caption{\textbf{Immigrant inflows and the popularity of right-wing populist movements---a non-linear threshold.} Shown is the annualized immigrant inflow into a given country (horizontal axis) as a percentage of that country's population, as well as the corresponding percentage change in RW populist votes (vertical axis). In parentheses are the fractions of immigrants in the total population of the corresponding country. For a group of countries in which the annualized increase in the percentage of RW voters exceeded 2\%, this increase is virtually independent on the inflow of immigrants. Such a result may reflect the EU's political organization, i.e., the lack of internal borders whereby if one country decides to accept immigrants, the decision may have repercussions for all the other member states. We also observe a threshold indicated by a dashed line at which the immigrant inflow into a given country is sufficiently high to invariably provoke an increase in the percentage of RW populist voters. In the model construction, this threshold suggests $\alpha=0.004$ on a annual basis.}
\label{f3}
\end{figure}
In an attempt to probe deeper into the internal dynamics of RW populism in the EU as a function of the inflow of immigrants, next we analyze how the immigration rate affects the rise in RW populist voters. Surprisingly, for a group of countries in which the annualized increase in the percentage of RW voters exceeded 2\%, Fig.~\ref{f3} shows that this increase is virtually independent on the inflow of immigrants. Why would countries with a relatively high and a relatively low inflow of immigrants exhibit about the same increase in the percentage of RW voters? This result may be a consequence of the EU's political organization. Because the EU functions practically as a supranational state with no internal borders, if one country decides to accept immigrants, this decision may have repercussions for all the other member states. The increase in the percentage of RW populist voters may therefore more systematically depend on the total inflow of immigrants into the entire EU, expressed here as a percentage of the total EU population, than the inflow in any individual country.
Some, albeit anecdotal, evidence to the effect that the decision of one country may affect the situation in another is seen in the case of Sweden and Norway. The former country was among those that were hit the hardest by the recent migrant crisis, yet the latter country saw practically the same annualized increase in the percentage of RW voters. Another interesting pair in this context is Germany and Poland. Again it was the former country that experienced a high inflow of immigrants, yet it is in Poland that 53\% of the population thinks that their government should refuse asylum seekers from the Middle East and North Africa (and only 33\% thinking Poland should do the opposite). The Polish example may contain another important lesson. Namely, this country seems to have already transitioned from the tolerant mode of democracy associated with globalization to a mode dominated by RW populism. If so, the implication is that the fraction of immigrants at which the Polish population is pushed beyond the tipping point is much lower than in western EU countries. Poland---and similarly Hungary, both of which share decades of socialist experience---is among the toughest opponents of immigration into the EU, strongly debating against the quotas that the EU imposed with a goal to more evenly spread the shock of recent migrant crisis.
Because Fig.~\ref{f3} implies that the interplay of factors influencing the popularity of RW populism is more complex than a simple bivariate regression could reveal, we turn to econometric analysis and multivariate regression. Based on the results of simple regression in Fig.~\ref{f1}, we take that the fraction of RW voters (response variable, $RW_t$) in a given country is primarily determined by the fraction of immigrants ($IM^L_t$) in this country. To account for the assumption that the percentage of RW voters is affected by the overall inflow of immigrants into the EU ($IM^{EU}_t$), this variable is also included in the model. Finally, to control for the possibility that violent incidents involving immigrants may sway the popular vote, we include into the model the total number of injuries ($I$) and casualties ($D$) recorded in such incidents across the EU~\cite{wiki}. Thus, the regression model becomes
\begin{equation}
RW_t = \beta_0 + \beta_{IM}^L IM^L_t + \beta_{IM}^{EU} IM^{EU}_t + \beta^D_{ter} D_t + \beta^I_{ter} I_t + e_t,
\label{e1}
\end{equation}
where $e_t$ is random error. In three countries that were hit the hardest by the recent migrant crisis (Germany, Austria, and Sweden), we find a significant relationship between the fraction of RW voters and the fraction of immigrants (Table~\ref{t1}). In Germany and Sweden, furthermore, the support for RW populism is positively related to the immigration inflow into the EU. Somewhat surprisingly, there is no significant dependence of the response variable on the number of injuries and casualties in violent incidents.
\begin{table}[t]
\begin{center}
\tabcolsep=0.11cm
\begin{tabular}{ccccccc}
\textit{state} & ${\beta_0}$ & ${\beta^{L}_{IM}}$ & ${\beta^{EU}_{IM}}$ & ${\beta_{ter}^{D}}$ & ${\beta_{ter}^{I}}$ & ${R^2}$ \\
\hline
\textit{Ger} & -1.17$^*$ & 10.29$^*$ & -4.3e-07$^*$ & 1.4e-04 & 1.563e-05 & 0.852 \\
$ $ & (-11.82) & (12.31) & (-5.36) & (0.67) & (0.19) & \\
\hline
\textit{Aus} & -1.47$^*$ & 10.11$^*$ & 7.9e-07 & -3.1e-04 & 2.1e-05 & 0.452 \\
$ $ & (-2.11) & (2.24) & (1.23) & (-0.20) & (0.03) & \\
\hline
\textit{Swe} & -0.34$^*$ & 2.60$^*$ & 5.8e-07$^*$ & -1.9e-04 & 2.1e-05 & 0.751 \\
$ $ & (-2.88) & (3.53) & (3.37) & (-0.47) & (0.13) & \\
\hline
\hline
\end{tabular}
\end{center}
\caption{Multiple regression as defined in Eq.~(\ref{e1}) using the data on the three EU countries most gravely affected by the recent migrant crisis. Star denotes the statistically significant regression coefficients at the 5\% significance level. Parentheses hold the value of t-statistic.\label{t1}}
\end{table}
\begin{table}[t]
\begin{center}
\tabcolsep=0.11cm
\begin{tabular}{ccccc}
$ $ & {Coeff.} & {Std. Err.} & {t Stat.} & $P>t$ \\
\hline
$\beta_{IM}^{L}$ & 3.48 & 0.486 & 7.17 & 0.000 \\
\hline
$\beta_{IM}^{EU}$ & 126.4 & 41.6 & 3.04 & 0.002 \\
\hline
$\beta_{ter}^D$ & 8.3e-05 & 7.7e-05 & 1.07 & 0.284 \\
\hline
$\beta_{ter}^I$ & -1.2e-04 & 1.9e-04 & -0.64 & 0.525 \\
\hline
\textit{MIPEX} & 0.120 & 0.244 & 0.49 & 0.622 \\
\hline
$\beta_0$ & -0.353 & 0.109 & -3.24 & 0.001 \\
\hline
\hline
\end{tabular}
\end{center}
\caption{Pooled time series cross-section analysis (TSCS) with random-effects GLS regression as defined in Eq.~(\ref{e2}). Test statistics:
Wald $\chi^2(5)=139.39$ and $Prob > \chi^2=0.000$.\label{t2}}
\end{table}
\begin{table}[t]
\begin{center}
\tabcolsep=0.11cm
\begin{tabular}{ccccc}
$ $ & {Coeff.} & {Std. Err.} & {t Stat.} & $P>t$ \\
\hline
$\beta_{IM}^{L}$ & 3.99 & 0.557 & 7.18 & 0.000 \\
\hline
$\beta_{IM}^{EU}$ & 163.2 & 42.1 & 3.88 & 0.000 \\
\hline
$\beta_0$ & -0.432 & 0.089 & -4.84 & 0.000 \\
\hline
\hline
\end{tabular}
\end{center}
\caption{Pooled time series cross-section analysis (TSCS) with random-effects GLS regression. Test statistics: Wald $\chi^2(5)=169.9$ and $Prob > \chi^2=0.000$.\label{t3}}
\end{table}
We extend the above econometric analysis with a pooled time-series cross-section (TSCS) method that combines the cross-sectional data on multiple countries. Here, the number of countries is $N=7$: Germany, Austria, the Netherlands, Sweden, France, Norway, and Denmark. Because for each country there are $T_i$ observations along the temporal dimension, the whole dataset has $\sum_{i=1}^N T_i=142$
observations. Compared to the model in Eq.~(\ref{e1}), the TSCS model has an extra variable, $M_{it}=(1 - MIPEX/100)$, where MIPEX is the Migrant Integration Policy Index (MIPEX)~\cite{mipex}, a proxy for the integration rate---a higher MIPEX implies better integration---and an extra index $i=1, 2,..., N$ that refers to a cross-sectional unit:
\begin{equation}
RW_{it} = \beta_0 + \beta_{IM}^L IM^L_{it} + \beta_{IM}^{EU} IM^{EU}_{it} + \beta^D_{ter} D_{it} + \beta^I_{ter} I_{it} + M_{it} + e_{it}.
\label{e2}
\end{equation}
The results of the TSCS regression model emphasize the fraction of immigrants in the general population and the inflow of immigrants entering the entire EU as the significant predictor variables (Table~\ref{t2}). The results of the TSCS regression model without MIPEX and violent incidents are shown in Table~\ref{t3}. Interestingly, even if we neglect the total inflow of immigrants into the EU, coefficient values in Table~\ref{t3} suggest that between 23\% and 24\% of immigrants in the total population of a country are sufficient to cause larger than 50\% support for RW populist parties, which is similar to the result obtained from Fig.~\ref{f1}.
Based on the analyzed data it is unclear whether the rise of RW populism is just a transient phenomenon or if a change in political orientation is accompanied with longer term memory. A glimpse into the persistence of voters' memory is offered by the sequence of events in Austria following the above-mentioned local Vienna election held in October 2015 wherein the RW populist party won 33\% of the popular vote. Namely, during the presidential election a few months later, just as the migrant crisis was at its peak, the RW populist movement received another boost and its candidate secured almost 50\% of votes, narrowly losing to a leftist rival. However, these results were contested and the re-vote is supposed to take place in early December 2016. Despite the migrant crisis having subsided since, polls have the RW populist party holding steady at around 35\% and its candidate at almost 51\%, thus indicating that the rise of RW populism is indeed more than just a craze. This indication seems to have some support in the literature as well. A substantial increase in the number of refugees and illegal immigrants in European countries during the 1980s provoked a similar wave of radical RW populism as described herein~\cite{Betz93}. In the wake of these events, already in the early 1990s, there were still between 11 and 14\% of the Europeans who categorized other nationalities, races, or religions as unsettling~\cite{Betz93}.
\section{Model}~Human interactions are for many purposes heterogeneous and prone to abrupt non-linear responses. Precisely such a response is seen in the rise of the RW populist party and its candidate in the Austrian elections. Accordingly, linear approaches as exemplified by regression Eq.~(\ref{e1}) are bound to get us only so far. To offer a mechanistic perspective on the rise of RW populism and account for the existence of tipping points in social dynamics, we adopt a complex network approach in the spirit of Refs.~\cite{BP16, NP14, BP15, Lee15}. The benefit of relying on complex networks, due to their ability to emulate the stated heterogeneity in human interactions, goes beyond just capturing the dynamics in the vicinity of tipping points. Heterogeneity seems important when considering immigration and integration issues because integrating immigrants who live in ghettos (or hubs in a network-theoretic jargon) may be substantially more difficult than if immigrants mixed uniformly with the native population.
In accordance with our interest in the increase of the immigrant population relative to the native one, without any loss of generality, we begin the model construction by setting a constant number of native (hereafter also insider) agents. These agents are arranged in an Erd\H{o}s-R\'{e}nyi random network of business and personal contacts. Immigrants (hereafter often outsiders) populate the network subsequently. Every insider observes the fraction of outsiders in their neighborhood and based on this fraction decides whether to be supportive of globalization or to turn to populism. Because insider agents refer to their neighborhood for information, the interaction is local, but also essential for generating and understanding tipping points~\cite{Watts}. There are, however, other non-local types of interactions, as well as the possibility that information is misperceived or misinterpreted. In the following, we formalize these concepts with three model assumptions.
\paragraph{Assumption (i): media and economic influences.} At each time $t$, representing a period of one month, every insider agent is influenced by media at relative rate $p$ (i.e., a probability per unit of time) and stays influenced for period $\tau$. We assume that this influence turns insiders into RW populist supporters. Again there is no loss of generality because, although media affect insiders both ways, we are interested only in the net rate (negative $p$ would indicate the opposite effect). The effect of media is global and reflects the possibility that just hearing about immigration may turn some insiders into RW populist supporters whatever the true local situation. More generally, rate $p$ could also reflect economic factors (e.g., unemployment). For a randomly chosen insider agent, we readily calculate the probability that this agent is influenced by media using equation~\cite{NP14} $p*=1-\exp(-p\tau)$.
\paragraph{Assumption (ii): local influence of outsiders.} In Greek local elections in Nov 2010, a far-right party Golden Dawn got 5.3\% of the vote, yet in some neighborhoods of Athens with large immigrant communities, the party won up to 20\%. This kind of election results suggest that contacts between insiders and outsiders matter. In our network model, a constant number of $N$ insiders is supplemented with $I(0)$ outsiders initially, where the latter number increases at each moment $t$ due to inflow $J_t$. Newly arriving outsiders are placed randomly between existing agents who at the beginning have an average number of connections (i.e., degree) $k$. The total number of outsiders, $I(t)$, is obtained by summing monthly $J$'s according to $I(t)=I(0)+\Sigma_{s=1}^t J_s$. At any moment, the fraction of outsiders equals $f_I(t)=I(t)/(N+I(t))$. To account for the above-mentioned effect due to contacts between insiders and outsiders, we assume that any agent $i$ with $k_i$ total connections turns to RW populism at rate $p'$ if this agent is surrounded by at least $m_i=f_I' k_i$ outsiders~\cite{Watts, NP14, BP15b}, where $0<f_I'<1$ is a constant model parameter quantifying how tolerant insiders are. This assumption merits a few additional comments.
First, the probability that randomly chosen insider agent $i$ with $k_i$ connections is surrounded by $m_i$ outsiders, and therefore prone to RW populism, equals $p_1(k_i,m_i,f_I)\equiv\Sigma_{j=m_i}^{k_i} f_I^{j}(1-f_I)^{k_i-j}{k_i\choose k_i-j}$. In this formula, $f_I$ is the true current state of the network. The reality may, however, be such that information is biased and insiders perceive more outsiders than there really are. If the bias is $\Delta f_I$, then $p_1(k_i,m_i,f_I+\Delta f_I)>p(k_i,m_i,f_I)$. This increased probability $p(k_i,m_i,f_I+\Delta f_I)$ implies that the tolerance parameter, $f_I'$, must decrease by amount $\Delta f_I'$ estimable using condition $p(k_i,m_i-\Delta f_I' k_i,f_I)=p(k_i,m_i,f_I+\Delta f_I)$. An implicit assumption here is that all insider agents are equally tolerant to immigrants because the tolerance parameter, $f_I'$, is defined as a global network property rather than an individual agent property. An alternative would be to assume a distribution of tolerance in which case $f_I'$ represents the mean~\cite{BP16}.
To accommodate the econometric results that immigration inflows affect the support for RW populism, we extend assumption (ii) by requiring that (A) if the immigration inflow, $J$, is below some threshold $J'$, society is gradually becoming more tolerant. Consequently, if $J<J'$ at a given time moment, then the tolerance parameter, $f_I'$, increases at this moment by amount $\Delta f_I'=\delta>0$. When extension (A) is met, there is a balance between immigration and immigrant integration---outsiders are successfully integrated---and insiders are capable to acclimate to changes in their society. Judging naively from Fig.~\ref{f2}(b), $J'$ for Germany should be close to 10\,000 people per month. The problem of finding a maximum $J'$ at which insiders acclimate to an increasing presence of outsiders may be interpreted as highly relevant for the success of globalization. Namely, in the context of his trilemma set, Rodrik argued~\cite{Dani00, Dani16} that the notion of democracy is compatible with national sovereignty only if globalization proceeds in a carefully balanced manner. One such balance, achievable in the model if $J<J'$, concerns immigration and integration as inevitable consequences of globalization. Otherwise, further progress is threatened by the rise of RW populism as an intolerant mode of democracy wherein cooperation between nations is, to put it mildly, downgraded on the list of political priorities.
Empirical evidence suggests that in addition to extension (A), we should consider an opposite extension (B). Specifically, as the inflow of outsiders, $J$, crosses some threshold $J''$ (not necessarily equal to $J'$), society gradually becomes less tolerant. A mathematical representation of extension (B) would be that if $J>J''$, then the tolerance parameter, $f_I'$, decreases by amount
\begin{equation}
\Delta f'_I=-\gamma J.
\label{e3}
\end{equation}
In view of the econometric models in Eqs.~(\ref{e1}) and (\ref{e2}), we make the decrease of the tolerance parameter proportional to inflow $J$, where $\gamma$ is a proportionality coefficient expressing the sensitivity of insiders to large outsider inflows. Furthermore, motivated by Fig.~\ref{f3}(b), we express threshold $J''$ in terms of the total population, i.e., $J''=\alpha N$, where $\alpha$ is another proportionality coefficient. Fig.~\ref{f3}(b), in fact, hints what an estimate of $\alpha$ would be in the case of the EU countries hit by the recent migrant crisis. The dashed line in this figure delineates the annual inflow below which countries responded to immigration in a mixed fashion, but above which support for RW populism always increased. Because $\alpha=J''/N$ and, based on Fig.~\ref{f3}(b), $12J''/N\approx 0.004$, we obtain
\begin{equation}
\alpha\approx 0.00033.
\label{e4}
\end{equation}
Extensions (A) and (B) represent two opposite limiting cases, one in which immigration is a relatively slow process and the other in which immigration is a relatively fast process. We were motivated to introduce these two extensions by the empirical evidence, yet some support can be found in other sciences too. Brain science, for example, offers a physiological interpretation stating that political attitudes have a counterpart in the brain structure~\cite{Kanai11, Zamboni09, Oxley08}. If outsiders increase in a manner perceived as controlled by insiders, the prefrontal cortex, a part of human brain responsible for decision making and for moderating social behavior, acclimates to the new circumstances. If, however, insiders start perceiving outsiders as invaders, the prefrontal cortex is overcome by the amygdala which in turn induces a fighting reaction and thus suppresses tolerance.
The model development so far, via assumptions (i) and (ii), accounts for processes that affect an individual insider's opinion. The rise of RW populism, however, would be most explosive if insiders could affect each other's opinions. Influence by peers is a well-documented phenomenon in human interactions, further catalyzed by the popularity of social media. Accordingly, RW populism has the potential to spread in a highly non-linear way, much like a virulent contagious disease. We include such a non-linear collective spreading mechanism in the third (and last) model assumption.
\paragraph{Assumption (iii): mutual insider contagion.} At any given moment $t$, an insider agent $i$ with $K_i$ connections to other insiders, turns to RW populism at rate $p''$ if at time $t-1$ this agent had at least $M_i=K_i/2$ RW populist supporters in the immediate neighborhood. Note that for simplicity, factor 1/2 plays a role analogous to the tolerance parameter in assumption (ii). The essence here is that, due to connections between insider agents, once RW populism emerges somewhere in the network, the whole populist movement can spread like a contagion. Such a collective spreading is essential because an insider agent can become RW populist supporter even if there are no outsiders in the immediate neighborhood, thus potentially explaining why some EU countries with almost no immigration oppose to accept even a small group of immigrants. A similar effect may have been important for the results of the recent presidential election in the US wherein the winning candidate was more often than not ridiculed in mainstream media, which are represented in our model with assumption (i).
\begin{figure}[t]
\centering \includegraphics[scale=0.99]{Fig04.pdf}
\caption{\textbf{Non-linearity, a tipping point, and the rise of right-wing populism.} Using a network of $N=5\,000$ agents, each with an average of 15 connections, we examine the effect of a constant inflow of outsiders at rate $J=2$ at each time step. In this setup, the total number of outsiders at any moment in time is $I(t)=\Sigma_i J_i=\langle J\rangle t$. As the fraction of outsiders, $f_I=I/(N+I)$, approaches the tolerance parameter, $f_I'=0.15$, the presence of a tipping point causes the fraction of RW populist supporters to start increasing non-linearly and eventually undergo a sudden jump (i.e., a discontinuous change) at about 37\,years (450 months) into the simulation (black curve). The sudden jump happens much earlier if the inflow of outsiders experiences shocks at times $t_1$ and $t_2$ at which $J=200$ outsiders enter the network. In particular, as the network approaches the tipping point, the effect of exactly the same shock becomes disproportionately higher (red curve). In this case, however, the tolerance parameter is still kept constant. Finally, we also examine the case in which shocks at times $t_1$ and $t_2$ affect the tolerance parameter, where responsiveness is controlled by parameter $\gamma=0.0001$. Here, the second shock at $t_2$ is sufficient to instantly tip the network into RW populism (green curve). Other parameters are $p=0.007$, $\tau=15$, $p'=0.5$, $p''=0.5$, and $\alpha=0.001$.}
\label{f4}
\end{figure}
Assumption (iii) concludes model construction, thus letting us to refocus on the simulation results and the implications thereof. In Fig.~\ref{f4}, starting with a network of 5\,000 agents, we numerically show how the fraction of RW populist supporters increases with a constant inflow of outsiders, here $J=2$ per month. This inflow corresponds annually to almost 0.5\% of the total population, which is slightly more than the threshold value implied by Fig.~\ref{f3}(b). Simulations with $J=2$ per month are supposed to emulate the fast limit represented in extension (B) above. After approximately 37 years of fast globalization, the network reaches a tipping point and undergoes an abrupt shift to a mode dominated by RW populism. Stating that RW populism dominates implies more than 50\% of RW populist supporters in the network (i.e., $P>0.5$, where $P$ denotes the fraction of RW populists). The 50\% threshold is due to democracy as the majority rule principle.
In Fig.~\ref{f4}, we furthermore report how the simulated network under assumptions (i)-(iii) responds to shocks (red curve). At this stage, extension (B) is not yet allowed to operate. The constant annual inflow of outsiders, $J=2$, is supplemented with two events at times $t_1$ and $t_2$, at which $J=200$. The network's state, characterized by the proportion of RW populists ($P$), exhibits a much stronger response at $t_2$ than $t_1$, although the shock inflow ($J=200$) is the same. The reason for this stronger response is that in $t_2$ the system is closer to the tipping point and consequently more unstable than in $t_1$. Because in reality the value of $J$ may be biased as a consequence of erroneous estimation or some other form of information misinterpretation, the described results suggest that approaching the tipping point is concurrent with strengthening nonlinear effects such that even a small shock may trigger the final push to a mode dominated by RW populism. The situation is even more explosive (in terms of $P$) if extension (B) is allowed to operate, i.e., if tolerance parameter $f_I'$ changes with $J$.
\begin{figure}[t]
\centering \includegraphics[scale=0.99]{Fig05.pdf}
\caption{\textbf{Breakdown of the causes of right-wing populism}. Fig.~\ref{f4} shows that the probability of RW populism, $P$, suddenly increases as society approaches a tipping point, but remains silent on the underlying causes. Here we discern between the contributions of local outsider influence (assumption (ii)) and mutual insider contagion (assumption (iii)). Far from the tipping point, $P$ mainly responds to local outsider influence (ii). By contrast, as the network approaches its tipping point, mutual insider contagion (iii) takes over and accelerates the transition to RW populist dominance. Parameter values are $N=5\,000$ with an average degree of 15, $J=2$, $p=0.007$, $\tau=15$, $p'=0.7$, and $p''=0.8$.}
\label{f5}
\end{figure}
The third simulation in Fig.~\ref{f4} (green curve), illustrates the dynamics under assumptions (i)-(iii) with extension (B) operating. Accordingly, tolerance parameter $f_I$ changes with $J$ as prescribed by Eq.~(\ref{e3}). Due to decreasing tolerance, the second shock at $t_2$ is now sufficient to push the system beyond the tipping point and thus cause an even earlier dominance of RW populism than in previous two simulations.
From simulations in Fig.~\ref{f4} alone, it is unclear how much local influence of outsiders (assumption (ii)) contributes to the rise of RW populism relative to mutual insider contagion (assumption (iii)). We examine these contributions in Fig.~\ref{f5}. After the initial transients fade, it is the local influence of outsiders that drives the increase of RW populists in the network. The contribution of mutual insider contagion is relatively small until the system approaches a tipping point. Near the tipping point, contagion moves fast and overtakes the local outsider influence as the main contributor to the rise of RW populism. Thereafter, the RW populist movement can practically sustain itself without much support from the outside.
\begin{figure}[t]
\centering \includegraphics[scale=0.99]{Fig06.pdf}
\caption{\textbf{Predicting the timing of RW populism}. We find that for a broad range of outsider inflows ($J$) and tolerance parameter values ($f_I'$), Eq.~(\ref{e5}) predicts the moment at which $P>0.5$ in a manner that agrees favorably with the simulation results. Except for $\gamma = 0$, other parameters are the same as in Fig.~\ref{f4}.}
\label{f6}
\end{figure}
In the regime of moderate immigration inflows ($J'<J<J''$), we can learn about the dynamics of our complex network using an analytical technique known in physics as the mean-field theory (MFT). As long as the number of agent connections does not deviate wildly from the network average, probability $P$ that a randomly chosen insider agent $i$ is an RW populist supporter due to any of the processes underlying assumptions (i)-(iii) is given by
\begin{eqnarray}
\nonumber
P &=& p^* + p' p_1(k_i,m_i,f_I) + p'' p_1(K_i,M_i,P) \\
\nonumber
&-& p^* p' p_1(k_i,m_i,f_I) - p^* p'' p_1(K_i,M_i,P) \\
\nonumber
&-& p' p_1(k_i,m_i,f_I) p'' p_1(K_i,M_i,P),
\end{eqnarray}
where the last three terms avoid double counting in accordance with the probability theory formula $P(A\cup B\cup C)=P(A)+P(B)+P(C)-P(A) P(B)-P(A) P(C)-P(B) P(C)$ for three mutually independent events $A$, $B$, and $C$ that cannot occur simultanously. In the MFT approximation, we can drop index $i$ because no single agent is considerably different from the collective average. Previously we set $M=K/2$ for simplicity, but the peer pressure measured by the value of the proportionality factor between $M$ and $K$ may easily differ between countries or regions. Furthermore, parameters $p'$ and $p''$ are constants just in theory. The real social dynamics is such that these parameters may change in response to rumors, political manipulations, or outside shocks. If parameters $p'$ and $p''$ substantially increase, their effect is to increase the value of $P$ as well, thus further improving the prospects for the dominance of RW populism. In our framework, to reflect democracy as the majority rule principle, when $P$ approaches 0.5, the non-linear processes embedded in assumptions (ii) and (iii) lead to a sudden transition to RW populist mode.
\begin{figure}[t]
\centering \includegraphics[scale=0.99]{Fig07.pdf}
\caption{\textbf{Interconnected networks or why ``somebody else's problem'' easily turns into ``my problem''.} In (a) we show the case when there are no inter-links between networks. The tolerance parameters between the two networks differ, $f_{I,1} = 0.2$ and $f_{I,1} = 0.4$, while the inflows into both networks are the same, $J_1=J_2=2$. (b) More tolerant network is now exposed to a higher inflow, $J_2=4$, and a shock at $t_1=500$. The average number of connections for intra-connections (inter-connections) in both networks equals 15 (10). The other parameters are the same as in Fig.\ref{f4}.}
\label{f7}
\end{figure}
A theoretical model is more valuable if it possesses predictive power~\cite{Galam06}. We therefore demonstrate that a network of agents under assumptions (i)-(iii) leads to a simple formula for the timing at which RW populism starts to dominate. The formula in question is
\begin{equation}
t_{th}=\frac{N f_I'}{J (1-f_I')}-\frac{I(0)}{J}.
\label{e5}
\end{equation}
We obtain this result in three steps. First, if the immigration inflow is constant, then the number of outsiders in the network after $t$ time steps is $I(0)+J t$. Second, the total population size thus equals $N+I(0)+J t$. Finally, Eq.~(\ref{e5}) follows if the current fraction of outsiders $(I(0)+J t)/(N+I(0)+J t)$ is equated with the critical parameter $f_I'$. In Fig.~\ref{f6}, we show for a number of immigration inflow--tolerance parameter pairs, $(J,f_I')$, that the simulated timing of the shift to RW populist mode (i.e., $P>0.5$) favorably fits theoretical predictions. In conjunction with the empirical data on tolerance towards immigrants in the EU countries, the formula in Eq.~(\ref{e5}) could be used to provide an estimate of when a given country might be approaching a possible tipping point at which RW populism becomes dominant.
Numerical simulations allow us not only to look at one network in isolation, but also to examine interdependence between two or even more networks. Especially interesting is the potential for a cascade effect whereby an RW populist movement in one network affects the rise of RW populism in another network. Such a cascade effect is interesting in a globalized world because globalization makes countries more similar to one another. This similarity should be even more present in supranational organizations like the EU wherein borders between nation states are all but erased. To examine how interdependence affects the rise of RW populism, we set up two random ER networks equivalent from an economic viewpoint (equal $p$ in the model), but different with respect to tolerance towards outsiders (different $f_I'$ in the model). An addition to the model is that, besides the usual intra-connections within one network, agents have inter-connections with their counterparts from the other network. Except this addition, the same model assumptions hold as before.
Numerical simulations reveal several interesting effects of interconnectedness. For easier comparison, we first run simulations in which inter-connections are erased (i.e., we just have two independent networks; Fig.~\ref{f7}(a)). As expected under the same inflows of outsiders ($J=2$), the network with a higher tolerance parameter ($f_{I,1}'=0.4$) approaches the tipping point much later than the network with a smaller tolerance parameter ($f_{I,2}'=0.2$). When the networks are interconnected and the more tolerant one experiences a higher inflow ($J_2=4$) and a shock ($J=500$) at time $t_1=500$, not only this network becomes more prone to RW populism, but it also pulls the other network, causing the transition to RW populism to occur sooner than it otherwise would (Fig.~\ref{f7}(b)). In essence, no one wants to be the first to cross the line, but in an interconnected world many may wait to be the second.
\section{Discussion and Conclusion}~Why some countries (e.g., the ex-socialist EU countries) strongly oppose receiving almost any immigrants, while others (e.g., the USA) have mostly welcomed immigration throughout history is an important topic in social sciences and, more recently, a major issue for the EU. Arguably one of the reasons why immigration has been welcomed in the USA is a lack of a single dominant ethnicity, leading to a clear distinction between the country's identity and the origin of its citizens. The second reason is that the USA has accepted people from all of the world, thus securing that there is no one large group of immigrants sharing religion, language, and/or ethnicity which could serve as a catalyst to mobilize this group and threaten the established social order. The opposite situation is in France where many immigrants share the same language and religion, both of which differ from the language and religion practiced by the French majority. A large homogeneous group of immigrants may instill fear among the current majority. Fear leads to a volatile situation, often exacerbated by the inflow rate of immigrants exceeding the speed of their integration. Such a situation is commonly resolved in one of the following two ways. Either there is an uprising as exemplified by the Visigoth immigrants and their ex-Roman commander Alaric who plundered Rome in 410, or the majority reacts to suppress the inflow which is nowadays accomplished through the support for populist right-wing parties.
The concept of globalization was conceived with cooperation between nations in mind, specifically to allow capital and labor to move freely across borders. In reality, however, globalization is affected by a multitude of non-economic factors such as ethnicity, culture, religion, and other human traits. For example, according to Eurobarometer 65 published in 2006, the main concerns of European citizens were related to unemployment (49\%), crime (24\%), the economic situation (23\%), immigration (14\%), and terrorism (10\%). However, a survey in the UK in 2006 lists race and immigration as a top issue mentioned by 38\% of the respondents, which may explain why populism showed up first in the UK. With so many factors at play, globalization is an enormously complex process and it should not come as a surprise if some of the non-economic factors fail to align with the purely economic ones. Such a misalignment is likely to lead to frustrations that feed populist movements around the world. By opposing collaboration in the spirit of globalization, populist movements may be creating a positive feedback mechanism whereby the general population becomes more ideologically rigid which, in turn, spurs populism even further~\cite{Jusup14}. A strengthened populist movement may even trigger tectonic shifts in world affairs as exemplified by BREXIT in the UK and the recent 2016 US presidential election.
What is sometimes forgotten is that in a globalized world problems rarely strike in one place only. Interdependence makes the developed countries more alike each other, thus synchronizing their social dynamics. This synchronization may help spill over political shifts in one country to other countries, eventually causing the rise of RW populism across large regions of the world. After BREXIT and the 2016 US presidential election, political elites should not expect to continue business as usual because being the first to undergo a major political shift (with almost certain unpleasant economic consequences) is difficult, but once someone has crossed that line in an interdependent world, a cascade (domino) effect may ensue---no one wants to be the first, but many wait to be the second.
A tipping point for the rise of RW populism may be easier to reach when voters face a binary choice. For example, such was the choice in the UK during the BREXIT referendum and, for all practical purposes, in the recent US presidential elections. In both cases the populist option secured a narrow victory. We also mentioned that in Austria, the RW populist party is currently receiving 34\% of votes in the polls, but its candidate in the presidential race is close to 51\%. A simple way to understand these percentages is to assume that political attitudes of voters are almost symmetrically distributed across the political spectrum from left to right. Consequently, if leftist voters comprise $\psi_L\%$ of the total population, then RW populist voters should maintain a similar presence, i.e., $\psi_R\%\approx\psi_L\%$. Facing only a binary choice, the remaining centrist voters are left without a clear representative of their views. These centrist voters are therefore likely to vote evenly between the options that are available. An implication is that around $\psi_L\%\approx\psi_R\%\approx33\%$, even a slight (statistically significant) imbalance in favor of $\psi_R\%$ over $\psi_L\%$ may tip a society towards RW populism. In Austria $\psi_R\%\approx34\%$ seems to be enough to promote the RW populist candidate into a front runner of the presidential race. In this context, polls from August 2016 indicate that in France the RW populist candidate for president has the support of about 28\% of voters. This percentage is very close to highlighted $\psi_R\%\approx33\%$. If the RW populist candidate managed to increase his support to $\psi_R\%\approx33\%$ before the first round of elections, he would be all but guaranteed to receive close to 50\% in the second round, illustrating the tipping point at work.
The final outcome in a ``battle'' between conflicting factors surrounding globalization will almost surely have tremendous economic implications. If a country approaches its tipping point, how the ensuing volatility will affect this country's credit rating in the long run? In the case of a domino effect, either in the entire EU or its large part, what will be the consequences for the Euro and the common banking system? Without a proper resolution of the migrant crisis, what will be the impact on systemic risk? In an attempt to shed light on some of the factors and underlying processes that affect the success of globalization, we offer an empirically motivated theoretical model of when to expect the rise of RW populism in response to unsustainable immigration inflows.
It is an ongoing debate under what conditions globalization, democracy, and national sovereignty are able to coexist~\cite{Dani00, Dani16}. In our model, the coexistence of this trilemma set is predicated on such controlled globalization that a delicate balance is struck between immigration inflows and the ability of a society to integrate immigrants. This ability is arguably improved if immigrants mix more uniformly with the native population, which is a principle practiced in Singapore where immigrant hubs are discouraged and tenants in government-built housing (comprising 88\% of all housing) must
be of mixed ethnic origin~\cite{BP16}. Because tolerance towards immigrants is conditional, if immigration inflows overshadow integration rates, then democracy might push the society into RW populism as an intolerant mode of functioning. The rise of RW populism is possible because elections are stochastic in their very nature, implying that the society resembles a mathematical random walker. Left to its own devices, a random walker will eventually hit an absorbing barrier. This barrier, of course, is used here as a metaphor for the demise of globalization at the very hands of a progressive system (i.e., democracy) that made globalization possible in the first place.
\vspace{3mm}
\noindent We are grateful to S. Galam, D. Kovac, and T. Lipic for helpful suggestions. B.~P. received support from the University of Rijeka. B.~P. and H.~E.~S. received support from the Defense Threat Reduction Agency (DTRA), the Office of Naval Research (ONR), and the National Science Foundation (NSF) Grant CMMI 1125290. M.~J. was supported by the Japan Science and Technology Agency (JST) Program to Disseminate Tenure Tracking System.
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Der Neue Verband der Lohnsteuerhilfevereine e.V. (NVL) wurde 1993 als Interessenvertretung der Lohnsteuerhilfevereine gegründet.
Rund 130 Lohnsteuerhilfevereine sind deutschlandweit Mitglied im Dachverband. Die Mitgliedsvereine betreuen in rund 6000 Beratungsstellen mehr als 1,5 Millionen Arbeitnehmer.
Der Neue Verband der Lohnsteuerhilfevereine e.V. fusionierte zum 1. Januar 2017 mit dem Bundesverband der Lohnsteuerhilfevereine e.V. zum Bundesverband Lohnsteuerhilfevereine e.V. (BVL).
Siehe auch
Bundesverband der Lohnsteuerhilfevereine (BDL)
Bundesverband Lohnsteuerhilfevereine (BVL)
Einzelnachweise
Weblinks
www.nvl.de – Offizielle Webseite des NVL
Dachverband (Deutschland)
Steuerrecht (Deutschland)
Gegründet 1993
Aufgelöst 2016
|
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MYRNA LORRIE Vinyl Records and CDs
Canadian country singer and songwriter, born on August 6, 1940 in Fort William, Ontario, Canada. Lorrie first sang publicly at age 11 on Fort William radio station CKPR on a program called School of the Air, which was hosted by Jack Master. At 12, Master's gave Lorrie her own radio show called Harmony Trails. At age 14 she wrote and recorded the song "Are You Mine" with Buddy DeVal, which was released on Abbott Records and reached number 6 on the Billboard Chart. The song became a hit in both Canada and the United States and was recorded by several artists; it peaked at No. 2 on the Billboard and Cashbox charts in early 1956. She was voted Best New Female Singer by fan polls in both Billboard and Cashbox magazines in 1955. On January 10, 1957 she had her first recording session for RCA, at the RCA Victor Studio 1, in New York. Her first producer was Steve Sholes who had signed Jim Reeves and The Browns in addition to Hank Snow, Eddy Arnold, Chet Atkins and Elvis Presley. Nine months later, on October 31, 1957, Myrna recorded at RCA Victor Studio on McGavock Street in Nashville, with Chester B. Atkins which included a song written by Myrna called "Tradewinds." Lorrie's first tour in the United States had her opening for country stars such as Hank Snow,... Read More
... Marty Robbins, Johnny Cash, Kitty Wells, and Sonny James. She appeared on the Grand Ole Opry broadcast as a guest of her idol Hank Snow. Later she toured with Faron Young, Skeeter Davis, Ferlin Huskey, and Porter Wagoner Shortly after, at the age of 23, she formed The Myrna Lorrie Show which successfully toured the Canadian country music circuit performing at fairs, small towns, and the Calgary Stampede until it disbanded in December of 1968. In 1970, Lorrie received a Juno, the Canadian recording industry award, for Top Country Singer, Female and in 1972 for Female Country Singer of the Year (Melhuish 196). She won a Big Country Award for outstanding performance by a female singer in 1977. On November 1st, 1989, Myrna Lorrie, along with her mentor, Hank Snow, was one of several inaugural inductees honoured by the Canadian Country Music Hall of Fame at the Centre in the Square Theatre in Kitchener, Ontario.
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Biopolitics
The concept of biopolitics has been one of the most important and widely used in recent years in disciplines across the humanities and social sciences. In Biopolitics, Mills provides a wide-ranging and insightful introduction to the field of biopolitical studies. The first part of the book provides a much-needed philosophical introduction to key theoretical approaches to the concept in contemporary usage. This includes discussions of the work of Michel Foucault, Giorgio Agamben, Hannah Arendt, Roberto Esposito, and Antonio Negri. In the second part of the book, Mills discusses various topics across the categories of politics, life, and subjectivity. These include questions of sovereignty and governmentality, violence, rights, technology, reproduction, race and sexual difference.
This book will be an indispensable guide for those wishing to gain an understanding of the central theories and issues in biopolitical studies. For those already working with the concept of biopolitics, it provides challenging and provocative insights and argues for a ground-breaking reorientation of the field.
Catherine Mills is Associate Professor of Bioethics and an Australian Research Council Future Fellow at Monash University, Australia.
Biopolitics
Catherine Mills
First published 2018
by Routledge
2 Park Square, Milton Park, Abingdon, Oxon OX14 4RN
and by Routledge
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Routledge is an imprint of the Taylor & Francis Group, an informa business
© 2018 Catherine Mills
The right of Catherine Mills to be identified as author of this work has been asserted by her in accordance with sections 77 and 78 of the Copyright, Designs and Patents Act 1988.
All rights reserved. No part of this book may be reprinted or reproduced or utilised in any form or by any electronic, mechanical, or other means, now known or hereafter invented, including photocopying and recording, or in any information storage or retrieval system, without permission in writing from the publishers.
Trademark notice: Product or corporate names may be trademarks or registered trademarks, and are used only for identification and explanation without intent to infringe.
British Library Cataloguing-in-Publication Data
A catalogue record for this book is available from the British Library
Library of Congress Cataloging-in-Publication Data
Names: Mills, Catherine, 1972- author.
Title: Biopolitics / Catherine Mills.
Description: First edition. | Abingdon, Oxon; New York, NY: Routledge, 2018. | Includes bibliographical references and index.
Identifiers: LCCN 2017020758 (print) | LCCN 2017037629 (ebook) | ISBN 9780203732588 (E-book) | ISBN 9781844656042 (hardback: alk. paper) | ISBN 9781844656059 (pbk.: alk. paper) | ISBN 9780203732588 (ebook)
Subjects: LCSH: Biopolitics-Philosophy.
Classification: LCC JA80 (ebook) | LCC JA80 .M55 2018 (print) |
DDC 320.01-dc23
LC record available at <https://lccn.loc.gov/2017020758>
ISBN: 978-1-844-65604-2 (hbk)
ISBN: 978-1-844-65605-9 (pbk)
ISBN: 978-0-203-73258-8 (ebk)
Typeset in Sabon
by Deanta Global Publishing Services, Chennai, India
Contents
Acknowledgements
Introduction
PART I
1A new regime of power: Foucault
2Biopolitics as thanatopolitics: Agamben
3Totalitarianism and the political animal: Arendt
4Affirmative biopolitics: Negri and Esposito
PART II
5Politics: Sovereignty, violence, rights
6Life: Biology, technology, reproduction
7Subjectivity: Persons, race, gender
Concluding remarks
Index
Acknowledgements
As I have been reading, thinking and writing about biopolitics for my entire academic career, too many people have contributed – directly or obliquely – to this project to thank them all by name. That said, I do wish to acknowledge my professional and intellectual debt to several people who have been especially supportive of or inspiring for me over the years. These are: Robert Bernasconi, Judith Butler, Penelope Deutscher, Rosalyn Diprose, Fiona Jenkins, Paul Patton, Catherine Waldby and Elizabeth Wilson. In addition, I have particularly enjoyed and benefited from conversations with, and the written work of, Melinda Cooper, Mick Dillon, Thomas Lemke, Chris Mayes, Daniel McLoughlin, Sergei Prozorov, Miguel Vatter and Jess Whyte. I am particularly indebted to Niamh Stephenson for her generosity and intellectual openness. My partner, Rob Sparrow, read the entire first draft – a true labour of love in which there was little reward or pleasure for him. I am grateful for his critical comments; I am also grateful for his support and encouragement in (academic) life more generally. I wish to dedicate this book to my daughters, who interrupted it with their births – and then every day henceforth. Because of them, this book is a testament to life rearranged.
Chapters 6 and incorporate material previously published in the following places: Mills, C. (2017). Biopolitics and Human Reproduction. The Routledge Handbook of Biopolitics, Eds. Prozorov, S. and Rentea, S. London, Routledge: 281–94. Mills, C. (2016). Biopolitics and the Concept of Life. Biopower: Foucault and Beyond. Eds. Morar, N. and Cisney, V. Chicago, University of Chicago Press: 82–101. Mills, C. Review of Agamben and Colonialism. Eds. Svirsky, M. and Bignall, S. Edinburgh: Edinburgh University Press. in Critical Philosophy of Race, 4:1, pp.139–142.
Finally, the completion of this book has been made possible by the grant of an Australian Research Council Future Fellowship (FT120100026).
Introduction
Since Michel Foucault's crucial articulation in the 1970s, and the subsequent publication of Giorgio Agamben's Homo Sacer (1998) in English, biopolitics has become indispensable as a theoretical point of reference in disciplines across the humanities and social sciences. Now, it seems that few fields are immune to the biopolitical. Further, derivations of the term, such as biolegalities, biocultures, biosociality and biocapital, are instigating and defining new fields of scholarship. This analytic fecundity no doubt arises at least partly from the fact that the concept seems to capture crucial aspects of the workings of the modern – and, perhaps, pre-modern – world. It is also enhanced by the fact that there are multiple theoretical accounts of biopolitics: the term itself may entail different theoretical commitments, different interpretive emphases and different problems to be resolved or overcome at the levels of philosophy and political activism. This allows for a certain analytic flexibility while also encouraging scholarship that attempts to resolve conflicts in favour of one view or another, or that attempts to redress and rectify the blind spots and shortcomings of the various and competing dominant theoretical approaches.
However, while undoubtedly productive, the dissemination of biopolitics and its cognate terms has also led to a situation of considerable conceptual confusion. With so many different inflections of the term in currency, it is difficult to know just what 'biopolitics' actually refers to, what it means, and, indeed, just what kind of concept it is. To get a clearer sense of this, it is possible to broadly categorize the different ways of understanding and using the concept of biopolitics in contemporary literature as either empirical-descriptive or critical-normative in their basic orientation. These broad categorizations are not strictly mutually exclusive; rather, they inter-relate in various ways, in, for instance, the way that an empirical-descriptive account of biopolitics might also lead to or suggest some critical-normative conclusions or commitments. Even so, identifying these different inflections helps to articulate the theoretical commitments of various contributors to the contemporary debates, and, at times, provides leverage for a critical assessment of their contribution. Further, within each broad categorization, there are also several variant uses of the term.
The empirical-descriptive inflection of the concept rests on the claim that there is an identifiable historical phenomenon, or set of phenomena, that is well-described and analyzed with reference to the concept of biopolitics. While most, if not all, versions of biopolitics have some empirical resonance, the most well-known example of this approach is Michel Foucault's early formulation of biopolitics, and the related notion of biopower, in his work in the mid-1970s. Thus, in Will to Knowledge (1990) – the first volume of the History of Sexuality series – and contemporaneous lectures at the Collège de France, Foucault posits that biopower emerged as a particular rationality of power in the eighteenth century. This rationality of power is distinguished from sovereign power in various ways, but in particular, it places the new political subject of the population at the centre of governmental calculations, and one of its key problematics is the fostering of life through the political conjunction of the individual and the population. In this and other aspects, Foucault claimed that biopower was deeply integrated with the governmental rationality of liberalism. In this view, biopolitics or biopower is something that happened, and is perhaps still happening, and its machinations and effects can be traced through detailed historical, social and political studies.
While most frequently treated as an empirical-descriptive concept, there are nevertheless variations within this approach in the way that the concept of biopolitics is applied and developed. One of the key axes of differentiation within the biopolitics debate is in fact that of historical duration – while Foucault seemed to see biopower as a particularly modern political formation, others have traced its origin back further. For example, Giorgio Agamben, who is perhaps the most well known of contemporary theorists of the biopolitical, has argued that far from being a modern phenomenon, a biopolitical rationality informs Western politics from its inception. Agamben's founding source for this claim is Aristotle, and particularly his discussion of the emergence of the polis from the household in his Politics (1998). Agamben argues that the central biopolitical relation is that established within life itself by the distinction made between political life and natural life – not the relation between the individual and the population as Foucault had posited. Interestingly, Mika Ojakangas (2016) has recently argued that the biopolitical dynamic of population-individual can itself be found in Aristotle and Plato, thereby extending Foucault's account back to the Ancient Greek polis. However, the problem of historical duration is not the only one that differentiates empirical approaches to biopolitics; another is that of scale.
The question here is what is appropriately described as biopolitical – that is, are institutions, or societies, or epochs rightly considered biopolitical? For some, the answer is that the arrangement of political institutions and the logics of the state – and these alone – are most properly biopolitical. For others, non-state institutions may also operate according to a biopolitical logic of fostering life or disallowing it, or of dividing life from itself or rendering it as 'bare life'. Thus, arguably, non-state institutions, such as hospitals and medical interventions, may instantiate biopolitical rationality as much as more obviously state-based mechanisms, such as immigration and border control, for example. Drilling down further, it might be argued that while some public health interventions, for instance, are informed by biopolitical rationality, others are not. The problem also goes in the opposite direction, in that rather than claiming that particular institutions are biopolitical, it might also be argued that whole societies or political systems are biopolitical. For instance, while Foucault saw a particularly deep connection between biopower and liberalism, other theorists of biopolitics such as Agamben and Roberto Esposito treat German Nazism – with its reliance on race as a political device – as the apotheosis or paradigm of biopolitics, but one which also infiltrates liberal democracy.
These questions of scale are significant if we are to clarify the referent of the term 'biopolitics'. However, they are often obscured in the work of contemporary theorists of biopolitics, for whom the focus rarely falls on specific instantiations of biopolitics, but on how to describe a more general political logic or philosophy that is said to inform social and political organization at a deeper level. Theorists such as Agamben and Esposito are concerned to elaborate not the ontic instantiations of biopolitics so much as the ontological foundations of Western politics, which they construe in a biopolitical register. As noted above, Agamben sees the fundamental division of life from itself and its qualification in the constitution of the political as a foundational logic of the political. While the specific manifestations of this logic may be different in modernity than in ancient Greece, his thesis is that they are nevertheless underpinned by the same political ontology of the division and qualification of life. While Esposito is sympathetic to Foucault's sense that biopolitics is a specifically modern phenomenon, like Agamben, he nevertheless wants to articulate an underlying logic of immunization that is manifest across various domains of political and social life. Again, the concern is not to show how a logic of immunization is realized in any particular domain, but to show that it constitutes a basic or foundational political rationality for the modern West.
In addition to these accounts of what biopolitics is, and how it manifests in different domains in social and political systems, the notion of biopolitics has a critical and normative function that is often taken for granted and not clearly recognized as such. Foucault's account of biopower allowed for both a positive and negative valence in his dictum that this new regime of power operated to both 'foster life or disallow it' (Foucault 1990, 138). Furthermore, the genealogical approach to the historical actuality of biopower that his work has given rise to allows for the possibility that not all manifestations of biopower are bad – biopower is not necessarily or inherently evil. The more ontological formulations of biopolitics do not lend themselves to this critical differentiation. Instead, these approaches tend to foreground the negative dimension of biopolitics; indeed, for Agamben, it is appropriate to say that biopolitics is better understood as thanatopolitics, or a politics of death. Consequently, the designation of a phenomenon as biopolitical in itself acts as a form of critique insofar as it evaluates a phenomenon as negative, as something to be resisted, transformed or overcome.
This evaluation of biopolitics then lends itself to a normative orientation that requires the formulation of a positive alternative to the current negative manifestation of biopolitics. This project is exemplified in the recent turn in biopolitical studies toward the development of an 'affirmative' or positive biopolitics. While Agamben sees biopolitics as inherently death-driven, and urges an overcoming of biopolitics into what he calls 'form-of-life', other theorists such as Esposito and Antonio Negri argue for fostering a positive form of biopolitics, one that does not rest on the oppressive and exclusionary presumptions of the modern West. While their theorizations of an affirmative biopolitics are far from action-guiding, they can be identified as normative insofar as they rest on claims about questions of value and the way the world ought to be. Interestingly, one of the key themes to emerge from this turn to an affirmative biopolitics in recent years is the idea that what is required in order to bring about a new way of living that is not captured, oppressed or constituted biopolitically is a new conception of life. However, this points to one of the fundamental problems in the field of biopolitical studies, and this is just what 'life' means.
To get a sense of the difficulty with the concept of life, it is notable that the Oxford English Dictionary lists 13 different meanings for the term when used as a noun, all with several different inflections and modifications. Not all of these have resonance within biopolitical studies, but many do. We can broadly characterize the different ways in which the term 'life' is used in biopolitical studies as: (1) attributive, in the sense that something is alive or animate rather than dead or inanimate – i.e. that life is an attribute of a thing; (2) possessive, insofar as life is something of which one is deprived by death; (3) collective, in the sense that a set of things is said to be 'life' insofar as they are animated by life – as in, the set of things that constitute 'life on Earth'; (4) vitalistic, in that the term 'life' indicates the source of living or vitality, as an animating or vivifying principle; (5) biological, insofar as life and the living are the objects of biological science; (6) existential, in the way that the term indicates the individual and collective conditions of existence or experience, such as 'life was very different in the nineteenth century'; (7) temporal, insofar as a life is constituted by the period between conception – or birth – and death; (8) biographical, in that a life can be told as a narrative; (9) subjective, in the sense of different manners of living or ways of life. While all these senses of the term can be found in contemporary biopolitical literature, several of them are of especial significance; these are the attributive, the vitalistic, the biological, the subjective, and, to a lesser extent, the biographical senses.
These senses of the term can be seen in the ways in which major theorists of biopolitics use it, or cognates of it. For instance, Giorgio Agamben's appropriation of the distinction between bios and zoē from ancient Greek sources such as Aristotle can be seen as a distinction between the subjective sense of a way of life on the one hand, and on the other, as mobilizing an attributive sense in the notion of zoē, though at times Agamben also uses this latter term in a more vitalistic sense. Foucault also seems to use both an attributive sense and a vitalistic sense of life at times, though this is also cross-cut with a biologistic sense. Further, in later work on the ethics of the self, he primarily uses life in the sense of a way of life. The biologistic sense of life as the object of study of the discipline of biology and the biological sciences is of primary significance for contemporary scholars of biopolitics such as Nikolas Rose and Paul Rabinow, who are concerned with the constitution of particular ways of thinking about life in contemporary biosciences – such as molecular biology and genetics. Both also recognize the ways in which this biologistic conception of life is (perhaps increasingly) integrated with subjective conceptions of life, indicated in the notions of 'biological citizenship' and 'biosociality' that each proposes. Finally, Hannah Arendt, who I read here as both an innovative contributor to thinking about biopolitics, despite the fact that she did not use the term, and a profound point of reference for later theorists – tends to draw on the subjective and biographical senses of life. As this indicates, then, life means many different things; while this has undoubtedly been productive, it can also lead to significant conceptual confusion, and I believe we do well to keep in mind the different valences of the term when trying to understand and assess the contributions of various commentators in debates on biopolitics.
It is also worth clarifying here another point of terminological confusion. As we shall see in more detail in the next chapter, Foucault actually introduced two terms in Will to Knowledge to describe the new regime of power focused on life. The more general of these was 'biopower', which he argued encompassed both a disciplinary power focused on the individual body, and 'biopolitics', which focused on the new political subject of the population. Thus, biopolitics was for him a more specific term that referenced the emergence and development of a governmental rationality focused on the vital phenomena of the population and the correlative techniques used to manage them. However, in contemporary debates the term 'biopower' has largely been abandoned, and 'biopolitics' is taken as a more general term that means both the state and non-state (or quasi-state) management of life. Throughout this book, for the sake of simplicity, I follow contemporary usage; I typically use biopolitics, except when specifically referring to Foucault's comments on 'biopower'. Finally, also for clarity and simplicity, I use the term 'biopolitical studies' to refer broadly to the body of literature that has arisen around the concept of biopolitics. This has the danger of implying a greater unity to this body of literature than exists in reality, since the concept crosses disciplines and methods and is not always used to mean the same thing. However, I found this necessary in order to distinguish between biopolitics as a phenomenon in the world, and the body of literature that claims to illuminate that phenomenon.
At this point, I would like to make several comments about the aims and structure of the book. First, I do not provide a history of the term 'biopolitics' here; for that, I recommend Thomas Lemke's (2011) introduction to biopolitics, in which he offers a catalogue of different variations of usage, both preceding and following Foucault. To give an overview of this work, Lemke argues that the Swedish political scientist Rudolf Kjellen first formulated the term in 1924, to encapsulate organicist conceptions of the state. In this usage, the term emerged against the backdrop of earlier philosophical interest in lebensphilosophie, especially that of Schopenhauer, Nietzsche and Bergson in Germany and France, which adopted the concept of life as a 'fundamental category and normative criterion of the healthy, the good, and the true' (Lemke 2011, 9). Further, Lemke traces the development of several different ways of understanding biopolitics beyond its point of emergence. One of these is the European National Socialist inflection, in which biological characteristics were seen as the origin or cause of social inequalities, as well as their justification, and in which race played a particularly significant role. Another was the biopoliticians of North American political science, who saw biological factors as the cause or origin of political behaviour and thus explained political action in evolutionary and sociobiological terms (Lemke 2011, 19). In the 1960s and 1970s, the term 'biopolitics' acquired different accents again, when it became connected to concerns about environmentalism and new technological capacities such as genetics. However, according to Lemke, in using the term to indicate an historical rupture or break in modern political rationality, Foucault broke significantly with prior formulations of biopolitics (Lemke 2011, 33). In his formulation, biology was no longer the cause of political behaviour, but was diagnosed as the object and target of political power in the modern era. It is this critical vein of biopolitical theory as it emerged in the work of Foucault and other theorists that I am concerned with in this book.
What I try to do, then, is give an overview of the contemporary field of biopolitical studies, which entails introducing the main theoretical frameworks and approaches, as well as outlining some of the ways in which the concept has been put to work, and, in that, often developed in different ways. My hope is that after reading this book, you will have a broad (though not comprehensive) knowledge of the field of biopolitical studies that will provide a foundation for further investigation of particular philosophical problems, or empirical areas of study. Of course, the mobility of the concept, and its breadth of application across the social sciences and humanities, means that it is not possible to provide an exhaustive analysis of the significant themes or points of disputation in the field. Furthermore, my own interests and expertise naturally influence the topics I address throughout. This means that there are some topics worthy of attention that I have been unable to discuss in any depth, including the increasingly important themes of the Anthropocene, and animals. This notwithstanding, I have tried to stake out some of the main lines of analysis, and identify some areas where further conceptual development is required, as well as further empirical or historical research.
Finally, in regards to the purpose of this book, the style and form of it has largely been determined by the audience for whom I hope it will be most useful. First and foremost, it is written for readers who may be interested in the concept of biopolitics, but who may have little knowledge of the various competing theorizations and different approaches to issues raised in the literature. As such, much of the work here is a matter of providing clear explication of the main insights of important theorists and contributors. In this, in relation to Negri and Esposito, I have stayed reasonably close to the text, and the arguments made. This is because there is as yet little critical literature on their work that undertakes this kind of systematization of their reflections on biopolitics. In regards to Arendt, there is a considerable secondary literature, but very little that assesses her work from the perspective of its contribution to thinking about biopolitics. In regards to Foucault and Agamben, I have allowed myself freer rein, since there are already many other excellent sources that one could consult for a broad overview. In these chapters, then, I have tried to develop a particular interpretative point of view, as much as I have given an overview of their work on biopolitics. As this suggests, I also hope that there are enough novel critical points made throughout the book to satisfy the reader already more au fait with biopolitical studies.
This may be particularly so in the second part of the book, where I approach topics and problems in biopolitical studies from the perspective of a critical feminist bioethicist. This perspective means that I put emphasis on the biomedical and technological aspects of the biopolitical management of life, particularly through the integration of medical and biological knowledge with political reason. Moreover, it means that I am especially concerned with the ways in which axes of subjectivation such as sex and gender contribute to biopolitics. Throughout the book, then, I argue for the centrality of reproduction in modern biopolitics and make a case for a stronger theoretical focus on the generative female body and associated concepts, such as natality, in the development and operation of biopolitics. Without this, I suggest, the intersection of biopolitics and patriarchy remains undisclosed.
Structure
The book is organized in two main parts. The first part of the book addresses the main contemporary theoretical or philosophical approaches to the concepts of biopower and biopolitics. This provides the necessary background for understanding and engaging with theoretical debates in the field, as well as conceptual resources that are then taken up to investigate particular topics as they have been addressed in contemporary biopolitics literature. In this part of the book, I outline and discuss the theoretical interventions of Michel Foucault, Giorgio Agamben, Hannah Arendt, Michael Hardt and Antonio Negri, and, finally, Roberto Esposito. In the second part of the book, I turn to a thematic analysis of the literature, rather than a 'figure-based' interpretative analysis. I am interested here in topics which have already received some coverage in the debates, and which may indicate some of the main flashpoints of disagreement, but which also lend themselves to further discussion. Oftentimes, I claim that the existing literature on these topics is underdeveloped, or that the phenomenon needs to be thematized in more significant ways in order to bring out the import of it for biopolitical thinking. This part of the book is more significantly shaped by my own interests than the former, in which the choice of figures discussed is largely dictated by their standing in the field. I structure this part of the book around three themes that are unavoidable in thinking about biopolitics – that is, politics, life and subjectivity. The implicit drift of this structure is that subjectivity is the mode through which politics and life are brought into contact. To state the point more explicitly, I advance the view that subjectivity is, to paraphrase Agamben, the battleground of biopolitics.
In Chapter 1, I discuss the seminal work of Michel Foucault. Despite the fact that Foucault actually said relatively little about the concepts of biopower and biopolitics, his work is a touchstone in subsequent debates. In this chapter, I argue that we need to read across Foucault's work to fully appreciate his contribution to thinking about biopolitics today. This is not to enforce a kind of conceptual continuity upon Foucault, but to bring out the deep thematic interest that he had in the knowledge and power of life even before this was conceptualized under the term 'biopower'.
In Chapter 2, I turn to the revisionist work of Giorgio Agamben. While Agamben claims to be 'correcting and completing' Foucault's initial theorization, we ought to see his work as a substantial philosophical revision that draws heavily on the philosophy of Martin Heidegger, the theory of Walter Benjamin, as well as the political science of Hannah Arendt. My aim in this discussion is to bring out the novelty of Agamben's contribution, while putting pressure on what I see as some of the more problematic elements of it, particularly the central notion of bare life.
In Chapter 3, I make what might be a surprising turn to the work of Hannah Arendt. While Arendt is infrequently referenced as a theorist or analyst of biopolitics, and she herself does not use the term, I show that philosophers of biopolitics, especially Agamben, draw heavily on her work, and understanding the influence of it is important for assessing and understanding the novelty of these later contributions. More importantly, I argue that Arendt's analysis of totalitarianism and racism as a political technology – while highly contested – is important in its own right for understanding modern biopolitics, as is her philosophical focus on natality.
In Chapter 4, I take up attempts to develop an 'affirmative biopolitics' that try to go beyond the negative 'thanatopolitical' formulation of Agamben. In this chapter, I first discuss the work of Antonio Negri and Michael Hardt, whose formulations of Empire and the multitude stand as the clearest formulation of an affirmative biopower in opposition to the thanatopolitical power of biopolitics. Following this, I take up the more recent work of Roberto Esposito, which is becoming increasingly influential within biopolitical studies and beyond. I outline his main thesis about immunization as the central paradigm of biopolitics, and raise some criticisms against it.
Chapter 5 marks the turn toward the more thematic analysis in Part two of the book. In this chapter, I develop an analysis of the 'political' aspect of biopolitics, working across the different approaches discussed in the previous chapters. In particular, I discuss questions of sovereignty, violence and rights. In this, I discuss Foucault's lecture series presented around the time of the publication of Will to Knowledge, as well as Agamben's book, The Kingdom and the Glory (2011), to consider the contrasting views they develop on the phenomenon of government. I also touch briefly on questions of economy and capital, and the ways in which the imbrication of biopolitics and economy may be understood.
In Chapter 6, I return to the 'bio' part of the term 'biopolitics'. I explicate recent attempts to develop an alternative conception of life not beholden to the terms set by biopolitics, especially the notion of form of life elaborated by Agamben and to some extent by Esposito in their comments on a late essay by Gilles Deleuze. I also consider the problem of technology, and how it might relate to a conception of life anchored in biopolitics. In this section, I discuss the important contributions of Nikolas Rose and Paul Rabinow, whose work both together and separately provides a novel and influential way of addressing the nexus of power, knowledge and technology that constitutes life. Finally, I address the issue of reproduction; despite its relative neglect by the theorists discussed in Part one, I argue that reproduction ought to be understood as a central axis of biopolitics.
In Chapter 7, I directly address the question of subjectivity, and the ways in which life and politics may be mediated in the process of subjectification. After a brief discussion of the ways in which Foucault and Agamben conceive of subjectivity and its relation to biopolitics, I elaborate Esposito's arguments around the concept of the person, as well as introduce Judith Butler's work on humanization and precarity into the biopolitical frame. Following this, I focus more specifically on two axes of subjectification – gender and race. I argue that these axes of subjectification and the technologies of power in which they are imbricated have been insufficiently addressed in recent biopolitical studies. I conclude this chapter with some brief comments on disability.
References
Agamben, G. (1998). Homo Sacer: Sovereign Power and Bare Life. Trans. Heller-Roazen, D. Stanford, Stanford University Press.
Agamben, G. (2011). The Kingdom and the Glory: For a Theological Genealogy of Economy and Government. Trans. Chiesa, L. and Mandarini, M. Stanford, Stanford University Press.
Aristotle (1998). Politics. Trans. Reeve, C. D. C. Indianapolis, Hackett Publishing Company.
Foucault, M. (1990). The History of Sexuality: An Introduction, Volume 1. Trans. Hurley, R. New York, Vintage Books.
Lemke, T. (2011). Biopolitics: An Advanced Introduction. Trans. Trump, E. F. New York, New York University Press.
Ojakangas, M. (2016). On the Greek Origins of Biopolitics: A Reinterpretation of the History of Biopower. New York, Routledge.
Part I
1A new regime of power
Foucault
Michel Foucault is widely regarded as the most important figure within debates on biopolitics, since while he did not invent the terms 'biopower' or 'biopolitics', his work is a touchstone for contemporary debates on these political rationalities. Of central importance is his claim in The Will to Knowledge (Foucault 1990), that the emergence of 'life' as an object of politics at the end of the eighteenth century marked a definitive shift in political rationality. Perhaps surprisingly, though, given the influence of his work within biopolitical studies, Foucault himself spent little time directly discussing the concepts of biopower and biopolitics. This has been taken by some as an indication that these were not especially important to him as analytic tools, and played only minor roles in the development of his thought.
In contrast to this view, one of the guiding presumptions of this chapter is that, while only explicitly discussed at a few points, the concept of biopower is an important point of conjunction for a number of Foucault's concerns, from his early interest in medicine to his later concentration on ethical subjectivity. As Claire Blencowe (2012, 3) claims, '[t]he concern with "life" as a historically produced category and with the role of limits in the constitution of life and experience stretches across Foucault's oeuvre, from The Birth of the Clinic to The Care of the Self'. This is not to say that there is an overriding conceptual continuity in Foucault's work that inevitably leads to the notion of biopower, and subsequently, resistance to it. Rather, as Timothy Campbell and Adam Sitze (2013, 7) note, Foucault's thinking about biopower is replete with 'shifts, feints, changes in focus and direction', a comment that could easily be extended to Foucault's work as a whole. Further, the place that the notion of biopower occupies in Foucault's oeuvre is itself ambivalent in that it both draws upon earlier threads and turns them to new ends, ones which, it is probably fair to say, he never fully realized. However, at the level of 'problematology' (Osborne 2003), a greater continuity begins to appear, in which the notion of biopower can be seen as a particular refraction of a problem or question that Foucault addressed throughout his work, namely the problem of knowing ourselves (understood in a very particular sense). Thus, 'biopower' is not an anomaly in his work, but is fundamentally tied to both earlier and later concerns. To establish this claim, I provide an overview of Foucault's account of the genealogy of biopower, and its emergence as a particular rationality of power that 'fosters life or disallows it' in the modern era. Focusing on Will to Knowledge, but with reference to the more recently published texts of his Collège de France lectures especially Society Must Be Defended (2003b), in this chapter, I want to bring forth both the continuity and disruption that biopower entails for Foucault's thinking.
To this end, in this chapter I outline several aspects of Foucault's work, with a focus on the way that the notion of 'biopower' ties together a number of different strands in his thought. In particular, I trace connections between his concern with biopower and his earlier work on the history of medicine and the epistemic status of 'life', as well as with his later work on ethics of the self. Following a brief overview of Foucault's principal accounts of the emergence of biopower, I investigate his approach more closely along the three axes of knowledge, power and subjectivity. In regards to the first of these, in the second section of the chapter, I outline connections between biopower and Foucault's earlier work in the so-called 'archaeological' phase, on the episteme that underlies modern knowledge of life and medicine. In the third section of the chapter, I outline the key theoretical claims that Foucault makes in his genealogy of power; within this, I focus on the question of norms and their operation, since norms are not only central to biopower, but also help tie conceptual aspects of Foucault's oeuvre together. Finally, I consider the status of Foucault's later work on the ethics of the self in light of the apparently thwarted approach to sexuality and power initiated in Will to Knowledge.
Right of death and power over life
The most influential statement of what Foucault sought to capture by the term 'biopower' is the brief chapter in Will to Knowledge subtitled 'Right of Death and Power over Life'. Foucault's basic point in Will to Knowledge is that rather than discourse about sex and sexuality being repressed in the Victorian age, it was continuously produced and incited, requiring infinitesimal detail and giving rise to constant anxiety. He argues that sexuality is not a 'natural given' held in check by a power that operates through interdiction and rule, nor the secret and obscure domain of our selves that knowledge gradually discovers. Rather, sexuality is the name of a 'historical construct... a great surface network' (Foucault 1990, 105) that links the body and its pleasures to the operation of power and knowledge, in continual circuits of incitement, intensification, regulation and discursive elaboration. As an historical construct, sexuality is deployed not simply as a means of prohibition and control, but as a means of harnessing the forces of the body, both of the individual and the population. In this, the deployment of sexuality was fundamentally integrated with a shift in the rationality and operation of power, from sovereignty to biopower.
Foucault begins his account of the emergence of biopower in the final chapter of Will to Knowledge by contrasting it sharply with the sovereign right of death that characterized political power up until the nineteenth century. Sovereign power, he argues, operated deductively, as a 'subtraction mechanism', such that it was 'essentially a right of seizure: of things, time, bodies, and ultimately life itself; it culminated in the privilege to seize hold of life in order to suppress it' (Foucault 1990, 136). At the end of the Classical period, however, this form of power underwent a profound demotion, such that deduction was no longer the predominant form of power but merely 'one element among others' that collectively worked to 'incite, reinforce, control, monitor, optimize and organize the forces under it'. Further, the right of death of the sovereign underwent a correlative transfiguration to 'align itself with the exigencies of a life-administering power' (Foucault 1990, 136). This new 'life-administering power' incorporated death into its functioning, but in the process, transformed its political significance. Thus, Foucault writes, 'the ancient right to take life or let live was replaced by a power to foster life or disallow it to the point of death' (Foucault 1990, 138). While death does not disappear from the horizon of power's operation, then, its status is profoundly transformed from being the emblem and right of power, to a mere 'counterpart' of a power that administers and fosters life.
Foucault argues that this new life-administering power emerged in two basic forms, beginning from the late seventeenth century and extending through to the nineteenth. The first of these forms to emerge at the end of the seventeenth century was that of the disciplines, which treated the human body as a machine in order to optimize and control its capacities through the 'parallel increase of its usefulness and its docility' (Foucault 1990, 139). This was the form of power that Foucault analyzed in detail in Discipline and Punish. However, in the eighteenth century, he argues, this 'anatomo-politics of the human body' came to be complemented by the second form, a 'biopolitics of the population', which focused on the species-body and its biological characteristics of mortality, birth rates, morbidity, longevity et cetera in order to subject them to measurement and regulatory control. These two forms of power thus operate as the two poles of biopower, where one focuses on the body in order to individualize and manipulate the forces of it and the other is 'centred not upon the body but upon life', that is, in which 'bodies are replaced by general biological processes' (Foucault 2003b, 249). These two poles, Foucault insists, are tied together through a 'whole intermediary cluster of relations' at the level not of speculative discourse but of 'concrete arrangements that would go to make up the great technology of power in the nineteenth century' (Foucault 1990, 139, 140). The notion of bio-power thus combines the earlier work on disciplinary power in Discipline and Punish (Foucault 1977) with a new form of power that Foucault identifies as bio-politics.
One of the principle mechanisms that tied these two poles together was, in Foucault's view, the deployment of sexuality. Sexuality, Foucault argues, emerged in the nineteenth century as one of the most significant vectors of the new formation of power. He writes, '[i]t was at the pivot of the two axes along which developed the entire political technology of life', tied both to the intensification and subjugation of the forces of the individual body in discipline and applied to populations because of its consequences (Foucault 1990, 145). 'Sex was a means of access both to the life of the body and the life of the species' (Foucault 1990, 146). Foucault argues that while sovereign power had prioritized the blood relation as one of its fundamental values, the regime of biopower that emerged in the eighteenth and nineteenth century focused instead on sexuality. This shift of focus was associated with the emergence of a congeries of concepts such as heredity, progeny, degeneracy and perversion, within which sexuality was not a symbol of power, but its object and target. Again, Foucault is careful to note that the transition from a society of blood to one of sexuality was not a distinct rupture, but entailed a series of 'overlappings, interactions and echoes' (Foucault 1990, 149). Interestingly, one of the major axes of this overlapping between a society of blood and one of sexuality was race. However, Foucault's comments in Will to Knowledge are brief, especially on the issue of race, and at this point, it is helpful to turn to his more extensive discussion in the lecture series presented at the Collège de France in 1976, and published some years later under the title, Society Must Be Defended.
In the March 17 lecture in this course, Foucault identifies a number of key characteristics of biopolitics as the arm of biopower addressed to populations. He offers several contrasts between disciplinary power that operates at the level of the individual body, and a biopolitics that concerns itself with the new political subject of the population, or 'man-as-a-species'. Foucault argues that this new technology of power that addresses itself to 'man-as-species' is primarily concerned with and attempts to control the phenomena of the mass, that is, with characteristics such as birth rates, rates of mortality and morbidity and longevity across a population. The late eighteenth and early nineteenth century thus saw the emergence of natalist policies and concerns with birth control, culminating in eugenics, as well as a concern with what Foucault calls 'endemics' – not the sudden mass deaths caused by epidemics, but the ordinary and permanent factors that caused illness and weakened the population, cost money and wasted resources and energy. This gives rise to public hygiene and institutions to 'coordinate medical care, centralize power and normalize knowledge' (Foucault 2003b, 244). Further, biopolitics concerns itself with the accidents that befall and incapacitate individuals insofar as they are universal and ineradicable – thus we see the introduction of insurance, the rise of concern with public safety and an emphasis on collective and individual savings. Also central to biopolitics was a concern with the relation between humans as a species and their environment, or the milieux in which they live. This includes both a concern with natural environmental features and their impacts upon populations – the link between swamps and epidemics, for instance – and the rapidly expanding urban environments and their capacity to harbour disease and foster depravity. Several elements cut across these new arrangements and political concerns: the emergence of the new political subject of the population, the focus on aleatory events that while unpredictable at an individual level are constant at the collective level and over a period of time, and the establishment of regulatory controls or security mechanisms that 'have to be installed around the random element inherent in a population of living beings so as to optimize a state of life' (Foucault 2003b, 246).
Interestingly, in Society Must Be Defended, Foucault clearly relates the emergence of biopower to the inscription of mechanisms of race within the operation of the state. Indeed, here racism was posited by Foucault as fundamental to the operation of the state, such that 'the modern State can scarcely function without becoming involved with racism at some point, within certain limits and subject to certain conditions' (Foucault 2003b, 254). The introduction of the concept of race into the increasing knowledge of the human as species plays a significant role in that it is 'a way of fragmenting the field of the biological that power controls... a way of establishing a biological-type caesura within a population' (Foucault 2003b, 255). Moreover, this caesura or rupture of the biological field of the population allows for a particular configuration of life and death, whereby in order for one race to flourish and live, another must die. This is not exactly a warlike relation of confrontation, Foucault says, but a kind of biological relationship, whereby the health and strength – the purity – of one race demands the demise of the other. This involves, then, a shift in the configuration of the target of destruction: no longer an enemy, exactly, but a threat, 'either external or internal, to the population and for the population' (Foucault 2003b, 256). And since the improvement of the species – of one race thereof – requires the elimination of the biological threat presented by another race, this shift legitimizes the mobilization of death within a power that manages life: '[i]n a normalizing society, race or racism is the precondition that makes killing acceptable', where 'killing' is not limited to direct murder but also includes forms of 'indirect murder' such as increased risk of death or even 'political death' in the form of expulsion and rejection, for instance (Foucault 2003b, 256). Of course, Nazi eugenics presents one of the more obvious examples of this kind of state racism, but Foucault is also cognizant of the fact that as a mechanism of the State, racism was central to colonialism. Further, though Foucault does not mention this, the integration of the deployment of sexuality with the institutionalization of racism in biopower makes it apparent why miscegenation was of such intense political and social concern in the late nineteenth and early twentieth centuries.
Though this apparent integration of biopower and racism is downplayed in Will to Knowledge, this explicit contextualization is useful in that it brings to the fore two inter-related aspects of the operation of biopower; the first of these is eugenics and the second is warfare. As a particular way of thinking about humans as subject to evolutionary pressures that either led to the degeneration of certain races or their strengthening, eugenics emerged in the nineteenth century and reached its apotheosis in the early twentieth century. Conceptually, it drew strength from Darwinian theories of evolution applied to social circumstance; institutionally, it took the forms of state control of reproduction, for instance, through enforced sterilization, and in the Nazi state in particular, the decimation of populations deemed to weaken the 'Aryan' race (Kevles 1995). Central to eugenic thinking, then, was the concern with race, which drove attempted interventions in populations in order to eliminate biological threats, either directly through murder or through the elimination of certain characteristics from the pool of heritance. Within this, eugenics addressed itself both to populations through techniques such as statistics, and the individual body through measures such as sterilization. Thus, the individual body and the population necessarily referred to each other in the phenomena of birth and death, and, as such, eugenics provides a privileged example of the operation of biopower. Further, insofar as the population and individual were tied together through the management of sexuality, eugenics required the emergence of a particular set of ideas in the understanding of human biology – namely, those of heredity and reproduction. As I will discuss further in the next section, this highlights the centrality to biopower of the emergence of new episteme or 'style of reasoning' (Hacking 1992) about the human as species. Finally, as a particular expression of or formation within the operation of biopower, eugenics was also tied, for Foucault, to a specific understanding of warfare.
Inasmuch as racism was the condition for the mobilization of death within biopower, it was necessarily implicated in the forms that warfare took in the era of biopower. Foucault argues that with the mobilization of death through racism warfare was essentially about two things – first, eliminating the biological threat to a population posed by the 'enemy race', and second, the exposure of one's own race to the 'absolute and universal threat of death' in order to 'truly constitute itself as a superior race' (Foucault 2003b, 259, 260). Foucault's primary example of this logic of warfare is again the Nazi state, which he describes as the 'paroxysmal development of the new power mechanisms' (Foucault 2003b, 259). The first of these dimensions of warfare is obviously evinced by the Final Solution and, even prior to that, the elimination of sections of German society such as Gypsies, homosexuals, the mentally ill and people with disabilities. To give credence to the second dimension, Foucault points to the order given by Hitler to destroy the living condition of the German people in early 1945, issued in the infamous Telegraph 71. More generally, though, one might consider that in the Second World War, the populations of countries themselves became the target of war, not only in Europe but also in England and Japan. At least in some of these cases, this exposure of the population was seen as the necessary counterpart of the defeat through destruction of the enemy. Today, we can wonder if this is still the case, when the USA, for instance, is heavily invested in and uses drones in the various theatres of war in which it is currently engaged, which means that not even soldiers, let alone its own population, are exposed to death in warfare. Further, despite the large numbers of civilian deaths in Afghanistan and Iraq, for instance, civilians are not the intended target of warfare – rather, civilian deaths are considered 'collateral damage'. In short, war is no longer waged upon populations (See Dillon and Reid 2009).
At this point, then, a difficult question about the empirical purchase of the concepts of biopower and biopolitics emerges. This question is difficult because it is in fact two questions: first, does the concept of biopower as articulated by Foucault accurately reflect the operations of power today, in the early twenty-first century? And second, is this what the concept should do, or in other words, is its utility exhausted by its empirical accuracy, or does it have a critical function that is not limited to this? Several responses are thus possible to the question of the empirical grasp and continued relevance of Foucault's concept of biopower today. One such response has been to suggest that, as its empirical accuracy is limited and misses significant aspects of the operation of power today, it should be abandoned in favour of more accurate conceptualizations. Gilles Deleuze's (1990, 177–82) short essay on control societies, in which he argues that the confinement of disciplinary power has given way to mechanisms of control that rely on codes and the modulation of access is probably the best known of this kind of response. Another vein of response is to retain the concept, but acknowledge that the phenomena that it elucidates are different today than they were in the nineteenth and early twentieth centuries that Foucault focuses on. Thus, Melinda Cooper (2008, 9–10) has argued that Foucault's account of biopower needs to be revised in certain ways to take account of the forms of integration of neoliberalism and the life sciences in the wake of both the welfare state and classical neoliberal ideas of market equilibrium, forms of integration that drive the economy of biotechnology today. In the same vein, bioethicist Jonathan Moreno postulates that we have entered a new era of biopolitics, which is no longer about bodies and populations; rather, 'the new biopolitics has to do with control over the tissues, systems and information that are the basis and manifestation of life in its various forms' (Moreno 2011, 20). The third form of response, in which the critical value of the concept is not limited to its descriptive capacity, can be seen in Giorgio Agamben's paradigmatic approach to biopolitics, in which empirical points of reference elucidate the concept but do not in themselves ensure or exhaust its critical functionality. We will have occasions to return to these questions of the continued empirical relevance of the concepts of biopower and biopolitics throughout this book; for now, let me continue the investigation of Foucault's arguments about the emergence and development of this regime of power. To do this, I first want to return to Foucault's earlier work, to situate the emergence of a particular conception of life in modern biology, which arguably provides a condition of possibility for biopower.
Knowledge of life: The birth of modern biology
Foucault's early work is rarely discussed in contemporary biopolitical studies. There is certainly good reason for this. He did not use the concept at that time, nor was he especially concerned with power and forms of political rationality. Instead, he pursued complex works in what may be called historical epistemology, on the rise of modern knowledge, especially of the disciplines of the human sciences and of medicine. Central to this was the emergence of the apparatus of the medical clinic – itself inextricably linked to a certain way of knowing the human body as a medical object. Nevertheless, the concept of biopower as it emerges in his work during the 1970s is intimately related to this earlier work. For one, while the discussion in Will to Knowledge emphasizes the shifts in political power that the emergence of biopower entailed, the shifts in knowledge about life are equally important. Foucault (1990, 142) writes:
Western man was gradually learning what it meant to be a living species in a living world, to have a body, conditions of existence, probabilities of life, and individual and collective welfare... the fact of living was no longer an inaccessible substrate that only emerged from time to time, amid the randomness of death and its fatality; part of it passed into knowledge's field of control and power's sphere of intervention.
(Emphasis added)
As this suggests, the emergence of biopower was as much a matter of changing conceptions of life, of what it was to be human, as it was of changing rationalities of power. And, arguably, such changes in the knowledge of life were made possible through the epistemic shift that Foucault analyzed in The Order of Things (1994c).
In this book, Foucault sets out to trace the rules that made knowledge possible in the Classical and Modern epistemes, one significant strand of which was a transformation in the way that Western culture understood life. He argues that the nineteenth century saw a radical transformation of the way in which life was understood, involving a break from the taxonomical approach of the Classical era and the inauguration of a 'synthetic' approach instead. This involved a fundamental shift from ordering organisms according to their visible characteristics (called organs), to a new ordering based on function, wherein organs were subordinate to the four largely shared functions of respiration, digestion, circulation and locomotion. Foucault largely attributes the instigation of this shift to the work of French naturalist, Georges Cuvier. Foucault writes that in the Classical
deployment of the visible, life appeared as the effect of a patterning process – a mere classifying boundary. From Cuvier onward, it is life in its non-perceptible, purely functional aspect that provides the basis for the exterior possibility of a classification. The classification of living beings is no longer to be found in the great expanse of order; the possibility of classification now arises from the depths of life, from those elements most hidden from view.
(Foucault 1994c, 268)
The importance of this shift is that it provided the condition of possibility for biology as a new way of understanding living organisms, distinct from the natural history of the Classical age (Foucault 1994c, 269). Somewhat controversially, Foucault (1994c, 275) goes on to postulate that in introducing a radical break from the Classical ontological continuity of beings, Cuvier also gave rise to several of the conditions that made possible the theory of evolution, which is so central to modern biology – a point that is affirmed by François Jacob (1993, 12–13) in his history of heredity.
The method that Foucault deploys in The Order of Things also underpins his analysis of the emergence of modern medicine at the end of the eighteenth century in The Birth of the Clinic. In this book, Foucault traces the shift from nosological medicine to modern clinical medicine, arguing that it involves a transformation of the relationship between the 'visible and the invisible'. 'The clinic', he states, 'is both a new "carving up" of things and the principle of their verbalization in a form which we have become accustomed to recognizing as the language of a "positive science"' (Foucault 1994a, xviii). The role played by the individual patient is a central aspect of this reorganization. According to Foucault, pre-modern 'nosological' medicine abstracted from the patient in order to identify a disease in its essence, apart from the contingencies and distractions presented by the concrete body of the individual patient. In contrast, clinical medicine 'establishes the individual in his irreducible quality', such that it became possible to 'organize a rational language around it': 'one could at last hold a scientifically structured discourse about an individual' (Foucault 1994a, xiv). However, Birth of the Clinic does not only make evident the ways in which medical knowledge of the patient underwent a transformation. It also brings out the necessity and effect of institutional transformation, including the reorganization of the hospital system, the redefinition of the patient and the establishment of a relationship between public and state assistance and the maintenance of health.
While it would be overstating the case to say that Foucault's analysis of these transformations can be read directly into the conception of biopower that he develops later, there are points of connection. Indeed, he himself provides one in a brief discussion of eighteenth century 'noso-politics'. In this, Foucault argues that while all societies may well have practised some kind of noso-politics, the eighteenth century saw the development of new rules and more explicit practices, which cast 'the problem of the health of all as a priority for all, the state of health of a population as a general objective of policy' (Foucault, 1980, 168). Specifically, he identifies the transformation of the hospital and the embedding of clinical medicine – where the hospital performs both therapeutic and teaching functions – as important within the 'set of problems relating to urban space, the mass of the population with its biological characteristics, the close-knit family cell and the bodies of individuals' (Foucault 1980, 182). Further, he argues that the family came to be understood as the central target of 'medical acculturation' (Foucault, 1980, 173), and linked the concern with the well-being of the population and care of the individual. Arguably, we have here the idea of biopower without the concept.
While this points toward the historical institutionalization of biopower, within Foucault's own work a methodological shift from 'archaeology' to 'genealogy' was crucial to allow him to properly analyze the material apparatuses involved in the modern regime of power. As he conceived of it, archaeology as a method focused on statements and rules of discourse. Genealogy, in contrast, focuses on the 'effective formation' of discourses by non-discursive or material elements: it seeks to show the integration of knowledge with the operations of power in the formation of truth, which is thereby characterized by a fundamental historicity. It was only with the method of genealogy and the attendant development of a concept of power that it became possible for Foucault to articulate a notion of biopower. As I discuss further in the following section, Foucault eschewed the claim to be offering a theory of power, but nevertheless set out several general precepts for the 'analytics of power' that he favoured instead. One of these is the claim that power is fundamentally integrated with knowledge in discourse, where discursive elements are deployed strategically in ways that may either foster or hinder the exercise of power. As Foucault writes,
discourse can be both an instrument and an effect of power, but also a hindrance, a stumbling block, a point of resistance and a starting point for an opposing strategy. Discourse transmits and produces power; it reinforces it, but also undermines and exposes it, renders it fragile and makes it possible to thwart it.
(Foucault 1990, 101)
In the context of this integration – though not identification – of power and knowledge, Foucault's notion of dispositif or 'apparatus', becomes especially important for understanding biopower. It not only bridges between the 'methods' of archaeology and genealogy, but also brings out the strategic alliance between discursive and non-discursive elements in its operation. Foucault says of the notion that it picks out 'a thoroughly heterogeneous ensemble consisting of discourses, institutions, architectural forms, regulatory decisions, laws, administrative measures, scientific statements, philosophical, moral and philanthropic propositions – in short, the said as much as the unsaid' and the apparatus itself is the 'system of relations that can be established between these elements' (Foucault 1980, 194). Further, Foucault suggests such apparatuses arise as a response to an 'urgent need', and thus have a 'dominant strategic function' (Foucault 1980, 195). In regards to biopower, then, one might speak of the apparatus of clinical medicine, in which the transformation of the hospital system in France responded to the need of ensuring population health and utility. Similarly, one could speak of the dispositif of the family, the role of which in the project of population health and hygiene Foucault elaborates at various points (e.g. Foucault 1980, 173–5; Foucault 1990, 107–12).
One could no doubt multiply examples of the apparatuses in operation in biopower, but the more general point to be taken from the idea of the dispositif is that biopower fundamentally operates through knowledge. It was precisely through knowledge of a population, and the individuals that are intermixed in that population, that biopower was able to take hold of life in order to foster or disallow it. Hence the significance, then, of practices such as statistics, which generated unprecedented knowledge of the characteristics of the mass and subsequently allowed for the ordering of individuals within a given mass in relation to norms deduced from it. As Ian Hacking (2016) argues, the years from 1820 to 1840 initiated an 'avalanche of numbers' in the enthusiastic development of means for demographic data collection, record keeping and analysis. Across Europe, the United Kingdom and North America, around this time the collection of information about populations came to constitute the 'fortuitous diamond that is the hallmark of the modern state' (Hacking 2016, 74). Nevertheless, as Foucault indicated in the quote in the previous paragraph, knowledge in and of itself is not enough – what is important is the mobilization of discursive elements in particular strategies of power. It is to this mobilization that I now turn.
Genealogies of power: Sovereignty, biopower, governmentality
Foucault's works published during the early 1970s are often characterized as belonging to the 'genealogical' phase of his oeuvre, and it is during this phase that he most explicitly develops an approach to, or analytics of, power and its relation to knowledge. As this implies, in order to grasp the stakes of Foucault's understanding of biopower, we must first situate it within the broader understanding of power that he develops during the 1970s. This understanding of power itself undergoes a number of transformations. Initially, Foucault opposes the traditional understanding of political power in terms of sovereignty, and instead, adopts a 'quasi-military' vocabulary to construe power as an inherently relational, unceasing battle of forces. Subsequently, he moves away from the militaristic characterization while still maintaining the view that one must be basically nominalist in analyzing power – it is not a thing as such, but always a mobile set of relations that coagulate into particular modes of organization of the world. Within this broad frame or 'analytics' of power, he then goes on to identify several different configurations of power, namely, sovereignty, biopower (including both discipline and biopolitics), security and governmentality. One of the central problems in interpretation of Foucault's work on biopower is how to characterize the relations between these different rationalities of power – if, indeed, they are different rationalities. At different times, Foucault variously proffered conflicting views or remained silent on the matter. For instance, with regard to sovereignty and biopower, even within the short discussion in Will to Knowledge, he posits both that biopower does and does not replace sovereignty (Foucault 1990, 136, 138), and the relationship between governmentality and biopower remains implicit at best. In this section, I touch briefly on this problem, but take it up in more detail in Chapter 5. Here, as a necessary prelude to this later discussion, I look at what Foucault means by the term 'power'.
There is perhaps no other part of Foucault's work that better displays the mobility of his thought than the attention he gave to the concept of power, roughly between the years 1970 and 1982. In general terms, during this time, Foucault's thought went through several distinct moments of conceptual change, all of which are underwritten more or less explicitly by a rejection of a traditional conception of power in terms of sovereignty. In Will to Knowledge, Foucault characterizes the sovereign conception of power as 'juridico-discursive', and identifies several key claims that it entails about power. First, in the sovereign model, power is understood as essentially repressive, imposed upon subjects from 'above'; this view typically presupposes that the individual exists independently or prior to power and is then subject to its effects in the form of limitation, exclusion, rejection and so on. Second, it assumes that power is both totalizing and uniform, in the sense that it operates in the same manner across all aspects of the social field. Third, this model of power supposes that the principal tool or instrument of power's operation is the law, and the mark of power is interdiction and the distinction between licit and illicit. Foucault argues that despite the evident realities of how power operates in the West, this theory of power retains its hold on contemporary attempts to understand power, insofar as these attempts are captivated by problems of 'right and violence, law and illegality, freedom and will, and especially the state and sovereignty' (Foucault 1990, 89). As such, power is still understood on the basis of juridical monarchy, despite the fact that this form of power has been infiltrated by new mechanisms that are irreducible to law and sovereignty.
Against this backdrop, Foucault proposes what he calls an 'analytics' of power or a 'grid for historical decipherment' (Foucault 1990, 90) that offers several postulations about power that more or less directly contrast with the sovereign model. First, power is not negative and repressive but positive and productive. This means that rather than presupposing the existence of subjects with natural rights in a relationship of domination, one should question the means by which subjects are brought into existence as a counterpart of power. Foucault makes this point at various times and in different ways, but perhaps most bluntly in the second lecture of Society Must Be Defended, where he argues that it is:
a mistake to think of the individual as a sort of elementary nucleus, a primitive atom or some multiple inert matter to which power is applied, or which is struck by a power that subordinates or destroys individuals. In actual fact, one of the first effects of power is that it allows bodies, gestures, discourses, and desires to be identified and constituted as something individual. The individual is not, in other words, power's opposite number; the individual is one of power's first effects. The individual is in fact a power-effect, and at the same time, and to the extent that he is a power-effect, the individual is a relay: power passes through the individuals it has constituted.
(Foucault 2003b, 29–30)
As we will see in the following section, Foucault appears to temper this account of subject-formation in his later work, or at least allows room for a dimension of self-constitution. But here, the point is obvious: power does not repress subjects, it produces them.
Second, Foucault insists on a multiplicity of force relations and techniques at work in different fields of power's operation, which do not necessarily constitute a unity. As he argued, power is always relational, such that the term 'power' is only a 'name that one attributes to a complex strategical situation in a particular society' (Foucault 1990, 93), or shorthand for a network of relations (Foucault 1997a, 291) that do not emanate from a centre, nor are organized according to a single over-riding strategy. Rather, the ways in which power operates in different fields, such as the economic, the domestic, sexual and so on, may vary at a local level, and should be analyzed at that level. Further, relations of power are immanent to those fields, and points of resistance are an internal element of relations of power. This conception of a multiplicity of immanent relations of power that must be analyzed at a local tactical and strategic level should give us pause in relation to positing an overall coherence to biopower, and it is significant that Foucault only offers a minimal definition of biopower's coherence in terms of fostering life or disallowing it. This should not, I think, be understood as a paradigm of power's operation that will be manifest in all domains of biopower's operation, but as a looser principle of coherence that takes different forms in different arenas or fields, the management of which may require different apparatuses or techniques of power. It is, I think, against this backdrop that one should understand Foucault's apparent grappling with the various ways in which techniques of power are mobilized, rationalized and put to work.
Throughout the years in which he sought to analyze power, Foucault proposed numerous names for the various technologies of power at work in modernity, including discipline, biopower, biopolitics and governmentality. The insistence on different force relations and a multiplicity of techniques and apparatuses at work in the operation of power provides a rationale for this multiplication of names. This indicates that these terminologies are not competing in the sense that the positing of one denies prior articulations – for example, governmentality does not deny or entail the rejection of the existence of discipline. These different terminologies describe different technologies of power – but they are not necessarily indicative of either historical succession in regimes of power or of conceptual succession in Foucault's work. Of course, this in itself does not resolve the question of how each of these technologies of power interact with each other, but to a large extent, this is an empirical question that could only be addressed through detailed genealogies. Even given this, though, within this general picture of broad compatibility, there remain a number of theoretical or interpretive complexities, especially regarding the relationship between biopower and governmentality, which I discuss in more detail in Chapter 5.
Finally, the principal mechanism of power is not law, but the norm, and the mark of power is no longer interdiction but normalization: not licit and illicit, but normal and abnormal. In his account of biopower, Foucault gives a central role to norms and normalization as the principal form of social and political regulation, suggesting at one point that '[a] normalizing society is the historical outcome of a technology of power centred on life' (Foucault 1990, 144). Biopower operates simultaneously at the level of both individual bodies and the concurrently emergent political subject of the population; it is concerned both with anatomical singularity and the vagaries of the group. The notion of normalization that Foucault develops strives to capture both these dimensions. Normalization ties together the two poles of biopower, these being discipline and biopolitics, while norms provide the central axis through which the individual and the group are brought into relation with each other. In this, as a technique of biopower, normalization is irreducible to the institutions and force of the law, though fundamentally intertwined with them. According to Foucault (1990, 144), within a normalizing society legal apparatuses are 'increasingly incorporated into a continuum of institutions (medical, administrative and so on),' the function of which are 'for the most part regulatory', such that the mode by which the law operates is increasingly that of the norm. This clearly does not mean that law itself is superseded; rather, Foucault argues that as a regulatory apparatus, the law continues to operate within the regime of biopower, but in a different mode than previously. Norms effectively become the operative condition of law in biopower, since they allow the law to operate in conjunction with apparatuses such as medicine, which are themselves increasingly regulatory. In effect, the norm gives the law access to the body in an unprecedented way, that is, as a continuous regulatory force rather than as a repressive and constraining instrument of sovereignty.
However, just as Foucault appears to have struggled to formulate the relation between various technologies of power in a way with which he was satisfied, so he did with his account of normalization. Thus, across his discussions of normalization, from The Abnormal (2003a) through to at least Security, Territory and Population (2007) and including both Discipline and Punish and Will to Knowledge, he presented various accounts of the specific role and operation of normalization within biopower. For instance, in Discipline and Punish, he suggests that disciplinary techniques such as norms have 'swarmed' from their originating institutions, such as the prison, to take over the entire social field (Foucault 1977, 211). In Society Must Be Defended, though, he rejects this as a 'first and inadequate interpretation' and argues instead that 'the normalizing society is a society in which the norm of discipline and the norm of regulation intersect along an orthogonal articulation' (Foucault 2003b, 253). Later, Foucault further refines his understanding of normalization to suggest that it works in opposing ways in discipline and a biopolitics of population. In the former, infractions of the norm are produced as a consequence of the application of the norm understood as an idealization; this means that the phenomenal particularity of an individual is identified and calibrated through the application of an ideal. Consequently, as Foucault states, normalization produces individuals as the necessary mode and counterpart of the operation of norms, that is, as a material artefact of power (see Foucault 1977, 184). In a biopolitics of population, though, norms are mobilized in exactly the opposite way, insofar as 'the normal comes first and the norm is deduced from it' (Foucault 2007, 63). The biopolitics of populations, and the apparatuses of security that Foucault identifies as crucial to it, involves 'a plotting of the normal and the abnormal, of different curves of normality, and the operation of normalization consists in establishing an interplay between these different distributions of normality and in acting to bring the most unfavourable into line with the more favourable' (Foucault 2007, 63). This is a more strictly statistical concept of the norm, in which the norm is derived from empirical phenomena, though it may not be strictly identifiable with any particular empirical datum.
Subjectivity and freedom: Bios as a work of art
Given the centrality of norms in biopower, a question arises about the status of Foucault's subsequent work in The History of Sexuality series, where he turns away from the genealogy of biopower to a genealogy of the 'desiring subject', understood as a history of the ethical subject. Does this amount to a rejection of his earlier work, or is there another way of understanding this shift that highlights its essential continuity with the work on biopower? Two forms of the claim that Foucault did not, after all, find much of value in the concept of biopower can be identified. The first, made by critics such as Charles Taylor (1984), suggests that Foucault came to reject his earlier attempts to understand power as distinct from oppression, and retreated to more traditional conceptions of power, truth, freedom and the subject. The second version of the claim is more sympathetic to Foucault's overall approach, and suggests a more limited retraction. The idea here is that he came to recognize the confusions inherent in the notion of biopower, and dropped this as an unpromising avenue of research (Patton 2007, 206; Dean 2013). Both these claims posit a conceptual break between the projects of the first and second volumes of The History of Sexuality series. In doing so, though, neither pay sufficient attention to the points of conceptual continuity between Foucault's understanding of power/knowledge and the genealogy of biopower that it supports, and his later work on the autopoietic practice of ethical subjectivity.
The task of this section is to bring out this continuity between the projects of a genealogy of biopower and the genealogy of 'desiring man'. In this, it is worth noting that at no point does Foucault explicitly say that the concept of biopower is a theoretical dead end. To the contrary, in response to a query put by Paul Rabinow that, given his concerns with ethics, ought not Foucault be writing a genealogy of biopower, he responds, 'I have no time for that now, but it could be done. In fact, I have to do it.' (Foucault 1997b, 256). Further, as Georges Canguilhem (1997, 32) points out, a concern with norms and normalization almost necessarily lends itself to a consideration of ethics, insofar as both are concerned with matters of value and its determination. Thus, Foucault's later work can be seen not as a rejection of earlier efforts, but as a shift in focus in the analysis of the problematic of the relation between the subject and truth, a shift that nevertheless entails significant continuity with earlier concerns. Granted this continuity, though, there remains a further question to address, and this is the extent to which Foucault saw an ethics of the self as yielding or in some way constituting a form of resistance to modern biopower, as has sometimes been supposed. In the following, I suggest that while Foucault did see some transformative potential or value in an ethics of the self, his understanding of this aspect of subject-formation cannot straightforwardly be understood as outlining a form of resistance to biopower.
In the introductory chapter to the second volume of History of Sexuality, Foucault attempts to explain the analytic shift required in his 'history of desiring man'. While in earlier texts, he had been concerned with the production of subjects through techniques of power, Foucault claims that in the investigations he undertook for Use of Pleasure (1987), he became increasingly aware of the significance of a different type of technology involved in the process of subject-formation or subjectivation. Specifically, he became aware of 'the forms and modalities of the relation to self by which the individual constitutes and recognizes himself qua subject' (Foucault 1987, 6). He calls these 'technologies of the self', which he defines as:
techniques which permit individuals to effect, by their own means, a certain number of operations on their own bodies, on their own souls, on their own thoughts, on their own conduct... so as to transform themselves, modify themselves, and to attain a certain state of perfection, of happiness, of purity, of supernatural power and so on.
(Foucault 1993, 203)
In short, technologies of the self are the practices and means by which individuals subjectify themselves as ethical subjects, in making themselves subject to normative codes or aesthetic and ethical criteria that shape modes of being. Foucault thus initiates a genealogical analysis of the practices by which individuals bring themselves into relation with normative or moral codifications and values, and thereby constitute themselves as subjects of particular ethical codes, that is, as ethical subjects.
As this makes clear, the central dimension of technologies of the self is the relation that individuals establish with themselves. Foucault claims that the practices of ethical self-formation that he came to recognize through the focus on the various problematizations of sex in Antiquity were closely related to a technique or art of living, an aesthetics of existence. In classical Greece, he argues, 'sexual activity and sexual pleasures were problematized through practices of the self, bringing into play the criteria of an aesthetics of existence' (Foucault 1987, 12). Foucault describes the aesthetics or 'arts of existence' as 'those intentional and voluntary actions by which men not only set themselves rules of conduct, but also seek to transform themselves, to change themselves in their singular being, and to make their life into an oeuvre, that carries certain aesthetic values and meets certain stylistic criteria' (Foucault 1987, 10–11). In other words, an arts or aesthetics of existence involves establishing a particular relation to oneself through the adoption of certain ethical principles and associated practices. These allow and encourage individuals to act upon their bodies, souls, thoughts and conduct in order to transform themselves and attain a certain state of happiness, wisdom, purity, health or personal fulfilment and so on. This entails developing a certain reflexive relation to oneself, in which one constitutes oneself as a subject of one's own actions, through the selection of a certain action or form of being as the object of ethical concern and transformation through voluntarily applied aesthetico-ethical criteria.
The centrality of this self-reflexive relation has led to the critique that Foucault's ethics are overly focused on the self at the expense of the other. Fueling this critique, Foucault in fact states that concern for the self is 'ethically prior in that the relationship with oneself is ontologically prior' (Foucault 1997a, 287). Concern for the other is then a (contingent) effect of concern for the self. Foucault explains that for the ancient Greeks, the care of oneself produces an ethical subject who is then able to act correctly toward others, since self-rule moderates rule over others (Foucault 1987, 81). What underpins this correct acting toward another is the question of domination and liberty – the ancient Greek ethics of the self involves a self-reflexive relation to one's own freedom that makes of that freedom both a practical exercise and the object of ethical concern. In short, an ethics of the self requires the 'problematization of freedom' (Foucault 1997a, 286). By this, Foucault means that the ethos or aesthetics of existence is predicated on and directed toward the elaboration of one's own freedom. Concern for the other emerges as an epiphenomenon of this concern for one's own freedom, since acting tyrannically toward another is indicative of the slavishness of one's own person vis-à-vis one's desires and will. Thus, the relation that one has with others is not the object of ethical concern per se. It is an effect of and limit upon the care that one has for oneself, such that ruling another is not unethical in itself but modes of ruling might be indicative of a failure to care for oneself.
This explanation relies too heavily on the substance of ancient Greek ethics to operate as a defense against the philosophical critique that Foucault underestimates the ethical importance of the other. Nevertheless, it alerts us to several concerns. For one, it points toward the issue of community and the existence and nature of human collectivities that are not simply biopolitically defined populations and the kinds of relations that may obtain in those collectivities; this is discussed in more detail by theorists of biopolitics such as Agamben and Esposito, and I touch on it again briefly in the relevant chapters. Further, though, it prompts us to ask what Foucault actually wants from this understanding of self-formation as a matter of the ethical shaping of one's own being as a subject. On the one hand, his explanation of the analytic shift in his genealogy identifies this as a descriptive element of subject-formation, one that broadly complements his earlier analysis of the production of subjects by power. On the other, the ethics of the self also appear to have a kind of normative value, whereby Foucault sees in them an element of transformative potential. It is this more normative reading of an ethics of the self that leads to the view that the ethics of the self provide at least an outline of a possible form of resistance to biopower. This view is given credence through Foucault's comments in 'On the Genealogy of Ethics' and elsewhere that he is fascinated by 'the idea of the bios as a material for an aesthetic piece of art' (Foucault 1997b, 260) and that creating ourselves as a work of art may open possibilities for different ways of living that are not strictly beholden to biopolitical apparatuses and concomitant forms of subjectivity.
However, to assess the veracity of this view, we must consider the issue of freedom in more detail, as it pertains to an ethics or aesthetics of the self and to biopower. As the foregoing makes clear, freedom is at the heart of an ethics of the self. Indeed, Foucault spoke of an ethics of the self as a 'practice of liberty', which highlights the way that freedom is not given once and for all but requires a practical exercise upon oneself to be maintained and elaborated. Thus, one is not simply free as opposed to unfree, but must continually practice and perfect one's freedom. This centrality of freedom is reinforced in interview discussions of his genealogy of desiring man, in which Foucault claims that '[f]reedom is the ontological condition of ethics. But ethics is the considered form that freedom takes when it is informed by reflection' (Foucault 1997a, 284). By this, we might understand Foucault to be claiming that freedom (perhaps understood as opposed to ontological determinism) is necessary for the possibility of ethics, but ethics itself is the matter of what one does with that freedom. Finally, while Foucault demurs that an ethics of the self can be understood as process of liberation, he concedes that political liberty may be a necessary condition of a subjective practice of liberty (Foucault 1997a, 282–4). While it is not always easy to understand exactly what Foucault means by the terms of freedom and liberty, one implication of these various comments is that in order to engage in a practice of liberty, the subject so engaged must already be free in some sense or another.
If this is so, it is significant for understanding an ethics of the self vis-à-vis biopower, and to elaborate on this it is worth considering some of Foucault's comments on power, freedom and subjectivity made in the interregnum between Will to Knowledge and Use of Pleasure. In a reflection characteristic of his self-interpretations during this time, Foucault claims that the 'modus operandi' of his work was to analyze the relation of experiences like madness, death, sexuality and crime to technologies of power, and the problem that emerged during the 1970s was that of individuality or the experience of 'self-identity in relation to the problem of "individualizing power"' (Foucault 2000, 300). At the end of this same lecture – 'Omnes Et Singulatim' – he proclaims that the state is both individualizing and totalizing 'right from the start' and the only possibility of liberation comes from attacking 'not just one of these two effects but political rationality's very roots' (Foucault 2000, 325). The provocation of this lecture, presented at Stanford in 1979, is reinforced by a similar formulation in the crucial essay 'The Subject and Power', published in 1982. Here, Foucault claims that:
[t]he political, ethical, social, philosophical problem of our days is not to try to liberate the individual from the state, and from the state's institutions, but to liberate us both from the state and from the type of individualization which is linked to the state. We have to promote new forms of subjectivity through the refusal of this kind of individuality which has been imposed on us for several centuries.
(Foucault 1982, 216)
These statements lend weight to the view that in the late 1970s and early 1980s, Foucault was casting about for a way of thinking that would lead away from the biopolitical trap of normalization, which we have seen is both totalizing and individualizing, and further, that the ethics of self came to constitute this path. To be sure, Foucault demonstrates some reluctance to embrace this view when, for instance, he claims that we cannot find the solutions for our own times in those of another age (Foucault 1997b, 256). Yet it is also clear that he does see political potential in an ethics of the self, insofar as it allows for a kind of dislodgement of oneself from the games of truth that have hitherto determined the shape of one's subjectivity.
This ambivalence may be explained away by reference to Foucault's reluctance to take the position of philosophical vanguardism, wherein the intellectual or philosopher tells other what to do. But there is also an underlying theoretical difficulty here, in that Foucault cannot straightforwardly pose an ethics of the self, understood as a practice of liberty, as a form of resistance because freedom is not for him in opposition to power. Rather, Foucault insisted that freedom provides a permanent support for power. In 'The Subject and Power', he writes:
Power is exercised only over free subjects, and only insofar as they are free... there is no face to face confrontation of power and freedom which is mutually exclusive... In this game freedom may well appear as the condition for the exercise of power (at the same time its precondition, since freedom must exist for power to be exerted, and also its permanent support, since without the possibility of recalcitrance, power would be equivalent to a physical determination [and therefore not a relationship of power])... At the very heart of the power relationship, and constantly provoking it, are the recalcitrance of the will and the intransigence of freedom.
(Foucault 1982, 221–2)
Thus, freedom is posited as the permanent support and necessary precondition for the exercise of power, not only in the sense that a subject must be free in order to exercise power, but also and more importantly, that power can only be exercised over free subjects.
Furthermore, in the context of biopower, it has been argued that it is precisely through freedom that we are governed. As Nikolas Rose (1999, 67) writes in relation to liberalism, '[f]reedom has been an objective of government, freedom has been an instrument or means of government, freedom has inspired the invention of a variety of technologies for governing'. As Rose and others elaborate, liberalism and its latest manifestation as neo-liberalism, relies upon forms of self-conduct predicated upon the freedom of the political and economic subject, especially evident in the contemporary repetition of the idea of individual choice across all domains of life – including death. As Rose (1999, 87) incisively puts the point, 'modern individuals are not merely "free to choose", but obliged to be free, to understand and enact their lives in terms of choice'. This idea of being obliged to be free has a striking resonance across various fields of contemporary biopolitics, especially evident in healthcare where respect for individual choice and autonomy is seen as a key value in opposition to the paternalism of experts.
These formulations suggest that in biopower, subjects are not only governed by virtue of their being free, but are in fact governed through the instantiation of freedom as choice. This poses an interesting paradox for an ethics of the self if it is understood as a potential form of resistance to biopower. For what would be at stake is not merely the need to constitute oneself through the practice of one's liberty, but, more complexly, the need to find ways to practice freedom differently. Thus, when Foucault insists on the importance of 'getting free of oneself', of dis-assembling oneself (Rabinow 1997), or taking distance from the self or subjectivity given to one through games of truth, this must also take the form of getting free of oneself understood as a particular kind of free subject. This necessity of getting free of freedom helps to clarify the ambivalence that Foucault maintained in relation to an ethics of the self, and tempers the thesis that ethical practices of the self effectively oppose the forms of subjectivity made possible in biopower.
Conclusion
As I have shown in this chapter, Foucault's work sets out a genealogical approach to biopower that links the emergence of this new rationality of power with the development of new kinds of knowledge, such as biology and statistics, which were made possible by a transformation in the episteme at the end of the Classical era. This transformation constituted the conditions of possibility for a regime of power that took life, both at the level of the individual body and at the level of the population, as its primary concern and object. Thus, the management of life geared toward the regulation of the individual body in discipline and the management of collective well-being at the level of the population in a biopolitics of population fundamentally shapes modern power relations. This has two broad implications. First, it necessitates a reconsideration of the central tenets of political philosophy and theory's approach to the question of political power. Power, Foucault contends, is no longer primarily exercised through the hierarchical and deductive mechanisms of the sovereign, but through productive networks that include non-state institutions. Second, it requires a rethinking of subjectivity, such that the free subject no longer stands in theoretic opposition to the sovereign, but is instead an artefact of power's operation. Foucault can hardly be said to have resolved all the questions raised by these genealogical and philosophical revisions. Yet, his work on biopolitics and biopower continues to resonate across various fields, insofar as it yields critical tools for the local analysis of particular technologies or dispositifs of power. Further, this work has been enormously productive for subsequent theorists of biopolitics, providing both a starting point for fundamental revision, and a set of guiding theoretical and methodological precepts that continue to underpin new ways of understanding the contemporary world. In the following chapter, I discuss the work of the Italian philosopher, Giorgio Agamben, whose revisions of Foucault have done a great deal to ignite recent interest in biopolitics.
Notes
Note that in English translations, the French title of La volonté de savoir is typically dropped in favor of the generic title of the series of books, The History of Sexuality. In order to allow a clearer differentiation of the volumes under discussion, throughout I refer to the first volume of the History of Sexuality project under its translated title of The Will to Knowledge, which is more consistent with the second and third volumes being referred to as The Use of Pleasure and The Care of the Self respectively. The entire project I refer to as History of Sexuality.
In Society Must Be Defended, Foucault rejects the idea that the sovereign right was replaced, suggesting that it 'came to be complemented by a new right which does not erase the old right but which does penetrate it, permeate it'. This new right is formulated as 'the right to make life and to let die'. See Foucault 2003b, 241. I discuss this relation between sovereignty and biopower further in Chapter 5.
Throughout his discussion of biopower in Will to Knowledge and elsewhere, Foucault tends to run together terms such as 'population' and 'species' without clarifying the relation of one to the other. But we can imagine a governmental concern with species-being that is not mediated through the concept of population; ditto, a concern with population well-being is possible without that also referring to species-being. Further, the genealogies of the emergence of population as a political subject, and the emergence and mobilization of the concept of species may differ. It is important, then, to keep in mind that while species and population interlock in various ways in biopower, as concepts they do not stand in for one another and may in fact entail opposing as much as complementary biopolitical strategies.
Ann Stoler notes that early literature responding to Will to Knowledge typically failed to engage at all or in any substantive way with the problem of race and its role in the emergence of biopower. As she also mentions, Foucault himself expresses some disappointment at the lack of response to the final chapter of Will to Knowledge in an interview in 1977. (Stoler 1995, 19–22) (Foucault 1980, 222).
As a starting point for this field of scholarship, see Elhers 2012; Bernasconi 2010; McWhorter 2009; for historical documents relating to the USA see, Bernasconi 2005.
Foucault's essay 'Nietzsche, Genealogy, History' is crucial in understanding the stakes of genealogy. See Foucault 1994b.
For further discussion of the history of statistics, see in particular, Hacking 1990 and Porter 1986; for a philosophical discussion of the interpenetration of statistics and biopower, and the concept of the norm, see Mader 2011.
For a fuller discussion of the way in which this rejection of sovereignty structures Foucault's thinking on power, see Dean 2013. Also, note that Foucault is writing against the Marxist understanding of power as class domination.
Foucault is relying upon an expanded sense of ethics as ethos or way of being, rather than the more modern conception of determining right and wrong conduct.
This comment has been the target of much interpretation, but I think it is possible to argue that it is both more prosaic and more Kantian than is often allowed. It is worth remembering that the interview in which it appears was undertaken in the midst of Foucault's 'return' to Kant and specifically to the question of the Enlightenment.
I take this formulation from Paul Rabinow, in (Rabinow 1997, xxxviii).
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2Biopolitics as thanatopolitics
Agamben
While Foucault's work has been foundational to contemporary debates, no other single text has done as much to contribute to the proliferation of literature on biopolitics than Giorgio Agamben's Homo Sacer (1998). Initially published in English translation in 1998, it was not for several years that the book really gained prominence. In retrospect, it appears that the popularity and prominence of the book was at least partially circumstantial, insofar as the hijacking of planes and destruction of the World Trade Center in New York City appeared to give rise to a new kind of politics – or at least brought to the fore latent tendencies within constitutional democracies such as the United States. In the wake of September 11, the politico-cultural scene of the early twenty-first century has been dominated by the so-called 'War on Terror', including events such as new and increased populations of displaced persons and refugees, indefinite detention, Guantanamo Bay, Abu Ghraib, extraordinary renditions, extra-judicial trials and revelations of state-sanctioned torture on the part of major democracies. In the light of these events, Agamben's analysis of contemporary politics as a politics of the exception, in which the decisional capacities of the sovereign – especially the capacity to suspend the normal rule – are highlighted, seemed to have a particular pertinence. Of course, that Homo Sacer seemed to foretell of the post-September 11 political climate was not a matter of prediction on Agamben's part; the apparent perspicacity of his discussions of the bleaker, 'hidden' side of Western politics was less a matter of knowing what would happen, as a matter of being sensitive to what could possibly happen, given the conceptual architecture that subtends Western political institutions.
Agamben sees this architecture as fundamentally biopolitical, and much of his analysis in Homo Sacer is dedicated to illuminating the way in which the 'life' of political subjects was captured within the political sphere. Starting from Aristotle, and with reference to both Roman and Anglo-Saxon political mythologemes, he claims that 'life' constituted the 'inclusive exclusion' that provided the foundation for the political sphere, strictly understood. Given this, he argued that biopolitics, of which 'life' provides both the object and subject, was intimately linked to the structure of sovereignty, to such an extent that Western politics has always been biopolitical. Indeed, he claims that the original task of the sovereign was the production of the biopolitical body. In focusing on sovereignty in this way, Agamben developed a strongly juridical interpretation of a form of power that Michel Foucault had previously argued involved the displacement of the law as the primary instrument of power, and its replacement by the more discreet and anonymous operations of norms. In developing this interpretation, one of the major tasks of Homo Sacer is to show how law relates to life, and, further, to show how violence is central to this relation. The twin 'enigmas' of sovereignty and sacrality are central to Agamben's way of responding to this task. In the first section of this chapter, I trace Agamben's construal of sovereignty and sacrality, to discern the outlines of his conception of biopolitics.
In addition to this, though, the inclusive exclusion of life in politics ensures that the figure of 'bare life' takes a central place in the logics of Agamben's argument. Bare life lies, Agamben suggests, at the foundation of the polis. The notion of bare life has also been of central importance to the reception and use of his work in fields such as sociology, cultural studies and literary studies, among others. Even so, the notion of bare life remains obscure, and I spend some time in the second section of this chapter teasing out its various resonances and implications. In the sections that follow from this, I turn to Agamben's reflections on the possibilities for living beyond biopolitics. At several moments throughout his work, he suggests that new conceptions of life and ethics are required to move beyond the aporias that characterize the biopolitical capture of life in law. In order to outline these suggestions, I consider his reflections on 'form-of-life' in section three, and in the final section, his suggestions for an ethics beyond law. Overall, then, the aim of the chapter is to outline key aspects of Agamben's theorization of biopolitics through the Homo Sacer series of texts, especially his initial formulation in Homo Sacer, but also more recent work that to some extent provokes revision of his initial thesis.
Sovereignty and sacredness: The twin evils of biopolitics
While Foucault's account of biopower entails a displacement and demotion of sovereign forms of power, Agamben's account in Homo Sacer and the associated text, State of Exception (2005), definitively reverses this and places sovereignty at the heart of biopower. Indeed, for Agamben there is little distinction to be made between sovereignty and biopower, since in his view the Western political tradition has been biopolitical from its inception. In a sense, then, there is no historical moment at which Western politics became biopolitical, since it has never been anything other than biopolitical. This is because Western politics is founded on a particular form of relation, which Agamben variously describes as a state of exception, a ban or a relation of inclusive exclusion. Essentially, the idea here is that Western politics is founded on that which is excluded, but that exclusion is no simple matter, since the excluded is nevertheless captured within politics. These formulations provoke questions such as: what is captured in the state of exception, what is included or excluded and what is subject to the ban? The answer to these questions is life – or more specifically, the biological life of human being is excluded from politics, and then included in it in the form of what Agamben calls bare life. How does this capture take place? The answer to this question appears to be through the doctrine of sacrality, or the sacredness of life. Thus, a particularly intimate relation between sovereignty and sacredness appears – and it is this that condemns us to the modern biopolitical condition. In this section, I trace the duality of sovereignty and sacredness to make sense of Agamben's basic theses regarding biopolitics, especially in Homo Sacer.
In the introduction to Homo Sacer, Agamben claims that Foucault's previous investigations into biopower must be 'corrected, or, at least, completed' (Agamben 1998, 9) and casts Homo Sacer as an attempt to do this. As we saw in the first chapter, there is an ambiguity in Foucault's account around the relation between biopolitical forms of power and sovereign forms. Agamben argues that the point of intersection between biopower and sovereignty remains curiously obscure in Foucault, and it is this that he seeks to illuminate in Homo Sacer. He reaches the conclusion that the analyses cannot be separated, since the inclusion of life in the political realm constitutes the 'original – if concealed – nucleus of sovereign power'; consequently, 'the production of a biopolitical body is the original activity of sovereign power' (Agamben 1998, 6). To arrive at this conclusion, he points to the distinction drawn by Aristotle in his thesis on the origins of the city-state in Politics that while the city-state comes into being for the sake of living, it continues to exist for the sake of living well (Aristotle 1998, 1252b–30). Reflecting on this canonical statement, Agamben argues that this indicates that Western politics is founded on a particular form of differentiation and exclusion, whereby while the good life provides its telos, 'simple natural life is excluded from the polis in the strict sense and remains confined – as merely reproductive life – to the oikos, "home"' (Agamben 1998, 2). In this formulation, 'life' is simultaneously excluded from, and included in, the sphere of politics and, further, this 'inclusive exclusion' forms the foundation of Western politics.
Agamben goes on that sovereign power is itself founded on a similar exclusion, whereby the sovereign both includes and excludes itself from the rule of law. Agamben's conception of sovereignty draws heavily on the decisionistic thesis of the German right-wing jurist, Carl Schmitt (1985, 5), who infamously argued that '[s]overeign is he who decides on the exception'. In this formulation, at stake in the exception is the very possibility of juridical rule and the meaning of state authority. According to Agamben, in deciding on the state of exception – a process in which the sovereign both includes and excludes itself from the law – 'the sovereign "creates and guarantees the situation" that the law needs for its own validity' (Agamben 1998, 17; citing Schmitt 1985, 5). Or, as he also puts it, 'what is at issue in the sovereign exception is... the creation and definition of the very space in which the juridico-political order can have validity' (Agamben 1998, 19). The sovereign thus operates as the threshold of order and exception, determining the purview of the law. The sovereign determines the suspension of the law vis-à-vis an individual or extraordinary case and simultaneously constitutes the efficacy of the law in that determination. However, the state of exception is such that what is excluded from the law continues to maintain a relation to the rule precisely through the suspension of that rule.
Agamben suggests that the term most appropriate to the capacity of the law to apply in no longer applying is that of the ban (Agamben 1998, 28). That which is excluded is not simply set outside the law and made indifferent or irrelevant to it, but rather abandoned by it, where to be abandoned means to be subjected to the unremitting force of the law while the law simultaneously withdraws from its subject. In reference to Jean-Luc Nancy's (1993, 44) complex discussion of abandonment, Agamben claims that the position of being in abandonment correlates to the structural relation of the exception: 'the relation of exception is a relation of ban' (Agamben 1998, 28). Just as with the exception that is included only through its exclusion, the subject of the ban is not simply excluded from the law, but is given to the law in its withdrawal. This correlation between the exception and abandonment means that it is impossible to say clearly whether that which has been banned is inside or outside the juridical order (Agamben 1998, 28–9). The importance of this concept of abandonment for Agamben's analysis is suggested when he claims that uncovering this relation is the key to understanding the origin and present condition of our political predicament. This is because sovereignty 'is the originary structure in which law refers to life and includes it in itself by suspending it', and the 'originary relation of law to life is abandonment' (Agamben 1998, 28–9). In short, in Western politics, life and sovereignty (or law) are both structured by and intersect according to the logic of abandonment.
Agamben's use of notions such as inclusive exclusion and abandonment thereby goes some way to establishing the link between law and life that he sees as characteristic of biopolitics; however, it does not yet make it clear why this relation should be seen as one of unremitting violence. In order to establish the centrality of violence in the relation between law and life, Agamben needs to show that sovereignty – and law – is itself constitutively violent. He makes two key points to establish this, the first entailing a reference to Pindar's fragment on 'nomos basileus' and the second in reference to the work of Walter Benjamin. In regards to the first, Agamben claims that Pindar is the first great thinker of sovereignty, and his fragment contains the 'hidden paradigm' that guides every definition of sovereignty thereafter. The fragment reads: 'The nomos, sovereign of all, / Of mortals and immortals, / Leads with the strongest hand, / Justifying the most violent. / I judge this from the work of Hercules.' (as translated in Agamben 1998, 30). The significance of the fragment, in Agamben's view, lies in the essential connection it draws between nomos and violence, of which he argues that sovereignty appears as the 'threshold' or point of intersection between law and violence. The implication of this is that sovereignty is integrally violent insofar as it is the meeting point of law and violence. Thus, Agamben's use of Pindar's fragment provides a crucial foundation for figuring the political centrality of violence in Homo Sacer (also see Benjamin 2005).
The second key point builds on this through reference to Walter Benjamin's essay 'Critique of Violence'. Agamben (1998, 63) describes this essay as proving the 'indispensable premise of every inquiry into sovereignty' because – like Pindar's fragment – it establishes the 'irreducibility' of the link between law and violence. Furthermore, it identifies the carrier or bearer of this link, namely, sacred or bare life. In the essay, Benjamin's category of mythic violence identifies the violence of law in both its law-positing and law-preserving forms. He also posits the necessity of a 'divine violence' in order to break apart the oscillations between law preserving and law constituting violence and to thereby depose law itself. This idea of divine violence has been the target of much interpretation, but what is important here is the connection that Benjamin posits between divine violence and life. In the final few – rather cryptic – pages of the essay, Benjamin claims that, 'mythic violence is bloody power over mere life for its own sake; divine violence is pure power over all life for the sake of the living' (Benjamin 1996, 250). Further, whereas mythic violence brings guilt and retribution, the latter 'only expiates'. From this, he goes on to suggest that the idea of the sacredness of human life is intimately related to matters of guilt, and that the bearer of guilt is 'life itself'. Finally, Benjamin suggests that one of the key questions that contemporary thought must confront in order to move beyond legal or mythic violence is that of the origin of the 'dogma' of the sacredness of life (Benjamin 1996, 251).
In responding to this exhortation, Agamben draws together Benjamin's thesis on sacrality with the forgoing thesis about sovereignty, to arrive at the conclusion that the sacredness of life emerges only to the extent that life is incorporated into the sovereign exception; as he writes, '[l]ife is sacred only insofar as it is taken into the sovereign exception' (Agamben 1998, 85). To outline this further, Agamben invokes the figure of 'sacred man' or homo sacer from Roman law, which, he argues, is characterized by a 'double exclusion and a double capture' (Agamben 1998, 82). Sacred man is simultaneously excluded from both human and divine law – while also included in each – and is thus singularly exposed to violence. The idea is that in being abandoned by (and to) the law, homo sacer is exposed absolutely to violence; homo sacer is simultaneously 'free, open to all' and the object without protection of potential violence. Agamben hypothesizes from this that 'homo sacer presents the originary figure of life taken into the sovereign ban and preserves the memory of the originary exclusion through which the political dimension was first constituted' (Agamben 1998, 83).
The upshot of Agamben's discussion of sacrality, then, is that homo sacer expresses the original political relation, that is, the relation of abandonment; further, life thus exposed to violence and death is the founding political element. To give more content to this second claim, Agamben elides the terms 'sacred' and 'bare', proposing to 'give the name bare life or sacred life to the first content of sovereign power' (Agamben 1998, 83). Thus, in response to the question of 'what is excepted and captured in sovereignty, and who is the bearer of the sovereign ban?' (Agamben 1998, 67), Agamben begins to identify what he sees as the (otherwise hidden) point of intersection between the 'juridico-institutional' and biopolitical models of power: bare life. As he states, '[n]ot simple natural life, but life exposed to death (bare life or sacred life) is the originary political element' (Agamben 1998, 88). Or, to put the point another way, '[i]n Western politics, bare life has the peculiar privilege of being that whose exclusion founds the city of men' (Agamben 1998, 7). I will return to a fuller discussion of bare life in a moment, but first, let me make several further points about Agamben's outline of sovereignty and biopolitics so far.
As is evident, Agamben places sovereignty at the centre of biopolitics, since these forms of power are fundamentally intertwined and are said to constitute the originary matrix of Western politics. In direct contrast to Foucault, Agamben claims that 'the inclusion of bare life in the political realm constitutes the original – if concealed – nucleus of sovereign power. It can even be said that the production of a biopolitical body is the original activity of sovereign power. In this sense, biopolitics is at least as old as the sovereign exception' (Agamben 1998, 6). The central role granted to sovereignty in this formulation sets Agamben's theorization of biopolitics somewhat apart from other accounts; but this formulation also points to a number of issues that could be pressed on further. For one, we might ask, just how old is the sovereign exception? The claim that biopolitics is at least as old as the sovereign exception is typically taken to support the thesis that Western politics is originally biopolitical – but this has to assume that sovereignty itself is an original aspect of Western politics. This would mean that there was no politics prior to sovereignty, and, moreover, sovereignty is essentially exceptional – this is not simply a particular historical manifestation of sovereignty, but a characteristic without which sovereignty does not appear. These suppositions seem to give credence to the charge of conceptual fundamentalism, in which 'the meaning of concepts is irrevocably determined by their origin' (Patton 2007, 218), made against Agamben and raise significant doubts about the historical veracity and overall plausibility of his account of biopolitics.
These doubts go to the heart of Agamben's philosophical methodology. In short, read as an historical account of the emergence of biopolitics at the origin of Western politics and surviving through the ages up until the modern era, Agamben's thesis makes little sense. No doubt, this reading is encouraged by some of Agamben's comments such as that Western politics is biopolitical from its inception, and finds its origin in ancient Greek thinking around life, nomos and political authority. But this would entail an outrageous de-differentiation of the forms of politics and law in the intervening centuries. Instead, then, Agamben should be understood as using fragments from the past to understand the present, through illuminating the effect that they may still have within the contemporary political imaginary. As such, Homo Sacer is not an attempt to articulate a historical process, but offers an understanding of the present that attempts to grasp the ongoing effect of previous legal and political formulations, conceptions and techniques within contemporary conditions.
As Agamben (2009, 32) explains, his philosophical method is one of 'paradigmatology', in which what is at stake is the recognition and articulation of 'paradigms' that elucidate the present without positing causal or historical claims. This means that figures such as that of homo sacer, or the concentration camp and so on, are understood as paradigms, where a paradigm is a 'singular object that, standing equally for all others of the same class, defines the intelligibility of the group of which it is a part and which, at the same time, it constitutes' (Agamben 2009, 17). Interestingly, Agamben most consistently and explicitly aligns his paradigmatic method with that of Foucault, especially the archaeological method developed in Archaeology of Knowledge (1972), but also the genealogical approach of books such as Discipline and Punish (1977). In this, he particularly points to Foucault's use of Bentham's panopticon, which is described in Discipline and Punish as a 'diagram of a mechanism of power reduced to its ideal form' (Foucault 1977, 205). However, one should also see here the influence of thinkers such as Walter Benjamin and Aby Warburg, both of whom in different ways fundamentally undermined the intellectual supremacy of historical narrativity and temporal continuity. Thus, Agamben should not be read as offering any kind of historical thesis or explanation; rather, his paradigmatology fundamentally intermingles diachrony and synchrony in a kind of intentional anachronism, with the aim of elucidating the present.
Perhaps the most controversial aspect of Agamben's paradigmatic approach in Homo Sacer is the analysis of concentration camps, including those of the Boer War and the Second World War. He maintains that the camps were 'born out of the state of exception and martial law' such that the camp is the 'materialization of the state of exception' (Agamben 2000, 38, 41); and insofar as the exception has become the rule, the camp is the 'hidden matrix' of Western biopolitics. This logic extends not only from the Nazi concentration camps and the refugee camps that have arisen around the globe, but also to apparently innocuous spaces such as airport lounges, gated communities and soccer stadiums, which are, or can become, zones of indeterminacy that are politically equivalent to concentration camps (Agamben 2000, 42). For many, this figuration of the camp as the 'hidden matrix' of biopolitics or the 'nomos of the modern' has stretched Agamben's thesis beyond the bounds of credibility. Further, it has given rise to important critiques of the Eurocentrism of Agamben's account, insofar as it fails to consider the violence of colonialism or slavery, and repeats an oft-made identification of the Nazi death camps as the defining moment of Western politics.
I will return to these concerns in a moment, but first it is important to note that the centrality of the camp in Homo Sacer makes evident another key feature of his theorization of biopolitics. This is the way that the supposed politics of life is in fact a politics of death – not biopolitics, but thanatopolitics. This collapsing of biopolitics into thanatopolitics has generated significant critiques of Agamben, and limits the usefulness of his account of biopolitics for engaging contemporary phenomena, such as genetics and biomedicine, which are so clearly about extending and fostering life. Paul Rabinow and Nikolas Rose have thus argued that biopower is not about 'making die' but 'making live'; they write, 'central to the configuration of contemporary biopower are all those endeavours that have life, not death, as their telos' (Rabinow and Rose 2006, 203). Consequently, Rabinow and Rose oppose any association of contemporary biopower with Nazism and the Holocaust. A similar kind of critique has been made by Mika Ojakangas, who challenges Agamben's characterization of the inter-relation between sovereignty and biopower and the consequent construal of biopower as inherently violent. For him, while it is accurate to characterize sovereign power as a power of violence and death, biopower is distinct from this and may even counteract the force of sovereignty. This is because '[i]t is precisely care, the Christian power of love (agape), as the opposite of all violence that is at issue in biopower' (Ojakangas 2005, 20). Ojakangas is not suggesting that biopolitical societies are simply driven by love and care; rather, biopower and sovereign power become entangled in a 'demonic combination' such that violence is nevertheless deployed within them in various ways. Thus, Nazi Germany did not entail the absolutization of biopower, but of sovereign power.
Interestingly, the arguments pressed in The Kingdom and the Glory (2011) may be read as Agamben's own recognition of having overstated the case in Homo Sacer, and thus seen as a corrective of these more controversial claims. In this book, Agamben sets out his argument in terms of two 'political paradigms' derived from Christian theology that are 'antinomical but functionally related to one another' (Agamben 2011, 1). These are political theology, which, Agamben argues, bases the transcendence of sovereign power on God, and economic theology, which centres on the idea of oikonomia, understood as 'an immanent ordering – domestic and not political in a strict sense – of both divine and human life' (Agamben 2011, 1). Surprisingly, Agamben then suggests that while political philosophy and the modern theory of sovereignty derive from the paradigm of political theology, 'modern biopolitics up to the current triumph of economy and government over every other aspect of social life derive from the second paradigm' of oikonomia. On the face of it at least, this alignment of biopolitics and oikonomia against political theology and sovereignty stands in sharp contrast to the thesis presented in Homo Sacer, that sovereignty is foundational to Western biopolitics. This apparent shift in Agamben's thinking may give us cause to reconsider the plausibility of the arguments made in the earlier book, a reconsideration that would benefit from careful explication of the key terms of Agamben's analysis of biopolitics. Toward that end, in the following section, I look more closely at the central term, 'bare life', upon which Agamben's analysis of life and law hinges.
The living and/or the dead: Bare life
The notion of bare life has been compelling within literature that seeks to adopt Agamben's theorization of biopolitics, or parts thereof. Despite (or, because of) its evident vagueness, the notion has appealed to many as a way of describing the lives of those people who are marginalized, oppressed or otherwise de-legitimated within contemporary socio-political arrangements. Even so, the notion is the source of much confusion, not least because Agamben himself never provided a clear definition nor maintained a clear referent for the term. What we see instead throughout Homo Sacer and associated texts is a significant slippage, whereby bare life is used at times to specifically refer to life defined by its own negation, and at others to indicate something more like a vitalistic sense of natural life. For example, he explicitly defines bare life as life exposed to death, and states, 'not simple natural life, but life exposed to death (bare life or sacred life) is the originary political element' (Agamben 1998, 88). However, he also appears to use the term to refer to natural life, evident in his use of the phrase 'bare natural life', for instance. In this section, then, I consider the proposed relation that bare life bears to both natural life and legal violence in order to get a clearer sense of the parameters of this concept. Further, I begin to develop several critical points that will be explored further in later chapters, especially concerning the force of law in relation to life, and the role of technology in the constitution of forms of life.
As mentioned above, Agamben notes that the qualitative distinction made by Aristotle in his treatise on the formation of the city-state between biological life (zoē) and political life (bios) effectively excluded natural life from the polis in the strict sense, relegating it entirely to the private sphere as the basic life of reproduction. The category of bare life emerges from within this distinction, in that it is neither bios nor zoē, but rather the politicized form of natural life. Immediately politicized but nevertheless excluded from the polis, bare life is cast as the limit-concept between the polis and the oikos. And in being that which is caught in the sovereign ban, bare life indicates the exposure of natural life to the force of the law in abandonment, the ultimate expression of which is the sovereign's right of death. Thus, neither bios nor zoē, bare life emerges through the irreparable exposure of life to death in the sovereign ban, such that the politicization of life is ultimately nothing other than its exposure to death, particularly in and through sovereign violence.
Throughout Homo Sacer, Agamben points to numerous figures that he sees as revealing the structure and presence of bare life in biopolitics, two of the most important of which are homo sacer and the wolf-man or wargus. As we saw in the previous section, a lot of the argumentative weight of Agamben's construal of biopower falls on the figure of homo sacer. Given this, what purpose does the wolf-man play in the text? At one level, the purpose of this figure is simply to broaden the scope of bare life beyond the Roman tradition of law, to show its presence within the Germanic and Anglo-Saxon traditions as well. However, there is also more to it than this.
Whereas homo sacer establishes a connection between life and violence, the wolf-man provides a connection between bare life and natural life. The wargus is cast as the threshold or point of indistinction between 'the animal and man, physis and nomos, exclusion and inclusion... precisely neither man nor beast, and who dwells paradoxically within both while belonging to neither.' (Agamben 1998, 105). With the figuration of bare life in the wolf-man, Agamben adds another dimension to the notion of bare life, which is now no longer simply indicative of a legal threshold, but an ontological one between nature and culture, or man and beast. However, if bare life emerges from within the distinction between bios and zoē, we can see that this figuration aligns bios with 'man' – or perhaps more generously, the human – and zoē with the animal. Thus, while Agamben appears to reject this alignment in his initial borrowing of the distinction from Aristotle, it nevertheless comes to condition his understanding of bare life. Even so, as several commentators have noted, this alignment is not supported in Aristotle's writings, and it is highly questionable what significance it has for 'the Greeks' more generally (Dubreuil 2008, 86; also see Derrida 2009, 325–441). This raises several questions, the answers to which are often taken for granted: what, if anything, does the term zoē mean today (in the midst of the genetic and biotech revolution)? Can we assume a semantic continuity in the idea of natural life from the ancient Greeks to the contemporary? And what is at stake in the contemporary inscription of zoē (as opposed, perhaps, to life as an object of the biological sciences) in social and political theory? We consider these questions further in Chapter 6, but for now, more can be said about the wargus.
To elaborate on the significance of the figure of wargus and the perceived connection between it and homo sacer, Agamben draws on one of the lais written by Marie de France in the twelfth century, entitled Bisclavret. In this tale, the titular nobleman turns into a werewolf each week; he is eventually betrayed by his wife and her lover, who steal the clothes necessary for his transformation back into human form, thereby preventing his re-humanization. While in the guise of a werewolf, the nobleman is captured by a king, who takes the werewolf home to keep as a pet. The wife's actions are later discovered by the king, who insists she return the clothing to Bisclavret, who then becomes human once again. The king banishes the wife and her new husband and restores Bisclavret to his title and estate. Of this story and the liminal figure of the werewolf, Agamben comments that '[t]he transformation into a werewolf corresponds perfectly to the state of exception, during which (necessarily limited) time the city is dissolved and men enter into a zone in which they are no longer distinct from beasts' (Agamben 1998, 107). This suggests a close parallel between wargus and homo sacer. However, the story of Bisclavret is actually not obviously one of sovereign violence; rather, in the lai the sovereign becomes the protector of the animal, and, ultimately, the mediator of the transformation into the human. Bisclavret is not cast out by the sovereign, but taken in and sheltered by him. Thus, while wargus and homo sacer are presented as consistent figures of bare life – for example, when Agamben writes that the sovereign decision 'refers immediately to the life (and not the free will) of the citizens... [which is] the bare life of homo sacer and the wargus, a zone of indistinction and continuous transition between man and beast, nature and culture' (Agamben 1998, 109) – they in fact work against each other. Indeed, the lay seems to indicate more the way in which the law is implicated in the constitution of the human, rather than its destruction as in homo sacer.
To get at this point and elaborate it further, let me consider Agamben's attempt to relate his account of biopower to advents in contemporary medicine and politics. In the final section of Homo Sacer, Agamben proffers a number of examples – which he himself concedes may appear to be 'extreme, if not arbitrary'(Agamben 1998, 186) – to illustrate his thesis on biopolitics, claiming that it is on the terrain of these 'difficult zones of indistinction, that the ways and the forms of a new politics must be thought' (Agamben 1998, 187). One of these examples, often subsequently used to exemplify the contemporary operations of biopolitics, is the case of Karen Quinlan, a young woman in a persistent vegetative state who was kept on life support machinery for a number of years. According to Agamben, this case reveals a life stripped of all individuation – 'pure zoē ' as he says - but it is a kind of natural life that is wholly exposed to death and the law. He writes, 'Karen Quinlan's body – which wavers between life and death according to the progress of medicine and the changes in legal decisions – is a legal being as much as it is a biological being' (Agamben 1998, 186). In this, she supposedly appears as a pure manifestation of bare life.
However, this characterization obscures important aspects of such cases – and highlights Agamben's penchant for generalization at the expense of the details of his examples – since Karen Quinlan's life did not simply depend on a legal decision as Agamben suggests. The legal events surrounding her sought to address questions of whether a ventilation tube constitutes 'extraordinary means' for maintaining life, and whether she should be 'allowed to die' following its removal. The legal decision was not one of whether to kill her, as could be read from Agamben's analysis. Indeed, in the event, following the removal of the ventilation tube, Quinlan continued to breath – to live in some sense – for a further nine years before dying of pneumonia. This suggests that her life exceeded the legal decisions that surrounded its continuance, that it was not wholly dependent on that legal decision. Certainly, Quinlan had lost all consciousness – but if this renders her 'pure zoē', then there is a danger that the bios/zoē distinction is simply being mapped onto Cartesian mind/body dualism. Pressing on this further, we could argue that, just as with the wargus, the exposure to the law does not simply render her bare life (if, in fact, it does that at all) – it may also be that which renders her human, as having rights and interests above and beyond mere survival. At the very least, she maintained a right not to be killed.
John Protevi (2009) makes a similar point somewhat differently in his discussion of another PVS case, namely that of Terry Schiavo. Protevi's critique of Agamben's handling of cases of PVS and 'rights to die' is two-pronged: first, he argues that Agamben focuses too exclusively on the law's role in constituting 'incorporeal transformations' at the expense of the material transformations effected within and by the body itself, such as the events of cardiac or respiratory failure that initially lead to a PVS diagnosis (Protevi 2009, 122–5). This neglect of the body itself is, as I have suggested, significant in relation to Quinlan as well. The second part of his critique argues that Agamben focuses on 'exposed' bare life at the expense of 'trapped' bare life – or life that is not allowed to die. This is particularly evident in relation to Schiavo, where state interventions typically took the form of enforcing the extension of her life through the reinsertion of the feeding tube that was keeping her alive. From the point of view of Agamben's biopolitics, this state insistence on living is paradoxical, especially given that the eventual legality of allowing Schiavo to die was predicated on her express wishes while conscious to be allowed to die in such circumstances. In this sense, the legal decision that supposedly rendered her 'bare life' simultaneously affirmed her personhood and the continued legal relevance of her bios.
There is, however, an even more obvious point to be made of Agamben's characterization of the Quinlan case, as well as that of what he calls the 'extreme vicissitude' of organ transplantation following diagnosis of brain death. This is that he is oddly insensitive to the role that technology plays in these scenarios, for we could go so far as to say that the legal decision that Agamben focuses on is itself dependent on technology: it is only because of the development of life support technologies that such decisions are either possible or necessary. Thus, the life of PVS patients is not simply a matter of either legal or biological being as Agamben suggests; they are, perhaps more than anything, technological beings. To continue this thought, it is worth recalling that in Bisclavret it is his clothing that marks the transition between man and wolf. Given this, while Agamben highlights the 'special proximity' between the wolf and the sovereign, it may be possible to pose an alternative reading that gives greater significance to the clothing, understood as a metonym for technology, as what makes possible and marks the transition between human and animal, or between culture and nature. Then, the sovereign might appear as that which protects the human, even when it does not explicitly appear as such. Not entirely dissimilarly, in the cases of PVS mentioned above, 'life support' technology might be said to maintain a body on the threshold between life and death, or perhaps even between nature and culture, while the law seeks to protect – and potentially enforce – the humanness that is no longer readily apparent. While these comments are merely suggestive, they at least indicate that the picture that Agamben is presenting can be made more complex with a greater recognition of the role that technology plays in the mediation of the human, or of bios and zoē. We return to a discussion of technology and biopolitics later, in Chapter 6, but here it is important to consider a further angle of Agamben's approach to conceptualizing life.
Sacredness versus beatitude; or bare life and form-of-life
If bare life is the bête noir of Western biopolitics, there is also a kind of redemptive hero, which appears in the notions of happy life, or 'form-of-life'. If modern politics indicates any kind of break from the politics of previous eras, it is not because of the emergence of a form of power focused on life as Foucault claims. Rather, Agamben suggests that the modern era is not distinguished by the fact that it seeks to manage the exigencies of life, but rather because it 'presents itself... as a vindication and liberation of zoē', and in doing so, constantly attempts to 'transform its own bare life into a way of life and to find, so to speak, the bios of zoē' (Agamben 1998, 9). What is at issue here, then, is the particular relation established between bios and zoē, where modern biopolitics renders them simultaneously fractured and indistinct, with bios increasingly conflated with zoē (Prozorov 2014, 97–8). This, Agamben contests, produces a number of dangerous aporia for modern liberal democratic politics, and there is no way beyond biopolitics that does not wrestle with these aporia. According to Agamben, the aporetic violence of modern democracy stymies any attempt to oppose biopolitical regimes from within the framework of bios and zoē and the capture of bare life. Such projects will merely repeat the aporia of the exception. Thus Agamben rejects Foucault's call at the end of Will to Knowledge for a 'new economy of bodies and their pleasures', claiming that 'the "body" is always already a biopolitical body and bare life, and nothing in it or the economy of its pleasure seems to allow us to find solid ground on which to oppose the demands of sovereign power' (Agamben 1998, 187). Instead, the only means of escape from contemporary biopolitics is through the constitution of a form of life that no longer partakes of the separation of bios and zoē, and, in that, overturns the logics of sacredness and sovereign violence.
This is the light in which Agamben's reflections on happy life or 'form-of-life' must be read. He explains that such a form-of-life is a life that is wholly immanent to itself and no longer reliant on any transcendental (such as juridical law or the sovereign) for its sense and purpose. As he writes:
The "happy life" on which political philosophy should be founded thus cannot be either the naked life that sovereignty posits as a presupposition so as to turn it into its own subject or the impenetrable extraneity of science and of modern biopolitics that everybody today tries in vain to sacralize. This "happy life" should be rather, an absolutely profane "sufficient life" that has reached the perfection of its own power and its own communicability – a life over which sovereignty and right no longer have any hold.
(Agamben 2000, 114–115)
In other words, 'happy life' or a 'form-of-life' allows neither a separation of bios and zoē, nor the collapse of one into the other. Rather, a reciprocal tension is maintained between bios and zoē, and the resulting form-of-life is characterized by an absolute immanence to itself, in its own potentiality of 'being-thus' (Agamben 1993, 93). Further, and most importantly, form-of-life is life realized wholly beyond the reach of the law (Agamben 2000, 114–15).
While hinted at in earlier work such as Means without End (2000) and Homo Sacer, elaborating this idea of 'form-of-life' is the focus of Agamben's recent work, The Highest Poverty (2013a). In this book, Agamben undertakes a detailed study of the foundations of Western monasticism in order to 'construct a form-of-life, that is to say, a life that is linked so closely to its form that it proves to be inseparable from it' (Agamben 2013a, xi). The first task in this is to understand the dialectic established between life and rule in monasticism, where life and rule do not collapse into each other so much as 'lose their familiar meaning in order to point in the direction of a third thing', that is, to form-of-life. From this perspective, Franciscanism appears as the 'decisive moment' in the history of monasticism, since it comes closest to realizing a 'human life and practice absolutely outside the determinations of the law' (Agamben 2013a, 110), though it still ultimately fails in this task. According to Agamben, central to both their achievement and failure is the doctrine of poverty, through which Franciscans renounced all rights, including the right to own property or use it, but maintained the possibility of a de facto use of things – a use without right.
The problem, though, was that the Franciscan's remained unable to articulate a notion of use that did not refer to the law. Through this doctrine, the Franciscan order attempted to set itself outside the sphere of positive law, but their failure lies in seeking to do so using juridical arguments to establish and legitimate the distinction between de jure use and de facto use. Agamben writes that the doctrine of usus facti
is obviously founded on the possibility of distinguishing de facto and de jure use... The force of the argument is in laying bare the nature of ownership, which is thus revealed to have a reality that is only psychological (uti re ut sua, intention to possess the thing as one's own) and procedural (power to claim in court). However, instead of insisting on these aspects, which would have called into question the very ground of property law..., the Franciscans prefer to take refuge in the doctrine of the juridical validity of the separation of de facto use and right... What is lacking in the Franciscan literature is a definition of use in itself and not only in opposition to the law.
(Agamben 2013a, 138, 139)
This failure to develop a conception of use, independent of law, and more specifically, to relate use to the monastic form of life or habitus, means that they were ultimately stymied in the task of developing a form-of-life. Finally, then, in the closing pages of The Highest Poverty, Agamben hints toward a more radical way of understanding use, particularly through the eschatological reflections of Franciscan theologian Peter Olivi. Connecting these reflections with the doctrine of use would, he suggests, constitute a full challenge to the current paradigm of operativity, and open into an idea of life in common and world that is used but not appropriated. This task, however, is yet to be completed.
Toward an ethics beyond guilt
While I discuss Agamben's conception of 'form-of-life' further in Chapter 6, at this point it is important to consider another aspect of his reflections on biopolitics. For if the Western biopolitical situation requires a new understanding of life, and the formation of form-of-life, then this in turn requires a new understanding of ethics, one which eschews the language of norms and rights and related concepts such as dignity, responsibility and culpability or guilt, and instead institutes a new vocabulary appropriate to a life without guilt. Interestingly, Agamben argues that we can glean material for this new conception of ethics from the writings of survivors of Nazi death camps. Agamben first addressed the task of developing a new approach to ethics in any detail in Remnants of Auschwitz (1999), published as the third volume of the Homo Sacer series. He proclaims, 'Auschwitz marks the end and the ruin of every ethics of dignity and conformity to a norm' (Agamben 1999, 69), and on the basis of this ruin, he uses survivor testimony as means to erect 'signposts' for a new ethics. His strong critique of the tradition of Western ethics is continued in Opus Dei (2013b), published as a kind of addendum to the fourth volume of the Homo Sacer series, The Highest Poverty. Here, Agamben proffers a genealogy of the concept of duty, and claims that ethics must be 'entirely liberated' from concepts of duty and will if it is to escape the conceptual apparatuses of biopolitics. In this section, then, I sketch out some aspects of Agamben's formulation of ethics beyond biopolitics.
While Agamben's writings on ethics have received far less attention than those on sovereignty, they are of a piece with the critique of Western politics developed in Homo Sacer and other works, and rely on that critique and related concepts such as form-of-life discussed previously. For instance, if Benjamin's suggestion that the doctrine of sacredness is implicated in the violence of the law is a motivating point for Homo Sacer, his related point about the expiation of guilt by divine violence plays a similar role in Remnants. Here, Agamben argues for a new conception of ethics that is not bound to the terms of guilt and responsibility and is therefore appropriate to life beyond biopolitics. Sketching out the broad frame of his approach to ethics, Agamben claims that Western ethics must break from the juridification of ethics to move to a new territory 'before good and evil'. He writes, 'ethics is the sphere that recognizes neither guilt nor responsibility; it is, as Spinoza knew, the doctrine of happy life. To assume guilt and responsibility... is to leave the territory of ethics and enter that of law' (Agamben 1999, 24). Agamben thus directly links the question of ethics with that of life beyond biopolitics, suggesting that ethics is only truly possible in the condition of a happy life expunged not only of guilt, but also of law.
This scathing critique of the central terms of Western ethics is continued in Opus Dei, published as a companion volume to The Kingdom and the Glory and The Highest Poverty. In it, Agamben undertakes a complex genealogy of the concept of duty from its origins in Christian liturgy through to Kantian ethics, which opposed and ultimately replaced the Classical understanding of ethics as virtue and character. The central thesis is that in the paradigm of office, understood as acts of duty, being and praxis – or what one is and what one does – are brought into a zone of indistinction in which 'being dissolves into its practical effects'. Moreover, 'being and acting today have for us no representation other than effectiveness. Only what is effective, and as such governable and efficacious, is real' (Agamben 2013b, xiii). The central problem here is that within the conceptual framework of office and duty, being can only be thought as a manifestation of law or norm, encapsulated in the concept of will. In a forthright statement of his opposition to the form that ethics then takes, Agamben concludes this study with the claim that a philosophy that is not already caught within the conceptual apparatuses of biopolitics must formulate an 'ethics and a politics entirely liberated from the concepts of duty and will' (Agamben 2013b, 129). Given this critique, what does Agamben propose instead for an ethics adequate to form-of-life?
In fact, to date, Agamben's most advanced attempt to develop a new ethics remains Remnants, where he proposes the inscrutable notion of an unassumable 'non-responsibility' or irresponsibility as the core idea of ethics. In setting out signposts for this new ethics, Agamben outlines an account of ethics as witnessing, and more specifically, witnessing the appearance of the inhuman in the human. The backdrop for this account is the Nazi concentration camps of the Second World War, of which Auschwitz stands in for the camp system as a whole, and the key figure is the Muselmänner. This term is used by survivors and others in regards to those beings in the camps who had reached such a state of physical and existential indifference that '[o]ne hesitates to call them living: one hesitates to call their death death' (Levi cited in Agamben 1999, 44). To bring out the significance of the Muselmann for ethics, Agamben starts from the paradox identified by Primo Levi, that the Muselmann, the one who cannot speak, is the true or 'complete witness' of the camps (Levi cited in Agamben 1999, 33). This paradox is quickly cross-cut with another, though, when Agamben argues that Auschwitz is the site of an extreme biopolitical experiment, 'beyond life and death in which the Jew is transformed into a Muselmann and the human into a non-human'(Agamben 1999, 52). In this, the Muselmann indicates a more fundamental indistinction between the human and the inhuman, in which it becomes impossible to distinguish one from the other; as Agamben (1999, 82; see also 56–69) writes, the Muselmann is 'the non-human who obstinately appears as human: he is the human that cannot be told apart from the inhuman'. Thus, the Muselmänner are indefinite beings in whom the distinction between humanity and non-humanity, as well as the moral categories that attend the distinction, are brought to crisis.
It is from this crisis that Agamben suggests a new ethics might emerge, specifically as an ethics of bearing witness to the inhuman. He proposes that the human being exists as the nodal point for 'currents of the human and inhuman' and states that '[h]uman power borders on the inhuman; the human also endures the inhuman... humans bear within themselves the mark of the inhuman... their spirit contains at its very center the wound of non-spirit, non-human chaos atrociously consigned to its own being capable of everything' (Agamben 1999, 77). This means that being human is fundamentally conditioned by an indefinite potentiality for being non-human, for being capable of everything and of enduring the inhuman. Being human is a question of enduring, of 'bearing all that one could bear,' and surviving the inhuman capacity to bear everything. And ultimately, in Agamben's account, the endurance that remaining human requires takes the form of testimony or bearing witness. Testimony plays a constitutive role in the circulation of the human and inhuman, since remaining human is ultimately a question of bearing witness to the inhuman: 'human beings are human insofar as they bear witness to the inhuman' (Agamben 1999, 121). In short, to endure the inhuman is to bear witness to it. It is in this sense that Levi speaks of the Muselmänner as the true witnesses, for they have endured the inhuman, borne more than they should ever have had to bear, and, in doing so, remained fundamentally human. Correlatively, the survivor is human to the extent that they bear witness to an impossibility of bearing witness, that is, of being inhuman. Hence, testimony arises in the non-coincidental currents of the human and the inhuman, as the human being's bearing witness to the inhuman.
This brief overview does not bring out the full complexity of Agamben's theorization of an ethics of testimony – not least because it does not touch on the central role of language in subjectification. Even so, several points can be made here about this approach to ethics beyond biopolitics. The first point goes to the logical coherence of Agamben's approach to the critique of politics understood as biopolitics and ethics understood as law. The focus on the Muselmänner helps to reveal the connections between Agamben's critique of politics and of ethics, since the Muselmänner provide the supreme figure of bare life: they are nonhuman humans stripped of all characteristics of subjectivity, including even the very capacity to enter into language and appropriate the 'I' of the subject. Further, the Muselmänner are wholly abandoned in the manner of homo sacer, singularly exposed to violence and unable to either be murdered or sacrificed. Given this, though, it is curious that Agamben then casts the Muselmann as 'the guard on the threshold of a new ethics, an ethics of a form of life that begins where dignity ends' (Agamben 1999, 69). For, somewhat contrary to the earlier claim that there is nothing in bare life that provides ground for opposition to biopolitics understood as sovereign power, here it appears that bare life does indeed give rise to a new ethics, one appropriate to a happy lived beyond the shadow of the law. We might see this as an example of the Hölderlinian logic that the saving power emerges where danger grows that Agamben draws on at times. But even if this is the right approach, this figuration raises significant questions about the particular relations that are supposed to hold between the different conceptions of life that Agamben works with. For how is it that Agamben's Muselmänner ethics are to become an ethics appropriate to form-of-life? Is it the supposed 'expiation, not merely of guilt, but of law' that divine violence effects that makes this transposition possible? If so, much remains to be said about such expiation.
Second, this prompts doubt about the thorough rejection of ethical concepts such as responsibility, dignity, duty et cetera, and the unassumable non-responsibility that Agamben instead posits as the guiding idea of ethics. Agamben's reasoning for this rejection in the case of responsibility is that it is at core a juridical concept, a point that he makes from the origins of the term in the Latin 'sponsor'. This reasoning displays both a kind of conceptual fundamentalism – this is what the term originally meant, and therefore, what it must always mean – and an unwillingness to entertain more positive, if more fractured and complex, relations between ethics and law. Agamben's rejection of juridicism in ethics is premised on the characterization of law as inherently violent that we saw in Homo Sacer, and the genuinely ethical must be wholly separate from that violence. But if this is the case, how is it that the genuinely ethical arises from that very violence – that is, from the radical desubjectification of the Muselmann, the non-human human, the witness who cannot speak?
Finally, while it seems right that the Holocaust necessitate a radical questioning of the terms of ethics – and in particular, the moral reliance on reason – and may itself elude understanding in those terms, it does not necessarily follow that the conditions of the camps should be taken as providing the requisite material for a new ethics. In fact, there are good reasons to be very wary of attempts to theorize a general approach to ethics from the extreme situation of the camps. As Debarati Sanyal (2002, 6) has argued, Agamben's conflation of the camps with everyday life renders the specific events of the camps – such as the soccer match described by Primo Levi – as '[allegories] for a recurrent, unlocatable and transhistorical violence, one contaminating the civilian world of even a liberal democracy and its daily rituals and spectacles'. Further, this leads to a fundamental indistinction between victim and perpetrator in endless circuits of guilt, and, especially in light of the claim that we are all 'virtually homines sacri' [bare life] (Agamben 1998, 115), this means that we are all always already both victim and perpetrator. This generates a context of constant complicity and undifferentiated moral reversibility, which in Agamben's view renders the attribution or assumption of moral responsibility and judgement impossible. It is in this context that something like the expiation of guilt takes on a real moral value: in the moral world of biopolitics – which is irremediably juridical – we are always already guilty. Thus, a life beyond biopolitics must entail an ethics from which guilt has been excised; in form-of-life, the expiation of guilt is a precondition for the emergence of a genuine ethics of non-responsibility.
Conclusion
While Agamben initially casts his approach to biopolitics as an attempt to 'correct and complete' Foucault's investigations, we can see from this discussion that his eventual approach to biopolitics substantially revises central aspects of Foucault's. Central to this revision is the repositioning of sovereignty in relation to biopolitics, such that sovereignty is foundational to biopolitics. Consequently, Western politics is biopolitical from its inception. Along with this, Agamben proposes the figure of bare life, as a means of grasping the life that is simultaneously expelled from the political sphere and integrated into it. Bare life is caught in this complex logic of abandonment, and, according to Agamben, cannot provide a means to resist biopolitical capture. Instead, Agamben argues that overcoming or escaping biopolitics requires the formation of what he calls 'form-of-life', that is, a life in which the separation of bios and zoē is no longer possible. Only then will the machine of biopolitical violence be stilled. This conception of biopolitics has been enormously productive since publication of the first volume of the Homo Sacer series, itself entitled Homo Sacer. However, Agamben has also been the target of considerable criticism. In particular, his formulation of biopolitics as essentially thanatopolitical has come under attack from various directions, as has his portrayal of concentration camps and the alignment of these with other mundane spaces, such as airports. In the following chapters, I first discuss the work of Hannah Arendt, which is profoundly influential for Agamben, but which also proposed a strikingly different conception of politics as based on the principle of natality or birth rather than death. Following this, I take up contemporary theorists of biopolitics who have moved decisively away from the death-driven view of biopolitics offered by Agamben, toward an 'affirmative biopolitics'.
Notes
Also see Agamben's discussion of the anomic status of sovereignty in State of Exception (2005, 69–71).
Agamben cites Levi 1988, 63–4, see also 82.
References
Agamben, G. (1993). The Coming Community. Trans. Hardt, M. Minneapolis, University of Minnesota Press.
Agamben, G. (1998). Homo Sacer: Sovereign Power and Bare Life. Trans. Heller-Roazen, D. Stanford, Stanford University Press.
Agamben, G. (1999). Remnants of Auschwitz: The Witness and the Archive. Trans. Heller-Roazen, D. New York, Zone Books
Agamben, G. (2000). Means without End: Notes on Politics. Trans. Casarino, C. and Binetti, V. Minneapolis ; London, University of Minnesota Press.
Agamben, G. (2005). State of Exception. Trans. Attell, K. Chicago, University of Chicago Press.
Agamben, G. (2009). The Signature of All Things: On Method. Trans. D'isanto, L. New York, Zone Books.
Agamben, G. (2011). The Kingdom and the Glory: For a Theological Genealogy of Economy and Government. Trans. Chiesa, L. and Mandarini, M. Stanford, Stanford University Press.
Agamben, G. (2013a). The Highest Poverty: Monastic Rules and Form-of-Life. Trans. Kotsko, A. Stanford, Stanford University Press.
Agamben, G. (2013b). Opus Dei: An Archaeology of Duty. Stanford, Stanford University Press.
Aristotle (1998). Politics. Trans. Reeve, C. D. C. Indianapolis, Hackett Publishing Company.
Benjamin, A. (2005). Spacing as the Shared: Heraclitus, Pindar, Agamben. Politics, Metaphysics and Death: Essays on Giorgio Agamben's Homo Sacer. Ed. Norris, A. Durham, Duke University Press: 145–172.
Benjamin, W. (1996). Critique of Violence. Selected Writings, Volume 1, 1913–26. Eds. Bullock, M. and Jennings, M. W. Cambridge, Harvard University Press: 236–252.
Derrida, J. (2009). The Beast and the Sovereign, Volume 1. Eds. Lisse, M., Mallet, M. L. and Michaud, G. Trans. Bennington, G. Chicago, University of Chicago Press.
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Foucault, M. (1972). The Archaeology of Knowledge. New York, Pantheon.
Foucault, M. (1977). Discipline and Punish: The Birth of the Prison. Trans. Sheridan, A. London, Penguin.
Levi, P. (1988). The Drowned and the Saved. Trans. Rosenthal, R. London, Abacus.
Nancy, J. L. (1993). Abandoned Being. The Birth to Presence. Trans. Holmes, B. Stanford, Stanford University Press: 36–47.
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Protevi, J. (2009). Political Affect: Connecting the Social and the Somatic. Minneapolis, Minnesota University Press.
Prozorov, S. (2014). Agamben and Politics: A Critical Introduction. Edinburgh, Edinburgh University Press.
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3Totalitarianism and the political animal
Arendt
Until relatively recently, little serious attention was paid to Hannah Arendt's work in contemporary debates on biopolitics. From one perspective, this is not surprising, since she does not use the term herself. From another, though, this is a true oversight, since as Julia Kristeva argues, life is the 'essential domain' of Arendt's thought, and this thematic is the central guide to her discussions of political history and metaphysics (Kristeva 2001a, 3–4). For instance, in The Origins of Totalitarianism, Arendt proposes a critique of Nazism and Stalinism that sees in them a shared contempt for human life, culminating in the destruction of all that is truly human in a life. In contrast, and deeply connected to this analysis, The Human Condition develops a critical defence of the human over and against the historical reduction of it to nothing more than biological necessity, whether of the individual or of the species. Further, her work is explicitly influential for some contemporary theorists, and ignored at a cost when interpreting others. For instance, Agamben acknowledges the profound influence of Arendt's work on his own, but claims in Homo Sacer that Arendt was unable to see the true political implications of her own analyses. Foucault, on the contrary, disavows any Arendtian influence in his work, but a fruitful comparison can be made of their thinking about life and politics, especially in terms of racism. Given this, this chapter makes a case for greater consideration of her work within debates on biopolitics.
I aim to draw out her contribution to the theorization of the intersection of life and politics through a discussion primarily of Origins of Totalitarianism and The Human Condition, in which she is most explicitly engaged with what are now central themes of biopolitics. In the first section of this chapter, I discuss The Origins of Totalitarianism, to provide a brief outline of her main thesis about the emergence of totalitarianism as a particular configuration of political power. I also discuss her influential comments on statelessness and human rights. In the second section, I focus on her alternative formulation of politics as action developed in The Human Condition. Throughout this discussion, I trace her revised Aristotelianism and the implications of this for thinking about the intersection of politics and life. Further, I reflect on her valorization of action over and against the perceived necessities of natural life and labour. In the final section of the chapter, I consider the implications of Arendt's analysis of totalitarianism, and her approach to politics developed in The Human Condition, for contemporary debates about biopolitics. I suggest that Arendt's work provides a key counterpoint to discussions by figures such as Agamben, Esposito and Foucault. I also suggest that critical discussions of these theorists would be enriched by greater attention to the prior work of Arendt.
The origins of totalitarianism
Arendt's Origins of Totalitarianism is neither a straightforward history of totalitarian states, nor a standard theorization of totalitarianism as a political ideology. Instead, it provides a complex map of various factors that contributed to the historical possibility of totalitarianism, including large-scale historical shifts such as the imperialist era, small scale details of the peculiar position of Jews in relation to fin de siècle high society, and the ideas that underlay and emerged from these. As she describes it, her method was to give an 'historical account of the elements which crystallized into totalitarianism... followed by an analysis of the elemental structure of totalitarian movements and domination itself' (Arendt 1953, 78). None of these elements cause, or provide the origins of, totalitarianism in themselves; rather, this singular form of political organization emerges from the circumstantial crystallization or amalgamation of them. Arendt provides a genealogy of totalitarianism that elucidates the conditions of possibility of the contemporary without strictly identifying causes or origins. She sets out in Totalitarianism to provide an overview of a complex of factors or elements in modernity, the particular configuration of which made possible the emergence of totalitarian governments – in the form of the Nazi regime in Germany and the Stalin regime in Russia – and more specifically, the project of the destruction of European Jewry undertaken by the Nazis. Broadly speaking, these elements fall into three categories that are interconnected in various ways: imperialism and the ideology of expansion; the emergence of racism as a political doctrine; and the destruction of the nation–state. Since it is not possible to give a thorough overview of her account of the contributing factors and their interconnections here, I will instead briefly describe some of the main threads of her argument, especially those that have bearing on subsequent discussions of biopolitics.
For Arendt, the imperialist era contributed several key factors to the eventual emergence of totalitarianism, although it did not directly lead there. Nevertheless, she suggests that prior to the imperialist era, 'there was no such thing as world politics, and without it, the totalitarian claim to global rule would not have made sense' (Arendt 1968b, ix). Of particular importance is the notion of 'expansion for expansion's sake', which Arendt claims was epitomized by the approaches of British imperialists such as Cecil Rhodes in Africa, and to a lesser extent, Lord Cromer in Egypt. While the imperialist aspirations and methods of each of these men differed in various ways, they shared, according to Arendt, the 'discovery of an expansion which was not driven by the specific appetite for a specific country but conceived as an endless process in which every country would serve only as [a] stepping-stone for further expansion' (Arendt 1968b, 95). What matters, then, is expansion per se, which is significantly different from notions such as the expansion of national territories through the annexation of neighbouring areas for instance. What is significant about this is that expansion is really only limited by the size of the planet, a limitation that Rhodes is said to have lamented. For Arendt, this notion of expansion 'as a permanent and supreme aim of politics' was entirely new, and had immense consequence for modern politics, since it introduced the possibility of a world politics that ultimately contributed to the disintegration of the nation–state system. Before considering this process of disintegration further, though, imperialism generated two further elements that made totalitarianism possible.
One of these was bureaucracy as a system of governance. The other was racism. According to Arendt, the first of these 'political devices' was 'discovered' in Algeria, Egypt and India, while the second developed initially in Africa. In her provocative but incomplete analysis, Arendt argues that race and bureaucracy were interrelated in numerous ways where the British were confronted with the problem of ruling foreign populations. Whereas race emerged as a 'principle of the body politic' to explain perceived natural hierarchies within humanity, bureaucracy was a response to the governmental problem of the simultaneous domination and protection of peoples perceived to be inferior. As a form of imperial governance, bureaucracy had several characteristics that foreshadowed certain traits of totalitarianism. First, a profound reliance on secrecy undermined the need for public accountability and gave priority to personal influence and rule by temporary decree. Second, bureaucratic administration was instrumentalizing insofar as the governed were merely stepping stones in the process of expansion, and those involved in administration were required to identify themselves wholly with the law of expansion. These factors contributed to a kind of 'aloofness' of the British imperial services, breaking all links between the governed and the governing (Arendt 1968b, 92–6).
The emergence of racism as a political principle contributed greatly to this break, since it not only legitimized it but also extended it into a break within humanity itself. In Arendt's account, race-thinking significantly pre-dates the consolidation of racism as a political principle, and derives from eighteenth-century French thinkers such as Henri de Boulainvilliers and especially Arthur de Gobineau, who sought to naturalize the social hierarchy of the aristocracy. Race-thinking was also advanced by the interventions of Charles Darwin and the extrapolation of his evolutionary theory to social circumstance. While these figures were significant in the formation of European race-thinking, though, racism proper did not come into effect until the imperial era, and indeed, may not have come into existence but for the experience of imperial colonialism. Controversially, Arendt argues that in this context racism was a comprehensible response on the part of English and Europeans who found themselves confronted by aspects of humanity with whom they could not identify. It was also an extremely effective device for social and political organization and subsequent domination, if not obliteration, of one race by another. She writes, '[r]ace was the Boer's answer to the overwhelming monstrosity of Africa... an explanation of the madness which grasped and illuminated them like "a flash of lightning in a serene sky: 'Exterminate all the brutes!'"' (Arendt 1968b, 65; citing Conrad 2006).
Yet, though racism developed as a political device in the context of imperialism, it extended beyond this context to take a different form in continental Europe, typified by the Pan-Germanic and Pan-Slav movements of the early twentieth century. Of these, Arendt writes that continental imperialism 'started with a much closer affinity to race concepts, enthusiastically absorbed the tradition of race-thinking, and relied very little on specific experiences. Its race concepts were completely ideological in basis and developed much more quickly into a convenient political weapon' (Arendt 1968b, 104). Interestingly, Arendt argues that despite the lack of attention paid to homegrown Pan movements in scholarly discussion of imperialism, these movements were fundamental to the later development of totalitarianism in the forms of Nazism and Stalinism (for further discussion of this, see Moses 2011). Indeed, in her view, continental imperialism shared several characteristics with international imperialism that proved central to the subsequent development of totalitarianism.
In addition to the emphasis on expansion and the mobilization of racism as a political device discussed already, this included a contempt for the political formation of the nation–state. Arendt argues that 'continental imperialism was and remained unequivocally hostile to all existing political bodies' (Arendt 1968b, 105), including state formations and party politics. Hence, while Pan movements were centrally motivated by a kind of nationalism, this was not the nationalism that underpinned the notion of nation–states and the system of state sovereignties that emerged throughout the nineteenth century. It was rather a 'tribal nationalism' of those people 'who had not participated in national emancipation and had not achieved the sovereignty of the nation–state' (Arendt 1968b, 107). Growing out of an experience of rootlessness and the lack of political community achieved through state sovereignty, tribal nationalism emphasized the inner characteristics of people and 'absolute claims to chosenness' (Arendt 1968b, 113). The origins of this chosen-ness could be conceived of as divine or natural, and bore an inherent affinity with racism, which determined that one race was chosen over and against another, and consolidated in ideas of a 'master race'.
For Arendt, the catalytic event that crystallized these elements into the foundational conditions of totalitarianism was the First World War and the destruction of the European nation–state system that followed in its wake. In her view, post-war politics were characterized by a 'vague, pervasive hatred' (Arendt 1968b, 148) that ultimately undermined the new states formed after the downfall of the Hapsburg Empire and Czarist Russia. Further, post-war politics saw the emergence of minorities within the nation–state system, as well as a massive increase in the numbers of stateless people. Arendt postulates that the mood of hatred turned into a fully-fledged political rationality around these two groups.
In regards to minorities, Arendt points to a problem in the notion of the nation–state that is implicated in the subsequent fate of the nation–state system established after the First World War. In short, the nation–state system seeks to give political organization to a group of people identified through heritage and territorial claims; this means, however, that considerable swathes of people within a sovereign territory – a state – might not actually belong to the nation. The response of the League of Nations to the problem of protecting the rights of minorities within newly created states took the form of Minority Treaties, which generally came into force after the Paris Peace Conference in 1919, as a condition of diplomatic recognition. While minorities in newly created states such as Yugoslavia and Poland (as well as some previously existing states) made subject to the Minority Treaties were supposed to be governed and protected by the treaties, these were in Arendt's view wholly inadequate for dealing with the situation of combined nationalities within a state territory. Further, they actually introduced a legal exception into the nation–state system, thereby making apparent the fact that 'only nationals could be citizens, only people of the same national origin could enjoy the full protection of legal institutions, that persons of different nationality needed some law of exception until or unless they were completely assimilated and divorced from their origin' (Arendt 1968b, 155). This situation of legal ambiguity and fragility – and, hence, the fragility of rights – was not thoroughgoing, however; in Arendt's words, 'minorities were only half stateless' (Arendt 1968b, 156). Much more consequential for the nation–state system, and for tools such as the articles of human rights treaties, was the massive increase in refugees in the era following the First World War, who Arendt describes as 'the most symptomatic group in contemporary politics' (Arendt 1968b, 157).
In Arendt's account, the rise of numbers of refugees in the era after the First World War had an etiolating effect on the nation–state system in several ways. First, the response of many states to this situation led to the abolition of the right of asylum – the 'symbol of the Rights of Man in international relationships' (Arendt 1968b, 160). Of this Arendt argues that while the right of asylum continued to function in the new nation–state system, it was felt to be anachronistic, and in conflict with the rights of the state in the international sphere. Thus, it was never written into law or international agreements, or the Covenant of the League of Nations. Second, the options for dealing with refugees, in the forms of renaturalization and repatriation, were ultimately revealed to be impossible. Of the former, many refugees simply refused to give up their national identity in order to be assimilated into another nationality. Further, in the face of the prospect of mass naturalizations, the states, rather than extending naturalization to at least some, actually cancelled prior naturalizations, thereby undermining confidence in the entire process of naturalization. Repatriation failed because neither the country of origin nor any other state agreed to accept the stateless person, effectively leaving them in a legal limbo – the stateless were not recognized legal subjects of any state that could or would therefore provide at least a modicum of legal protection. This meant that stateless persons remained a fundamental anomaly within the state system – neither able to be assimilated nor deported, protected neither by the state of residence nor by any other. This status of legal anomaly ultimately revealed a fundamental problem in the structures of rights that supposedly protected humans qua humans.
Arendt's comments on statelessness and the implications of it for doctrines of human rights are now probably the most widely recognized from her various discussions in Totalitarianism. Provocatively, she argues that the condition of statelessness revealed the essential link between human rights and citizenship rights, since even the former required a state to give them authority. Thus, while minorities made clear the connection between citizenship and national belonging, the stateless revealed a more radical problem in that even those rights that were supposed to hold simply by virtue of being human actually proved to be a version of citizenship rights, since they could only be upheld or enforced by states. As Arendt (1968b, 172) writes, human rights are
defined as 'inalienable' because they were supposed to be independent of all governments; but it turned out that the moment human beings lacked their own government and had to fall back upon their minimum rights, no authority was left to protect them and no institution was willing to guarantee them.
The crux of the issue here, though, is not simply that the stateless had lost the protection of a given state, but that it was impossible for them to find another state that would offer such rights: it was not the loss of home and polity that was unprecedented, but the impossibility of finding a new home or polity.
This gives rise to a stringent critique of human rights from Arendt. In her view, this impossibility effectively set the stateless outside the human community, and deprived them not only of rights, but of 'the right to have rights, or the right of every individual to belong to humanity' (Arendt 1968b, 178; emphasis added). She argues that human rights are supposed to pertain to humans merely by virtue of being human. However, when faced with a situation in which growing numbers of people had nothing to fall back on but their mere humanity, '[t]he world found nothing sacred in the abstract nakedness of being human' (Arendt 1968b, 179), and the system of human rights broke down. This situation of mass denaturalization and the suspension of human rights ultimately paved the way to one of the central legal manifestations of totalitarianism, whereby large swathes of populations were effectively set outside the legal system of states and rights, or in Agamben's terms, abandoned to and by the law.
However, before considering totalitarianism in more detail, it is first important to account for one further 'element' in Arendt's analysis. One of the principal challenges that Arendt sets for herself in the Totalitarianism project is to avoid an explanation of the Nazi's attempted extermination of European Jewry that renders it either historically inevitable or, conversely, entirely contingent that Jews in particular were targeted in this way. This challenge means that the Holocaust cannot be positioned as the result solely of a dynamic intrinsic to modernity, even if it would not have been possible without certain characteristics of modern politics and rationality in place. Nor, however, can its emergence be seen as strictly related to the idiosyncrasies of German politics and culture after the First World War. Nazism was world-historical in a way that this interpretation would deny. This requires a careful analysis of the particularities of anti-Semitism as a political ideology, and of the relationship between the European Jewry and the nation–state, since in Arendt's analysis the fate of the Jews was intimately tied to the fate of the nation–state system itself. Consequently, Arendt sketches out a genealogy of modern anti-Semitism and its role in the emergence of totalitarianism, which covers the function and relationship of Jews to the nation–state and the social role of the Jew in fin de siècle high society, culminating in the Dreyfus affair in late nineteenth-century France, in Part One of Origins of Totalitarianism, entitled Antisemitism (Arendt 1968a).
In her discussion of the Jewish relation to the state in this book, Arendt highlights the reliance of the nation–state system on Jewish financiers, who acted as genuine 'Europeans' in that they could negotiate between states without being identified with one or another. In this way, Jews were able to exert considerable influence through their financial interests, but did not take up the mantle of political power and engagement. Effectively, even while propping up its existence, Jewish financiers remained somewhat aloof from the state and the public display of power entailed by it. This alliance between the Jewish financiers and the state was further inflected by the social position of Jews, of which, Arendt argues, Jews were to a large extent outside society, a position that they fostered for themselves in various ways. This outsider status was actually reinforced by the arrival of a small number of Jews in high society, since, Arendt compellingly argues, such Jewish arrivals were only able to enter into society by virtue of a double movement – they were allowed entrance because of their Jewishness (in a manner similar to other exotic 'vices'), but at the same time, they had to erase or deny their connection to Jewish society. This double logic is well illustrated in her discussions of Marcel Proust in French fin de siècle high society, as well as Benjamin Disraeli in Britain, and captured in her description of Jews entering into Gentile society as either 'pariah or parvenu'.
The effect of these positions in relation to the state and society was that the Jews appeared as a group unto itself, 'ruled by mysterious laws, held together by mysterious ties, and aspiring to a mysterious rule "behind the scenes"' (Arendt, cited in Canovan 1994, 44). This made the Jews particularly vulnerable to scapegoating in the condition of generalized hatred in the era following the First World War. Even so, Arendt is at pains to argue that Jews were not simply victims within this; rather, she controversially argues that they were at least partly culpable for their fate at the hands of the Nazis. This is because of their perhaps wilful political naivety and failure to read the mood of the times. Further, the Jewish notion of themselves as a chosen people integrated all too well with the racial notions in circulation, and, moreover, came into direct conflict with the Nazi ideology of Aryan racial superiority and predestination to world domination. In this and other ways, anti-Semitism ultimately acted as the 'amalgamator' of the various elements described above to produce conditions ripe for totalitarianism, and Nazism in particular.
For Arendt, totalitarianism is a very specific political formation that cannot be seen as co-extensive with ordinary dictatorship, despotism or authoritarianism, including the contemporaneous Fascist regime of Mussolini in Italy. One of the things that set totalitarianism apart from other forms of political domination is its global aspiration, a tendency in line with the imperialist drive to expansion for expansion's sake. Importantly, this means that totalitarian regimes were not nationalistic in a true sense, though they may have used nationalistic rhetoric a times. For example, Arendt argues that Hitler was not honestly concerned with Germanic domination of the world, but with the domination by one non-national race – the Aryan – of all others. In effect, all countries, whether enemies or allies, were only stepping stones in the ultimate aim of global domination. Further, totalitarianism, as its name suggests, drives at total domination, which itself has several central characteristics in Arendt's account. Especially notable is the belief that 'everything is possible', which is intimately linked to the destruction of human nature through the suppression of spontaneity or contingency. The belief that everything is possible that marks totalitarian regimes cannot be reduced to the nihilistic slogan that 'everything is permissible'; rather, it signals something unprecedented about totalitarian regimes.
The key to understanding totalitarianism in Arendt's view is the concentration camp, especially its peculiar, unprecedented, social world, where destruction took precedence over 'normal' utilitarian political and economic goals and the standard parameters of human existence were fundamentally shattered. In particular, the camps constituted an experimental field in which the totalitarian regimes of Nazism and Stalinism were able to most rigorously pursue and enact the belief that 'everything is possible' (Arendt 1968c, 135). This does not mean as closed worlds the camps were the most susceptible to propaganda, or ideological indoctrination into the world of the totalitarian blurring of fact and fiction; in fact, propaganda had little if any place in the camps themselves. Instead, the camps verified the totalitarian ideology, allowing for the realization of total domination, which was only partially realizable outside of them. As an 'indecent experimental inquiry into what is possible' (Arendt 1968c, 134) the camps were unprecedented and without analogue – the attempt to draw parallels between the concentration camps of the Nazis and Stalinist regimes and other forms of camps, including the concentration camps of the Boer War, serve only to mislead and confuse. For such parallels miss the essential characteristic of totalitarian camps, which aim at the destruction of human nature itself. They not only served to isolate some people from the normal social world, ultimately making mass extermination possible, but also served to destroy the characteristics that make human beings human, as opposed to mere biological things. Central to this destructive project was the total suppression of spontaneity as an expression of human uniqueness. Thus, Arendt writes, the camps 'are meant not only to exterminate people and degrade human beings, but also serve the ghastly experiment of eliminating, under scientifically controlled conditions, spontaneity itself as an expression of human behaviour and of transforming the human personality into a mere thing' (Arendt 1968c, 136).
The project of turning the human being into a mere thing proceeded through three steps. The first of these was the destruction of the juridical person, by virtue of which a human being is both constrained and protected by the law. This takes place through two related mechanisms – first, certain categories of people such as the stateless are put outside the juridical system and recognition of their essential lawlessness is enforced; second, the camps themselves were placed outside the normal juridical sphere – as exceptional spaces that cannot be ruled by normal order – and those interned in them selected outside the normal juridical procedure, without reference to any crime, for instance. Arendt argues that the intermixing of criminals with political prisoners and, eventually, people who were entirely innocent of any crime established everyone as deserving of camp internment, regardless of their individual actions. Further, the predominance of the innocent, for whom there is no link between actions and arrest, within the camp reveals the essential feature of the camp in the suspension of the law, and therefore of all and any rights, and the destruction of legal personhood as a prerequisite to total domination (Arendt 1968c, 145–9).
The second stage of the destruction of human nature focuses on the moral dimension of personhood. The principle mechanisms for this are, first, the anonymization of death, and, second, the mass production of moral complicity. Of the first, Arendt argues that '[t]he concentration camps, by making death itself anonymous... robbed death of its meaning as the end of a fulfilled life. In a sense, they took away the individual's own death, proving that henceforth nothing belonged to him and he belonged to no one. His death merely set a seal on the fact that he had never really existed' (Arendt 1968c, 150). This elimination of existential meaning was further reinforced by the elimination of conscience in the production of complicity. By confronting prisoners with intolerable and impossible moral dilemmas between killing some or killing some others – epitomized in Sophie's Choice, in which a mother is forced to choose which of her children should die and which live – the Nazi camps ultimately undermined moral capacity altogether. Further, this production of complicity was systematized in the Kapos: camp prisoners enlisted to usher other prisoners into the gas chambers and tend to the corpses, supervise forced labour and undertake other camp administration. This system of enforced complicity 'constantly blurred' the 'distinguishing line between persecutor and persecuted, between the murder and his victim' (Arendt 1968c, 151) and rendered individual conscience irrelevant.
The third and final stage of the destruction of human nature entailed the destruction of personal uniqueness, and along with it, the absolute suppression of spontaneity. Arendt argues that this third phase follows easily from the preceding two, but is even more central to understanding the operation of totalitarian power. Importantly, the destruction of personal uniqueness is integrally related to the suppression of human spontaneity, and thus freedom. As Arendt writes,
to destroy individuality is to destroy spontaneity, man's power to begin something new out of his own resources, something that cannot be explained on the basis of reactions to environment and events. Nothing then remains but ghastly marionettes with human faces, which all behave like the dog in Pavlov's experiments, which all react with perfect reliability even when going to their own death, and which do nothing but react.
(Arendt 1968c, 153)
The destruction of individuality and spontaneity is the final and central horror of totalitarianism, since '[t]otal power can be achieved and safeguarded only in a world of conditioned reflexes, of marionettes without the slightest trace of spontaneity' (Arendt 1968c, 155). The importance of this is that it means that totalitarianism is not only a machine for transforming the external world, but also for transforming human nature itself: as she writes in the second edition published in 1958, '[h]uman nature as such is at stake' (Arendt 1968c, 157). This understanding of totalitarianism has profound implications for Arendt's positive construal of politics, developed in her subsequent book, The Human Condition, to which I now turn.
Politics as action: _The Human Condition_
Initially published in 1958, Arendt's work The Human Condition constitutes an 'impassioned defense' of human life in the face of its destruction by totalitarianism. Initially conceived as an extended critical engagement with Marx to supplement the analysis of Stalinism in Origins, the book Arendt ultimately wrote presents a broader analysis of the theoretico-historical trajectory of Western political thought. Methodologically, the book is a significant move away from Origins of Totalitarianism; however, its central concern with human activity and its relation to life derives directly from her analysis of totalitarianism and the destruction of human spontaneity that it entailed. The central concept of The Human Condition is that of vita activa, which encompasses what Arendt sees as the 'three fundamental human activities' of labour, work and action. Of these, the third is most important, since action affords for Arendt a fundamental rebuttal of the potential for total domination as outlined in Origins. At the same time, she recognizes a basic ambiguity in the idea, in that totalitarianism is itself a new beginning, one that may itself be encapsulated in the idea of action. Thus, she proposes a revaluation of action, but in a way that is sensitive to both the promise and danger of it. In the course of discussing action in this section, we will also have occasion to consider several other notions that operate within The Human Condition. These are: the insistence on plurality as the fundamental condition of humanity; the importance of natality or new beginnings; the distinction between what and who one is; and, finally, the story that Arendt tells about the distinction between the public and private spheres and the rise of the social.
For Arendt, the concept of vita activa stands in contrast to vita contemplativa and designates a basic 'unquiet'. From this general starting point, she breaks it down into the three categories of labour, work and action, each of which will be discussed in turn here. Labour is in a sense the most basic form of human activity, and Arendt defines it as 'the activity which corresponds to the biological process of the human body, whose spontaneous growth, metabolism, and eventual decay are bound to the vital necessities produced and fed into the life process by labor. The human condition of labor is life itself ' (Arendt 1998, 7). As this indicates, labour is essentially directed toward the satisfaction of biological needs. It has no lasting effect, and its results are 'as quickly consumed as the effort is spent'; it 'produces objects only incidentally and is primarily concerned with the means of its own reproduction... it never "produces" anything but life' (Arendt 1998, 87, 88). Further, labour is inherently repetitive and menial, aspires to nothing beyond itself and reveals nothing about the nature of the actor beyond the existence within them of animal needs. In this, Arendt figures it as principally the activity of the oikos, or domestic sphere. As this indicates, Arendt sees labour as having little value beyond that of meeting the needs of humans as animal beings. It is not creative, and cannot rise above animal conditions to contribute to world-making. As we shall see in a moment, this understanding of labour underpins her critique of modernity, which she sees as inappropriately valuing labour at the expense of genuinely political activity.
The second category of work is in Arendt's view typically and unfortunately subsumed into the category of labour, and one of her tasks in The Human Condition is to attempt to establish a distinction – one she concedes is 'unusual' – between labour and work. Arendt wants to separate out the category of work in order to allow for a differentiation between homo faber and animal laborans – or, more pointedly, to begin to analyze the conditions that make humans human, over and above their animal being. As this suggests, these conditions are essentially fabricated, and Arendt defines work as 'the activity which corresponds to the unnaturalness of human existence' (Arendt 1998, 7). She goes on to claim that work 'provides an "artificial" world of things, distinctly different from all natural surroundings. Within its borders each individual life is housed, while this world itself is meant to outlast and transcend them all. The human condition of work is worldliness' (Arendt 1998, 7). As this indicates, work entails fabrication of durable things, but more specifically and importantly, it contributes to the fabrication of the world. This touches on the distinction that Arendt makes between earth and world, where the former refers to the natural conditions of existence and the latter points to the artefactual, and indeed, artificial, aspects of human existence. Here, though, artificial does not have the negative connotations that it often has; rather, it highlights the way that humans are invested in the creation of the conditions of their lives beyond natural necessity. Further, this suggests that the humanness of the human is itself artefactual, insofar as the humanity of the human comes into force through institutions and things that are themselves manmade. However, Arendt also views work as driven by utilitarian concerns and means-ends logic. While these obviously have a place in human affairs, it becomes problematic if all domains of human experience are subject to this logic, since this would make all things mere means to another thing.
Arendt poses the final category of action against the extension of this utilitarian logic of work to politics, to reveal an authentic politics that is not reducible to government and bureaucratic management. Action makes the essential artifice of the human especially apparent, for it is in action that humanity is most fully realized and revealed. Arendt (1998, 174) writes, 'the measure [of humanity] can be neither the driving necessity of biological life and labor nor the utilitarian instrumentalism of fabrication and usage'. Instead, while these are requisite, only action can reveal the true character of humans as political beings, which it does through an essential connection to plurality. Arendt summarizes her understanding of action and its significance thus:
Action, the only activity that goes on directly between men without the intermediary of things or matter, corresponds to the human condition of plurality, to the fact that men, not Man, live on the earth and inhabit the world. While all aspects of the human condition are somehow related to politics, this plurality is specifically the condition – not only the conditio sine qua non, but the conditio per quam – of all political life.
(Arendt 1998, 7)
In a simple sense, plurality means only that there is more than one. However, beyond this, human plurality cannot be reduced to sheer multiplicity since this would leave humans at the level of all other organic things.
Instead, what is at stake in human plurality is the appearance of unique beings to each other, an appearance that Arendt argues could not take place without speech and action. As she writes,
human plurality is the paradoxical plurality of unique beings. Speech and action reveal this unique distinctness... they are the modes in which human beings appear to each other, not indeed as physical objects, but qua men... This appearance, as distinguished from mere bodily existence, rests on initiative, but it is an initiative from which no human can refrain and still be human.
(Arendt 1998, 176)
As this suggests, Arendt posits a very tight relation between action and speech – she often mentions them together – though this does not mean that they are interchangeable. Just as action may take forms that do not partake of speech, so not all manifestations of speech constitute action (think here of Heidegger's notion of idle talk in Being and Time, for instance). Thus, while deeply imbricated, speech and action have distinct roles in plurality and politics, insofar as action is more strongly associated with natality, and speech with the appearance of uniqueness (though, again, these are not mutually exclusive associations).
For Arendt, one of the central features of action is the capacity therein for new beginnings, which, she suggests 'corresponds to the fact of birth' and which consequently makes natality – not mortality – the central category of politics (Arendt 1998, 178). While of considerable significance in The Human Condition, Arendt's discussions of the concept of natality remained fragmentary and suggestive. As Miguel Vatter (2006, 138) notes, natality is rarely subject to sustained analysis, either by Arendt herself or by commentators, who take the relation between action and natality to be either too self-evident or too obscure to elaborate in detail. In essence, though, it refers to the fact that a human life begins with the event of birth, an event that can be neither controlled by the one who is born, nor entirely anticipated by others involved (new genetic technologies and the desire for control enacted in them notwithstanding). This surprise or unexpected beginning of birth is understood by Arendt as a kind of 'miracle', which introduces an element of 'weak messianism' into her work (Young-Ah Gottleib 2003). In terms of its political significance, natality's implications are manifold. First, it underpins the somewhat idiosyncratic conception of freedom that Arendt proposes, whereby freedom is dissociated from notions of will and instead seen as a kind of virtù, or excellence manifest in action, that links freedom directly with spontaneity. Second, natality is necessarily linked to plurality since birth requires more than one, and, as I will discuss further in a moment, this makes it foundational to the appearance of individual uniqueness in the public sphere.
I will return to the implications of the concept of natality later, but now it is time to consider the role of speech in plurality and politics. In this regard, at first glance, it appears that Arendt remains committed to the basic Aristotelean claim that it is by virtue of having speech that 'man' is a political animal. However, for Arendt it is not so much the mere having of speech that is important, but what speech does – and this is that it makes possible the appearance of unique beings in the public sphere or what she alternatively refers to as the 'space of appearance'. Thus, speech is significant not because of a specific capacity for persuasion or rational communication but because of its revelatory quality; speech reveals the singularity of plural beings, and, further, without this revelatory capacity action and speech would lose their human relevance. The idea of singularity that Arendt is using is encapsulated in the distinction that she makes between 'what' and 'who' someone is, where the former refers to 'qualities, gifts, talents and shortcomings', including physical characteristics of the body, that may be shared with others, the description of which amounts to describing a 'character' or 'type'. The latter notion of 'who' someone is gestures instead at the absolute uniqueness of a person, or what Arendt refers to as 'the living essence of the person as it shows itself in the flux of action and speech' (Arendt 1998, 181).
The problem of the notion of who, though, is that it is in a sense revealed but nevertheless resistant to description; Arendt (1998, 181) writes,
[t]he manifestation of who the speaker and doer unexchangeably is, though it is plainly visible, retains a curious intangibility that confounds all efforts toward unequivocal verbal expression. The moment we want to say who somebody is, our very vocabulary leads us astray into saying what he is.
Interestingly, Arendt ultimately reaches the conclusion that who someone is can only be revealed in biographical narrative, which is itself an activity that can only be fully realized after death, and thus necessarily requires others to tell that story (Cavarero 2000). Consequently, action is centrally concerned with relations with and between others, or what Arendt calls the 'inter-est, which lies between people and therefore can relate and bind them together' (Arendt 1998, 182).
As we can see, then, Arendt's concept of action integrates a number of other notions in a tightly bound web, and the relations between these are not always clear. Given this, it is worth quoting her at length to summarize the imbrication of speech, action, natality and uniqueness in plurality, itself both the necessary and sufficient condition for politics. She writes,
The new always happens against the overwhelming odds of statistical laws and their probability, which for all practical, everyday purposes amounts to certainty; the new therefore always appears in the guise of a miracle. The fact that man is capable of action means that the unexpected can be expected from him, that he is able to perform what is infinitely improbable. And this again is possible only because each man is unique, so that with each birth something uniquely new comes into the world. With respect to this somebody who is unique it can be truly said that nobody was there before. If action as beginning corresponds to the fact of birth, if it is the actualization of the human condition of natality, then speech corresponds to the fact of distinctness and is the actualization of the human condition of plurality, that is, of living as a distinct and unique being among equals.
(Arendt 1998, 178)
Thus, while not directed at the production of durable goods – indeed, action may at times be the most ephemeral of all the activities – it is nevertheless key to understanding what is distinctive about humans in Arendt's view. For what this ultimately means is that men, not 'Man', are the subject of politics – that is, men in their plural singularity rather than in their species-being. As such, action is also a crucial link between the Origins of Totalitarianism project, and The Human Condition. For the final horror of totalitarianism in Arendt's view was the destruction of the possibility for human action, and along with that, the total suppression of the appearance of plural singularities within a public sphere, epitomized in narrative. This dual suppression of human spontaneity and the appearance of uniqueness essentially erased the very conditions of bios, or specifically human life beyond biological necessity; in short, totalitarianism reduced men to their species-being in Man. It is in this sense that Arendt argued that human nature was at stake in totalitarianism.
With this triumvirate of labor, work and action in hand, Arendt's critical thesis about modernity comes to the fore. The driving thesis of The Human Condition is that Western political history has been marked by two tendencies, the first of which is the devaluation of action and correlative over-valuation of labour. According to Arendt, the relative valuations of labour and action have effectively been reversed since the ancient Greek prioritization of action in the polis, such that in modernity labour takes precedence. Interestingly, this forms the crux of Arendt's critique of Marx elaborated in The Human Condition, since she sees in his theory of capital an uncritical valuation of labour and animal laborans. The second tendency is the rise of the social as a specific category of human experience, which has ultimately obscured the political as the authentic space of appearance. Arendt argues in the first chapter of The Human Condition that whereas the ancient Greeks operated with a strong distinction between the public and private spheres that made possible authentic political appearance, the distinction has since been over-ridden by the notion of the 'social', which is associated with activities formerly restricted to the private sphere and pertaining to the reproduction of the life of the species (see Pitkin 1998). In conjunction with the expansion of the economy (oikonomia, the ordering of the oikos) in the eighteenth century, this has transformed the public sphere into a realm of the satisfaction of material needs, rather than the space of appearance.
Against these tendencies, Arendt urges the revaluation of action, which she sees as a positive defence against the totalitarian suppression of spontaneity and the appearance of uniqueness, that is, of action. Even so, her revaluation of action is tempered by the recognition of dangers within action itself, since it is by definition unpredictable and its consequent effects unknown. Because of this thesis of the precedence of the conditions of life at the expense of the political, it has been suggested that Arendt is the first contemporary theorist of biopolitics. While this may help to restore her to the centre for these debates, in itself this statement does little to illuminate the specific ways in which her work anticipates, foreshadows and, in some cases at least, sets out problems for more recent interventions. In order to begin to trace this influence, in the following section I will briefly mention some points of contact between her work and that of Agamben, Foucault and others.
Arendt and biopolitics
There are clear traces of the influence of Arendt's thought in Agamben's theorization of biopolitics, and he readily admits this, in, for instance, comments in the introduction to Homo Sacer. This influence is also evidenced by an early letter from Agamben to Arendt, recently published alongside his essay, 'On the Limits of Violence' (Agamben 2009). Despite this, remarkably little critical energy has been spent on disentangling Agamben's debt to Arendt – in contrast to the attention paid to his intellectual relationship to Foucault, Benjamin, Derrida, Heidegger and Schmitt for instance. This may be because his comments in Homo Sacer appear to dismiss her project as adding little to the conceptualization of biopolitics. But this is misleading, if not disingenuous, since significant aspects of his account of biopolitics are foreshadowed in The Origins of Totalitarianism, not least his central claim about the appearance of bare life in modern politics. In fact, there are several especially notable crossovers in Arendt and Agamben's respective discussions, where her views are deeply formative for his. For instance, as we will discuss in Chapter 5, Agamben's critique of human rights emerges directly from Arendt's critique presented in Origins. Further, his characterization of the camps is very closely related to Arendt's presentation, though this is not to say that their views are ultimately consistent.
In his discussion of the Nazi concentration camps in Homo Sacer, Agamben notes Arendt's insight that the totalitarian principle that 'everything is possible' is central to understanding the camps, and makes two claims that echo the insights of Arendt outlined above: first, the camps emerged from a state of exception, wherein normal legal rule is suspended and extra-juridical confinement is normalized; and, second, the inhabitants of the camps were 'stripped of every political status and wholly reduced to bare life' (Agamben 1998, 171). Further, in Remnants of Auschwitz, Agamben extends on Arendt's portrayal of the camps as an 'experiment on the possible', suggesting that they constitute a biopolitical experiment on the 'operators of Being'. By this he means that modal categories such as possibility, impossibility, contingency and necessity are 'ontological operators, that is, the devastating weapons used in the biopolitical struggle for Being' and, where the battlefield for this struggle is subjectivity itself (Agamben 1999, 146–147). Agamben argues that possibility ('to be able to be') and contingency ('to be able not to be') should be understood as 'operators of subjectification' which indicate 'the point at which something possible passes into existence'. Opposing these are the 'operators of desubjectification', impossibility and necessity, both of which entail negation in that the former indicates the negation of being able to be and the latter, the negation of being able not to be. As tools in the biopolitical battle for Being, these operators isolate and divide the possible and the impossible in subjectivity; more specifically, they 'divide and separate' the 'living being and the speaking being'. From this perspective, the devastating novelty of the concentration camps is that they represent the field in which 'the impossible is forced into the real' and contingency is radically negated (Agamben 1999, 148). To summarize, Nazi camps such as Auschwitz were a biopolitical experiment in making the impossible possible through the negation of the possibility of not being.
As this reveals, Agamben's interpretation of the camps leans more toward a metaphysical rather than political account (although his view of the interrelation of the metaphysical and political should be kept in mind). It is in this tendency and the generalization it precipitates that Agamben and Arendt often part ways. For instance, Agamben also argues that the camps constitute the 'nomos of the modern' and sees a similar logic of legal indeterminacy and exception operating in numerous topological spaces such as the internment camps of the Boer War, Guantanamo Bay and even airport holding zones. In sharp contrast to this, Arendt explicitly rejects such extrapolation on the basis that it occludes the specificity of the camps as a totalitarian device. Further, whereas Arendt sees the destruction of the person as occurring in three stages in the camps, culminating in the destruction of uniqueness, the biographical dimension of human life, Agamben rests his case in Homo Sacer of the suspension of legal status and the consequent destruction of legal personhood alone. That is, for Arendt the destruction of legal personhood is a precondition for total domination; for Agamben, it is itself total domination. As this suggests, then, there are ways in which Agamben constitutes a philosophical radicalization of Arendt, since despite her qualms about modern politics, she maintained faith in the promise of politics, and of state structures such as law and right. Agamben maintains no such faith. Further, while Arendt remained faithful to Aristotle's construal of the centrality of speech in the formation of the political, Agamben seeks instead to undo the link established here between humanity, the good life and 'having' language. But while this might generate a stronger critique of the modern state, and of the Western traditions of philosophy and politics more generally, it may also be worth asking what is lost in Agamben's analysis. For arguably, his philosophical extrapolation comes at the cost of empirical specificity and analytic nuance.
In particular, this highlights the need for a more detailed and nuanced understanding of the formation and mobilization of ideologies or dispositifs racism and eugenics than Agamben is wont to offer. At the outset of Origins, Arendt insists that the explanation for the Holocaust ought not make it either an historical inevitability or entirely a matter of contingency that the Jews in particular were targeted for annihilation. In his abstraction from historical detail, Agamben manages to commit both these analytic sins. In his view, the camps arose as an inevitable outgrowth of a logic inherent in the Western political tradition, and, at the same time, he is unable to provide any explanation as to why it was Jews in particular that were persecuted. It is important to keep in mind of course, that it was not only Jews systematically murdered by the Nazis and persecuted in the concentration camp system. Nazism also targeted persons considered disabled or mentally ill, for instance. But whereas eugenic ideologies provided a certain rationale for these latter killings (which is not to say that these were in the least bit correct or defensible), the same does not hold for the attempted destruction of the European Jewry. For this, an account of racism, and more specifically anti-Semitism, is required. However, Agamben offers no such account and nor does his conception of biopolitics readily allow for one. Indeed, Agamben's lack of reference to Darwinian evolutionism and its extension into sociobiology and eugenics also makes it difficult to see why the mentally ill or the disabled also fell outside the boundaries of acceptable life according to the Nazis. In short, for Agamben (1998, 115), 'we are all virtually homines sacri'; however, it is unclear why that virtual exclusion and abandonment is actualized in some cases and not others.
Whatever one concludes about the account of totalitarianism that Arendt's study yields, then, it remains the case that the methodology that she adopts of tracing the elements that combine to make it historically possible allows for such detailed analysis. This is not to say that Arendt's own account of race-thinking and racism is without problems. In fact, several commentators have pointed out that she herself has blindspots in relation to race, especially evident in regards to her treatment of the emergence of racist thinking in African imperialism in Origins, as well as her treatment of Black American slavery and the civil rights movement (e.g. Gines 2007; Gines 2014). Further, her account of racism, imperialism and totalitarianism in Origins has been the criticized for overemphasizing ideas of expansion at the expense of ideas of race and providing a misleading history of the concept of race (Bernasconi 2007), misunderstanding the relation between imperialism and totalitarianism (Benhabib 2000, 75–6; Mantena 2010) and underestimating the historical force of Social Darwinism and eugenics (Barta 2007; Bernasconi 2007, 52) among other charges. From the perspective of analyzing biopolitics, though, what may nevertheless be important is her central claim that racism became a political device, that is, a means or technique for governing, in the nineteenth century, and particularly in the context of colonialism.
In this, her thinking converges with the similar claim from Foucault that racism constituted a central political technique of biopower. In Society Must Be Defended, Foucault proposed to trace the history of the idea of race war, and the emergence of racism as a political technique or device. He argues of European racism that it emerged from revolutionary discourses of class war and race war, especially through the transformation of the latter into a concern with racial purity. This concern with purity, Foucault argues, drives the emergence of state racism in particular, in which 'sovereignty was able to invest or take over the discourse of race struggle and reutilize it for its own strategy. State sovereignty thus becomes the imperative to protect the race' (Foucault 2003, 81) Foucault goes on to argue that the twentieth century saw two modifications of this political mobilization of race – first, the Nazi transformation, which takes up the thematic of the state as the protector of racial purity within an 'ideologico-mythical landscape' that recalled elements of the earlier emphasis on race struggle, and second, the 'Soviet-style' transformation, in which elements of race struggle were articulated with 'the management and the policing that ensure the hygiene of an orderly society' (Foucault 2003, 83). In the final lecture of this course, Foucault ties racism to biopower, insofar as it is a mechanism by which death can be mobilized within a power aimed at fostering life; as he writes, 'racism alone can justify the murderous function of the State' since it is 'the precondition for exercising the right to kill' (Foucault 2003, 256). Interestingly, Foucault also goes on to link racism to colonialism, arguing that it 'first develops with colonization, or in other words, with colonizing genocide. If you are functioning in the biopower mode, how can you justify the need to kill people, to kill populations, and to kill civilizations? By using the themes of evolutionism, by appealing to racism' (Foucault 2003, 257). Thus, while Arendt and Foucault's insights into the emergence of racism as a political technique are not necessarily consistent in their entirety, they do share a concern with racism as a political technique, and not just an ideology. This suggests that it would be worth investigating these themes in their work more thoroughly, and considering the implications of this point of crossover for thinking about biopolitics.
Finally, it is worth briefly considering the ways in which recent interventions in biopolitical studies take up Arendt's notion of natality, particularly with the aim of developing a positive or affirmative biopolitics. One such intervention is that of Miguel Vatter, who argues that the concept of natality proposed by Arendt leads her to posit a biological foundation to freedom. Vatter arrives at this conclusion through a number of deft interpretive steps, beginning from the basic claim that some account must be given of the caesura within biological life if an affirmative biopolitics is to be possible. On the basis of this, he rejects a common reading of natality as reiterating the distinction between bios and zoē, and claims instead that 'natality is irreducible to the bios politikos: natality is essentially antecedent with respect to the common world... one must bring the concept of natality back to the level of zoē" such that natality entails a politicization of zoē, not the 'always already "political" bios' (Vatter 2006, 152). From this, he goes on to argue for an interpretation of natality as the only notion that meets the criteria that Arendt sets for freedom – that it be simultaneously automatic (like natural processes) and undetermined (that is, counter-natural). Freedom is thus 'the automatism of an interruption', which is just the status of natality vis-à-vis zoē (Vatter 2006, 153).
As illuminating as Vatter's interpretation is, though, I remain unconvinced about the connection he posits between freedom and biological life. To grasp the problem in Vatter's intervention, a significant ambiguity in Arendt's notion of natality must be registered. For while the notion draws on the 'fact of birth', it is not obviously biological birth that Arendt has in mind as central to politics; rather she extrapolates from the ontic fact of birth such that natality also encompasses a kind of 'second birth', by which by an individual is 'inserted' into the public sphere, and which takes place in speech and action. As Adriana Cavarero (2014, 14) points out, 'we are born twice: the first time as newborns; a second time, and then time and again, as "actors" on the political scene that confirms us as unique beings and as beginners'. This second, and recurrent, birth is more closely associated with politics and the public sphere through the emphasis on beginnings, and further, through the appearance of uniqueness in speech and action. Even so, Arendt cannot completely do away with biological birth, which is firmly associated with labour and the reproduction of life and is therefore restricted to the private sphere.
This appears to introduce into Arendt's work an indistinction between labour and action, categories that she is otherwise at pains to distinguish. Further, a question arises about what work this biological remnant actually does in Arendt's theorization, and whether it is as positive for her as Vatter suggests or, instead, a problematic inclination that she is unable to eradicate but also unable to assimilate into her conceptual frame. Two points suggest that the biological dimension of birth is more problematic than Vatter allows. For one, his interpretation appears to overstate the connection of freedom and biology in its failure to take account of the role of speech in the political sphere and its association with natality. In fact, he barely mentions speech and the related issues of uniqueness and narrative even though it is for Arendt almost inseparable from action and therefore freedom. As we saw, it is the revelatory capacity of speech in particular that makes possible the appearance of human uniqueness, that is, of plurality. It is not clear how this revelatory function would relate to the biologically based notion of freedom that Vatter is urging. Second, as Adriana Cavarero (2014) has recently reiterated, the maternal is curiously absent in Arendt's understanding of birth, though it is ineliminable from the scene of biological birth – all those born are born of a mother, and, yet, in Arendt, this is never the case for natality understood as the condition of action. Instead, she refers consistently to divine creation. Similarly, in Vatter's account the mother and questions about the maternal are absent – in Cavarero's words, those born are born of nothing. This absence of the maternal again suggests that the connection between natality and biological birth is weaker than he supposes. I return to this problem of natality and birth in Chapter 6.
Conclusion
In this chapter, I outlined Arendt's account of the emergence of totalitarianism as a political formation in the twentieth century, focusing on the elements of imperialist expansion, the destruction of the nation–state system and the development of racism as a political technique. Further, I considered Arendt's account of the political significance of the Nazi concentration camps of the Second World War. From this, I argued that this account of totalitarianism provides a crucial backdrop for Arendt's revaluation of action in The Human Condition, since action comes to operate as a defence against the reduction of human life to mere biology that Arendt argued was effected in the Nazi concentration camps. From this analysis of Arendt's major contributions to political theory, it should be clear that her work has much to offer contemporary thinking about biopolitics; indeed, her work has a deep influence on the thinking of Agamben. It is from this that he derives much of his analysis of the concentration camps, as well as his critique of human rights. Arendt's work thus raises a number of concerns and questions that we will return to throughout later chapters. In particular, questions about state sovereignty and rights, racism and reproduction have been touched on here but will also reappear in more detailed discussions later. In the next chapter, I turn to the work of Roberto Esposito, who, in contrast to Agamben but nevertheless taking some inspiration from Arendt, seeks to develop an affirmative biopolitics that overcomes the thanatopolitical violence of totalitarianism.
Notes
For example, Thomas Lemke (2011) only references her work twice in Biopolitics and then only in passing. Few of Agamben's interpreters make any significant reference to Arendt, though his work is very obviously in conversation with hers. Foucault disavows any similarity between his work and Arendt's, though this is somewhat disingenuous; there are certainly interesting parallels if not direct influence. See Foucault 1984, 377–379. For analyses of Arendt's contribution to biopolitical studies, especially in regards to Foucault, see Blencowe 2010, and Braun 2007.
Kristeva (2001b, 5) suggests that The Human Condition constitutes a 'passionate defense of life'; however, in this context, the concept of life is too broad for this claim to be illuminating. I argue that what Arendt is actually seeking to defend is a notion of the human, against its reduction to mere biological life.
While Arendt's discussion of totalitarianism develops with reference to both Nazism in Germany and Stalinism in Russia, in keeping with much of the biopolitics literature, the focus here will be on the former of these. For an important study of Stalinism and biopolitics, see Prozorov 2016.
Even so, her purpose is not to draw up a neat categorization of different kinds of activities as either labour or work. Indeed, the distinction is notoriously unclear when used in this way.
References
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4Affirmative biopolitics
Negri and Esposito
As is evident from the previous chapters, biopolitics has largely been seen as a problematic political formation, one that requires critique and resistance or overcoming in some way or another. This is starkest in Agamben's work, where biopolitics centrally involves the production of bare life, and is more accurately understood as a thanatopolitics, or politics of death. In this view, the only hope for living well is through the development of a new political formation that does not partake in the biopolitical capture of life. However, while this has been the dominant mode of thinking about biopolitics, in recent years a different approach has emerged, one which strives toward developing an affirmative biopolitics. There are two predominant models for an affirmative view of biopolitics. The first of these is developed by Antonio Negri and his co-author Michael Hardt in works such as Empire (2000) and Multitude (2004), in which they contrast the stifling hegemony of Empire and biopolitical production with the revolutionary significance of the multitude. The second form of affirmative biopolitics argues that although the modern formation of biopower has been deadly, resources can be found within it that turn it toward a more positive and life-affirming condition. This is most developed in the work of Italian political theorist Roberto Esposito, particularly in his books Immunitas (2011) and Bios (2008).
In this chapter, I provide an outline of these differing ways of thinking about an affirmative biopolitics and the possibilities for living that it is perceived to afford. I begin with Hardt and Negri's interventions, as they provide the starkest formulation of an affirmative biopolitics, which is most easily contrasted with the thanatopolitical approach of Agamben for instance. I will argue of this work that, although it corrects a tendency within prior formulations of biopolitics to see it as overwhelmingly negative, thereby essentially obviating Foucault's observation that biopower fosters life inasmuch as it allows for death, it is itself subject to a similar critique. That is, Agamben's approach to biopolitics is structured by an opposition between biopolitics and form-of-life; similarly, Hardt and Negri's approach is structured by an opposition between Empire and the creative power of the multitude. Despite their clear differences, then, these theorists nevertheless share the intuition that modern regimes of power are overwhelmingly negative and require a kind of (more or less messianic) overcoming.
From this perspective, Esposito's work offers an interesting alternative as it strives to work with both the negative and positive aspects of biopolitics. More specifically, in his account of the immunitary paradigm, which he sees as the central characteristic of modern biopolitics, he elucidates the paradoxical integration of the negative and positive valences of immunization in biopolitics. Thus, there is a kind of mutually reinforcing or constitutive relation between negativity and positivity in the operation of the immunitary paradigm. Esposito's account of biopolitics, which he develops across the books Immunitas and Bios, and to some extent, Communitas (2010), strives to work with this double edge of biopolitics, such that it contains within itself elements for its overcoming. In the second section of the chapter, I provide an outline of Esposito's account of the immunitary paradigm, with particular regard to the way in which he construes it as central to biopolitics. He provides an account that is simultaneously a political philosophy while also being, to some extent at least, responsive to historical actuality. As with the discussion of Hardt and Negri, my central concern will be with his formulation of the passage beyond the modern impasse of biopolitics, which he characterizes as a need to push the dispositifs of immunity to the point at which they turn into their opposite, this being community. Across both sections of the chapter, I will also be interested in the turn toward the philosophy of immanence by Hardt, Negri and Esposito, also evident in Agamben's formulation of form-of-life.
The power of the multitude
The contribution to debates on biopolitics made by Antonio Negri, along with his collaborator, Michael Hardt, primarily appear in the books Empire and Multitude. On its publication in 2000, Empire became enormously popular internationally among the intellectual Left, and especially in Left activist circles. This in part reflects Negri's own positioning as a leading figure in militant activism in Italy, as well as a philosopher engaged in more abstract reflection on philosophy of law and the state. As Timothy S. Murphy (2012) outlines in one of the very few book-length studies of Negri's intellectual contributions, Negri was heavily engaged in Workerist and Autonomist political activity in Italy from the 1950s, while employed at the University of Padua. Negri's political activism eventuated in him being charged with involvement in the assassination of former Italian Prime Minister, Aldo Moro, in 1978. After that, he spent the next 25 years either imprisoned in Italy or exiled in France, until he was granted full parole in 2003. No doubt, this radical political heritage contributed to the initial popularity of Empire.
However, in subsequent years, academic attention has diminished, and critical analysis of the contribution of Hardt and Negri to debates on biopolitics is now relatively limited. This is partly due to the fact that while they draw on Foucault's conception of biopolitics in texts such as Empire, they do not propose a full theory of biopower as such. Nevertheless, this work was significant for prompting a different way of thinking about biopolitics, one that was deeply indebted to the political theory and philosophy of Marx and Spinoza and emphasized the revolutionary potential within biopower. Consequently, my outline here is restricted to a general overview of Empire, especially insofar as it constitutes a significant moment in the formulation of an affirmative biopolitics. In the first section of the chapter, I will outline the key claims that Hardt and Negri made in Empire, with the aim of making clear the overall logic of the account, without engaging too much of the detail of it. Indeed, it should be noted that Empire synthesizes arguments made at length in previous books authored by Negri, such as his earlier philosophical studies of Descartes (Negri 2007), Hegel (Negri 2011) and Spinoza (Negri 1991), as well as his previous collaborations with Hardt (1994). As Negri says, 'Empire is a product of hybridization and metamorphosis: we hybridized all the different parts of our past research and life experience, and such hybridization produced a metamorphosis, a monster' (Casarino and Negri 2008, 70). As this indicates, much of the philosophical detail of the arguments made in the book is actually in other texts, and it is not possible to incorporate those in this brief discussion.
The central thesis of Empire involves four broad claims, the elaboration of which roughly coincides with the four parts of the book. The first of these is that a new paradigm of imperialism is emerging in postmodernity. This form of imperialism is not a repetition of earlier colonial imperial projects, but instead a global empire that supersedes the nation–state and co-opts its sovereignty. Second, while the sovereignty of nation–states has declined in conditions of globalization, this does not mean that sovereignty itself has; instead, sovereignty takes a new national and supra-national form under a 'single logic of rule'. This new imperial sovereignty does not rely on fixed territorial boundaries or on a particular centre of power (as the United States does); it is instead a 'decentered and deterritorializing apparatus of rule that progressively incorporates the entire global realm within its open, expanding frontiers' (Hardt and Negri 2000, xii). Third, this historic shift in global rule and the associated realization of a world market also indicates a shift within the capitalist mode of production to biopolitical production, that is, 'the production of social life itself, in which the economic, the political and the cultural increasingly overlap and invest one another' (Hardt and Negri 2000, xiii). And fourth, the supra-national, indeed global, reach of the new Empire does not render it beyond critique and resistance; rather, Hardt and Negri argue that it necessarily also affords new opportunities for resistance and transformation of processes of global flow and exchange. In particular, they argue that the 'creative forces of the multitude that sustain Empire are also capable of autonomously constructing a counter-Empire... the multitude will have to invent new democratic forms and a new constituent power that will one day take us through and beyond Empire' (Hardt and Negri 2000, xv). In the following paragraphs, I will elaborate on each of these claims in turn.
The first task then is to get a fuller understanding of the notion of Empire, which Hardt and Negri articulate in the first part of the book, approaching the concept from several angles. In terms of its juridical character, they argue that Empire can be conceived of as a combination of concepts from Hans Kelsen, John Rawls and Niklas Luhman's system's theory. In short, Empire is a global juridical order underpinned by reference to universal values of justice, especially as justificatory devices for interventions understood as responses to a permanent state of emergency. Within this, a notion of right understood as a right of the police, consisting of 'the juridical power to rule over the exception and the capacity to deploy police force' (Hardt and Negri 2000, 17) emerges as an essential coordinate of Empire. From the perspective of the actual material constitution of rule, though, this combination of the state of exception and the right of police does not lead to a form of global totalitarianism – indeed, Hardt and Negri are dismissive of this concept. Instead, they argue that 'right remains effective and (precisely by means of the state of exception and police techniques) becomes procedure' (Hardt and Negri 2000, 26). For Hardt and Negri (Hardt and Negri 2000, 26), this reveals an 'unmediated relationship between power and subjectivities', best understood in their view in the frame of the biopolitical production of social life.
Before elaborating their notion of biopolitical production, though, it is first worth getting clearer on the problem of sovereignty as it relates to Empire. In Part two of Empire, Hardt and Negri provide a brief history of the concept of sovereignty, beginning with its emergence from the revolutionary conditions of early European modernity (from the thirteenth century), which saw the transposition of the 'powers of creation' and authority from the transcendental to the immanent realm – that is, from God to humanity. In the emergent modernity, itself defined as a form of crisis according to Hardt and Negri, counter-revolutionary forces – including philosophers such as Descartes, Kant and Hegel – re-established transcendentalism at the heart of modern metaphysics, while Hobbes and Rousseau did the same for the political conception of sovereignty. Importantly, this form of sovereign authority is sustained by the development of capitalism and the 'affirmation of the market as the foundation of the values of social reproduction' (Hardt and Negri 2000, 85), which appears as its essential content. As they claim, Adam Smith's theory of value 'was the soul and substance of the concept of the modern sovereign state', such that '[m]odern European sovereignty is capitalist sovereignty' (Hardt and Negri 2000, 86, 87). At the same time, following the revolutions in France, England and the United States and the associated decline of feudal monarchism, the concept of sovereignty was gradually brought into alignment with the emergent concept of the nation. This ultimately gave rise to the concept of national sovereignty in the nineteenth century, and the subsequent entrenchment of national sovereignty saw it extend beyond its European epicentre in the period of colonial imperialism. Ultimately, however, these associations of sovereignty with capital on the one hand, and the nation on the other, have proved incompatible. The globalization of the market throughout the twentieth century, especially in the post-war period, has led to the severing of the link between sovereignty and the nation, such that sovereignty as such has not disappeared or declined, but has been re-inscribed at a supra-national level, that is, in Empire.
This leads to one of the claims that has been controversial in the reception of the Hardt and Negri thesis – that is, that the United States does not hold a position of pre-eminence equivalent to the European imperial powers of old. Left critics of the contemporary world order have typically understood the United States as a new imperial super-power, a position evident in critical views promulgated by leftist commentators of the various US military interventions in the Middle East, for instance. While Hardt and Negri do not deny that the United States holds a position of pre-eminence within Empire, they do argue that its role is simply not that of the traditional imperial power seeking to extend its territorial rule and thus market. Instead, they argue that the position of the United States is unique insofar as it is called upon as the 'only power able to manage international justice, not as a function of its own national motives but in the name of global right' (Hardt and Negri 2000, 180). Moreover, the US model of constitutional sovereignty is writ large into the supra-national sovereignty of Empire: 'the contemporary idea of Empire is born through the global expansion of the internal U.S. constitutional project'. Importantly, this constitutional project is not imperialist, but imperial, that is, not concerned with securing territory and subsuming subjects into itself, but 'constructed on the model of rearticulating an open space and reinventing incessantly diverse and singular relations in networks across an unbounded terrain'. (Hardt and Negri 2000, 182)
Thus, the United States is not an imperialist super-power, but the model for and fulcrum of the imperial project of Empire.
At this point, we must return to the notion of biopolitical production that Hardt and Negri posit. While they borrow the formulation of biopolitics from Foucault, they also argue that ultimately he remained tied to a structuralist epistemology that made it impossible for him to identify the systemic drivers or 'real dynamics of production in biopolitical society' (Hardt and Negri 2000, 28). Their task, then, is to identify these dynamics by building on earlier attempts – primarily those of Deleuze and Guattari and the Italian political theorists of immaterial labour. In the preface of the book, they characterize biopolitical production as 'the production of social life itself, in which the economic, the political and the cultural increasingly overlap and invest one another' (Hardt and Negri 2000, xiii). In the third part of Empire, they go on to elaborate their understanding of the mode of production in Empire in more detail, explicitly flagging this shift as replicating the invitation in Marx's Capital to 'leave the noisy sphere of exchange and descend into the hidden abode of production' (Hardt and Negri 2000, xvii). As Jason Read (2001) usefully summarizes, though, Hardt and Negri's turn to the sphere of production is not a turn to a narrow economic sense of the term. Rather, they expand the concept of capitalist production in Empire in two ways: first, biopolitical production entails an historical transformation whereby capitalist production comes to encompass 'language, subjectivity, affects and desire' (Read 2001, 25) along with more traditional economic production. Second, production is understood in an expanded ontological sense not as the production of things, but as the production of the world. According to Read, these two elements are inseparable and in a sense co-constitutive. At this point it is worth noting that while Foucault's conception of biopower picked out the ways in which phenomena associated with biological life, either of populations or individuals, came to be the targets of a new form of power, Hardt and Negri are specifically concerned in their conception of biopolitical production with social life and all it entails: language, culture, subjectivity and forms of relation or intersubjectivity. In Agambenian terms, their focus is on bios, not zoē. Importantly, this focus on the biopolitical production of social life allows them to articulate a vision of an affirmative biopolitics alongside the operation of Empire.
This vision rests on the somewhat controversial notion of the multitude that they use in Empire, and elaborate further in the companion volume, Multitude. In philosophy, the term 'multitude' derives from the work of Machiavelli and Spinoza, and refers to a population not yet constituted as a people, that is, who have not yet entered into a social contract with a sovereign body and are thus not constituted as a political subject of it. For Spinoza, the multitude presents the limit of the power of the sovereign, and this understanding provides the philosophical starting point for Hardt and Negri's own conception. While they do not provide a definition as such of the multitude in Empire, several comments give shape to the notion. In general, the multitude is understood as a plurality of 'productive, creative subjectivities of globalization' (Hardt and Negri 2000, 60) that 'work toward the liberation of living labor, creating constellations of powerful singularities' (Hardt and Negri 2000, 61). In contrast to a people, which is an ideological component of national sovereignty, '[t]he multitude is a multiplicity, a plane of singularities, an open set of relations, which is not homogeneous or identical with itself and bears an indistinct, inclusive relation to those outside of it' (Hardt and Negri 2000, 103). In its relation to Empire, '[t]he multitude is the real productive force of our social world, whereas Empire is a mere apparatus of capture that lives only off the vitality of the multitude' (Hardt and Negri 2000, 62). And as a creative force, '[t]he emancipation of humanity from every transcendent power is grounded on the multitude's power to construct its own political institutions and constitute society' (Hardt and Negri 2000, 165). Finally, the 'biopolitical existence of the multitude has the potential to be transformed into an autonomous mass of intelligent productivity, into an absolute democratic power, as Spinoza would say' (Hardt and Negri 2000, 344). In short, the multitude is a non-homogenous, non-exclusive creative force upon which Empire rests, but which also has the power to constitute alternative political and social forms of life.
As this indicates, Hardt and Negri do not simply position the multitude as an exteriority in relation to Empire, but rather as an internal and indeed constitutive element of it. Thus, commenting on the notion of biopolitical production and its relation to the multitude, Read (2001, 29) writes that '[t]he immersion of bodies, desires, language and communication into the global circuits of Empire means that the imperial order cannot be produced or reproduced without the actions of the multitude'. Further, though, as such, the multitude also yields the conditions for the overcoming of Empire. In essence, the dependency of Empire on the power of the multitude makes it susceptible to undoing by the same force. This positioning of the multitude in relation to Empire reflects a thesis initially proposed by Negri in his earlier reflections on Descartes, of 'two modernities'. As I mentioned earlier, for Hardt and Negri, Descartes, Kant and Hegel among others constitute the philosophical foundation of the hegemonic conception of modernity, but existing alongside this conception of modernity is a more radical, creative conception derived primarily from the work of Spinoza and the immanent ontology proposed therein (Hardt and Negri 2000, 70–83). For Hardt and Negri, the Spinozist conception constitutes a 'counter –history of modernity' that 'promotes escape from the reactionary, dialectical wing of modernity' (Murphy 2012, 177). Thus, rather than biopolitical production devastating the power of the multitude in the production of social life, the multitude is both an essential element of biopolitical production and an ineradicable counter-point to it.
This conception of the multitude brings into focus the specificity of Negri and Hardt's understanding of affirmative biopower and how it relates to accounts such as Agamben's. For underpinning this construal is an essential point of contention between Hardt and Negri and Agamben on biopolitics and its overcoming; this is the political philosophical problem of constituent power. The contrast here is that while Agamben sees constituent power as related intrinsically to sovereignty, Negri wants to see the multitude itself as constituent power (Negri 1999). Agamben discusses this attempt to separate constituent power from sovereignty in Homo Sacer, and makes the critical point that Negri fails to provide any satisfactory criterion by which constituent power can be differentiated from sovereign power (Agamben 1998, 44). For his part, Negri responds by suggesting that Agamben's 'most serious problem is that he does not allow for any kind of constitution of the political whatsoever' (Casarino and Negri 2008, 159).
These conflicting understandings of the relation between sovereignty and constituent power have significant implications for thinking about an affirmative biopolitics. As Hardt and Negri see it, the multitude is a 'powerful life' already existent within Empire as its foundation and limit, and therefore constitutes its potential overcoming. This is sharply opposed to Agamben's formulations of 'bare life' or 'naked life' – life stripped bare by the operation of biopower – and 'form-of-life'. Agamben is explicit in Homo Sacer that nothing in bare life 'seems to allow us to find solid ground on which to oppose the demands of sovereign power' (Agamben 1998, 187). Further, the overwhelming message in regards to form-of-life is that it is a life that is wholly immanent to itself but yet to be created. The problem for Agamben, then, is to provide some theoretical explanation of what makes possible the movement from bare life to form-of-life. He does this through the notion of inoperativity, in which the biopolitical machine that captures and constitutes life as bare life is said to be brought to a halt (for further discussion of inoperativity, see De La Durantaye 2009; Prozorov 2014). However, not all commentators have been convinced by this notion. For example, in a sharp critique of Agamben's conception of biopolitics as thanatopolitics, Hardt and Negri (2000, 366) write that Agamben's understanding of bare life merely exposes 'behind the political abysses that modern totalitarianism has constructed the (more or less) heroic conditions of human passivity' and suggest instead that Nazism amounts to an unsuccessful attempt to 'destroy the enormous power that naked life could become'.
To summarize, then, in Hardt and Negri's account, Empire and its kernel of biopolitical production is positioned against the affirmative and creative power of the multitude, although also dependent on it. The latter is understood as an immanent revolutionary force that presents the limit and possible transformation of sovereignty, including in its supra-national form of Empire. In short, the multitude is understood as constituent power, and Empire is merely constituted by that power. In terms of how this relates to the concept of biopolitics, Negri comments that it is necessary to conceive of an antagonism within the concept, where on the one hand biopolitics turns into biopotere, understood as the institutional 'dominion over life' (that is, Empire) and on the other, it turns into biopotenza, understood as 'the potentiality of constituent power' (the multitude). He concludes that 'in biopolitics intended as biopower [biopotenza], it is the bios that creates power, while in biopolitics intended as biopower [biopotere], it is power that creates the bios, that is, that tries alternately either to determine or to annul life, that posits itself as power against life' (Casarino and Negri 2004, 167). This reflects his attempt to articulate a particularly Spinozist understanding of biopolitics, in which 'life' is understood as a multitude of singularities, rather than as a renewed vitalism. In Esposito, we see a similar interest in the philosophy of immanence that is a trademark of Negri's work, but ultimately his understanding of the immanence of life does take a more vitalist form. Further, instead of working with the dualism of constituent and constituted power that structures Negri's approach to biopolitics, Esposito is more interested in a radical deconstruction of key dispositifs within biopolitics.
The immunization paradigm
Esposito's interventions in contemporary debates on biopolitics provide a new way of addressing the apparent opposition between thanatopolitics and an affirmative biopower or biopolitics. In particular, he proposes to understand biopolitics through the notion of an 'immunitary paradigm', in which the medico-scientific figure of immunity is extended into a political logic of self-defence that has both negative and positive valences. His first major foray into theorizing the immunitary paradigm and its political implications is his book, Immunitas, published in Italian in 2002 and translated into English in 2011. In the subsequent book, Bios, published in Italian in 2004 and in English in 2008, he links this account of immunization more explicitly with the concept of biopolitics, effectively intervening in contemporary debates on this concept and providing an innovative theorization that moves beyond Agamben's 'radically negative' account and Negri's 'absolutely euphoric' one, by returning to ambiguities in Foucault (Esposito 2008, 8). Part of the novelty and appeal of Esposito's work is that while rejecting major elements of each approach, he also seeks to integrate theoretical insights and motivations that are consistent with aspects of all three of these major figures. Ultimately, he provides an account of an affirmative biopolitics that is both philosophically oriented to identifying the core logic of biopolitics, while maintaining a commitment to seeing biopolitics as a specifically modern phenomenon. Further, his account does not reiterate the oppositional construal of biopolitics as either negative or positive; rather, he elaborates the ways in which the logic of immunization actually incorporates both the negative and positive in a paradoxical logic of protection through exposure. In outlining Esposito's contribution to debates on biopolitics – contributions which are likely to become more influential – I first discuss key aspects of his characterization of the immunitary paradigm in Immunitas. I then follow his linkage of immunization and biopolitics in Bios, particularly focusing on the ways in which he argues for an affirmative biopolitics that paradoxically arises from the thanatopolitics of Nazism.
The primary aim that Esposito has in Immunitas is to uncover and elucidate the paradigm of immunization that he argues structures modern politics. The main claim of his account of the immunitary paradigm is that our social and political systems have at their centre a self-defensive logic by which the danger to be defended against is incorporated into the system, in such a way as to generate an appropriate defence. To build this argument, Esposito begins with noting that the term 'immunity' has two especially important inflections. First, Esposito highlights that immunity bears a complicated relation to community, in which 'neither term is limited to negating the other but instead implicates the other... as its necessary presupposition' (Esposito 2011, 5). His point of reference here is Roman law, in which citizens could be granted immunity from the fulfilment of community obligations (see also Cohen 2009, 40–4). He identifies a 'primal juxtaposition' between immunity and community through an etymological analysis that suggests that immunitas is the proper or non-communal that interrupts the communal bonds of reciprocal gift-giving (munus). Thus, he writes that '[i]f members of the community are bound by the obligation to give back the munus that defines them as such, whoever is immune, by releasing him- or herself from the obligation, places himself or herself outside the community' (Esposito 2011, 6). This renders the immunitas somewhat akin to the exception in Agamben's thought – it is both privative and privileged, while also set outside the community against which it is defined. As Ed Cohen (2009, 40–5) points out in his more extended discussion of immunity, this political and legalistic inflection of the term has been the predominant one for centuries.
However, the second, more recent, inflection of the term 'immunity' forms the core of the argument in Immunitas, and derives from eighteenth and nineteenth century biological and medical discourses. In this view, immunity refers to the 'refractoriness of an organism to the danger of contracting a contagious disease' (Esposito 2011, 7), which can either be a natural capacity or an acquired one. In acquired immunity, which most interests Esposito, the capacity to resist a contagion is itself precipitated by infection with nonlethal quantities of it, since this stimulates the production of antibodies that are then able to ward off further infection. This logic of the pre-emptive inoculation of the body or the community entails that a positive refractoriness is stimulated by the incorporation of the negative threat in tolerable degrees. In effect, the positive self-defensive capacity for survival is dependent on the incorporation of the external threat into oneself; as Esposito puts it, life is prolonged 'only by continuously giving it a taste of death' (Esposito 2011, 9). As this makes evident, immunity is a matter of borders and their violation or trespass. In short, in order to ward off the potential threat, that threat is incorporated within the border, but only in order to be more effectively repelled. This logic of repulsion or defence via controlled incorporation is central to Esposito's thinking.
Esposito goes on from these general remarks on immunity to offer a 'deep genealogy' of the immunitary paradigm within Western thought, the outlines of which he traces across five domains of Western thought – law, theology, anthropology, politics and biology – which correspond to the five chapters of the book. As it is not possible to summarize Esposito's genealogy across all of these here, I will focus on the final two chapters on politics and biology. In regards to the first of these, Esposito's key concern is to outline the immunitary inflections that the influential analogy between the natural human body and the political body of the state gives to the modern Western political lexicon and its constitution as biopolitical. The chapter proceeds through four sections, the interrelations of which are not always clearly elaborated, but which together provide a picture of the ways in which the life of politics has historically come to be conceived of as immunized. Esposito's discussion is wide-ranging and summarizing his line of argumentation is no easy task. However, of particular importance for his discussion are: (1) the characterizations of the body politic by Hobbes, Rousseau and others; (2) concepts of pathology; (3) the cell theory of Rudolf Virchow; and (4) Foucault's account of biopolitics. Each of these contribute to immunological themes such as notions of political unity and social plurality, the externalization of threats to the body and the superimposition of therapeutic and political orders.
In the first section of the chapter, he sets up the theme of political unity and social plurality as a key interpretive frame of the immunitary paradigm through reference to Hobbes and others. He begins with the suggestion that the dynamic between the poles of life and politics entailed in biopolitics actually depends on a third term, that is, the body, since 'the bodily dimension is where life lends itself to being preserved as such by political immunization' (Esposito 2011, 112–13). The body is 'the instrument and terrain' of the immunitary paradigm, which then depends on the incorporation of the individual body in politics. According to Esposito, the founding theorization of modern sovereignty presented by Thomas Hobbes is the most extreme formulation of the incorporation of the individual body within the political body. Further, Esposito argues that rather than the shift to mechanism that Hobbes' philosophy entailed signalling a diminution of the analogy between the individual body and body politic, it actually signals 'a growth in the immune-oriented significance of the State-body analogy' (Esposito 2011, 114). This is because the machine lexicon does not replace that of the body, but actually supplements it and provides ground for the establishment of an order that pre-empts threats to the otherwise precarious body of the sovereign: it produces an 'artificial life' that exists beyond the death of any individual body. This conception of the state is then contrasted with other theorists in the contractarian tradition, particularly Jean-Jacques Rousseau and his concept of the general will. The significance of this discussion is that it provides two conflicting pictures of the relation between the individual body and body of the State. These are a conservative, authoritarian one in which the individual body is fully incorporated into the unity of the state, and a democratic and revolutionary one in which political unity derives from the plurality of individual bodies.
The following sections explore the implications of significant moments in the transmission between political ideas and medical-scientific ones for these contrasting visions and thus the development of the immunitary paradigm in politics. In a section entitled 'The pharmakon', Esposito focuses on the history of concepts of pathology to argue that the emergence of the immune paradigm required two shifts in the political metaphor of the body, particularly in terms of the localization of disease and its relation to health. The first of these meant a shift in thinking about disease and decay, from seeing it as emerging from within the body to seeing it as a matter of external threat that imposes itself upon the body. The second entails a shift from allopathic principles of therapy – in which disease is treated with its contrary – to homeopathic principles, in which a poison or disease is contradicted by a dose of itself. Esposito comments that from here, it is only a short step to the pre-emptive logic of inoculating against a disease or poison with a non-pathological dose of itself.
Following this, the cell theory of German physician Rudolph Virchow, who is often credited with founding modern pathology, marks a further transformation in the structure of the body-state analogy. It does so by shifting the level of comparison from parts of the body (classes or orders of the state) and totality (people or nation) to the elemental components of the body. What is at issue in this view is not a singular life force, but the mutual influence and complex interaction of different elements. The political importance of this is that it undermined the hierarchy of organs – the heart, the brain – that shaped conservative accounts of the body politic, and shifted focus to the individual lives that made up the unity of the people as the driving force of government. Thus, in this formulation, the body politic appears to refer to 'a community open to the constitutive difference of its members, rather than to a fully-fledged State' (Esposito 2011, 132). This re-conception of the analogy, which shifts emphasis away from political unity toward social plurality, contributes in Esposito's view to a crisis of sovereignty. Moreover, it contributes to the transposition of political power from the sovereign to the people, which, in biopolitical terms comes to be understood as a population.
Esposito concludes from this that Virchow's cell theory enmeshes with Foucault's conception of biopower. In particular, the idea of many lives contributing to the strength of political unity coalesces with Foucault's portrayal of the emergence of biopower as entailing a shift from the power of the sovereign exercised in death, to the power of the state deriving from fostering the life of the individual as part of a population. Thus, as Foucault suggests, life becomes central to biopower; however, Esposito supplements his account with the claim that the central mechanism for this transposition of life and death in their relation to political power is that of the immunitary paradigm. Interestingly, Esposito ends this chapter with a brief comment on norms and normativity, pointing to the centrality of norms in biopower according to Foucault's account, and in opposition to that, the conception of immanent biological norms that emerges in the work of Georges Canguilhem. I return to a fuller discussion of norms and normativity in Canguilhem later in this chapter, but here we need more of a sense of Esposito's engagement with biology, which focuses on the history of immunology.
While Esposito pointed to important moments in the transposition of medical ideas into politics in the chapter just discussed, in the final chapter of Immunitas he discusses moments of transmission from political discourse into medicine, particularly immunology. He begins the chapter with comments on the essays, 'The Biopolitics of Postmodern Bodies' by Donna Haraway and 'L'intrus' by Jean-Luc Nancy, which set up the centrality of the immune system within biopolitics, and the dismantling and reconstitution of the individual body through organ transplants. But more important is the analysis that follows of the transposition of political notions of warfare and defence into immunology. Esposito shows that the notion of self-defence against external enemies – a kind of politics of fear not unlike that of Hobbes – has been decisive in immunology (also see Esposito 2011, 154; Cohen 2009, esp. 26–9). Historically, the immune system was cast as the bulwark against the 'invading hordes' (Golub 1987, cited in Esposito 2011, 154) that threaten the order and integrity of the body with chaos and destruction. This characterization, refined over time in concord with developing notions of warfare, relies on a capacity for identifying what constitutes oneself, and what constitutes 'non-self' or 'other', for the enemy must be known in order to be defeated. This distinction between self and other is central to immunology, and for Esposito, gives rise to two special cases, the first of which is diseases of auto-immunity, and the second of which is maternal immune tolerance.
In regards to auto-immunity, Esposito argues that it constitutes a pathological form of immunization, in which the recognition of self and other breaks down in such a way that the self is no longer recognized as that which is to be protected rather than attacked. Auto-immunity seems to entail an othering of the self, a particularly pathological form of misrecognition that in the military configuration of the immune system is consistent with civil war. But, Esposito argues, what is most interesting about auto-immunity is not that it entails pathological misrecognition of the self, but that it is ultimately the 'non-pathological or normally pathological' expression of the 'logic of the immune system in its pure state' (Esposito 2011, 164). That is, '[i]f the immune system works by opposing everything that it recognizes, this means that it has to attack even the "self" whose recognition is the precondition of all other recognition: how could the immune system recognize the other without first knowing the self?' (Esposito 2011, 164). The puzzle of auto-immunity, then, is not that the immune system comes to attack the self, but that it does not – that is, that it can circumscribe an arena of non-aggression or tolerance. In other words, if 'the destructive rebellion against the self is not a temporary dysfunction but the natural impulse of every immune system' (Esposito 2011, 165), then how is it that that impulse is reined in, or rather, transmogrified into a protective mechanism? The question of tolerance becomes crucial here: what matters is the toleration mechanism that (more or less) effectively prevents dissolution into self-destructive chaos.
Tolerance is also crucial to the workings of the second special case, which is the operation of the immune system during mammalian pregnancy. Maternal immune tolerance is well recognized as a paradox within immunology, for it entails that the immune system refrain from attacking the implanted embryo and developing foetus, which can be understood as genotypically akin to an allograft. What is important here, though, is not tolerance of the self but of the other. Since the implanted embryo derives half of its genetic material paternally, it should be detected by the immune system as a foreign invader, but it is instead incorporated into the self for the purposes of immunity. Interestingly, maternal immune tolerance is a particularly privileged example for Esposito, of which he argues that the genetic foreignness introduced by 'the father' is the means by which immune tolerance on the part of 'the mother' is precipitated. He writes,
what allows the child to be preserved by the mother is not their "resemblance" but rather their diversity transmitted hereditarily by the father... contrary to the myth of symbiotic unity between the mother and the child she bears inside her, the mother is engaged in a furious battle with the fetus... This is the ultimate – and prime – issue around which the entire immune paradigm wraps itself until reaching... its opposite, "community": the force of the immune attack is precisely what keeps alive that which it should normally destroy. The mother is pitted against the child and the child against the mother, and yet what results from this conflict is the spark of life.
(Esposito 2011, 170–72)
This description of maternal immunity is striking for several reasons and I return to it in a moment, but first I want to consider the general emphasis on tolerance further.
As these cases make clear, the phenomenon of immune tolerance is of considerable significance for Esposito, since it alerts him to the potential for an affirmative biopolitical philosophy that emerges from within the immunitary paradigm. Such an approach would not deny the contradictions of immunity, but would instead deepen them further to the point at which the semantics are reversed 'in the direction of community' (Esposito 2011, 165). This is because insofar as the phenomenon of immune tolerance is a product of the immune system itself, it indicates that the immune system is not limited to the single response of 'rejecting other-than-self' but 'includes the other within itself, not only as its driving force but also as one of its effects' (Esposito 2011, 167). He argues that the theoretician of the immune system who has gone furthest in this direction is the American philosopher and historian of medicine, Alfred Tauber, especially in his classic text, The Immune Self. In this book, Tauber challenges the orthodoxy of self/non-self recognition as the basis of immune tolerance to argue that the immune system should be seen as a complex dynamic of relative reactivity and dormancy between an organism and its environment, in which the immune self is not a static identity that is simply defended as such, but is itself continually constituted in the operation of immune tolerance. As Esposito (2011, 169) puts it, '[t]he self is no longer a genetic constant or a preestablished repertoire, but rather a construct determined by a set of dynamic factors, compatible groupings, fortuitous encounters'. He concludes through the example of maternal immune tolerance, '[f]rom this perspective, nothing remains of the incompatibility between self and other. The other is the form the self takes where inside intersects with outside, the proper with the common, immunity with community' (Esposito 2011, 171). Thus, it is through the notion of tolerance that immunity is turned into its positive counterpart of community.
Two critical points against Esposito's uses of the phenomenon of immune tolerance as a starting metaphor for an affirmative biopolitical philosophy can be made. First, his emphasis on the notion of tolerance is revealing. For one, it allows us to position Esposito's understanding of the immunitary paradigm most clearly in relation to the notion of a general logic of auto-immunity earlier proposed by Jacques Derrida. In texts such as 'Faith and Knowledge', Rogues and elsewhere, Derrida argued for an understanding of certain political formations and events as exemplary of a 'quasi-suicidal' logic of auto-immunity, in which an organism destroys itself, specifically by 'protecting itself against its self-protection by destroying its own immune system' (Derrida 1998, 73). Of a piece with this critical emphasis on the self-destructive logic of auto-immunity, Derrida maintained considerable suspicion toward the notion of tolerance, arguing that it is a 'supplementary mark of sovereignty' (Derrida 2003, 127). He goes on to link it to a conditioned hospitality that is offered to the other by the dominant party in an encounter, so long as the other abides by pre-established norms and conditions. This, he postulates, is a hospitality of invitation, not of visitation, where the latter unconditional hospitality requires the risk of the suspension of immunity (Derrida 2003, 128–9).
A similar point is made by Wendy Brown (2006) in her critique of the liberal doctrine of tolerance. She argues that tolerance marks a conditional allowance for difference or deviance, and furthermore, operates as a 'civilizational discourse' that underpins the project of Western Imperialism insofar as the notion of tolerance is wielded as a justification for state violence. As she writes, '[t]olerance regulates the presence of the Other both inside and outside the liberal democratic nation–state, and often it... legitimates the most illiberal actions of the state by means of a term consummately associated with liberalism' (Brown 2006, 8). Now, in Esposito's defence, it may be argued that his account of tolerance attempts to push beyond the simple bestowal of recognition upon the other by the self and 'putting up with' difference. But even so, we might register some concern about the centrality of tolerance in this account, especially given that his later discussions of the immunitary paradigm avoid reference to the phenomenon of immune tolerance at all. At the very least, this indicates shifts in his thinking about the key features of immunity and the immunitary paradigm.
The second critical point targets Esposito's characterization of maternal immune tolerance and the gestational relationship involved in it. While potentially allowing recognition of the gendered inflection of the immunitary paradigm, Esposito's use of the example of maternal immune tolerance ultimately reveals certain problems in his thought. Esposito's characterization of the relation between a pregnant woman and her foetus is deeply problematic for several reasons. First, it relies on obviously pro-life rhetoric in describing the unborn as a 'child' (Mills 2017, 291) and presumes a model of maternal-foetal conflict. Indeed, despite the earlier critique of militaristic models in the immunitary paradigm, he readily falls back into a militaristic characterization of the gestational relation as one of necessary conflict, albeit conflict understood to generate life rather than destroy it. Further, one might also be sceptical of the phallocentric characterization of the paternal contribution as the precipitant of maternal immune tolerance. In this, the woman appears simply as the receptacle for the constitutive foreignness implanted by the man. This suggests that the criticism that Anne O'Byrne (2013, 125) levels at the companion volume, Communitas, that it is a 'phallic work' might also be relevant to Immunitas.
Finally, Esposito's characterization of the gestational relationship makes no mention of the placenta, despite the fact that it plays a primary role in mediating the relation between the gestational body and that of the developing embryo and foetus. Within immunology, it is now understood that the site of placentation is actually central to the maternal immune response. The placenta develops as a material border between the gestational body and the foetus such that the gestational body and foetal body are in fact kept separate from one another. Nevertheless, the purpose of this border is transmission as much as it is separation – indeed, the placenta is a material border or boundary that reveals the way in which separation is fundamentally entwined with transmission. As this suggests, the significance of the role of the placenta is not limited to a question of biological accuracy; rather, as feminists such as Anne-Maree Maher (2002) and Julie Palmer (2009) have argued, as a socio-political metaphor, the placenta provides a highly productive starting point for re-thinking the maternal body, and subjectivity more generally, in a manner that recognizes the constitutive entwinement of self and other. Thus, while Esposito wants to use the phenomenon of maternal immune tolerance as an exemplary metaphor for the affirmative logic of the immunitary paradigm, his failure to mention the placenta indicates that he is missing an opportunity for a more thorough reconception. This suggests a certain resistance on his part to the more radical implications of the generative capacities of the maternal body for his account of biopolitics. I will address this concern at several further moments in this book, but for now, it is necessary to get a clearer sense of how Esposito's theorization of the immunitary paradigm is extended and combined more thoroughly with the concept of biopolitics in the book, Bios.
While published slightly later than Immunitas, Bios was Esposito's first major entry into Anglo-American debates on biopolitics. In it, Esposito sets up his approach as a direct response to Foucault, but one which attempts to draw out the ambivalence of his formulation which had largely been passed over in silence by later theorists of biopower. Not altogether unlike Agamben's conclusion to Homo Sacer, Bios opens with a number of disparate and seemingly unconnected examples. These include: a wrongful life claim in the French Appeals court; simultaneous humanitarian food drops and bombings in Afghanistan after the September 11 attacks; the police raid on the Dubrovska Theatre in Moscow in 2002 against Chechen hostage-takers; the government's infection of villagers in China with the AIDS virus following donation of blood; and mass ethnic rape during the Rwandan genocide. Esposito argues of these examples that their commonality becomes intelligible through a biopolitical logic whereby the protection of life reverses into the production of death. Further, he argues that while Foucault was himself aware of this nexus within biopolitics, he did not provide a satisfactory answer to the seemingly inexorable link between life and death that modern biopolitics entails. Moreover, the 'radically negative' and 'absolutely euphoric' responses to Foucault also do little to address or resolve this nexus either, since they each focus on only one side of the problem. Surpassing these approaches, Esposito proposes to provide the 'missing link' required to resolve the nexus, primarily by unearthing not only what biopolitics signifies, but 'how it was born' (Esposito 2008, 8) This missing link is, according to Esposito, immunization.
In developing this thesis, Esposito affirms a central point of Foucault's genealogy of biopower, that it is principally a modern phenomenon. This is because the immunitary logic of defence through the controlled incorporation of a threat became a key feature of modern politics. Esposito (2008, 9) writes, 'only modernity makes of individual self-preservation the presupposition of all other political categories, from sovereignty to liberty' evident in, for instance, the political philosophy of Hobbes. However, as with Agamben, he also affirms the privileged position of Nazism, insofar is it represents the apotheosis of biopolitical dynamics. This is because the immediate translatability of life into politics, such that politics assumes an 'intrinsically biological' character, did not appear fully until the 1930s, in Nazism (Esposito 2008, 9). Esposito is explicit though that this does not mean that Nazism produced a biopolitical 'philosophy', since (in accordance with a rich vein of critique), Nazism in fact engendered the destruction of philosophy. He writes, 'Nazism does not, nor can it, carry out a philosophy because it is an actualized [realizzata] biology... Nazism's transcendental is life, its subject race, and its lexicon biological' (Esposito 2008, 112). Further, Nazism did not simply continue the metaphorical association of politics and biology. Rather, by collapsing every distinction between biology and politics, and demanding that 'politics be identified directly with biology', it constituted a 'completely new form of biocracy' (Esposito 2008, 113). The concrete realization of this biocracy lay in the tight symbiotic relation between the institutions of biology and politics, in which the legitimacy of the biomedical sciences gave strength to the political powers, and in return, the regime provided the bodies required for biomedical experimentation.
Underpinning this biocracy was an extreme immunitary logic in which life and death were fully transposed. Elaborating on this further, Esposito argues that the central logic of the Nazi biocracy entailed the superimposition of life and death, whereby the protection of the German people required putting to death all those that were seen to threaten its health and vitality. In effect, Nazism appears as a paroxysmal realization of the logic of the immunitary paradigm, since at its heart lies the mobilization of mass homicide with the end and rationale of protecting and regenerating the German people. In elaborating this dynamic, three features of Nazism stand out for Esposito, namely, the doctrine of degeneracy, the practice and thinking of eugenics, and the institutionalization of genocide. The doctrine of degeneracy is important since it was used to legitimate the need for many Nazi techniques and, further, linked Nazism to a more widely accepted discourse in philosophy, culture and medicine. As a popular psychophysical doctrine of the nineteenth and early twentieth centuries, degeneracy made sense of the apparent weakening of the German race and gave authority to the social ostracism of those considered degenerate or abnormal. Its correlation with notions of both heredity and contagion also gave apparent justification for strict measures to protect the German people.
These measures were further bolstered by eugenics, which Esposito interprets as the positive or generative counterpart of degeneracy. Eugenics enjoyed enormous popularity world-wide in the early twentieth century. In general, its practices entailed both a positive and negative component, the former attempting to foster breeding among populations deemed genetically fit or superior, and the latter through the suppression of the life of those deemed unfit or inferior, either through sterilization and the prevention of reproduction or in a more extreme manifestation, through murder. Notably, Esposito refers to such eugenic killing as euthanasia; while this confuses in the context of contemporary debates on the right to die, part of his point is that the lives subjected to killing by the Nazis were not considered life but mere existence in such a way that the putting to death was simply a realization of an internal aspect of that existence. This extreme negative eugenic practice was institutionalized in Nazi Germany in a way not taken up in other states – that is, in the mass killing of people considered unfit or unworthy of life.
While initially targeting groups such as the disabled and homosexuals, this institutionalized negative eugenics was soon extended through integration with doctrines of race to include homogenous populations such as the Romany and the European Jewry. In this integration, negative eugenic practices became strictly genocidal. Of these genocidal practices of Nazism, Esposito argues that while biopolitical theorists converge on the identification of a caesura in life introduced by its increasing implication in politics between those who must live and those who must die, the advantage of the immunitary perspective is that it reveals the specifically 'homeopathic tonality' of Nazi biopolitcs, in which '[t]he disease against which the Nazis fight to the death is none other than death itself. What they want to kill in the Jew and in all human types like them isn't life, but the presence in life of death... death became both the object and instrument of the cure, the sickness and its remedy' (Esposito 2008, 137–8). The identification of this as a homeopathic logic recalls the discussion in Immunitas, where Esposito claims that a shift from understanding the treatment of disease on allopathic principles to homeopathic ones was an important moment in the development of the immunitary paradigm. In the Nazi schema then, death was identified as an inherent aspect of the life of maligned populations, and the homeopathic treatment for that was more death.
At this point, it is important to recognize that while Esposito posits Nazism as the apotheosis of immunitary biopolitics, it also has further significance for him. For he argues that this biopolitical logic subsequently extends from Nazism to infect modernity more generally, such that it constitutes the fundamental matrix of modern politics and political thought. Consequently, engaging with Nazism is necessary for any new concept of politics, and more specifically, for developing an affirmative biopolitics that does not simply repeat the superimposition of life and death that it has so far entailed. In other words, as the apotheosis of the immunitary logic of biopolitics, Nazism may also provide the starting point for rethinking the life-affirming possibilities of biopolitics. Because Nazism works within the immunitary logic in such a 'paroxysmal manner as to turn the protective apparatus against its own body' (Esposito 2008, 10) and 'represents the culmination of biopolitics, at least in that qualified expression of being absolutely indistinct from its reversal into thanatopolitics... [it is] precisely for this reason the catastrophe in which it is immersed constitutes the occasion for an epochal rethinking' (Esposito 2008, 10). Hence, it is precisely from the negative biopolitical core of Nazism that Esposito seeks an affirmative biopolitics uncontaminated by the thanatology that emerges in modern immunitary biopolitics.
The development of such an affirmative biopolitics from Nazism entails tracing and radically deconstructing three principal 'immunitary dispositifs' through which Nazi biopolitics and the superimposition of life and death was operationalized. These are: (1) the double enclosure of the body; (2) the pre-emptive suppression of birth; and (3) the normativization of life. According to Esposito, the point of an affirmative biopolitics is not to simply eschew the logic of thanatopolitics as both Agamben and Negri arguably do, but rather to re-assume its categories – particularly those of 'body', 'birth' and 'life'. This re-appropriation seeks to convert 'their immunitary (which is to say their self-negating) declension in a direction that is open to a more originary and intense sense of communitas' (Esposito 2008, 157). In Esposito's view, this provides the only avenue toward a 'biopolitics that is finally affirmative' (Esposito 2008, 157). Given this, in the remainder of the chapter, I focus on these three key immunitary dispositifs, and the direction in which Esposito indicates that each of them may be taken in order to give rise to an affirmative biopolitics.
In regards to the dispositif of the double enclosure of the body, Esposito focuses on the politicization of the body within Nazism, especially through race and racism. Esposito first refers to the collapsing of distinctions between the corporeal body and notions of ego or self, such that 'the body is no longer only the place but the essence of the ego' (Esposito 2008, 141; emphasis added). However, he claims that this should not be understood as the reduction of bios to zoē or to 'bare life'; instead, this should be understood as a 'spiritualisation of zoē and the biologization of the spirit', which he notes 'constitutes the nucleus of Nazi biopolitics' (Esposito 2008, 142, 217 n83). Interestingly, Esposito claims that the name that should be given to this superimposition of the spiritual and biological is race, since within Nazism it is race that 'confers meaning on the identity of the body with itself, a meaning that exceeds the individual borders from birth to death' (Esposito 2008, 142). Moreover, race allows a further doubling of the body on itself. While the superimposition of the spiritual and the biological remains on the level of the individual, the concept of race allows the subsequent incorporation of every 'corporeal member' into the 'larger body that constitutes the organic totality of the German people' (Esposito 2008, 142). One consequence of this is that maintaining the – spiritual and biological – health of the German people requires the separation and elimination of pathological elements, namely, all those who were deemed to threaten the racial character of the German people.
In attempting a radical deconstruction of this dispositif, Esposito focuses on the second aspect, which is the incorporation of the individual into the communal or public body. His line of thinking is that all alignments of the body with the state – whether in the form of Hobbesian authoritarianism or Rousseauian contractualism – end up producing an 'immunitary short-circuit' that closes the political body onto itself in opposition to its outside. The Nazi racial-biological alignment of the body of individuals with the German nation and people was an extreme version of this, which managed to incorporate within the political body the immunitary 'line of distinction between inside and outside' (Esposito 2008, 158), ultimately producing a remnant understood as 'existence without life'. Esposito renames this remnant 'flesh', in reference to Merleau-Ponty; it then becomes the deconstructive mechanism by which the double enclosure of the body can give way to an affirmative biopolitics. By the term 'flesh', Merleau-Ponty indicates something along the lines of that which is in common between the body of the individual and the world. This is not simply matter or substance, but the ontological foundation or bedrock of being that underlies sensory experience. Flesh refers to the 'sensibility of things, the perceptibility both of the perceptual environment and of ourselves as perceivers (Carman 2008, 123). For Esposito, this lack of differentiation in the concept of flesh between body and world can lead to a positive construal of community, in which, because of the lack of exteriority, the immunitary enclosure of the body gives way to irreducible multiplicity. Esposito is aware of problems with the notion of flesh, and responds briefly to critiques by philosophers such as Deleuze, Derrida and Jean-Luc Nancy. He points out, though, that these critiques can be read as being directed more specifically to the Christian associations of the notion. Given this, he holds that 'the notion of flesh needs to be rethought outside of Christian language, namely, as the biopolitical possibility of the ontological and technological transmutation of the human body' (Esposito 2008, 168).
The Nazi immunitary dispositif of the pre-emptive suppression of birth is also related to matters of race and the notion of heredity that goes hand in hand with it. As Esposito points out, sterilization of large sections of the population was crucial to the operation of Nazi biopolitics, which included castration of homosexuals, sterilization of women over the age of 36 and tubal ligation or hysterectomy for populations of women deemed degenerate or mentally deficient. Along with the enormous emphasis on sterilization went juridical decrees on obligatory abortion in cases of unauthorized procreation. Interestingly, the negative eugenics of the suppression of birth operated alongside an intense pro-natalism that sought to ensure the 'regeneration' of the German people through promoting births in populations of sufficient racial quality, through for instance, enforced reproduction among the SS and authorized women and the kidnapping of children from surrounding countries such as Poland (see Clay and Leapman 1995). Thus, the value of birth was determined by its place within a 'political-racial calculation'. However, it was in the concentration camps that the suppression of birth took on its ultimate form, where birth and death are fully integrated into one another. Following Arendt's analysis of the camps, in which she points out that it was as if the inmates, who 'nobody is supposed to know if they are alive or dead', had 'never been born' (Arendt cited in Esposito 2008, 145), Esposito claims that the ultimate power of sovereignty within biopolitics is not so much the capacity to put a subject to death as 'to nullify life in advance' (Esposito 2008, 145).
In considering the need to deconstruct the dispositif of the suppression of birth in Nazism, Esposito first discusses the intersection of the notion of birth and nation, which he suggests finds its most 'exasperated expression in Nazism'. Of interest to Esposito is the modern reversal of priority between the biological inflections of birth in nativity, and the political ones in the nation, such that the latter came to have priority and 'defines the domain in which all births are connected to each other in a sort of parental identity that extends to the boundaries of the state' (Esposito 2008, 171). Nazism constituted both a continuation of this logic and its disruption, insofar as what was at issue was not simply the politicization of birth, but the co-extension of the biological and the political, in which 'politics is nothing other than the modality through which birth is affirmed as the only living force of history' (Esposito 2008, 171). At the same time, and because of this, birth became the 'fold along which life is separated from itself' and 'birth itself becomes the object of a sovereign decision that, precisely because it appears to originate directly from it, transcends it, traversing it along excluding lines' (Esposito 2008, 171). This, he argues, accounts for the ambivalence of the Nazi response to birth, simultaneously fostering some lives and pre-emptively excluding others. The ambivalent relation between the biological and political in the notion of birth also provides Esposito with an entry point to consider Arendt's revaluation of birth in the wake of Nazism, in which she emphasizes the centrality of natality to politics. Esposito points to the ambiguity in the Arendtian notion of natality, where birth simultaneously relates directly to the 'animality in man' and is also the thing that separates man most clearly from the animal in its political valence. He concludes that this Heideggerian tonality ensures that she remains 'on this side of the biopolitical paradigm' (Esposito 2008, 179) since she does not and perhaps cannot answer the question of how the vitality of life generates the political salience of action and the singularization it entails. In effect, she remains committed to a problematic chiasm in life, where birth becomes the point of intersection of the vital and political, and in Esposito's view, she is without theoretical means to integrate them beyond this biopolitical manifestation.
Given this, Esposito then turns to the work of Gilbert Simondon to find a way to push beyond the biopolitical suppression of birth. Two features of Simondon's work are of interest to Esposito – first, his dynamic ontology of life in which being is identified with becoming, and, second, the notion of individuation, by which Simondon means the necessarily incomplete and continuous process by which individuals emerge from a pre-individual foundation, and every individuation is also the occasion for another. The importance of this perspective for Esposito is two-fold. First, it avoids making a separation between the living and the human, or between animality and humanity, or vital life and political life; the human is only ever the living human, such that 'man never loses his relation with his living being' (Esposito 2008, 180). Further, the notion of birth is central to Simondon's understanding of individuation in the sense that each new individuation is effectively a form of birth and gives birth to another, such that life becomes understood as 'perpetual birth': as Simondon writes, 'to live is to perpetuate a birth that is permanent and relative' (Simondon cited in Esposito 2008, 181). From this superimposition of birth and life, Esposito concludes that Simondon effectively reverses the Nazi suppression of birth, in which life and death are superimposed, by 'guiding all life back to the innovative potential of birth' and making it the point of distinction from death. Thus, he writes, '[i]f one thinks about it, life and birth are both the contrary of death: the first synchronically and the second diachronically. The only way for life to defer death isn't to preserve it as such... but... to be reborn continually in different guises' (Esposito 2008, 181). We will have occasion to return to this characterization of birth later, but suffice to say here that it strikes me as more problematic than Esposito allows, insofar as birth becomes a metaphor for the new, and further, in that the superimposition of life and birth has the interesting effect of excluding the female body that gives birth: birth is not only a perpetual feature of life, but that life is ultimately born of nothing.
Finally, of the normativization of life, Esposito argues, contra Agamben, that the Nazi regime was characterized by an absolute normativization of life, such that this regime did not derive its power from the subjective decision in the shadow of the suspension of law, but rather, in the derivation of a normative framework from the very 'vital necessities of the German people'. The relation between law and life at stake in this, he argues, entails a double presupposition whereby the juridical norm presupposes the facticity of life, and life presupposes 'the caesura of the norm as its preventative definition' (Esposito 2008, 184). Thus, he concludes that Nazism created a 'norm of life' – not, however, in the sense that it adapted its own norms to the demands of life, but in the sense that it 'closed the entire extension of life within the borders of a norm that was destined to reverse it into its opposite' (Esposito 2008, 184), that is, into death.
To establish this argument, Esposito tracks the 'juridicization of medicine' and correlative biologization or medicalization of the law within the time of the German Third Reich. He claims that within this period, the biological and the juridical were 'for the first time... completely superimposed', evidenced by the pre-eminent authority of medical figures within the Reich, as well as the integration of medical science in every step of the eugenic killing orchestrated by it, from management of the T4 program, to selection of victims for gassing, and even its administration. In the course of this superimposition of the biological and juridical, doctors became little more than public functionaries, resulting in a 'clear-cut transformation of the relation between patient, doctor and state' (Esposito 2008, 139) whereby the first relation was increasingly loose and the second increasingly tight. Further, the integration of the juridical and biological reached its apotheosis in the concentration camps. Rather than simply destroying law, the Nazi regime 'extended it to the point of including within what also obviously exceeded it' (Esposito 2008, 140) insofar as all aspects of life were subordinated to norms. Ultimately, the camps constituted a form of pre-emptive detention, in which what was 'detained in advance... was life as such, subjected to a normative presupposition that left no way out' (Esposito 2008, 140). In effect, life and norm are deeply entwined in such a way that life is wholly subjected to the demands of a norm, itself an external form of arbitrary juridicism and life maintains no autonomy with regard to the norm; life becomes fundamentally and inescapably heteronomic.
The problem for Esposito at this point is to suggest a way forward to a genuine politics of life or an affirmative biopolitics that breaks this deadly knot in which life and norm are entwined and mutually presupposed. He argues that attempts to distinguish more clearly between life and norm, such as in transcendental normativism and juris-naturalism, are unsatisfactory responses, however, since neither the absolutization of the norm nor the primacy of nature can be considered external to Nazism. Instead, then, Esposito looks for resources in philosophical traditions that have emphasized the radical immanence of life and norm, and which in that way undermine the double presupposition that ties them together in Nazism. Of particular interest to him is the tradition of a philosophy of immanence, in which life and norm coincide in a continual process of becoming, initiated by Spinoza and continued in different forms in the work of Georges Canguilhem, Gilbert Simondon and Gilles Deleuze. Of these, and in resonance with his earlier comments in Immunitas, he suggests that the theorization of vital norms developed by Canguilhem may be especially valuable, since it allows for the 'maximum deconstruction of the immunitary paradigm and the opening to a different biopolitical lexicon' (Esposito 2008, 191).
To reach this conclusion, Esposito references the radical vitalization of the norm that Canguilhem proposes in his work on the concepts of the normal and the pathological in the history of medicine. Here, he argues that life is internally and necessarily normative, since even at the simplest level 'living means preference and exclusion' (Canguilhem 2008, 136). Living necessarily involves polarities of valuation, such that an organism cannot be understood as indifferent to the environment in which it finds itself. Esposito goes on to emphasize that this means that disease and health are both normative states in that both indicate new forms of life for the organism, and moreover, reveal the normal functioning of the body. Conditions of disease or biological abnormality are not simply deviations from a fixed prototype of the normal: they are instead normative forms of a qualitatively different order. Similarly, to be 'normal' is not to coincide with a pre-established norm, but rather, to be able to harness and maintain one's own normative power: to be normal is to be able to create new norms. In view of this radicalized immanence of life and norm, Esposito writes that '[i]f Nazism stripped away every form of life, nailing it to its nude material existence, Canguilhem reconsigns every life to its form, making of it something unique and unrepeatable' (Esposito 2008, 189). For Esposito, the productive power of Canguilhem's thinking is that the immanence of norms in life undermines the separation and mutual presupposition of the facticity of life and normative transcendentalism. Moreover, this analysis rejects an objectivist approach to life and emphasizes instead the vital potential in life, in terms of an internal capacity to generate norms.
Building on this position, Esposito turns to the essay by Deleuze on Charles Dickens' novel, Riderhood, in which Deleuze proposes a conception of life understood as pure immanence, an essay which was also significant for Agamben, and which I discuss in more detail in Chapter 6. Suffice to say here that for both Esposito and Agamben, Deleuze's essay marks a critical juncture in the development of a philosophy of life beyond biopolitics, particularly in its theorization of impersonal life, in which Esposito sees the construal of the norm as the immanent impulse of becoming. He concludes Bios with the comment that this means that 'no part of life can be destroyed in favour of another' since 'every life is a form of life and every form refers to life' (Esposito 2008, 194); this, he suggests, provides the minimal presupposition for the development of an affirmative biopolitics. In later chapters, we will have occasion to revisit this proposal of the immanence of the norm in life, and, in particular, to question whether the immanence of normativity and life is as well established by Canguilhem, or is as politically valuable, as Esposito supposes.
Conclusion
To conclude, in this chapter I have considered efforts to develop an affirmative or positive approach to biopolitics, that shifts away from the wholly negative, thanatopolitical conception proffered by Agamben. In the first section, I outlined the account of Empire and the transformative power of the multitude provided by Hardt and Negri. Their account of an affirmative biopolitics relies upon the immanent potentiality of the multitude, which is simultaneously constitutive of Empire and capable of overcoming it through producing alternative forms of living. In the second section, I outlined the recent work of Esposito, who attempts to integrate the negative and positive aspects of biopolitics in his conception of the immunitary paradigm. Esposito's work is complex, but the general argument is that modern biopolitics is characterized by an immunitary logic in which self-defence is precipitated by the controlled incorporation of a threat. In Bios, this is cast more specifically as the incorporation of death in life in order to protect life. And within this frame, Nazism appears as the most extreme manifestation of the entwining of life and death in biopolitics. However, Nazism also becomes a privileged site of intervention, insofar as it may also provide the point for the resolution of the biopolitical integration of life and death. For Esposito, an affirmative biopolitics must be sought through the maximum torsion of the three key dispositifs of Nazism. He urges a reappropriation of the biopolitical terms of body, birth and life, and their radical deconstruction to yield an affirmative biopolitics that breaks the immunitary logics of modernity.
This summary of recent conceptions of affirmative biopolitics completes the first part of this book. In the second part, I will address the theories outlined here from more specific angles to both give shape to the ways in which certain problems or questions have been addressed in the debate so far, and to show the failures of the principal contributors to address other questions or problems. Thus, I discuss a broad range of literature to bring into focus several key points of contention as well as some significant lacunae within the contemporary field of biopolitical studies.
Notes
Hardt and Negri capitalize 'Empire' throughout the book, to distinguish their concept from earlier imperial projects.
This relation between immunity and community is touched on throughout Immunitas, but is more thoroughly discussed in the text Communitas, in which Esposito examines the notion of community in depth.
References
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Canguilhem, G. (2008). The Normal and the Pathological. Knowledge of Life. Eds. Marrati, P. and Meyers, T. New York, Fordham University Press: 121–133.
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Part II
5Politics
Sovereignty, violence, rights
The chapters in this part of the book address more specific problems that traverse the work of the theorists discussed in Part one, and allow a more focused analysis of their contributions to thinking biopolitics. The first of the sets of problems that I take up relates broadly to the category of politics, and in this I develop a more detailed analysis of several key concepts and themes within biopolitical studies. Following this, I turn to the concepts of life that are produced by biopolitical configurations, and attempts in biopolitical studies to develop alternative conceptions that may challenge and undermine biopolitical life. In the final chapter, I take up the issue of subjectivity and explore the ways in which this has been understood as a mechanism of biopolitical management. In the course of these chapters, I touch on a number of phenomena and outline the way in which they have been discussed or ignored in the contemporary debates; these include sovereignty, governmentality, rights, violence, technology, reproduction, race and sex/gender. Of course, my discussion is by no means exhaustive, either of these topics or of the range of important topics taken up within biopolitical studies. However, in the following, I hope to trace the broad parameters of debates, and particularly identify lines of thinking that are in need of further analysis.
This chapter takes up the thematic of politics. In the first section, I focus on the question of the role of sovereign power in biopolitics, which is a deep and vexed one. Foucault was himself troubled by the relationship between sovereignty and biopower, and in lecture series that would have naturally extended on the concept of biopower, he appears to give it up in favour of concepts of governmentality and security. In Agamben, the relationship between sovereignty and biopolitics seems clear in Homo Sacer (1998), but is considerably revised in later volumes included in the Homo Sacer series, such as State of Exception (2005) and The Kingdom and the Glory (2011). As this suggests, there is considerable dissension over how to best characterize the historical and conceptual relationship between sovereignty and biopolitics, and the implications of this relationship. Following this, in the second section, I turn to the related problem of violence within biopolitics – the question here is whether biopolitics should be thought of as inherently violent, or whether the mobilization of violence indicates a remnant of sovereignty within biopower as Foucault suggested at times. In the course of this discussion of violence, I touch on phenomena of the state's involvement in the production and management of death, as well as the prolongation of life. In the third and final section, I briefly address the question of law and rights, especially human rights. I consider Arendt's critique of human rights and its influence on Agamben's subsequent rejection of human rights, as well as Foucault's provocative call for a new form of right.
Sovereignty and government
Foucault's approach to sovereignty as the traditionally central problem of political theory and philosophy was on the face of it straightforward, summed up in the much-cited claim that:
political theory has never ceased to be obsessed with the person of the sovereign... What we need, however, is a political philosophy that isn't erected around the problem of sovereignty, nor therefore around the problems of law and prohibition. We need to cut off the King's head: in political theory that has still to be done.
(Foucault 1980, 121)
In keeping with this claim, in the several discussions in which Foucault outlined general principles for the analysis of power, he consistently contrasted these with sovereign power. The upshot of this is that the model of sovereignty was no longer effective as a means of understanding the operation of power. Even so, Foucault himself did not unequivocally state that sovereign power had disappeared from the political horizon; in fact, at several points in his discussions of biopower, he suggests that these forms of power reticulate in various ways. Furthermore, to the extent that his own approach to analysing power did do away with the centrality of the figure of the king, he offers several alternate ways of conceiving of power, primarily through the concepts of disciplinary power and biopolitics of population. However, in a lecture series concurrent with and immediately after the publication of Will to Knowledge (1990) (in which Foucault introduced the concept of biopower into his published works), he seems to begin to extend on the concept of biopolitics but then replaces it with other terms, primarily that of governmentality.
The term 'governmentality' appeared in Foucault's work in 1978, in the series of lectures published under the title Security, Territory, Population (2007). Foucault opens this series of lectures with the statement that in them he intends to study 'the set of mechanisms through which the basic biological features of the human species became the object of a political strategy... roughly what I have called bio-power' (Foucault 2007, 1). Thus, the core problem of the lectures was the emergence of population as the key object of political reason and techniques. From this initial set-up, the first several lectures of the course explore the meaning and intersection between the three terms of the course title. In this, what Foucault says about security and its relation to territory and population is at least broadly consistent with what he says about the series 'population-biological processes-regulatory mechanisms-State' (Foucault 2003, 250) that he discussed in the final lecture of the previous series, published as Society Must Be Defended (2003). However, we also see emerge in these lectures an increasing use of the terminology of governing, until in the fourth lecture 'the problem of government' takes centre-stage and will then dominate the rest of the lecture series.
In this lecture, Foucault argues that throughout the sixteenth century, there was an explosion of the concern with the idea of government across diverse domains, including the question of how to govern oneself (involving a return to Stoicism), and the government of children in pedagogy. One of these concerns was that of 'the government of the state by the prince' (Foucault 2007, 88), which marks a crucial moment in what Foucault ultimately calls the 'governmentalization of the state' (Foucault 2007, 109) and its reorganization in a form that was distinct from that of sovereignty. Despite the emergence of this concern in the sixteenth century, its development was hampered by the 'military, economic and political emergencies' of the seventeenth century, along with the pre-eminence of sovereignty as both a theoretical problem and institutional arrangement. However, in the eighteenth century, the blockage of the 'arts of government' was overcome, largely due to the emergence of population as a political phenomenon, made possible through statistics (the 'science of the state'), which revealed aggregate effects and regularities relating to populations that could not be reduced to those of the family. Further, the emergence of population involved several specific transformations that led away from sovereignty and made the development of a different mode of governing possible. According to Foucault, these principally include the introduction of economy into the domain of politics, the displacement of the family as a model of economy to an instrument of governance and the repositioning of population as both end and object of governance. Through these complex and intermingled shifts, the central problem of the state is no longer that of ensuring the strength of the sovereign over its subjects, but gradually comes to be a matter of ensuring the well-being of the population as a central measure of the success of the governmental management of things and the strength of the state.
In attempting to situate this development of governmental reason in relation to alternate conceptions of power such as sovereignty and discipline, Foucault makes it clear that he does not see this as a matter of replacement. In regards to the sovereignty, Foucault is explicit that he does not suppose that government wholly displaced sovereignty as an organization of power in the eighteenth century. In fact, he suggests, the problem of sovereignty was never sharper, since the very nature of the sovereignty of the state – its juridical and institutional form as well as its legitimacy – had never been more strongly problematized (Foucault 2007, 106–7). Similarly, while discipline emerged alongside administrative monarchies, it was never more necessary than in the management of the population, since its techniques allowed access to the minutiae within the collective phenomena. Thus, Foucault concludes that 'we have a triangle: sovereignty, discipline, and governmental management, which has population as its main target and apparatuses of security as its essential mechanism' (Foucault 2007, 107–8). In the remainder of the lectures in this series, he goes on to trace aspects of the development of governmental management, from its origin in the early Christian pastorate, its reliance on 'diplomatic-military' techniques that reference the principle of raison d'état and limited the actions of states vis-à-vis other states, and its integration with the institution of the police that emerged contemporaneously with reason of state.
The tripartite arrangement of sovereignty, discipline and governmental management that Foucault posed has been much discussed, since it complicates the picture of the relation between biopower and sovereignty presented in Will to Knowledge, and because it leaves largely unclear the specific relations that hold between the three formations of power. Further, the shift from an explicit language of biopolitics to that of governmental management has led some scholars to argue that Foucault moved away from his initial inquiries into biopower, in favour of the more analytically specific notion of governmentality. For instance, one of the leading scholars of governmentality, Mitchell Dean, argues that 'the idea of biopower remains conspicuous here only by its absence' (Dean 2013, 46). For Dean, this absence is indicative of Foucault's ultimate rejection of this terminology, perhaps, he speculates, because Foucault came to see in it the tendency to unification that he had formerly criticized, and because of a 'political embarrassment' about the denunciatory valence of the term (Dean 2013, 40). Dean's detailed analysis of the shifts in Foucault's thought and his failure to return to the terminology of biopower offers a compelling view; however, it relies upon treating the terms 'biopower' and 'biopolitics' as synonymous. Conversely, maintaining a stronger distinction between these terms allows a different view of what is at stake in these shifts. For what is absent from Foucault's claim is not strictly biopower, but biopolitics. The former of these is recalled in the positioning of discipline, which it encompasses, within this triangulation. Biopolitics, has, however, been replaced by government. We might then conclude that Foucault renames the pole of biopower that addresses itself to the population as a political subject, but does not in fact reject the framework of biopower tout court.
Arguably, however, the lecture series of the following year presents another terminological, if not conceptual, challenge in its investigation of liberalism as a specific art of governing. Placing these lectures in relation to the foregoing, Foucault indicates that this lecture series, entitled The Birth of Biopolitics, extends on the central problem of population, 'on the basis of which something like biopolitics could be formed' (Foucault 2008, 22). However, Foucault also says that understanding biopolitics requires first getting clearer on liberalism; he states, 'only when we know what this governmental regime called liberalism was, will we be able to grasp what biopolitics is' (Foucault 2008, 22). From this starting point, Foucault traces the development of liberalism, understood broadly, from classical English liberal theorists such as that of Jeremy Bentham, through to more recent iterations in German Ordoliberalism and the Chicago School of American neoliberal thought. He argues that in the mid-eighteenth century, a new art of governing emerged that challenged the raison d'état of the previous century. What was specific about this new art of governing was its guiding concern with the perceived problem of governing too much, and the consequent elaboration of an internal limit on the growth of the state. Thus, rather than the state being limited in its activities by external public law, it became self-regulating and self-limiting, a development that Foucault argues was made possible by the consolidation of political economy in the mid-eighteenth century (Foucault 2008, 13–18).
Foucault goes on to outline several fundamental shifts internal to the possibility of a liberal art of government, some of which I mention here. The first of these was the emergence of the market as a 'mechanism of exchange and a site of veridication regarding the relationship between value and price' (Foucault 2008, 44). By this, he means that a view arose that the natural mechanisms of the market worked to reveal the 'true price' of a thing. Further, 'inasmuch as prices are determined in accordance with the natural mechanisms of the market they constitute a standard of truth which enables us to discern which governmental practices are correct and which are erroneous' (Foucault 2008, 32). Thus, the market becomes a means by which to verify or falsify governmental practice. The second relates to the principle by which public authorities could be regulated by laws, and of this Foucault argues for the historical primacy of the English 'radical' approach to governing, which urged that government is assessed with regard to utility. This means that '[g]overnment's limit of competence will be bounded by the utility of governmental intervention' (Foucault 2008, 49), its benefit or harm, without reference to fundamental rights that may be asserted against the sovereign. Third, as this indicates, the relationship between the sovereign and the subject is recast as a question of the relationship between the governed and the government. No longer a matter of the assertions of fundamental rights of subjects, governmental concern was rather with interests and their extension or truncation through harm or danger. Specifically, the concern was with the assertion of one set of interests at the expense of another, and especially the intersection of individual interests and the collective interest. Foucault concludes that the protection of collective interests against individual interests is 'the problem of security', such that the 'game of freedom and security is at the very heart of this new governmental reason' (Foucault 2008, 65) called liberalism.
Foucault's turn to the genealogy of liberalism in lectures ostensibly on the birth of biopolitics raises significant questions about the perceived relationship between liberalism and biopolitics. As both Thomas Lemke (2011, 48) and Mitchell Dean (2013, 39) discuss, Foucault appears to understand 'liberalism as the general framework of biopolitics' and suggests that biopolitics is part of the larger picture of 'this new governmental reason' (Foucault 2008, 21–22). Foucault draws the link between liberalism and biopolitics even tighter when he claims that the problems of biopolitics cannot be separated from liberalism, since 'it is in relation to liberalism that they assumed the form of a challenge', insofar as the political subject of the population was to be taken into account in 'a system concerned about respect for legal subjects and individual free enterprise' (Foucault 2008, 317). Making sense of the integration of liberalism and biopolitics, Lemke points out that as an arts of government, liberalism entailed a conception of nature that opened it up as a field of possible intervention, such that '[n]ature is not a material substratum to which governmental practices are applied but rather their permanent correlate' (Lemke 2011, 46–7). Further, liberal arts of government relied on technologies of security to manage the interaction of individuals and populations and ensure well-being, which they do through a normalization process that starts from the measurable reality of a phenomena and the manipulation of variability from the norm. From this, we can see that liberalism is a privileged object of analysis for the project of understanding biopolitics, for it is in liberalism that the non-reductive reticulation of the individual and the population became especially acute. Hence, while the shifts across the lecture series concurrent with the published statements about biopower and biopolitics well illustrate Foucault's tendency to refine, elaborate and rework his concepts and analyses, this need not indicate that he rejected the framework of biopower and biopolitics. Rather, we can see both the lecture series discussed here as working within the analytic and genealogical 'horizon' of biopower (Senellart 2007, 370) as set out in Will to Knowledge, and broadly continuous with the earlier lecture series. To be sure, they complexify the early, somewhat schematic analysis in, say, Society Must Be Defended, but ultimately these lectures extend on rather than break from it.
Interestingly, a similar shift toward the language of government is also evident in the work of Agamben. As we saw in Chapter 2, in the book Homo Sacer, Agamben advances the view that rather than biopolitics constituting a different regime of power (related to sovereignty in complex ways), sovereignty is itself biopolitical. As he puts it, 'the inclusion of bare life in the political realm constitutes the original – if concealed – nucleus of sovereign power. It can even be said that the production of a biopolitical body is the original activity of sovereign power' (Agamben 1998, 6). This constitutes a decisive refiguring of the position of sovereignty in modern politics from that explored by Foucault; rather than the sovereign right of death constituting one element among others in the multiple techniques of power mobilized in biopower, sovereignty here again fills the field. Political power is, then, sovereign power. Further, this power consists in the capacity to decide on the normal situation and the exception, as proposed by Carl Schmitt. This repositioning of sovereignty necessarily led to a very different conception of biopolitics – as we saw in earlier chapters, what was at issue was no longer social norms as regulatory devices, but the law and its integral relation to violence and the power of the sword.
While there was little ambiguity in Agamben's view of the centrality of sovereignty in Homo Sacer, what becomes apparent in books after this is that sovereignty loses its centrality until in The Kingdom and the Glory, it is almost entirely occluded by the paradigm of government. In State of Exception, Agamben claims, 'the state of exception tends to increasingly appear as the dominant paradigm of government in contemporary politics' (Agamben 2005, 2). The appearance of the concept of government here is interesting, but it remains un-theorized and the discussion throughout this text refers again primarily to the problem of sovereignty. Even so, this provokes questions about the relationship between government and sovereignty. These questions become even more pressing in regards to The Kingdom and The Glory. Here, Agamben argues that two political paradigms – which are 'antinomical but functionally related to one another' – can be traced from early Christian theology. He identifies these as 'political theology, which founds the transcendence of sovereign power on the single God, and economic theology, which replaces this transcendence with the idea of an oikonomia, conceived as an immanent ordering – domestic and not political in a strict sense – of both divine and human life' (Agamben 2011, 1). He then makes the somewhat surprising claim that '[p]olitical philosophy and the modern theory of sovereignty derive from the first paradigm; modern biopolitics up to the current triumph of economy and government over every other aspect of social life derive from the second paradigm' (Agamben 2011, 1). This looks like a significant revision of earlier claims, in which sovereignty and biopolitics appeared to be one and the same; now, sovereignty is sidelined and oikonomia – the government of the household that is excluded from politics proper – takes priority as the origin of biopolitics. Given this reconfiguration, it is worth considering Agamben's argument further.
Agamben's discussion in The Kingdom and the Glory is complex and wide-ranging; here I will simply make several points about the broad argument made therein. The first of these addresses the way in which Agamben locates his own argument vis-à-vis Foucault, as well as in regards to Carl Schmitt, whose work was also central to Homo Sacer. In relation to Foucault, Agamben locates Kingdom and the Glory 'in the wake' of Foucault's genealogy of governmentality, and an attempt to understand the 'internal reasons' for his failure to complete these (Agamben 2011, xi). In this, while Foucault saw the general problem of government emerging in the sixteenth century with governmental reason consolidated in the eighteenth, Agamben reads the history of government as far back as the early Church Fathers such as Tertullian. Further, Agamben claims that for all the perspicacity of Foucault's analyses of government, what he fails to address are the theological aspects of oikonomia, which then form the focus of his own analysis. In order to elaborate these, he turns to a disagreement between Schmitt and Eric Peterson on political theology and the nature of the katechon, which defers the coming of the Antichrist and, consequently, redemption. For Agamben, this debate is important primarily because Peterson's work allows him to challenge the pithy statement of Schmitt's that '[a]ll significant concepts of the modern theory of the state are secularized theological concepts' (Schmitt 1985, 36; see McLoughlin 2015, for further discussion). Against Schmitt, Agamben wants to show that the relation between politics and theology 'always runs in both directions' (Agamben 2011, 193). Furthermore, the point of intersection of politics and theology is glory; as Agamben puts it, glory is 'the secret point of contact through which theology and politics continuously communicate and exchange parts with one another' (Agamben 2011, 194). To understand the significance of this claim, though, we need to backtrack to Agamben's genealogy of oikonomia.
Agamben's aim in Kingdom and the Glory is to provide a genealogy of governmentality that traces its history beyond the early Christian pastorate discussed by Foucault, to the earliest formulations of the Christian Trinitarian doctrine of oikonomia. In this, Agamben first turns to a lexical analysis of the term 'oikonomia', the meaning and function of which he traces in early Christian theology, especially the nascent stages of the doctrine of the holy Trinity. While the term has a longer history – it was in use in Ancient Greece to indicate the ordering of the oikos or household – it was in the early Church Fathers that it developed the theological sense that most interests Agamben, which pertains to God's relation to the world and comes to be conjoined with the notion of providence. Further, he argues that one of the central purposes of the introduction of oikonomia into the discussions of the Trinity was to hold off the threat of polytheism, which it does by establishing a split within God, instead of ceding the necessity of many gods. However, this introduces a split that henceforth troubles Western philosophy, namely, that between being and acting, in which what is at issue is freedom and will. From this, Agamben shifts to a discussion of the distinction between Kingdom and Government, and the various authors that have contributed to the emergence and consolidation of it. Of this, he argues that the split introduced by Trinitarian theology underpins the notion of the king or sovereign 'who reigns but does not govern', and shifts the operative efficacy of power to the management of worldly affairs in government.
If this is the case, though, a question then arises as to why the more apparently ceremonial aspects of power persist. This problem of the persistence of glory and its relation to oikonomia, or the relation between 'power as government and effective management [oikonomia], and power as ceremonial and liturgical regality [Glory]' (Agamben 2011, xii) is in fact the central crux of Agamben's analysis. The importance of this relation, Agamben argues, is that it allows us to 'catch a glimpse of something like the ultimate structure of the governmental machine of the West' (xii). The exemplary case of glory or glorification that Agamben addresses in attempting to illuminate this relation is liturgical doxologies and acclamations, of which he makes several interesting claims. First, he shows that Schmitt and Peterson affirm the juridical and political significance of the liturgy and acclamation, though in different ways and with different ends. Second, he argues that oikonomia is internally related to glory in the Trinitarian doctrine insofar as it involves the 'reciprocal glorification between the Father and the Son' (Agamben 2011, 201). Moreover, glory, produced through liturgy and acclamation, is ultimately the means by which Kingdom and Government are brought into articulation, and as such it is the bridge between reign and administration, and between being and acting, and so on. Finally, he argues that while the ceremonial aspects of politics seem to have diminished in the contemporary era, what has actually happened is a displacement of acclamation, such that mass media and popular opinion now take the place of the 'present people' in the production of acclamation. He concludes, '[c]ontemporary democracy is a democracy that is entirely founded upon glory, that is, on the efficacy of acclamation, multiplied and disseminated by the media beyond all imagination', such that what is at issue is 'nothing less than a new and unheard of concentration, multiplication and dissemination of the function of glory as the center of the political system' (Agamben 2011, 256). Thus, rather than the governmentalization of politics in oikonomia gradually eliminating the need for and function of acclamation and glory, these have in fact become the defining and central feature of modern politics.
In a way, the shift toward government in Agamben's thinking is not surprising, since it mirrors the apparent shift in Foucault's thought. Indeed, as is clear from the above, the broad strokes of Agamben's analysis are set as a response to Foucault, even though they ultimately entail a substantial revision of Foucault's theses – a common move on Agamben's part. However, whereas for Foucault the shift to government need not entail an entire rejection of his earlier statements on biopolitics and biopower, the thesis Agamben presses in The Kingdom and the Glory may require considerable revision of his earlier claims about biopolitics in Homo Sacer. In other words, despite being identified as a volume in the Homo Sacer series, the thesis of The Kingdom and the Glory may not complement that of the initial volume, but may, in fact, contest it. While I am not able to explore this in detail here, Agamben's analysis of the genealogy of oikonomia and its implications for contemporary politics suggests not only a diminution of the significance of the sovereign as the decisional centre of politics, but also a splitting of the power of the sovereign, such that the capacity for rule at most sits alongside the capacity to govern. Further, it is this latter aspect of governing that has taken priority in the modern era, such that politics appears as a matter of economic ordering and administration, even if it nevertheless relies on the 'constitutive outside or negative foundation' of sovereignty (Prozorov 2014, 92). This reorientation in Agamben's thought raises significant questions that would bear further examination, not least of which would address the role of violence in a politics dominated by government and economic ordering. I return to the issue of violence in a moment, but first, let me make one further point about Foucault and Agamben's discussions of government.
Interestingly, both these discussions of the genealogy of government would seem to lead to significant reflections on the political role and significance of economy. As I mentioned above, Foucault's account of the governmentalization of the state posits that the consolidation of political economy was a crucial factor in the development of a liberal art of government. As Foucault argued, political economy gave rise to the idea of the self-limitation of government in accordance with the natural order of things, especially of price and value as determined in the market. From this, the later lectures in Birth of Biopolitics provide analyses of ways in which the German Ordoliberals and the American Chicago School conceived of the relationship between governance and economy. The first of these sought to 'define what a market could be, organized (but not planned or directed) within an institutional and legal framework' (Foucault 2008, 323) created by the state. The latter argued for the extension of market rationality to domains of life that are not primarily economic, such as family life, education, penal institutions and so on. In addition, Foucault comments that the Chicago School fostered a return to the classical figure of homo œconomicus, or economic man. However, this figure was not understood as a 'partner in exchange', but as an entrepreneur and particularly an 'entrepreneur of himself', meaning 'being for himself his own capital, being for himself his own producer, being for himself the source of (his) earnings' (Foucault 2008, 226). Throughout Kingdom and the Glory, Agamben presents a detailed genealogy of the concept of oikonomia as economic management and administration, and offers reflections on the 'economy of the moderns' in an appendix to the main argument. Commenting on numerous philosophers, including the Physiocrats as well as figures such as Adam Smith and his notion of the 'invisible hand', he states that political economy is a 'social rationalization of providential oikonomia' (Agamben 2011, 282). Further, he argues that modern liberalism is an extreme manifestation of the immanent ordering of oikonomia, but nevertheless remains tied to economic theology (Agamben 2011, 285).
These reflections on economy notwithstanding, one thematic that Foucault and Agamben, as well as Arendt and Esposito, fail to sufficiently address is that of how the politics of life is integrated with capitalism. To be sure, in Will to Knowledge, Foucault made the claim that 'biopower was without question an indispensable element in the development of capitalism: the latter would not have been possible without the controlled insertion of bodies into the machinery of production and the adjustment of the phenomena of population to economic processes' (Foucault 1990, 140–1). Agamben also makes occasional remarks about the dynamics of biopolitics and spectacular capitalism, in ways that suggest a strong critique of capital and its contribution to the modern 'zoē-fication' of bios. Unfortunately, these remarks are not developed in any depth. However, the link drawn between biopower and capital is expanded in the work of Hardt and Negri. Their notion of biopolitical production is a direct response to Foucault's claim, and in it they attempt to understand the elements of the relationship between capital and biopower that they claim Foucault was unable to broach. In particular, they expand the Marxian notion of economic production to incorporate the production of subjectivities and social life itself, that is, bios. For them, it is at this level that the multitude constitutes a fundamental resistance to biopolitical production in the service of Empire. Thus, Hardt and Negri propose a general framework for examining the relationship between capital, especially the domain of production, and biopower.
Beyond this, more specific analyses have also been developed in regards to the emerging economies of biotechnology, captured in the notion of 'biocapital'. In a book of this title, Kaushik Sunder Rajan examines the integration of capitalist economic processes and genetic biosciences in the late twentieth century, tracing the increasing corporatization of science on the one hand, and the emergence of biological markets on the other. According to Sunder Rajan, the life sciences such as genomics are 'overdetermined by the capitalist political economic structures in which they emerge', by which he means that these structures do not determine but 'disproportionately set the stage' (Sunder Rajan 2006, 6) within which contemporary biosciences emerged. However, the relationship between capital and the life sciences is not unidirectional, for capital itself is also changing as a result of this interaction: 'the life sciences represent a new face, and a new phase, of capitalism' (Sunder Rajan 2006, 3), that is, biocapital. He explicitly casts this project as an attempt to bring Foucault's work on biopower and biopolitics into closer conversation with Marxian concepts of political economy, such as commodity forms and processes of exchange. Further, he argues that the way in which it is possible to conceive of life is itself at stake in the conjunction of capital and the life sciences. He writes, 'the sorts of knowledge genomics provides allows us to grammatically conceive of life in certain ways, not in terms of an Aristotelian poesis, but rather as that whose futures we can calculate in terms of probabilities' (Sunder Rajan 2006, 14). In this, Sunder Rajan's work coincides with that of anthropologist Paul Rabinow, who has proposed that the life sciences are giving rise to new forms of social relations based on genetic information, captured in the notion of 'biosociality'. It is also complemented and extended by the work of sociologists such as Cathy Waldby (2006) and Melinda Cooper (2008; 2014), who examine the commodification of body tissue in systems of global tissue exchange and the political economy of the life sciences, including their reliance on laboratory labour. As these scholars make clear, the life sciences are having a fundamental impact on the ways in which it is possible to conceive of and live life, the implications of which I explore further in the chapter following this one.
Violence
The problem of government and the 'governmentalization of the state' raises questions about the incidence of violence in politics, for the shift away from sovereignty and the correlative right of death may be thought to lead to the supposition that violence is thereby diminished in politics as well. Looking around us today, however, it seems almost incontrovertible that politics and violence are tightly intermeshed, for instance, when what are understood as the interests of the state or nation are all too frequently sought through the means of war. Thus, Carl von Clausewitz's aphorism that 'war is the continuation of politics by other means' hardly surprises, and instead seems to express a deep truth about the nature of modern politics. In line with this apparent contradiction, the question of the status of violence in relation to biopolitics constitutes one of the major fault-lines between the main theoretical frameworks discussed in the previous section. Here, I consider several different approaches to the question of violence across biopolitical studies. This helps to illuminate some key suppositions about the nature of politics and the law, as well as acts as a prelude to the consideration of rights, and especially human rights, in the following section.
In contrast to the Clausewitz aphorism, and to the widespread contemporary acceptance of violence as a form of politics, Hannah Arendt makes controversial the assertion that violence and politics are one and the same. In a slim volume called On Violence, Arendt considers the virtually unquestioned acceptance of violence as a means of extending political aims, and argues that, contrary to popular belief, violence is not only distinct from politics, but also undermines it. She writes,
politically speaking, it is insufficient to say that power and violence are not the same. Power and violence are opposites; where the one rules absolutely, the other is absent. Violence appears where power is in jeopardy, but left to its own course it ends in power's disappearance... Violence can destroy power; it is utterly incapable of creating it.
(Arendt 1969, 56)
Arendt's view is premised on the rejection of a model of power as 'power over', and its redefinition as 'the human ability not just to act but to act in concert' (Arendt 1969, 44). As collective action, Arendt's understanding of power harks back to her theorization of action in The Human Condition, where it is integrally related to the condition of human plurality. As such, political power requires persuasion rather than violence, even though there may also be times when violence can be justified to attain political ends. Interestingly, as Richard Bernstein discusses, Arendt also suggests that violence is present in all fabrication insofar as homo faber, 'the creator of human artifice, has always been a destroyer of nature' (Arendt 1998, 139). Consequently, while violence may be unavoidable in world-making, it becomes problematic when the 'mentality [of homo faber] becomes all pervasive... because it "legitimizes" violence – especially in the founding and forming of states' (Bernstein 2011, 19). Arendt thus sets out a multifaceted view of violence, whereby it is both unavoidable in world-making and destructive of the political power of acting in concert, but also increasingly legitimated within the field of the political driven by the utilitarian means-end logic of homo faber.
While I have argued that Agamben's thought is heavily influenced by Arendt's, it is on the question of violence that we see considerable distance open between them. In short, this is because Agamben does not accept her account of the political as action, and furthermore, does not follow her lead in regards to the displacement of death as the foundational concept in political theory and its replacement with natality. Instead, as we saw in Chapter 2, in Homo Sacer, Agamben advances the view that violence is an intrinsic characteristic of law, and law is the principle regulatory means within biopolitics. Further, the placement of bare life as the originary figure of politics ensures that death remains the central and motivating force of political power. Moreover, he makes apparent that what initially appears as biopolitics is in fact more accurately considered a thanatopolitics, or politics of death. In this sense, what is at stake in Agamben's theory of biopolitics is not the protection and enhancement of life, but the production of a life that is irremediably imbued with death, such that the principle of differentiation between life and death is wholly undermined. This is why the Nazi concentration camps take such a position of priority in his formulation.
But given this, the later analysis of oikonomia, which states that modern biopolitics in the form of government derives from the paradigm of economic theology and not sovereignty, poses a puzzle. For this apparently does away with the claim about the political centrality of violence, and it is worth noting that Agamben barely mentions violence in Kingdom and the Glory. The question that has to be asked, then, is that if this account of biopolitics is to be given priority over the sovereign conception of Homo Sacer, how does Agamben explain the persistence of violence in contemporary politics? Perhaps we could argue that it is the sovereign who reigns (but does not govern) that provides the foundation for the continued mobilization of violence with government. However, this is merely speculative as there is little textual evidence in Kingdom and the Glory to substantiate this view. In any case, to this point, it is the arguments of Homo Sacer that have had the most impact on the field of biopolitical studies, and in this, responses to Agamben's construal of biopolitics as inherently violent have been mixed.
Agamben's way of framing the problem of violence in Homo Sacer has generated two opposed forms of critique. First, Nikolas Rose and Paul Rabinow argue that Agamben exaggerated thanatopolitical logic does not grasp the specificity of biopower today. For them, biopower is not about 'making die' so much as it is about 'making live'; they write, 'central to the configuration of contemporary biopower are all those endeavours that have life, not death, as their telos' (Rabinow and Rose 2006, 203). Or, as Rose puts it, biopolitics operates 'according to the logics of vitality, not those of mortality' (Rose 2007, 70). Consequently, Rabinow and Rose strongly oppose any association of contemporary biopower with the Holocaust, thus indicating their distance from Agamben in particular, for whom Hitler's Germany and concentration camps more generally are the nomos of modern biopolitics. Second, from an opposing perspective, François Debrix and Alexander Barder (2012) argue that the biopolitical framework, including Agamben's thanatopolitical formulation, are unable to grasp the full 'horror' of modern violence, which targets not life, and not even 'dead life' but the very meaning of humanity itself. In their view, the 'pulverization of lives and bodies' in post-September 11 politics exceeds the biopolitical imaginary and instead requires concepts such as Adriana Cavarero's notion of 'horrorism' and an idea of agonal sovereignty for critical analysis.
It seems to me that while each of these perspectives have some analytic merit for understanding aspects of contemporary world politics, both suffer from a narrow focus on one aspect, whether it be the augmentation of life in the biosciences, or the horror of spectacular violence. The difficulty, though, is that both of these exist together; both are aspects of the contemporary global order, and both put pressure on our ways of conceiving of the living value of humanity. In this regard, then, there is something to be said for an approach that entails sufficient flexibility to allow for analysis of both the phenomena of the life sciences and of genocide, for instance. Put another way, there is virtue in an approach that can encompass both the positive, life fostering, dimension of biopolitics, as well as the negative, life destroying, aspects. In this, both Foucault and Esposito have much to offer, though both also fall short in other ways. In regards to Esposito, despite the fact that he attempts to include both the positive and negative aspects of biopolitics in his framework, it may be that the elucidation of the paradigmatic logic of biopolitics in terms of immunization is analytically limiting. Why, for instance, must we understand the therapeutic and enhancing possibilities opened up by a technology such as CRISPR-Cas9 in the same immunitary terms as genocidal massacres? In regards to Foucault, the aphoristic formula of 'fostering life or letting die' allows greater analytic flexibility, but there are still some conceptual concerns.
As Foucault saw, the existence and mobilization of violence within a power dedicated to fostering life is on the face of it somewhat paradoxical, and requires explanation. One way in which he understood it was as the mobilization of sovereign power and a right of death within biopolitics. This kind of reactivation of sovereign power within biopower was conceivable for Foucault because of the way in which he allowed for a tripartite arrangement between different formations of power. Even so, this does seem to contradict the exhortation to 'cut off the king's head', with which we began this chapter. To reconcile Foucault's claims here, it has been suggested that it may be useful to make a distinction between sovereignty and sovereign power. In this, the former refers to the discursive figure of the sovereign (and especially the monarch), the persistence of which in political theory Foucault perceived as a blockage to an accurate analysis of actual configurations of power. The latter refers to one of these configurations of power relations, enacted through violence and the right of death (Edkins and Pin-Fat 2004, 3), and, Foucault argued, able to be mobilized because of the development of racism as a political technique. Building on this account of sovereign power and racism, Achille Mbembe (2003) has identified this mobilization of the sovereign right of death within biopower as a necropolitics: he claims that in modern biopower, we can actually see a triumvirate in the operation of modern power relations of discipline, biopolitics and necropolitics. Further, he argues, especially in relation to colonial violence, that theorists of biopolitics are insufficiently cognizant of the ways in which modern biopower incorporates necropolitics.
This view suggests that the violence of necropolitics is in fact an internal element in the workings of biopower, not simply the reactivated remnant of a sovereign right of death. Michael Dillon and Julian Read (2009) have extended this view in their attempt to make sense of Foucault's statements on war in biopolitics. War was in fact a phenomenon that interested Foucault greatly throughout his various discussions of power, including but not limited to those on biopower. In these, he most provocatively claims that 'massacres have become vital' (Foucault 1990, 137). Dillon and Read argue that this is because war has itself become an element of the liberal way of rule. By tracing the senses of Foucault's claim, they articulate the biopolitical logic of killing some for the sake of others. Further, Dillon and Reid consider Foucault's comments on the 'emergence of mankind as a species' (Foucault 2007, 75) to argue that from a governmental perspective, the liberal subject 'is a biological being defined instrumentally in terms of its species properties, the early referent object of which was population' (Dillon and Reid 2009, 19). Dillon and Reid encapsulate this species-orientation of the subject of liberal governance in the notion of the 'biohuman', and go on to argue that insofar as the liberal way of war is consistent with a liberal way of rule, war is waged on the human in the name of the biohuman. Further, as life itself has become increasingly understood in informational terms, such that it is no longer simply or securely biological, so the liberal way of war has also changed, in particular, through 'the grammar of biostrategization', whereby '[t]he military is as interested... in life-creating and life-adaptive processes as it is in killing, because... it locates the nature of the threat in the very becoming-dangerous of the vital signs of life itself' (Dillon and Reid 2009, 125). The upshot of this is that liberal warfare has been refigured in accordance with the constitution of the human as the biohuman. Further, this refiguration involves a dense chiasmus of life and death, whereby the life-fostering and creative processes are as of much interest to the military as are the life-destroying ones. Life and death are no longer opposed in warfare, but are in a sense, co-constitutive.
This argument about liberal biopolitical warfare makes clear the inseparability of life and death in biopolitical violence, whereby killing is not only undertaken for the sake of the living, but through life itself. However, Nikolas Rose is right to point out that 'letting die is not making die' (Rose 2007, 70); in other words, this logic of putting to death through life does not yet explain the ways in which life is simply allowed to perish in biopower, without being actively put to death. This targets Foucault's second way of rendering the negative, life-denying aspects of biopower, which he understands as 'letting die'. What is at issue here is a political rendering of the bioethical distinction often drawn between killing and letting die, which many see as entailing different moral culpabilities even though both end in death. Similarly, we can push the point that there are also different political logics at issue in killing or letting die, ones that Foucault gestures to but does not fully clarify; consequently, the rationality of letting die remains underexplored in his work. And insofar as necropolitics is based on the sovereign right to put to death, it does not capture this either.
What may be useful to make more sense of the biopolitical logic of 'letting die' is something like Agamben's notion of abandonment. Agamben draws this notion from a dense philosophical essay by Jean-Luc Nancy, 'Abandoned being', in which he writes that,
To abandon is to remit, entrust, or turn over to... a sovereign power, and to remit, entrust, or turn over to its ban, that is, to its proclaiming, to its convening, and to its sentencing... The destitution of abandoned being is measured by the limitless severity of the law to which it finds itself exposed... The law of abandonment requires that the law be applied through its withdrawal... Abandoned being finds itself deserted to the degree that it finds itself remitted, entrusted, or thrown to this law that constitutes the law.
(Nancy 1993, 44)
Agamben argues from this that abandonment is structurally akin to the inclusive exclusion that produces bare life in biopolitics. However, if we can disentangle Nancy's understanding of abandonment from Agamben's, we can find in it an eloquent characterization of life lived in the shadow of the law and of norms, where an abandoned being is abandoned by and to the law, both subject to regulation and open to violence in the law's withdrawal. Such lives of social abandonment, lived on the margins of social norms, wherein laws apply but do not protect, take different forms, as does the violence to which they are subject. For instance, they may be lives of highly monitored, regulated abandonment such as of poor and pregnant addicts in America (Knight 2015), the enduring abandonment of indigenous groups in Australia (Povinelli 2015) or the comprehensively occluded lives of those abandoned to death in Brazil (Biehl 2013). What a concept of abandonment, understood as a structural feature of global political economy, may help to do then is articulate the 'ordinary, chronic, and cruddy' (Povinelli 2015, 13) forms of suffering and hastened dying that underwrite so many lives in the contemporary world.
Beyond a conception of abandonment, the logic of letting die also has different manifestations, ones in which death appears not as an essential counterpart to life but as a symptom of techno-scientific failure. As Jeffrey Bishop (2011) argues though a nuanced consideration of intensive care units, palliative care and artificial respiration machinery, modern medicine is driven by the goal of immortality such that death appears as a failure to maintain function through medical expertise and technology. In the medical circumstances of care for the dying that he considers, the positive life-fostering dimension of biopolitics is tightly integrated with the negative life-denying aspect, such that the latter is only met when the former is understood to become 'futile'. More recently, Paolo Palladino (2016) has explored the philosophy of life and death in contemporary biogerontology and anti-ageing agendas, from which emerge a view of the life course as one of intrinsic ageing in the continual postponement of death – something like ageing without dying. This immortality imperative in modern medical sciences intersects in complex ways with debates on the 'right to die', otherwise known as 'physician assisted suicide' and 'assisted dying'. Among other things, at issue in these debates is the extent to which the state explicitly enacts the legal conditions for 'letting die', especially where that death incorporates active participation on the part of medical practitioners to bring about death. This clearly runs counter to the immortality imperative, and raises significant questions about the prerogative of the state to enforce the continuation of living. As this suggests, the state management of death is multifaceted and integrates in rich ways with the terms of analysis of biopolitics.
Rights
Arguably, the logic of abandonment, and perhaps that of letting die, is also at issue in the treatment of stateless persons, and the debate over the provenance of human rights instigated by Arendt's searing critique of the doctrine of human rights that she presents in The Origin of Totalitarianism. In the closing section of the second volume, Imperialism, entitled 'The Perplexities of the Rights of Man', Arendt proposes that the Declaration of the Rights of Man at the end of the eighteenth century was an historical turning point. This is because it meant that 'from then on Man, and not God's command or the customs of history, should be the source of Law' (Arendt 1968, 170). As 'inalienable', the rights ascribed to Man in the Declaration were supposed to be dependent on no external authority for their legitimacy – Man alone was supposed to be their guarantor as well as their subject. However, the supposed universality of human rights was quickly limited, to include only those citizens of nations not subject to imperial conquest. In effect, then, the inalienable rights of man became inextricably entwined with the question of national emancipation and the European nation–state system.
The full implications of this entwining only became apparent, Arendt argues, when Europe itself had to confront the crisis of statelessness in the 1920s to 1940s. Then, it 'turned out that the moment human beings lacked their own government and had to fall back upon their minimum rights, no authority was left to protect them and no institution was willing to guarantee them' (Arendt 1968, 172). Thus, the supposedly inalienable rights of man turned out to be unenforceable when people were no longer citizens of a state, such that human rights were ultimately a form of citizenship rights. The loss entailed in this, though, was not simply a loss of any particular rights, but of what Arendt calls 'the right to have rights', by which she means the loss of a polity in which 'one is judged by one's actions and opinions' (Arendt 1968, 176–7). The problem with this expulsion from polity, then, is that one is left without the features that ensure a belonging to humanity above and beyond mere species-membership – that is, a loss of personhood. As Arendt writes,
[n]ot the loss of specific rights, then, but the loss of a community willing and able to guarantee any rights whatsoever, has been the calamity which has befallen ever-increasing numbers of people. Man, it turns out, can lose all so-called Rights of Man without losing his essential quality as man, his human dignity. Only the loss of polity itself expels him from humanity.
(Arendt 1968, 177)
In essence, then, the right to have rights is a right to politics, and it is that belonging to the political that ensures humanity. The associations that Arendt thus draws between human rights, citizen rights and personhood, which are then undermined in statelessness, have been enormously productive for subsequent scholars of biopolitics. For instance, for Esposito, it provides critical leverage in his analysis of the concept of the person (Esposito 2012, 68–70), and for Agamben, Arendt provided the terms for a radical rejection of rights discourse and the contemporary formations of political sovereignty.
Two of Arendt's points are especially significant for Agamben. The first of these is the way Arendt questions the status of the human in relation to the citizen. She claims that to the extent that stateless peoples are expelled from political community by virtue of their lack of legal status, their belonging to the human species is akin to the way in which other animals belong to species, that is, merely in terms of biological facticity. In the terms of her political theory developed further in The Human Condition, the stateless or refugees are excluded from the realms of action and human artifice – hence from the political as such – and are instead reduced to a condition of 'mere existence' or 'abstract nakedness'. In Agamben's terms, this is roughly equivalent to the status of zoē, or mere biological life, though his account of the relation of bios (political life) emphasizes the ambiguity between these conditions more than Arendt does. He also extends on this point to emphasize the relation between nation–states and birth, claiming that '[n]ation–state means a state that makes nativity or birth (that is, naked human life) the foundation of its sovereignty' (Agamben 2000, 21). This not only highlights the indistinction between bios and zoē, but, in doing so, also illuminates for him the intrinsic relation between the natural and the political in nation–state formations and the conceptions such as rights that they generate and rely on. For Agamben, not only recourse to human rights but any reliance on citizenship rights will necessarily reinscribe this biopolitical relation between sovereignty and natural life. Further, political rights are ineffective as a limitation on the violence of biopolitical sovereignty, and 'every attempt to found political liberties in the rights of the citizen is... in vain' (Agamben 1998, 181).
The second significant point is closely related to this. As Agamben points out, Arendt's chapter on statelessness and human rights is entitled 'The Decline of the Nation–State and the End of the Rights of Man'. The significance of this is that insofar as rights and the nation–state are inextricably linked, the decline of one also entails the demise of the other. Moreover, the explosion of numbers of refugees – initially from the First World War but also at an apparently ever increasing rate – contributes to the end of the state by making its foundation in natural life apparent and opening possibilities for a new politics distinct from national sovereignty. That is, while the figure of the refugee should have consolidated human rights, it has instead marked a radical crisis of the concept 'by breaking the identity between the human and the citizen and that between nativity and nationality' (Agamben 2000, 21, 23). In short, the refugee is a 'limit-concept' that brings crisis to nation–state formations and thereby opens the way to a new politics. Citing Arendt's formulation that refugees constitute the 'vanguard of the people', Agamben suggests this is not because they presage the formation of a new state, but because they break the link between state, natality and territoriality, and thus inaugurate the possibility of a new 'aterritorial' or 'extraterritorial' space of topological indeterminacy. Such a space would entail perforating the interior and exterior [the regulation of which is key to the nation–state as the recent 'refugee crisis' in Western liberal democracies such as Australia and the UK made evident], thereby liberating politics from the nation–state formation and allowing every citizen to recognize the refugee that he or she is (Agamben 2000, 23–5).
Agamben's rejection of human rights as a means of responding to and protecting against sovereign violence is consistent with his broader account of biopolitics and the purported necessity of a new conception of politics and life beyond the terms of law. Consequently, acceptance of this critique of human rights is theoretically dependent on accepting his framework of biopolitics, including the notions of bare life and form of life. If, however, those seem problematic, then an alternative, less absolutist and more pragmatic, approach to human rights is hinted at in some brief comments of Foucault's on the status of refugees. Foucault takes various positions on the question of rights, but in doing so, does not reject rights tout court. Instead, he suggests that what is required is new form of right that is no longer tied to either sovereignty or discipline. A glimpse of what form this 'new right' might take as a concept is gleaned from a short document 'Confronting Governments: Human Rights', written on the occasion of the formation of an International Committee against Piracy in response to attacks on Vietnamese refugees in the Gulf of Thailand, which does not hesitate to evoke a conception of rights.
Written with a particular political purpose in mind, as a form of protest rather than a theoretical treatise, this document evokes a notion of an international 'community of the governed', which is 'obliged to show mutual solidarity' and 'always bring the testimony of people's suffering to the eyes and ears of governments'. Further, this suffering 'grounds an absolute right to stand up and speak to those who hold power' (Foucault 2000, 474, 475). What is interesting about this appeal to rights is that it does not require an idea of the human as its basis – rather, it evokes the figure of a community of the governed. Further, the right entailed is a right to resist power and government on the basis of the suffering of others; it is not a right of protection per se, but a right of resistance and bearing witness. It would not do to make too much of this mere glimpse of a new conception of right, but it is nevertheless suggestive in the context of biopolitics. More generally, we can see in Foucault's approach what has been described as a naturalism that recognizes the tactical value of rights discourse (Ivison 2014), without having to ground rights in either a humanistic philosophical anthropology or in political ontology. Instead, Foucault makes appeal to the relationship between the governed and the government that he saw as a feature of the modern state (as opposed to the sovereign/subject relation), such that the condition of being governed forms the basis of community, solidarity and rights.
Conclusion
In this chapter, I addressed the broad thematic of politics through key debates in biopolitical studies on concepts such as sovereignty and government, violence and rights. In the first section, I discussed the lecture series that Foucault presented around the time of the publication of Will to Knowledge. Rather than explicitly developing his thoughts on biopower articulated in Will to Knowledge, the lecture series of Security, Territory, Population and The Birth of Biopolitics appear to move away from the terminology of biopower and toward security and government as frames of analysis. I argued in this chapter that these lectures nevertheless do not break definitively with biopower, but extend on and make more complex the earlier schematic outlines. In contrast to this, Agamben's theological genealogy of oikonomia in The Kingdom and the Glory develops a thesis that sits in considerable tension with his earlier arguments in Homo Sacer, which have been so central to his status within biopolitical studies. I suggest that it is far from clear how his account of the economic theological basis of government, which he claims underpins modern biopolitics, interacts with the earlier thesis on sovereignty and political theology on important questions such as violence. Following on from this discussion of sovereignty and government, then, I addressed the question of violence more specifically. In this, I considered various phenomena of the state's involvement in violence and death – though the discussion is by no means exhaustive – and spent some time elaborating the biolopolitical logic of 'letting die'. In this, I suggested that the concept of abandonment might be usefully extended as a way of elucidating the structural distribution of suffering. In the final section of the chapter, I canvassed disagreements on rights, and especially human rights, that are evident among theorists of biopolitics, specifically Arendt, Agamben and Foucault. What this chapter makes evident is that while much of the literature on biopolitics focuses on the political aspects of it, there is nevertheless little agreement on how to characterize the relationship between traditional concepts and configurations of power as sovereignty and biopolitics. Consequent to these different approaches to sovereignty, theorists also hold significantly different views in regards to violence and the protective function of rights. In the following chapter, I use this discussion of political concepts as background to explore further the 'bio' part of the term 'biopolitics'.
Notes
In what could be read as a veiled critique of Arendt, Foucault claims that 'what is important for our modernity, that is to say, for our present, is not then the state's takeover (étatisation) of society, so much as what I call the "governmentalization" of the state'(Foucault 2007, 109).
This comment from Foucault is taken from the script of his lecture, but was not spoken by him in presenting the lecture (see the footnote in Foucault 2008, 20–22).
Dean suggests that liberalism, understood as an art of government, was important to Foucault because it promised to 'banish this ghost [of sovereignty] he has been fighting all these years' (Dean 2013, 41).
The brief discussion of will foreshadows the longer analysis in Opus Dei.
Other examples of the state's management of death could also be considered in this section, most notably the continued use of capital punishment by constitutional democracies such as the United States and Japan. For further discussion see Thurschwell 2008; Sitze 2009; Meranze 2011. Another example of the state management of death is the emergence of the modern death certificate, which can itself be seen as a biopolitical apparatus. To my knowledge, a biopolitical analysis of the institution of the death certificate is as yet wanting. For a brief non-academic history of the death certificate, see Schulz 2014.
References
Agamben, G. (1998). Homo Sacer: Sovereign Power and Bare Life. Trans. Heller-Roazen, D. Stanford, Stanford University Press.
Agamben, G. (2000). Means without End: Notes on Politics. Trans. Casarino, C. and Binetti, V. Minneapolis; London, University of Minnesota Press.
Agamben, G. (2005). State of Exception. Trans. Attell, K. Chicago, University of Chicago Press.
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6Life
Biology, technology, reproduction
As the previous chapter made evident, questions of political theory and philosophy have been central to the development of debates on biopolitics. Conflicts about how to conceptualize sovereignty and its relation to biopolitics, or how to think about government or rights, have been among the key lines of differentiation between the major theorists. Indeed, discussions of biopolitics have to a large extent been discussions of politics. This raises a question about the specific work that the 'bio' of biopolitics does, and whether it does or ought to serve to distinguish biopolitics from politics more generally. In this, it is interesting that there has been remarkably less engagement in debates on biopolitics with the 'bio' part of the term, a recent turn to philosophy of life notwithstanding. For instance, while several of the major theorists discuss Hobbes at length, there is barely any mention in their works of figures such as Antoine van Leeuwenhoek and John Ray, or even Charles Darwin and Gregor Mendel to name only several of the best-known figures in the development of modern biology. Indeed, of the major theorists, only Foucault offered any detailed account of the emergence of modern biology as a particular formation of knowledge, in The Order of Things (1994). Here, Foucault traces the emergence of a particular episteme or way of thinking about life and living organisms, which, I suggested in an earlier chapter, acts as a condition of possibility for the centring of life as an object of scientific knowledge and political power. In short, biopower, as Foucault understood it, emerges in tandem with the modern episteme that underwrote the development of disciplines such as biology, statistics and public health.
However, my primary task in this chapter is not to extend on this argument, but to consider the ways in which the concept of life has been taken up in recent debates. As I outlined in the introduction, there are a variety of senses of the term 'life' in circulation in biopolitical studies. In this chapter, I am primarily interested in two of those – first, the biological conception of life, in which it is an object of scientific knowledge, and second, a vitalistic sense. While the first of these is primarily of interest to Foucault and scholars working in the style established by him, the second sense is more important to other theorists of biopolitics. In this though, there has been a decided turn to a more standard philosophical register, away from the genealogy or 'historical ontology' of Foucault. More specifically, several theorists have urged the development of a new concept of life that evades the terms of biopolitics. In the first section of this chapter, then, I briefly sketch aspects of Foucault's approach, and then focus on the debate that has emerged on how best to develop an alternative conception of life not captured by or beholden to biopolitics. In this, I discuss the approach to life proposed by Gilles Deleuze in his essay, 'Immanence: A Life' (2001), which has given rise to interpretations by Agamben and Esposito among others.
In the second section, I build on this to consider the ways in which the question of technology has emerged within contemporary debates on biopolitics. I argue that for the most part, the explicit engagement with technology on the part of major theorists has been limited. This is somewhat puzzling in the context of the apparently thorough technologization of life today. While the theorists discussed in the first part of the book do not address the technological aspects of biopolitics directly, scholars such as Nikolas Rose and Paul Rabinow trace the ways in which contemporary biosciences are themselves producing new conceptions of life. I outline Rose and Rabinow's key contributions to examining the intersection of bioscience, technology and politics, and briefly reflect on this productive engagement between biopolitical studies and science and technology studies. In the third and final section of the chapter, I turn to discussing the role that sex, birth and reproduction play in biopolitics. I address this by first considering how reproduction has been addressed in biopolitical theory, and then briefly describing some of the empirical issues that arise in this area. The theoretical discussion focuses attention on the way in which biopolitical theory has privileged death as the marker of human finitude. In contrast, Hannah Arendt's category of natality shifts the emphasis to the beginning of life. With Arendt, I emphasize the biopolitical importance of birth or natality, understood not as a sui generis moment of emergence but as a process of reproduction.
Biology
As Eugene Thacker deftly summarizes, 'in an era of biopolitics, it seems that life is everywhere at stake and yet it is nowhere the same' (Thacker, 2010, ix). In accordance with this claim, one could point to the proliferation of notions of life – nuda vita or bare life (Agamben 1998), creaturely life (Santner 2006), surplus life (Cooper 2008) and eternal life (Vatter 2014) to name but a few. While the multivalence of the concept of life has undoubtedly been productive, though, just what the prefix 'bio' in biopolitics actually refers to has become all the more confusing. Indeed, while the focus of biopolitical studies has typically been on human life (see Wolfe 2013), it is often unclear if what is at issue is actually biological or vital life, or something more like subjective life. In this section, I explore the different approaches taken so far to the concept of life within biopolitical theory. First, I briefly discuss Foucault's archaeological approach to the concept of life, and then turn to more recent debates on the concept of life in biopolitical studies. One of the key texts in this area has been the essay by French philosopher, Gilles Deleuze, entitled 'Immanence: A Life'. Because of the centrality of this essay in subsequent discussions, I will use it here as a mechanism for bringing out the different conceptions of life suggested by the key theorists and scholars of biopolitics. In doing so, I will link this discussion back to earlier chapters, particularly to Agamben's notions of bare life and form-of-life, and Esposito's formulation of a life of immanent normativity.
First, to Foucault. In keeping with his archaeological and genealogical methods, Foucault does not propose to develop a new concept of life as such. Instead, he analyzes the historical emergence of particular ways of thinking about life within the framework of the conjunction of power and knowledge – that is, in the constitution of truth. This he undertakes primarily in The Order of Things, where he traces the epistemic shift in thinking about life, from the vast taxonomies of natural history to the classifications of modern biology. Of this, he argues that the shift to the modern episteme involved a fundamental re-orientation of knowledge about the natural world, and relatedly, the concept of life. No longer concerned with exterior similarities, the modern episteme focused on internal structures and their function; Foucault writes, 'from Cuvier onward, it is life in its non-perceptible purely functional aspect that provides the basis for the exterior possibility of a classification' (Foucault 1994, 269). This shift in episteme provided the conditions of possibility for the emergence of a modern discipline of biology, which has been centred on the problem of the vertical transmission of traits between generations. This means that the new biology is no longer concerned with notions of resemblance and contiguity, but is organized by concepts such as those of anatomic function, species, reproduction and heredity (Jacob 1993).
Given this context, it is of some significance that Foucault planned to undertake a genealogy of the concept of heredity at the Collège de France, but ultimately did not do so. Foucault indicated in his candidacy presentation that the 'privileged example' for his future research would be 'the knowledge of heredity' that developed throughout the nineteenth century in breeding programs and attempts to improve species, culminating in the emergence of genetics in the early twentieth century. Stuart Elden (2017, 18) points out that Foucault likely did not undertake this task as he considered it accomplished in François Jacob's book, The Logic of Life, published in France in 1970. Nevertheless, Elden (2017, 10) remarks, '[h]eredity was a recurrent theme in Foucault's work', from his discussion of it in relation to mental illness, through his comments on race and eugenics, to his work on sexuality and abnormality. Indeed, genealogy and heredity are conceptually and historically intertwined.
Interestingly, Jacob closes his history of the concept of heredity with the claim and question that '[t]oday, the world is messages, codes and information. Tomorrow what analysis will break down our objects to reconstitute them in a new space?' (Jacob 1973, 324). It could be said that this rhetorical question points to a methodological limitation of genealogy, insofar as it is limited to illuminating the modern conception of life and its correlates, and remains tied to the conceptions of life that it traces. The transformative power of genealogy lies in its capacity to reveal the basic knowledge structures of our existence as historically contingent, and, going a step further, in its opening up a potential for the retrieval of non-hegemonic conceptions or ideas – what Foucault calls at one point 'subjugated knowledges'. However, it does not necessarily yield a new conception of its object in itself, unless that can be retrieved from the subjugated elements of a discourse. In the terms of biopolitics, a genealogy of life allows us to see the ways in which it became an object of scientific and political knowledge, but it does not, and perhaps cannot, provide a conceptualization that evades or overcomes the terms of biopolitics.
Notably, it has been suggested that Foucault moves toward developing a new conception of life in his reflections on the work of his mentor, Georges Canguilhem, published posthumously as the essay 'Life: Experience and Science'. In this essay, he argues that at the centre of the problems which preoccupy Canguilhem resides 'a chance occurrence... like a disturbance in the information system, something like a "mistake"', in short, error; Foucault states, 'life – and this is its radical feature – is that which is capable of error' (Foucault 1998, 476). Further, he claims that the error that is borne within life as its necessary potentiality provides the radical contingency around which the history of life and the development of human beings is twined for Canguilhem, which enabled him to identify and draw out the relation of life and knowledge. Foucault writes,
if one grants that the concept is the reply that life itself has given to that chance process, one must agree that error is the root of what produces human thought and its history. The opposition of the true and the false, the values that are attributed to the one and the other, the power effects that different societies and different institutions link to that division – all this may be nothing but the most belated response to that possibility of error inherent in life.
(Foucault 1998, 476)
Thus, it is through the notion of error that life is placed in a relation of contiguity and contingency with truth and structures within which it is told. 'Error', or the inherent capacity of life to 'err' both establishes the relation of life to truth and undermines that relation by disentangling humanity from the structures of truth and power that respond to the potential for error. Hence, 'with man, life has led to a living being that is never completely in the right place, that is destined to "err" and to be "wrong"' (Foucault 1998, 476). Of course, that Foucault saw errancy as a productive aspect of Canguilhem's work does not mean that he would have adopted this concept for himself, or that he would have developed a new philosophical approach to the concept of life that diverged from the method of genealogy or historical ontology. Even so, it may be that the notion of errancy opens to the potential capacity of life to 'constantly' escape the 'techniques that govern and administer it' (Foucault 1990, 143).
Interestingly, Foucault's Italian successors have largely eschewed his genealogical approach to the concept of life. Instead, they have taken a more abstract philosophical approach that attempts to deconstruct (in a broad sense) the distinction between bios and zoē, not for its own sake, but in order to yield a new affirmative concept of life. In this project, Deleuze's essay, 'Immanence: A Life' is a crucial point of reference for both Agamben and Esposito. In this essay, Deleuze sketches the outline of what he calls a life of pure immanence, which is both inseparable from but irreducible to the life of an individual. He illustrates this with reference to a novel by Charles Dickens, entitled 'Our Mutual Friend', in which the scurrilous character Riderhood elicits fascination and concern from onlookers the nearer he is to death. For Deleuze, the point of fascination is not the individual characteristics of Riderhood per se, but the impersonal life that becomes apparent in the moment of confrontation and play with death. As Deleuze (2001, 28–9) puts it,
[t]he life of the individual gives way to an impersonal yet singular life that releases a pure event freed from the accidents of internal and external life, that is, from the subjectivity and objectivity of what happens: a "Homo tantum" with whom everyone empathizes and who attains a sort of beatitude. It is a haecceity no longer of individualization but of singularization: a life of pure immanence.
However, Deleuze does not want to imply that a life only becomes evident in this moment; rather, he states that an immanent life is 'everywhere', manifest in individual lives though it is not equivalent or reducible to them. Indeed, this singular life, he suggests, is most evident in small children who are as yet not individuals but nevertheless have singularities.
Commenting on this in his essay 'Absolute Immanence' (1999), Agamben notes that both Foucault and Deleuze turn toward a discussion of 'life' in the last of their publications during their lifetimes. This coincidence, he suggests, bequeaths to future philosophy the concept of life as a central subject. He writes that Foucault's essay aims at 'a different way of approaching the notion of life' in the non-subjective notion of error. What is more important for him though, is the way in which Deleuze seeks 'a life that does not consist only in its confrontation with death and an immanence that does not once again produce transcendence' (Agamben 1999, 238). Agamben argues that insofar as these essays provide a 'corrective and a stumbling block' for each other, they clear the ground for a genealogy that will 'demonstrate that "life" is not a medical and scientific notion but a philosophical, political and theological concept' (Agamben 1999, 239). Furthermore, he claims that such an inquiry would reveal the archaism and irrelevance of the various qualifications of life: animal life and organic life, biological life and contemplative life etc., and give way to a new conception of life that recognizes beatitude – blessedness or happiness – as the 'movement of absolute immanence' (Agamben 1999, 238). Without going into the details of Agamben's interpretation of Deleuze here, what is important to note is that the notion of a conception of a life of happiness as one of absolute immanence points to the core feature of Agamben's understanding of form-of-life. The central aspect of form-of-life is that it is a life in which the constitutive parts can no longer be separated from each other. This means that life can no longer be separated into the spheres of bios and zoē, and further, that it is no longer possible for one of those parts to be reified as a transcendental principle such as we find in the idea of the sacredness of life itself.
While he touches on the idea of form-of-life at various points throughout his work, the clearest and most complete articulation of this that we are likely to see from Agamben is in the recent book, The Use of Bodies (2016). In this, Agamben initially suggests a thesis not altogether unlike Arendt's claim that while labour was denigrated by the ancient Greeks, it nevertheless came to supersede the political category of action. Thus, he suggests that the work of Plotinus marks an important turning point in the trajectory of the concept of life in ancient Greek thought, since it precipitates the valorization of zoē over and against Aristotle's own demotion of this notion in favour of bios. Even more important though, Plotinus ultimately also moves toward a concept of form of life, in which zoē and bios can no longer be divided, since contemplation or intellectual life becomes synonymous with life itself. For Plotinus, life was no longer to be thought as 'an undifferentiated substrate (hypokeimenon) to which determinate qualities would come to be added (for example, rational or linguistic being) but as an indivisible whole, which he defines as eidos zoes, "form of life"' (Agamben 2016, 218).
In the condition of modern biopolitics, attaining toward such an indivisible life requires 'neutralizing' the biopolitical apparatus that splits bios from zoē in order to make the latter the target and object of politics. This, Agamben notes in a veiled critique of Derrida's method of deconstruction, is to be achieved not through re-valorizing bios, or rearticulating the relation between zoē and bios, but through rendering the machine that induces the split inoperative, 'so that form-of-life can appear as the tertium that will become thinkable only starting from this inoperativity, from this coinciding – which is to say, falling together – of bios and zoē" (Agamben 2016, 225). Ultimately, then, it is only such a life, or form-of-life, in which the biographical life of bios is again inseparable from the biological life of zoē that can be considered happy: 'only that life is happy in which the division disappears' (Agamben 2016, 226). Note, though, that this state of union cannot be achieved simply through reducing bios to zoē, nor by sacralizing zoē or turning it into a political principle, for the conjunction is more fundamental. As I have argued previously (Mills 2008, 134), the political-philosophical task, according to Agamben, is the appropriation of the arthron that divides and articulates the different modalities of life (bios/zoē), or in other words, by jamming the biopolitical machine that renders life divided in the first place.
In order to elucidate the sense of form-of-life further, Agamben goes on to urge the development of what he calls an 'ontology of style'. By this, he means that rather than thinking of style of living as a matter of aesthetics or ethics, it is necessary to think of it at the level of an ontology, one which highlights the manifestation of singular being in a particular manner of living, such that life and living are inseparable. Ultimately, Agamben claims that form-of-life can be understood as just this ontology of style: '[w]hat we call form-of-life corresponds to this ontology of style; it names the mode in which a singularity bears witness to itself in being and being expresses itself in the singular body' (Agamben 2016, 233). Agamben goes on to give more content to this idea by valorizing a style of being premised on exile, specifically the exile of oneself to oneself in an 'intimacy without relation' (Agamben 2016, 236). Here, the relation of the ban that was so central to the argument of Homo Sacer is turned in a positive direction – not simply as a casting out from political community (which thereby includes bare life in it), but as a casting into form-of-life (such that bare life is no longer possible). What Agamben is really getting at in this new conception of an ontology of style is somewhat unclear, and it may be worth considering how it articulates with Foucault's aesthetics of the self and the idea of a singularity (that is indescribable but expressed in speech and action) found in Arendt to get clearer on its implications.
However, instead of discussing this further here, let me backtrack to Deleuze's essay in order to consider briefly Esposito's response to it. This conception of life as pure immanence is also crucial to Esposito's call for an affirmative biopolitics, for which he states that the idea that 'every life is a form of life and every form refers to life' (Esposito 2008, 194) provides the minimal presupposition. What is interesting, though, is that while Agamben's essay poses Deleuze and Foucault's approaches to the concept of life as 'stumbling blocks' for each other, Esposito instead attempts to integrate the emphasis on norms that Foucault derives from Canguilhem with the notion of an absolute immanence proposed by Deleuze.
As we saw in Chapter 4, Esposito argues that the theorization of vital norms developed by Canguilhem may be especially valuable in developing an affirmative biopolitics, since it allows for the 'maximum deconstruction of the immunitary paradigm and the opening to a different biopolitical lexicon' (Esposito 2008, 191). To reach this conclusion, Esposito reads Canguilhem as proposing the radical vitalization of the norm in The Normal and the Pathological (1991), where he argues that life is internally and necessarily normative, since even at the simplest level 'living means preference and exclusion' (Canguilhem 1991, 136). Esposito takes this to generate a conception of life in which norms are immanent to it, and normativity means the capacity to generate new possibilities for living that are internal to, and derive from, the immanent potential of life itself. In view of this radicalized immanence of life and norm, Esposito writes that 'If Nazism stripped away every form of life, nailing it to its nude material existence, Canguilhem reconsigns every life to its form, making of it something unique and unrepeatable' (Esposito 2008, 189). In this way, Esposito also moves toward a conception something like Agamben's form-of-life, attained here through pushing the logics of biopolitics to an extreme at which they turn into their opposite. Thus, rather than separating norms and life in a quasi-transcendental relation, they are made to coincide such that life is normativity, and vice versa.
Interestingly, while Esposito is striving for a deeper reconciliation of the essays of Foucault and Deleuze through the idea of an immanent normativity, this comes at the cost of downplaying the very aspect of Canguilhem's work that Foucault finds most productive or commendable, that is, the notion of an errancy internal to life. In fact, Esposito has little to say about errancy, which is not unrelated to the fact that he also has little regard for the ways in which Canguilhem specifies the condition of the human organism as distinct from organisms in general, about which two further points can be made. The first goes to the fact that the environment in which human beings are located is necessarily social and, as such, cross-cut with the force of social norms. As Canguilhem writes, human norms are:
determined as an organism's possibilities for action in a social situation rather than as an organism's functions envisaged as a mechanism coupled with the physical environment. The form and functions of the human body are the expression not only of conditions imposed upon life by the environment but also of socially adopted modes of living in the environment.
(Canguilhem 1991, 269)
This means that the 'normal' is always an effect of a complex comingling and expression of vital norms in the midst of socially defined ways of living. Human life is never simply biological; and nor, for that matter, is it ever simply social or political. That Esposito leaves aside the necessary embeddedness of an organism in its environment means that he also risks obfuscating the ways that social norms cut across the vital norms of the living human being.
The second point derives from this, for while the existence of human beings is conditioned by social norms, it cannot be assumed that vital and social (or legal) norms are equivalent in the manner that Esposito treats them. Rather, what needs to be taken into account is the disjuncture between vital and social norms, and consequently, what requires explanation is the means by which they intermingle. In other words, vital and social norms may be empirically inseparable, but they are nevertheless analytically distinct. In the postscript to The Normal and the Pathological, Canguilhem argues that while physiological norms are immanent to the organism, social norms have no equivalent immanence. In a living organism, norms are 'presented without being represented, acting with neither deliberation nor calculation', such that there is 'no divergence, no distance, no delay between rule and regulation'. In contrast, rules in a social organization must be 'represented, learned, remembered, applied' (Canguilhem 1991, 250). In light of this insistence on the exteriority of social norms, we would do well to qualify Esposito's thesis on the 'vitalization of the norm'. While Canguilhem's work develops a philosophy of life that emphasizes the productive power of the living in terms of the capacity to create norms, he also resists a complete vitalization of the norm, insisting on a more differentiated approach to norms and normalization. This is important because while the exteriority – perhaps even transcendence – of social norms is indicated by the capacity to question those norms, it also opens them to such questioning and, ultimately, to transformation. No such route is possible for Esposito.
This brief map of some of the contributions to the current debate raises a number of theoretical questions that cannot be resolved here. However, I do wish to make two brief critical points of the general project of developing a new conception of life. The precept of this project seems to be that the generation of a new philosophy of life will effectively overcome the negative, death-driven, aspects of biopolitics today. However, in my view, this risks what we might call a fallacy of philosophical salvation, where the generation of philosophical concepts is supposed to be sufficient to resolve entrenched social and political problems. No doubt, new concepts – or new versions of old concepts – are important and often useful, but they are not efficacious in the manner suggested by some proponents of affirmative biopolitics. Further, as noted by social scientists, this project lacks any serious engagement with the history of biology, or developments in biotechnology and biomedicine. As Nikolas Rose and Paul Rabinow put it, Agamben and others fall into an abyss of overarching theories that 'describe everything but analyse nothing' (Rabinow and Rose 2006, 199). The consequence of this is not only that it 'produces fundamental flaws' in historical understanding (Meloni 2016, 23), but also that it fails to grasp the various ways in which these developments are actually generative of (our understanding) of life. As scholars of the history of the sciences make clear, 'objects of scientific knowledge are not given ready made in nature... rather, they are technically produced in a continuous process of assemblage, rectification and repetition' (Lenoir 2010, xii). This is also true of those objects that we now popularly think of as the basic building blocks of life, such as the gene. Building on this section, then, in the following I turn to the work of Nikolas Rose and Paul Rabinow, whose contributions have been indispensable in illuminating the intersections of biomedicine and biopolitics. Leading to this, though, I first provide a brief discussion of the way in which technology has been addressed in biopolitical theory.
Technology
Throughout his work, Agamben repeatedly asserts that life is not a scientific or medical concept, but a philosophical and theological one. In this, he may be right that contemporary biology does not always vex itself over developing a definition or theory of life that encompasses its various manifestations and phenomena. Even so, it would be too hasty to imagine that biology or medicine has little to do with how we think about life today, and the serious shortcoming of Agamben's approach is that it fails to engage with how modern bioscience and biomedicine shape contemporary conceptions of life, and further, how they may generate their own visions of life that are not necessarily constrained or determined by the distinction between bios and zoē. In fact, when we think about it, it is striking how little contemporary theorists of biopolitics engage with the phenomena of biomedicine, bioscience and biotechnology, as if these had little to do with governance or politics. However, as social scientists of technology have made amply clear, this is far from the case. The question this provokes, then, is whether, and if so, how, questions raised by and about science and technology can be responded to within a biopolitical frame. I will briefly consider how technology has been discussed within biopolitical theory, then I discuss the empirical-theoretical approaches to biopolitics, biomedicine and technology taken by scholars such as Nikolas Rose and Paul Rabinow.
The relative absence of explicit reflection in recent biopolitics literature on technology is surprising, given that contemporary life is so thoroughly imbued with technologies, ranging from the outpouring of artefacts in manufacturing and consumerism, through media technologies, personalized biomedicine and genomics and technologies of war. It is safe to say that no realm of human existence takes place in the absence of technology. Indeed, the capacity to interact competently with all manner of technologies is arguably a central attribute of the entrepreneurial subject of neoliberal governance, as is at least a cursory knowledge of genetics and the options presented by technologically driven medicine and bioscience. Indeed, the production of technological subjectivities can be seen as a central feature of neoliberal governance, and the practice of government is itself inherently technical and technological. Moreover, historically, biopolitical management of populations and individuals has necessarily relied on and has also provoked the development of new technologies (For example, see Black 2002). Given the 'centrality of technology to the reconfiguration of... the space of government' (Barry 2001, 2), it may be worth asking why there has not been more congress between scholars of biopolitics and of science and technology. For while there are notable exceptions, few scholars of biopolitics draw in depth on contemporary approaches to the productive aspects of contemporary science and technology, and vice versa. Here we might think of the Actor Network approach, developed by Bruno Latour and others, or the concept of 'co-production' devised by Sheila Jasanoff and co-authors, as having potential for alliance.
This absence of consideration of technology is also problematic from a theoretical perspective, given that in The Human Condition, the impact of technology on humanity is one of Arendt's central concerns. Arendt opens this work with a reflection on the launching of the first artificial satellite, the Sputnik, by Russia in 1957. In this, she sees a desire to escape the human condition of plurality on earth, also evidenced in attempts to 'create life in a test tube' and breed better people via technology; this she sees as making life artificial, and 'cutting the last tie' that binds humanity to nature and to the earth (Arendt 1998, 2). Technology, then, is alienating and threatens to undermine the conditions of existence of humanity. Of course, Arendt's concern is not idiosyncratic – it relates directly to the essay by Martin Heidegger, 'The Question Concerning Technology', and more generally to the thematic of techne in Aristotle. This indicates that the problem of technology is not irrelevant to that of biopolitics; even so, it is a thematic that remains significantly underdeveloped within recent biopolitical theory.
Perhaps the most in-depth philosophical analysis of technology as it pertains to biopolitical studies is Timothy Campbell's book, Improper Life (2011). In this, Campbell argues that theorists of biopolitics must reckon with Heidegger's essay on technology in order to overcome the thanatopolitical declension that comes from the distinction between the proper and the improper that structures Heidegger's analysis. Campbell argues that this thanatopolitical inclination derived from Heidegger is significant within Agamben's approach to biopolitics, as well as that of Esposito and, to a lesser extent, Foucault. Campbell elaborates the distinction between the proper and improper through the question of writing, and, according to Heidegger, the alienation from the immediacy of handwriting precipitated by the technology of the typewriter. In relation to Agamben, for instance, Campbell argues that this distinction underwrites that between bios and zoē, as well as those of Muselmann/witness and subjectification/desubjectification (Campbell 2011, 40).
As enlightening as this approach may be in terms of the intellectual relation between Agamben and Heidegger, though, it is not clear that this ontological critique of technology as Enframing, or the alienation of being from Being, tells us much about the particular roles that various technologies play in the operation of biopolitics. Philosophers of technology critique Heidegger's approach for being totalizing and overly negative. Similarly, while it may be that the distinction between the proper and the improper informs our self-understanding in regards to technology, it far from exhausts the effects of technology in the constitution of subjectivity and ways of living. It is difficult, for instance, to draw implications from Campbell's discussion of the typewriter for the contemporary technologies such as wearable fitness devices and personalized genetic testing (see Mayes 2016). For what is missing here is a sense of the ways in which technologies do not only desubjectify, but also make new forms of subjectivity possible.
The contemporary approach to biopower that has done most to bring out the connections between biopolitics and technology is that jointly and separately developed by Paul Rabinow and Nikolas Rose. Both scholars of various aspects of contemporary medicine and life sciences, Rabinow and Rose propose an approach to biopower that is empirically rich and conceptually provocative. In a co-authored paper (Rabinow and Rose 2006), they characterize their work as analytically focused on the diagnosis of the 'near future', in the sense given to this term by Deleuze, that is, 'what we are in the process of becoming' (Deleuze 1992, 164). They argue that
the concept of biopower designates a plane of actuality that must include... [o]ne or more truth discourse about the 'vital' character of living human beings, and an array of authorities considered competent to speak that truth; [s]trategies for intervention upon collective existence in the name of life and health; [and]... modes of subjectification, in which individuals can be brought to work on themselves... in the name of individual or collective life or health.
(Rabinow and Rose 2006, 197)
This account of biopower emphasizes the ways in which configurations of power and authoritative knowledge are brought to bear in the constitution of subjectivity. Indeed, as Rose and Rabinow go on to show in their individual works, the contemporary conceptions of biological life produced by the biomedical sciences are increasingly inseparable from the constitution of subjective life.
Rabinow's broad method of an 'anthropology of the contemporary' involves ethnographic research in unlikely sites such as biotechnology labs, in order to trace the ways that new knowledges about and techniques for intervening in the 'building blocks' of life are driving new ways of understanding ourselves, as persons and as a species, and our place in the world. Thus, in his book French DNA, Rabinow undertakes an anthropological analysis of events involving an American biotechnology company and France's premier human genomics lab, Centre d'Etude du Polymorphisme Humain (CEPH). He argues that we are witnessing a 'biologicalization of identity different from the older biological categories of the West (gender, age, race) in that it is understood as inherently manipulable and re-formable' (Rabinow 1999, 13). This transformation of the basic categories of identity and self-understanding led him to propose that we are in a kind of purgatorial space, since new technologies are making redundant the historical forms of self-understanding provided by the ancient Greeks and the Christian traditions, while new forms of understanding remain yet to be settled.
Indeed, these new developments in scientific knowledge are changing the very operation of biopower. In reflections on the Human Genome Project, Rabinow argues that the 'two poles of the body and the population are being re-articulated in what could be called a post-disciplinary rationality' (Rabinow 1996, 91). He maps this shift through the chiasmus of sociobiology and biosociality, where the former indicates social projects in which the social is modelled on the biological. In the latter condition of biosociality, however, nature or the biological is increasingly modelled on the social, and Rabinow contends that it will entail 'a circulation network of identity terms and restriction loci, around which and through which a truly new type of autoproduction will emerge' (Rabinow 1996, 99). To be clear, Rabinow did not suggest that biosociality would simply replace the older sociobiology, but that they would continue to interact in complex ways. Nevertheless, what he saw in the development of biosociality were the emergence of new forms of governance, through the reconfiguration of health risk and surveillance in terms of prevention rather than therapeutics, and the emergence of new forms of social life. These, he suggested, would entail the 'formation of new group and individual identities and practices arising out of these new truths' about the genetic basis of disease conditions and susceptibilities (Rabinow 1996, 102).
This diagnosis has been reiterated by Nikolas Rose in his recent analysis of the contemporary politics and sciences of life in the influential text, The Politics of Life Itself (2007). He argues that that we are shifting to a new understanding of life brought about by developments in the life sciences (2007, 11–12; passim). He describes this new scientific approach to life as 'molecular' as opposed to the 'molar' perspective that focuses on organs, limbs, bodies of traditional biopolitics. This opens up possibilities for 'the reverse engineering of life, its transformation into intelligible sequences of processes that can be modelled, reconstructed in vitro, tinkered with, and reoriented by molecular interventions to eliminate undesirable anomalies and enhance desirable outcomes' (Rose 2007, 83). Further, in a more obviously political register than Rabinow's biosociality, he coins the term 'biological citizenship' as a lens for examining the ways in which belonging to a 'community of the governed' may be organized around biological knowledge in ways that are different from the historical link in citizenship between national belonging and birth (Rose 2007, 131–6). Rose argues that biological citizenship brings to light new forms of individual and collective identity-formation based on corporeal and genetic responsibility. Further, they are underpinned by a 'political economy of hope' that emerges from the overlapping aspirations of patients, practitioners, scientists and biotech companies.
What the approach of scholars such as Rabinow and Rose makes clear is that technology is a central aspect of the operation of a contemporary politics of life. Technological apparatuses and assemblages make possible new forms of subjectivity and sociality, and rearrange the basic terms of political existence. Further, they are giving rise to new ways of thinking about life, in ways that make it evident that natural life is not opposed to technology, but is imbued through and through with it. One lesson we can draw from this is that natural life is itself constitutively related to technology and its own technologization; or, to put the point starkly, life itself is a technological artefact.
The strength of this approach lies in it allowing for recognition of the historical variability in biopolitical management, as well as the variability in the mechanisms and strategies that may be operationalized in it. In this, Rose and Rabinow both provide exemplary studies of biosciences that are sensitive to the biological realities of life today. Despite its richness, though, this approach also has its weakness, in that the empiricism urged by Rabinow and Rose may forestall any independent attempt to conceptualize life. For example, in Politics of Life Itself, at no point does Rose give any account of 'life itself', preferring instead to 'explore the philosophy of life that is embodied in the ways of thinking and acting espoused by the participants in [the] politics of life itself' (Rose 2007, 49). Thus, what is under investigation here is not life, but what is said about life. Within this, the term 'life' tends to operate as a signifier without referent, almost infinitely encompassing and divisible, with the consequence that 'life itself' is whatever is said about it (as long as the speaker is sufficiently authoritative), and the operations by which life is managed and directed are seen as almost inevitably efficacious.
This appears to leave us in something of a difficult position methodologically speaking. On the one hand, theorists of an affirmative biopolitics are insufficiently attentive to the ways in which contemporary conceptions of life are being produced by bioscience, and propose an abstract conception of life that is said to overcome negative biopolitics without any substantial empirical tethering. On the other hand, empirically focused scholars of biomedicine and the biosciences illuminate the ways in which these contribute to forming our self-understanding, including our understanding of life itself. However, they may be unwilling or unable to generate a conception of life independent of what the biosciences say about it. This apparent dilemma is somewhat false though, for in their production of objects of knowledge, contemporary biosciences do not produce just one concept of life, but yield a multivalent patchwork of concepts that may be put to use in different ways for different purposes. Further, the philosophical project of refining a concept of life that is not simply an artefact of biopolitical governance can be attentive to the implications of new assemblages of knowledge and technology without accepting uncritically all of the claims about life that are generated by them. Hence, as Maurizo Meloni recently states, an approach that brings 'the notions of biopolitics down from the heaven of abstraction to concrete historical practices and situated material contexts' (Meloni 2016, 23) seems necessary. However, this must also be married to a commitment to critique – whether through the identification of 'subjugated knowledges' within a discourse or derived from elsewhere – to realize the critical and transformative potential of genealogy. To my mind, the discussions in the following section and next chapter make the imperative of critique especially clear.
Reproduction
One area of contemporary life – though by no means the only one – in which the technological saturation of subjectivity discussed above is apparent is that of reproduction, wherein technology makes certain lives possible and others impossible, or at least increasingly difficult to imagine and make live. In this section, I first consider the way that reproduction has been discussed in biopolitical theory to date, and more specifically, demonstrate its relative neglect in theorizations of biopolitics. I make a case for a more extensive engagement with questions of reproduction if biopolitical theory is to accurately describe the workings of biopolitics. I discuss this thematic in relation to the major theorists discussed in earlier chapters, with an eye to the ways in which their accounts of biopolitics either signpost ways to think about reproduction or has been opened up by others to this domain of 'biopolitical experience' (Blencowe 2012). In particular, I discuss how the idea of natality proposed by Arendt has been seen to open into a new affirmative biopolitics, and discuss the ways that it imbricates with the biopolitics of birth. Following this, I briefly outline some of the key trends in empirically focused literature on biopolitics and reproduction.
As Foucault argued in Will to Knowledge, one of the principle mechanisms that tied the two poles – that is, the individual body and the population – of biopower together was the deployment of sexuality. Sexuality, Foucault argues, emerged in the nineteenth century as one of the most significant vectors of the new formation of power because of its 'privileged position... between organism and population, between the body and general phenomena' (Foucault 2003, 252), since '[s]ex was a means of access both to the life of the body and the life of the species' (Foucault 1990, 146). Foucault goes on to argue that the family unit was significant in the deployment of sexuality, for it is 'the interchange of sexuality and alliance: it conveys the law and the juridical dimension in the deployment of sexuality; and it conveys the economy of pleasure and the intensity of sensations in the regime of alliance' (Foucault 1990, 108; also see Donzelot 1979). This shift was related to the emergence of the concept of reproduction in the late eighteenth century (Jordanova 1995), and associated notions of sexual complementarity (Lettow 2015), degeneracy and perversion, within which sexuality became an object and target of power. However, while Foucault identified the 'socialization of procreative behavior' (Foucault 1990, 104) as one of the key axes of biopower, in the intervening period between the publication of the introductory volume of his History of Sexuality and the second volume, his project shifted such that a full analysis of this was never undertaken.
This has led some feminist scholars to question the value of Foucault's concept of biopolitics as a framework for analyzing the politics of reproduction today (Franklin 2013, 273). For others, the full fecundity of his conception of biopower is yet to be recognized (Deutscher 2008; Deutscher 2012), and his genealogical method has lent itself to exploring questions about the regulation and governance of reproduction (For example, see Weir 2006; Takeshita 2011; Murphy 2012). Further, his conceptual apparatus of biopower could be extended relatively easily to include questions of reproduction, since it does not entail a single overarching logic to biopower beyond fostering life and allowing death, and the centrality of operations of normalization. There are, of course, significant questions about whether his account of the emergence and operation of biopower and normalization in Europe adequately describes these in relation to reproduction (Murphy 2012, 11), or whether his understanding of the interconnection of biopower and neoliberalism is adequate to capture contemporary formations of biocapital (Cooper 2008). Despite these questions, though, there is 'something profoundly useful about the way Foucault initially posed the question of biopolitics as the history of governing living-being, its qualities, kinds, health, rates, deviations, productivities, evolution and so on' (Murphy 2012, 13).
The possibility of extending existing insights is more problematic in relation to Giorgio Agamben's account of biopolitics, to which the responses largely fall into two camps – those who see the idea of bare life as appropriable for feminist purposes, and those who see the theoretical unicity of Agamben's account of biopolitics as excluding feminist concerns and thus in need of more or less radical revision. In relation to the first of these, the notion of bare life has been appropriated for various purposes, and in relation to reproduction it has been useful for feminist scholars interested in the political figuration of the foetus in abortion politics. Penelope Deutscher (2008) argues that while the foetus cannot really be seen as either bios or zoē, it comes to be represented as each of these at different moments in abortion debates (also, see Latimer 2011; Weingarten 2014). More generally, the notion of bare life appears to lend itself to an anti-abortion politics, since it may generate an account of the way in which the threat of violence in abortion – here cast as a killing that is without legal consequence or threat of punishment – constitutes every embryo or foetus as bare life: life that can be killed without constituting homicide. The problem with this construal is that strictly speaking, bare life is life that is cast out of the political sphere – it is the citizen–subject abandoned to violence and thus stripped of legal status in that abandonment. This is not the case with the embryo or foetus, where the contestation lies in the point at which they enter into moral and political community. It is a question of when a foetus acquires a biography, which also means acquiring certain social properties such as gender, race and even class, as well as a legal status in rights.
A more radical response to Agamben's work would entail asking how the integration of matters of reproduction into his theoretical framework might necessitate a reworking of it. As we saw in the earlier summary, Agamben's construal of biopolitics sees it emerging in the distinction between zoē and bios, and the Aristotelian account of the foundation of the polis that works with this distinction (though perhaps not in the way that Agamben implies (Dubreuil 2008, 2; Finlayson 2010). Agamben (1998, 2) writes, '[i]n the classical world,..., simple natural life is excluded from the polis in the strict sense, and remains confined – as merely reproductive life – to the sphere of the oikos, "home'". What is interesting about this claim here is the description of the natural life that is relegated to the domestic sphere as 'merely reproductive life'. While Agamben thus mentions reproduction, he does not go on to thematize its role in the formation of the city-state, and therefore the foundation of politics. However, the problematic of sexual difference and reproduction works across Aristotle's account in interesting ways. In simple terms, in his account, the natural desire to reproduce gives rise to households, which subsequently give rise to and form parts of the city-state. In other words, the domestic sphere is not strictly opposed to the polis, but is a prior step in its emergence: the polis emerges naturally from the development of the household and village. But while the oikos is not strictly opposed to the polis, women are nevertheless excluded from participation in politics, because they are akin to slaves insofar as they require mastery by men. In short, while the deliberative part of the soul is present in women it lacks authority, and this justifies the natural rule of men over women.
This suggests that the 'simple natural life' that is relegated to the household is not so simple after all; in fact, it is sexed from the start. Interestingly, Adriana Caverero made a similar point in a book published in Italian in the same year as Agamben's Homo Sacer (1998) and translated into English four years after it, as Stately Bodies (2002). She argues that Western politics has simultaneously expelled the body from the political sphere and reincorporated the body into itself. However, for her, the distinction that both permits this and is reinforced by it is not a fracturing in life itself – between zoē and bios for instance – but that between male and female, and relatedly, between logos and corporeality. Thus, she writes, '[t]he constitutive nonpolitical – or rather antipolitical – nature of the body as the opposite of the logos within this tradition finally comes down to the basic opposition between female and male natures... the body expelled from the polis, is, in its full and true substance, a female body' (Cavarero 2002, ix). Cavarero's approach thus introduces questions of sexual difference into the question of the history of the politico-philosophical fate of the human body and what we understand as biological life.
But it also does more than this by bringing forth questions of reproduction and the figure of the mother. For her, the generative potency of the female body is particularly troubling for Western politics, a point that she makes through an interpretation of Antigone that places Jocasta – Oedipus's wife and mother – at the centre. This feminist reading would have a number of significant implications for an account of biopolitics that seeks to find its origins in ancient Greece. In particular, it suggests that a careful, gender-sensitive reading of Aristotle might yield a different account of the distinction between bios and zoē, in which this apparent fracturing of life per se has a different relevance and effect for men than it does for women. Further, this suggests that 'sexual difference [is] inescapable for the thought of biopolitics (that is, inescapable both for the critique of biopolitics and for the reconceptualization of life that such a critique ought to involve' (Casarino 2012, 98). I discuss sexual difference further in the following chapter, but for now I return to Esposito's characterization of pregnancy as a central metaphor of the immunization paradigm and Arendt's conception of natality.
While Agamben has seemingly been unable to incorporate questions of sexual difference and reproduction into his account of biopolitics, these do acquire a stronger profile in the intervention of Esposito, though as I mentioned earlier in Chapter 4, this is not without problem. As I discussed, pregnancy is one of Esposito's principle devices for explicating the affirmative logic of immunity. In short, he uses it to show the way that foreign-ness can work positively within immunization, since the maternal body does not destroy the foreign object that the developing embryo effectively is to it. Rather, her immune system reins itself in: 'by immunizing the other, it is also immunizing itself. It immunizes itself from an excess of immunization' (Esposito 2011, 170). Further, in response to the pre-emptive suppression of birth that Esposito identifies as one of the key axes of Nazism, he urges a contrary valorization of birth, achieved in particular through the work of Gilbert Simondon on individuation. Esposito writes in a reflection on Simondon, '[i]f one thinks about it, life and birth are both the contrary of death: the first synchronically and the second diachronically. The only way for life to defer death isn't to preserve it as such... but rather to be reborn continually in different guises' (Esposito 2008, 181). As I indicated earlier, the problem with this approach is that birth is entirely metaphoric and disembodied: birth is not only a perpetual feature of life, but that life is ultimately born of nothing. The birth at issue is simply a metaphor for the new, and is certainly not the bloody, painful and often highly technologized and medicalized separation of one body from another, specifically, from the body of a woman. In fact, birth is made distinct from the female body that gives birth – it happens in absentia of a female body. This particular portrayal of birth not only cuts across Esposito's work but is – it might be argued – central to the history of Western reproductive imaginaries and biopolitics (Duden 1993; McGrath 2002).
As we saw in Chapter 3, the ambivalent denial of the female body that gives birth is also evident in Arendt's notion of natality. As feminist interpreters of Arendt such as Cavarero (2014) and Julia Kristeva (2001) point out, although central to Arendt's framework, the concept of natality is strikingly ambiguous. On the one hand, natality provides Arendt with a way to challenge the basic foundations of Western philosophy that places death at the centre of politics and of existentialism – that is, of life. It also suggests a potentially fecund way of thinking freedom, not as a matter of will, but as arising from the basic capacity to begin anew that natality signals. On the other hand, though, Arendt is at pains to separate natality from physical birth and the gendering of politics that this perspective might entail. Indeed, she largely erases the maternal body from the site of birth. This appears to etiolate the concept. If, however, we recover an association between natality and the female body that gives birth, we may be in a position to elucidate some biopolitical phenomena that are otherwise opaque. For instance, Robin May Schott and others have used Arendt's reflection on natality to elucidate the logic of using rape as a weapon of war, of which Schott argues that war rape radically transforms the political principle of birth, in order to inscribe birth as a 'weapon of death' (Schott 2010, 63). Schott also comments on Agamben's reflections on war rape in Homo Sacer and elsewhere, arguing that for him the transformation of the principle of birth in war rape is more directly related to shifts in the nation–state than it is to changes in the 'violent use of sexual difference in wartime' (Schott 2010, 61). At the least, this suggests that a perspective that ties natality more tightly to the generative female body may offer important insight into the operation of biopolitics today.
In recent work, Miguel Vatter (2014) has probably gone furthest toward integrating Arendt's conception of natality into a theorization of biopolitics, and more specifically, into an affirmative conception of biopolitics. In his republican take on the problem of government and politics in The Republic of the Living, he makes a case for what he calls eternal life, in which life and thought can no longer be separated, as the foundation stone of an affirmative biopolitics. He develops this through a conjunction of natality and normativity. Somewhat counter-intuitively, for him, the former of these refers to species life or zoē, while the latter refers to the capacity for genuinely new beginnings. It is, he argues, necessary to conceive of a life that merges natality and normativity without the 'intermediation' of bios or what he also calls 'normality', that is, the rule-bound life of civil society (Vatter 2014, 7, 3). In his view, natality 'is encountered in the sphere of prepolitical or familial life (Greek: oikos) that governs the sexual reproduction of life, and then in the economy that governs living labor' (Vatter 2014, 3). As such, it provides the foundation or ground for normativity, 'that aspect of political life that is creative of novelty' (Vatter 2014, 3). Unlike Arendt, then, here Vatter seems to suggest that natality (species life) is unable to generate the new from within itself; rather, that generative capacity is located entirely in the sphere of normativity, and natality must be bound to this in order to give rise to an eternal life of contemplation. While Vatter's reading of natality and the thesis of its reconnection to normativity in eternal life is provocative, I have doubts about the parallel drawn between zoē and natality, and consequently, the role that sexual difference might play in the 'republic of the living'.
This brief discussion of the main contemporary theorizations of biopolitics suggests, then, that there is much work to be done at the 'as yet uncharted crossroads where biopolitics and patriarchy meet' (Casarino 2012, 98). Indeed, the question of the political capture of the generative female body would seem to have implications not only at the level of biopolitical theory, but also in terms of an analytics of contemporary life. For as Catherine Waldby and Melinda Cooper (2008, 58) point out, 'the compliance, negotiability and general agency of female populations is a central issue in the development of the reproductive bioeconomy'. This is further made clear in the empirical studies of Cooper and Waldby (2014) and Waldby and Mitchell (2006) which trace the ways that the vast biotechnology economy rests to a large extent on the bodies of women, who donate or sell oocytes and embryos for use in the production of stem cells, for the achievement of birth in oocyte markets and surrogacy, or as the raw material for research in reproductive sciences. The generative capacity of women's bodies is also captured in the bio-economy in other ways, for instance, in surrogacy arrangements in which women gestate babies under contract.
However, it is not only through the contemporary bioeconomy that the bodies of women have been subject to specific biopolitical measures to manage reproductive processes and contribute to population well-being. While the politicization of life by the German Third Reich has been significant within theorizations of biopolitics, greater attention has been paid to the obviously thanatopolitical aspects of it than has been shown to the reproductive policies of the Reich. As Esposito notes, these reproductive policies took both a 'negative' and 'positive' form – the first through the prevention of conception or of birth (through enforced sterilization and abortion) within some groups of people, and the second through the increased reproduction of those seen as necessary to creating the Aryan master race. In this, birth was tied very tightly to questions of nation, and the survival of it; in this case, as in many others, responses to those questions were shot through with ideologies of racial superiority and scientism that made biology and political power inextricable. Importantly, attention to reproductive politics highlights the continuities between the racial hygiene of the Reich, and broader eugenic ideologies and practices such as enforced sterilization in countries such as the UK and the United States. As these policies indicate, eugenics entails a kind of 'quality control' at the heart of the determination of the health of the population through both the positive and negative regulation of birth. Thus, birth control means controlling both who gives birth and who is born.
While older eugenic projects were largely restricted to controlling conception, today, the 'quality control' that attaches to reproduction in the biopolitical management of life extends throughout pregnancy, with prenatal testing becoming an ever-more routine aspect of pregnancy care. Such technologies, which include non-invasive foetal DNA testing, obstetric ultrasound, amniocentesis and chorionic villus sampling, provide a means to test the genetic profile and morphology of a foetus in utero, and allow prospective parents the opportunity to terminate a pregnancy where findings are in their view adverse. However, as the operation of such technologies makes clear, while the eugenics typical of nineteenth and early twentieth-century biopolitics involved state regulation of reproduction, this is no longer the predominant mode of biopolitical management. Rather, today individuals themselves are responsible for the enactment of biopolitics in reproduction. Foucault saw this trend emerging in the nineteenth century, where the family was integral to a 'political socialization achieved through the "responsibilization" of couples with regard to the social body as a whole' (Foucault 1990, 104–5). Thus, the contemporary 'flexible eugenics' (Taussig, Rapp and Heath 2005) enacted through individual decision-making may not be entirely new, but the extent of this individual responsibilization is probably unprecedented. If this is the case, then the current discursive arrangement or 'apparatus of choice' (Mills 2016) in operation in reproductive medicine ensures that, paradoxically, the enactment of freedom itself becomes a site for the biopolitical management of who comes into the world.
Conclusion
To a large extent, biopolitical studies have focused on questions of politics, explored in the previous chapter. It has only been relatively recently that there has been a turn to attempting to develop a new philosophy of life. This has been driven by responses to Deleuze's remarks on the notion of a life of pure immanence. While enormously provocative, one consequence of this approach has been a lack of engagement with the history of biopolitical knowledge formations of modern biology and central concepts such as heredity, and the ways in which these have played a part in biopolitics. In this chapter, I trace the outline of responses to Deleuze, as well as the ways in which biology has given rise to concepts of life, including new concepts of knowledge generated by genomic biosciences. In regards to this, I also discuss the work of scholars such as Nikolas Rose and Paul Rabinow, among others, who provide an empirically driven study of the knowledge formations of life today. This view starts from the premise that the objects of biological knowledge are technological and epistemic artefacts, and analyses the production of new forms of life. Finally, I have argued that reproduction plays a central role in the extension and maintenance of the biopolitical management of life, at least through the history of eugenics as well as in contemporary phenomena such as prenatal testing and surrogacy. Feminist scholars are increasingly engaging with questions of reproduction as they pertain to theories of biopolitics, and vice versa. This is at least partly due to the fact that the contemporary politics and practices of reproduction are increasingly subject to risk, uncertainty and the neoliberal commodification of life processes. Even so, major theorists of biopolitics have largely failed to reckon with the implications of reproduction and, hence, sexual difference. In my view, both understanding the operation of biopower or biopolitics and developing a more positive conceptualization of life that is not beholden to neoliberal commercialization and governance requires addressing the question of what a critical account of biopolitics that takes reproduction and sexual difference seriously would look like. In the following chapter, I elaborate on this further, particularly through questions of subjectivity, race and gender.
Notes
Van Leeuwenhoek crucially improved magnifying lenses and was the founder of microbiology; Ray was the first to propose a biological definition of the concept of species, which is the basic level of biological classification.
Though even this is controversial, as philosophers of biology and some biologists do concern themselves with the question of 'the meaning of life', but answering it is not a prerequisite of doing biological science.
Rabinow and Rose do specify one limit, which is that a discourse about life must be 'in the true' to have authoritative force. They go on to argue that biological claims about race were no longer 'in the true' in either political or biological discourse in the late twentieth century. They are, however, re-entering the domain of biological truth through a 'molecular gaze' (Rabinow and Rose 2006, 205–6).
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7Subjectivity
Persons, race, gender
In the previous chapter, I argued that reproduction played, and continues to play, a central role in biopolitics. This claim can be understood in a number of ways. First, it may be understood to mean that as a site or locus of the biopolitical management of life, reproduction is and has been especially important. I think this is probably right, but making this point is not meant to discount other significant areas of biopolitical power that may have little to do with reproduction per se. The claim may also be taken to mean that the concept of reproduction is significantly related to the emergence of the modern episteme that provided the conditions of possibility for a new form of power, and was also integrally related to the formation of that power. This rendition of the claim is made by historians of philosophy and of the life sciences such as Susanne Lettow, Ludmilla Jordanova and Hans-Jörg Rheinberger, among others. As these scholars show, the concept of reproduction was first used in the modern sense by Buffon, in 1749. Interestingly, Lettow (2014) makes the further claim that the concept of reproduction was also deeply connected to the development of ideas of race and sexual difference. Further, she argues that the emergence of these concepts was intricately related to the emergence of biopolitics as a modern phenomenon. As she writes, the 'concern with reproduction, genealogy, and the belonging of individuals to supra-individual entities like the species, the sex or the race also contributed to the emergence of a biopolitical gaze that addressed humans as subjugated to these new biosocial entities and to a new understanding of kinship relations' (Lettow 2014, 23). Following the broad direction of this argument, in this chapter I investigate questions to do with the biopolitical significance of race and sexual difference.
The basis contention underlying the discussions here is that biopolitics made possible and gave rise to new forms of subjectivity, which is to say that being in the world could be comprehended in new ways. At the same time, biopolitics renders some other ways of being in the world illegitimate, socially unintelligible or otherwise unrecognizable and devalued. I begin this discussion by outlining the ways that theorists of biopolitics have understood subjectivity and subject formation. From Agamben, I take the point that subjectivity is the battleground of biopolitics, but argue that this should not be understood solely in terms of modal operators such as contingency and necessity, as he argues (Agamben 1999, 146–7). Instead, this insight requires engagement with the contemporary conditions under which subjectivity is actively constituted and lived. I suggest that the approach of Foucault to questions of subjectivity better allows for this kind of analysis. From this discussion, I consider recent interventions that address the critical aspects of subjectivity at the broad level of the concept of the person, or of humanity. In relation to personhood, Roberto Esposito argues that the concept of person operates as a biopolitical dispositif that gives value to some lives and not others. Judith Butler focuses on the notion of humanity, and the ways in which contemporary iterations of it work to exclude some populations and individuals from the category of the human, thereby allowing for the differential valuation of different lives and forms of subjectivity.
Building on this, in the second and third sections of the chapter, I take up two axes of subjectivation that have been central to the differential valuation of life. The first of these is race, and the second is sex/gender, which I approach through the frame of sexual difference. My main line of questioning here is, to what extent do the accounts of biopolitics discussed in the first part of this book lend themselves to, or make possible, a critical analysis of these axes of subjectivation and their imbrication in biopolitics? In the second section of the chapter, I provide an outline of thinking about race, especially in relation to the political phenomena of colonialism, slavery and immigration. As I discuss, race is analyzed in significant ways in the work of both Arendt and Foucault. However, these discussions remain problematic, and more recently, other scholars immersed in biopolitical studies have proffered strong critiques of the ways in which this field has approached – or failed to approach – questions of race. Interestingly, questions of gender and sexual difference have received far less attention in biopolitical studies – except in relation to the social and cultural analysis of human reproduction discussed in the previous chapter. Building on my remarks throughout this book, in the third section of this chapter, I turn to one attempt to think sexual difference explicitly in a biopolitical frame, and use this to propose an alternative way of approaching the problem. As part of this section, I also discuss a third central axis of biopolitical subjectivation. This is the axis of disability or abnormality. Here I comment briefly on the 'biocultural' approach to disability proposed by Lennard Davis, as well as the work of other scholars inspired by Foucault's work on normality and abnormality.
Persons and humans
Subjectivity was sufficiently important to Foucault that at one point he described it as the central problematic of his oeuvre; he claims, 'it is not power, but the subject, which is the general theme of my research' (Foucault 1982, 209). In broad terms, Foucault's approach to the question of subjectivity took two forms. In his genealogical work in Discipline and Punishand Will to Knowledge, his attention was drawn to the ways in which subjectivity itself was an effect of the operation of a productive power. Here, he argued that the individual ought not be seen as pre-existing power and therefore repressed by it; rather, individuals were to be seen as an artefact of the operation of power/knowledge. As such, the analysis of subjectivity required a critique of the rationality of the state and other technologies of power. This analysis attempts to account for the specificity and generality of different operations of power, by tracing its points of operation and the means by which it infiltrates, produces and constrains our experience, tying subjects to power, and specifically biopower, through the imposition of individuality.
In his later work in the second and third volumes of the History of Sexuality project, Foucault turned instead to the ways in which subjects engage in practices of self-making. Specifically, he initiated a genealogy of the ways in which individuals act upon themselves as ethical subjects, that is, the self-work that they do in order to enact certain principles and modes of ethical being in their own lives. Foucault's view in these texts was that a 'history of desiring man' required that he focus not only on the ways in which subjection is produced through the operations of regimes of power and knowledge, but also on 'the forms and modalities of the relation to self by which the individual constitutes and recognizes himself qua subject'. (Foucault 1987, 6). He called these ways of acting upon oneself 'technologies of the self', a term that encompasses the practices and means by which individuals subjectify themselves as ethical subjects, in making themselves subject to particular moral codes, modes of being, or aesthetic or ethical criteria. This work upon oneself as a subject entails establishing a particular relation to oneself through the adoption of certain principles and practices to act upon one's own body, thoughts and conduct in order to transform oneself and attain a desired state of happiness, wisdom, purity, health or personal fulfilment and so on. Thus, it requires a certain reflexive relation to oneself, in which one effectively makes oneself subject to oneself, specifically, as a certain kind of ethical subject.
The apparent shift in thinking about subjectivity from seeing it as an effect of power, to seeing it as an effect of one's work upon oneself, naturally, prompted considerable discussion about the relationship between these two views. I argued in the first chapter of this book that it is probably best to see these as complementary dimensions of subjectivation rather than contradictory accounts of the entire process of becoming a subject. Further, there is some justification for thinking that the latter account of self-formation provides an important check to the former account, insofar as it makes the place of freedom in self-formation considerably clearer. From the study of ancient Greek ethics of the self in The Use of Pleasure (1987), Foucault emphasized the centrality of freedom in self-formation, claiming, in fact, that an ethics of the self was a 'practice of liberty'. Even so, his concern is not with theorizing about freedom and its moral and political value (though he clearly thought it had such value); rather, his interest was in how people practiced freedom in their everyday lives. In short, then, Foucault was not interested in elaborating principles of duty or moral right, but he did provide a method for examining how such principles might be adopted by and embedded within the lives of ethical subjects.
This particular formulation of freedom may have implications for the extent to which such an ethics of the self might be considered to provide an ethico-political response to biopolitical subjection. The ambiguity of Foucault's position on the contemporary relevance of an ethics of the self is well indicated in the comments he makes in the interview 'The Ethics of Concern for Self as a Practice of Freedom', where he suggests both that a practice of freedom is a necessary accompaniment to post-colonial liberation and that the relevance for a care of the self for contemporary politics is a problem he has not made much progress with, but would 'like to come back to' (Foucault 1997a, 282, 294). However, Foucault was certainly not urging a simple return to the Greeks as a liberation in itself, nor urging the full adoption of their ethics, which he explicitly rejected (Foucault 1997a; Foucault 1997c). Yet, he apparently also saw an ethics of self-formation as potentially opening an avenue for challenging the forms of subjectivity available to us today and for creating new ways of living and relating. This is amply clear in comments he made in regards to homosexuality, for instance, where he exhorted that the work that one does on oneself to transform oneself may allow one to 'invent... I do not say discover – a manner of being that is still improbable' (Foucault 1997b, 137). Perhaps we could say then that while Foucault rejected the substance or content of the Greek ethics, he nevertheless saw ethical and political potential in their form, as a way of undertaking the political task to 'get free of oneself' (Foucault 1987, 8).
Foucault's account of subjectivity has been enormously influential across the social sciences and humanities since the 1980s. Even so, philosophers, especially feminist philosophers, have been split over the theoretical value of this approach, with contention particularly over Foucault's treatment of the body and corporeality. The particular importance of Foucault's approach was that he made the body a central aspect of his account of the operation of power, whether it be disciplinary power or a biopolitics of populations. Thus, the body was understood as both an object and target of power, as providing the raw materiality upon which power attached itself and through which it worked. Some feminists read Foucault's account of the body as saying that the materiality of the body was inscribed by power (Grosz 1994); however, this led to a view that the body was essentially passive or 'docile' – and not simply rendered docile by its capture within relations of power. Others emphasized the ways in which Foucault presented the body as a possible locus of resistance, often encapsulated in his claim in Will to Knowledge that a new 'economy of bodies and pleasures' was required to resist the mobilization of sexuality as an axis of power. Feminists have also taken a different line, examining the ways in which the body is implicated in practices of self-making and transformation, such that contemporary subjectivity is rendered 'somatic' (Heyes 2007). Whatever one makes of the specifics of Foucault's approach to subjectivity and the body, it is undeniable that these feature centrally in his account of biopower. Interestingly, this is not the case with more recent theorists of biopolitics.
As has often been noted, Agamben opens Homo Sacer with a provocative positioning of his account of biopolitics as an attempt to 'correct and complete' that previously proposed by Foucault. Specifically, Agamben argues that Foucault's account is unable to broach the hidden point of intersection between technologies of the self on the one hand, and the political techniques through which the 'State assumes and integrates the care of the natural life of individuals into its very center' (Agamben 1998, 5) on the other. Of this, Agamben (1998, 6) asks, 'is it legitimate or even possible to hold subjective technologies and political techniques apart?' Disclosing this 'zone of indistinction' is then the task that he sets for himself in Homo Sacer. However, in further outlining his project, he almost immediately redefines the problem: only one paragraph later, the zone of indistinction to be disclosed is no longer that between technologies of the self and political techniques, but between the 'juridico-institutional and the biopolitical models of power' (Agamben 1998, 6). Consequently, technologies of the self silently drop out of the analysis. This is not to say that Agamben has nothing to say about subjectivity – in fact, he discusses subjectivation and desubjectivation at length in Remnants of Auschwitz. However, the account he gives focuses on the appropriation of the first-person pronoun as an act of subjecting oneself – that is, both becoming a subject and becoming subject to – in language. While this account links to the discussion of biopolitics in Homo Sacer in various ways, what remains unclear is how power actually functions in relation to subjectivation.
Agamben's account of the process of subjectivation in language is complex and I will do no more than briefly summarize it here. In brief, he posits that subjectivity should be understood as the 'production of consciousness in the event of discourse'(Agamben 1999, 123), by which he specifically means the appropriation of the first-person pronoun, 'I' within the event of speech. In reference to Emile Benveniste's analysis of pronouns, Agamben argues that terms such as 'I' and 'you' indicate an appropriation of language, without referring to a reality outside of discourse. Instead, their sole point of reference is to language itself, and particularly the taking place of enunciation. Furthermore, the pronoun reveals that a double movement of subjectivation and desubjectivation structures the relation of the subject to the language. Put simply, while the appropriation of language allows for the constitution of the subject in language, it also requires that the psychosomatic individual erase or desubjectify itself as an individual in its identification with the pronoun 'I'. Paradoxically, then, the assumption of the position of the subject of enunciation does not open into the possibility of speaking. Instead, it illuminates the impossibility of it – it is not the psychosomatic individual who speaks, since the 'I' is always separate from them, but nor can the pronoun itself be identified as the speaking subject. Agamben (1999, 129) states, 'the living individual appropriates language in a full expropriation alone, becoming a speaking being only on condition of falling into silence'. At stake here according to Agamben is nothing less than the traditional philosophical definition of the human as a speaking being, or the living being that has language: as 'zōon logon echōn'. In particular, the nature of the having of language by a living being is brought into question and shown to be conditioned by a full expropriation. This ultimately means that 'the fragile text of consciousness incessantly crumbles and erases itself, bringing to light the disjunction on which it is erected: the constitutive desubjectification in every subjectification' (Agamben 1999, 123). Agamben makes much of this logic of desubjectivation and subjectivation in language in Remnants of Auschwitz, insofar as he builds his account of an ethics of testimony upon it.
There are, however, significant difficulties in reading Agamben as offering an account of subjectivation at work within biopolitics. For one, it becomes remarkably difficult in his framework to say anything about bodies and the means by which the somatic self is rendered as a subject through, for instance, apparatuses of embodiment and self-understanding. The tools for such a focus are simply not available in his work. What we have is an abstruse elaboration of the ontology of biopolitics at the level of (human) life in general and its perceived fracturing into bios and zoē. However, it is oddly unclear how this relates to the somatic self and its formation as a particular kind of subject, as well as its recognition and treatment by others. Let me clarify with an example: at several points, Agamben comments on the rape of (mostly) Bosniak women in the Yugoslav wars, indicating that women were raped as bare life. However, this analysis obscures the point that women raped in war are raped as women and, moreover, women of a particular ethnic grouping. What Agamben fails to recognize, then, is that 'bare life is implicated in the gendered, sexist, colonial and racist configurations of biopolitics' (Ziarek 2012, 147). In short, his account of subjectivation does not throw light on the role of power in producing certain kinds of subjects, and in rendering other forms of subjectivity illegitimate or unliveable. Hence, while Agamben charges that the intersection between technologies of the self and political techniques is the 'vanishing point' of biopolitics in Foucault's enquiries, it is all the more so in his own. And insofar as this is true, Agamben misses a crucial aspect of modern biopolitics, that is, the relentless pursuit of some forms of subjectivity, which, at its most severe, eradicates the possibility of escape from the biopolitical casting of oneself as an instantiation of a particular identity.
Esposito allows more purchase on this aspect of biopolitics in his critical analysis of the concept of the person, through which he argues that much of the biopolitical valuation of lives is channelled. He argues that the concept of the person constitutes a biopolitical dispositif that permits a differential valuation of lives, and excludes some from the position of normative recognition as worthy of rights, including a right to life (Esposito 2012a). Esposito argues that while the concept of the person was promoted as a necessary antidote to the Nazi reduction of human life to mere biology during the Second World War, it has never really achieved the task that was set for it, which was to heal the breach opened up between the notion of rights and life. The reason why it has not been able to do this, is precisely because the notion of the person itself produces this breach: as Esposito puts it, 'the dispositif of the person, intended by the creators of the Declaration of Human Rights to fill in the chasm between man and citizen left gaping since 1789, produced an equally profound gap between rights and life' (Esposito 2012b, 74). This gap emerges because all attempts to reintegrate rights and life have posited the body as a thing in contrast to the spiritual or reasonable person of consciousness. This separation between body and person has also then allowed for the mobilization of a break between person and human, whereby not all persons are human, and equally, not all humans are persons.
While having more ancient origins, this breach between humans and persons is well-illustrated in some approaches to contemporary bioethical issues, and here Esposito particularly targets the utilitarian Peter Singer. For what the separation of person and human necessitates is an intermediation that identifies beings on a spectrum between the merely human, those who belongs to the species homo sapiens, and the person, who has self-consciousness and is able to exercise power over oneself and one's own body. In accordance with this, in contemporary bioethics discourse there is much discussion of the moral status of beings such as the human foetus, the cognitively disabled, the senile and the comatose, who do not meet the traditional philosophical criteria set for personhood. For Esposito, these debates reveal that 'the person-hood deciding machine marks the final difference between what must live and what can be legitimately cast to death' (Esposito 2012b, 13). Against this biopolitical dispositif of personhood and the differential valuations that it necessitates, Esposito urges what he calls a philosophy of the impersonal that specifically challenges the designation of the body as a thing in order to reintegrate the body with the normative category of the person. This has the effect, in his view, of re-integrating norms and life in something like what we discussed previously under the name 'form of life'.
While Esposito's work on the dispositif of the person is certainly interesting it provokes some doubts, two of which I mention here. First, while he argues that the reintegration of the person and the body is not a reactivation of the doctrine of the sanctity of life, his views on contemporary medicine and bioethics at times veer close to those that assert a normative value in life itself through, for instance, the notion of a 'right to life'. Second, and more importantly for this discussion, Esposito fails to consider the ways in which the division between persons and things is operationalized along particular lines such as race and sex. Indeed, in his book Persons and Things, he mentions race and sex only once each. In contrast, scholars such as Charles Mills and Catharine Mackinnon have argued that personhood is structured along racial and sexed lines. As Charles Mills argues, if race and gender are taken seriously, the 'political economy of personhood' is revealed as 'the normative vehicle of the justifiable absolutist rule of equal white male persons over morally inferior, gender- and racially-demarcated sub-persons' (Mills 2011). In this, then, while Esposito sees his task as addressing the problematic distinction between persons and things 'from the body's point of view', we have to ask, which – or whose – body?
A different approach to the question of subjectivation and biopolitics, which takes the differential socio-political valuation of the body in subjectivity as one of its starting presuppositions, is that proposed by Judith Butler. While Butler only infrequently uses the terms 'biopolitics' or 'biopower', her works such as Precarious Life and Frames of War are clearly deeply engaged with the biopolitical problem of the differential valuation of life in regimes of power. In these books, Butler builds on her earlier work on the production of the subject through heteronormative regimes of power to argue that the very appearance of the subject as human, and thus as having a claim to life, is itself regulated by norms and relations of power. Butler's early accounts of subjectivation drew significantly on Foucault's conception of the subject as an artefact of power. In later work, she ties this theory more explicitly to biopolitical conditions of precarity and corporeal vulnerability. While they have broader import, Butler developed her arguments specifically in relation to the indefinite detention of suspected terrorists in Guantanamo Bay, where the suspension of the laws of war regulating the treatment of prisoners meant that detainees were left in a purgatorial space, potentially without ever facing trial and consequently without any standing or rights before the law. From this situation of legal suspension – or more accurately, abandonment – Butler concludes that 'the humans who are imprisoned in Guantanamo do not count as human; they are not subjects protected by international law. They are not subjects in any legal or normative sense' (Butler 2004, xvi).
What Butler is getting at here is how a normative sense of the human can be mobilized to exclude human bodies from the category of the human. We might wonder here about the theoretical value of this torsion of the human as both biological and normative and whether there is more to be gained from a critical leveraging of the terminology of the person. Setting aside this worry here, though, what is important about Butler's interventions are the ways she makes plain that the social and political intelligibility or recognizability of a subject as a human subject is dependent on historically contingent formations of power. This means that even belonging to the basic category of the human can be put into question. This may sound like a move toward something like 'bare life' – a biological minimum abandoned to and by the law. However, Butler takes distance from this notion for its lack of specification; she argues that the general claims about bare life that Agamben makes cannot help elucidate the ways in which this power of abandonment operates differentially to 'derealize the humanity of subjects' (Butler 2004, 68) on, for instance, the basis of race or ethnicity. At this point, then, it is time to focus more specifically on the question of how race is integrated with and works within biopolitics.
Race
According to Foucault, the category of race and the political techniques of racism that developed from it were indispensable to the operation of biopower, since these made possible the mobilization of death within a power of life. If Foucault is right, then the centrality of racism in biopower must be examined from a number of perspectives – historical, anthropological and sociological, as well as philosophical and theoretical to be sure, but also as viewed and experienced from the periphery or from the position of those subject to racism. This spectrum of perspectives indicates that the issue of race and biopower eludes easy summary, and in a book such as this, I can do little more than provide a map of some of the issues that might be taken up in relation to race, racism and biopolitics, and some of the work available that addresses these questions. I have already touched on questions about race and colonialism throughout the book, especially in relation to Arendt; here I revisit the issue to elaborate on the critiques of biopolitical theorists yielded by scholars of race and colonial studies, particularly Ann Stoler, Achille Mbembe and Alexander Weheliye.
In the past twenty years or so, a number of scholars influenced by Foucault's genealogical method and his account of biopower have sought to elucidate the historical, philosophical and political significance of race and race-thinking. Ladelle McWhorter (2004) makes the point that while race discourses prior to the eighteenth century were undoubtedly 'political and polemical', they did not function in the same way as later race-thinking. She also points out that while there is some debate over the progenitor of the modern concept of race, Robert Bernasconi (2001) has convincingly identified Immanuel Kant as the inventor of it insofar as he proposed a scientific definition of the concept. Kant defined race as 'class distinctions between animals of one and the same line of descent (Stamm), which is unfailing transmitted by inheritance' (Kant cited in Bernasconi 2001, 14). Two specific features of this definition of race are to be highlighted. First, the concept of race was explicitly linked to emerging theories of heredity and debates about the continuation of characteristics through generations. Second, it articulated the concept of race as an explicitly hierarchical one, such that one race took priority over others. In his discussions of biopower, Foucault is less concerned with the emergence of race as an idea or concept – his lack of discussion of race and racism in The Order of Things is a notable failure – than with its use by the state in the management of bodies and populations, that is, with state racism. Of this, he claims – though presents no real argument or evidence for the claim – that '[state] racism first develops with colonization, or in other words, with colonizing genocide' (Foucault 2003b, 257). The implication of this is that a central aspect of biopower – that aspect that allowed the mobilization of a power of death within the power of life – actually derived from colonialism; in other words, colonialism was the origin of the thanatopolitical aspect of biopower.
The American anthropologist and historian, Ann Stoler, provided one of the earliest critiques of Foucault's approach to matters of race and racism within the colonial context. Stoler focused attention on the different treatments of colonialism in Will to Knowledge and the lecture series, Society Must Be Defended, in her ground-breaking book, Race and the Education of Desire. Stoler's basic claim was that despite occasional references to colonial imperialism, Foucault largely failed to take into account the ways in which the colonial projects of Europe during the nineteenth century constituted and were embedded in the biopolitical deployment of sexuality. At issue is not simply that he did not spend much time talking about race and sexuality in colonial contexts, or more broadly, contexts other than the French or European metropoles. Rather, her point is that nineteenth-century bourgeois discourses about sexuality in Europe were themselves deeply dependent on and structured by the racial categorizations and anxieties of the colonies. As she puts it,
the discursive and practical field in which nineteenth-century bourgeois sexuality emerged was situated on an imperial landscape where the cultural accoutrements of bourgeois distinction were partially shaped through contrasts forged in the politics and language of race.
(Stoler 1995, 5)
But while Foucault may have failed to appreciate the full extent of the ways in which race thinking and the management of populations in the colonies fed into European conceptions of sexuality and its deployment as a regulatory apparatus, he nevertheless grasped that concepts of race and racism, and their mobilization as state strategy in colonialism, were central to the rationality of biopower.
Unfortunately, the same cannot be said of more recent theoretical interventions. Agamben's subsequent revision and extension of Foucault's analytics of biopower largely ignored Foucault's hints that colonialism was central to the constitution of biopower: as Stewart Motha states, Agamben 'has said little or nothing about colonialism per se' (Motha 2012, 128). And while the editors of a recent collection on Agamben and colonialism argue that he provides 'an essential set of concepts for critical debates around colonialism and colonial states of exception' (Bignall and Svirsky 2012, 7), others have seen in his work a highly problematic blindness about colonialism in particular, and race more generally. One defence of Agamben's apparent lack of attention to colonialism and postcolonialism, is that while he says little about colonialism per se, he does propose a 'paradigm of colonialism' through reflections on the metropolis. As Leland de la Durantaye puts it,
Agamben is not concerned with colonialism per se – that is with the historical phenomenon of colonialism – but, instead, with the paradigm it provides for understanding life in all our cities, for its ability to illuminate a more vast and variegated space, its ability to articulate a set of more sweeping problems.
(De La Durantaye 2012, 233)
It has become standard to defend Agamben against criticisms of a lack of attention to empirical specificity and historical accuracy by pointing out that his treatment of any given phenomena is to see it as a paradigm. But, while that may be accurate as a matter of interpretation, what then needs to be assessed is the usefulness of this 'paradigmatic' method itself. Rather than attending to the historical specificity of complex colonial (and postcolonial) apparatuses and the experience of living in them, in this approach these phenomena are reduced to a single etymological moment and then used to illuminate other 'more sweeping problems'. However, it is difficult to see the intellectual or political virtue of this in a context that so requires analyses that do not trade in generalizations, and that do not simply mobilize a figure of the colony to understand something else. In short, to turn colonialism into a paradigm does very little to illuminate, let alone transform, the lives lived in the midst of that particular configuration of oppression and its enduring effects.
Given the centrality of colonialism to the contemporary formations of global politics, there is also a question about how accurate his account of biopolitics can be if it does not engage this at all. Indeed, starting from this failure of biopolitical theorists to address phenomena such as colonialism, apartheid and slavery, Achille Mbembe (2003) has put forth a vigorous critique of theories of biopolitics. In conversation with Foucault's comments on state racism and colonialism, Mbembe argues that the concept of biopower is inadequate to grasp 'contemporary forms of subjugation of life to the power of death' (Mbembe 2003, 39). Instead, he proposes the notion of necropower or necropolitics to describe conditions of life, such as in colonialism, where 'the sovereign right to kill is not subject to any rule' (Mbembe 2003, 25). This unconstrained power to put to death gives rise to a formation of terror in which race and racism are central, evident not only in the colonies but also in apartheid and slavery. Further, he argues that late modern conditions of occupation such as in Palestine involve a 'concatenation of multiple powers: disciplinary, biopolitical and necropolitical' (Mbembe 2003, 29). Mbembe's analysis is highly provocative in its identification of sovereignty in biopower as the 'generalized instrumentalization of human existence and the material destruction of human bodies and populations', and as such, the 'nomos of the political space in which we still live' (Mbembe 2003, 14). His account helps to focus attention on institutions of racialized violence in both their empirical specificity and biopolitical logics. However, it is also subject to the same form of critique as Agamben's thanatopolitical account in Homo Sacer, that it over-generalizes the logics of that violence and in doing so, obscures other aspects of biopower.
Alexander Weheliye has recently propounded one of the most far-reaching critiques of the ways in which biopolitics theorists have addressed issues of race and racism that extends beyond the focus on racial violence. He claims that 'concepts of bare life and biopolitics... are in dire need of recalibration' (Weheliye 2014, 1) if we are to understand and transform racialized global structures of oppression. Weheliye charges that Foucault and Agamben – the brunt of the critique is levelled against the latter – engage in a kind of 'philosophical unseeing of racializing assemblages' (Weheliye 2014, 65) that orchestrate and differentiate human beings into the normative categories of 'human', 'non-human' and 'not-quite humans'. This is most evident in Agamben's theorization of homo sacer and the Muselmann, which ultimately produces the conceptual fiction of a 'biological sphere above and beyond reach of racial hierarchies', that is, bare life (Weheliye 2014, 53). Against this attempt to conceive of a life that somehow transcends racialization and racism – paradoxically, through its debasement from the human – Weheliye urges greater attention on the ways in which race plays into the construction of the human subject. Further, he draws on Hortense Spillers' notion of the flesh to show how the political position described by the notion of bare life is historically transmitted and, moreover, is affixed to some bodies and not others. Here, flesh is understood as preceding the body, not as a biological substance, but as a kind of non-subjectivity forged through the cleavage of 'depravation and deprivation' (Weheliye 2014, 39). But while the idea of the flesh constitutes a 'liminal zone comprising legal and extralegal subjection, violence and torture' (Weheliye 2014, 132) it is not simply this; Weheliye argues that it may also present 'lines of flight from the world of Man', thereby opening possibilities for the recognition and incarnation of 'alternate forms of liberty and humanity' (Weheliye 2014, 132).
One aspect of both Stoler's and Weheliye's analyses that is particularly interesting is the connection drawn between racialized biopolitical subjectivities and sexuality. This connection is obviously consistent with Foucault's analysis at one level, in that he argues both that the deployment of sexuality is key to the conjunction of the individual body and population in biopower (in Will to Knowledge) and that racism, and specifically state racism, is key to the thanatopolitical aspects of it (in Society Must Be Defended). Both Stoler and Weheliye point out that Foucault does not fully integrate race and sexuality; but this dual positioning also points to an interesting ambiguity in Foucault's discussions of racism, wherein he occasionally suggests that 'racism' can be defined as a broader technology of differentiation than is typically understood, one that does not necessarily reference race per se. In the course lectures published under the title, Abnormal, Foucault makes the suggestion that there may be a racism that does not refer to races, but is instead a:
racism against the abnormal, against individuals who, as carriers of a condition, stigmata, or any defect whatsoever, may more or less randomly transmit to their heirs the unpredictable consequences of the evil, or rather of the non-normal that they carry within them... It is an internal racism that permits the screening of every individual within a given society.
(Foucault 2003a, 316–17)
According to this view, racism operates as a general principle of exclusion and division within a population that targets not race per se, but the abnormal.
This raises a question of whether this extension of racism is either justified or plausible. For some, this expanded usage is to be resisted, since it threatens to evacuate racism of any specificity. For others, such as Ladelle McWhorter, it has been a productive extension, one that makes 'good historical, analytic and in some contexts even politically strategic sense' (McWhorter 2009, 35). Her analysis of racism in the United States of America, then, is guided by the claim that sex and race is so intertwined as bases of oppression that it is not possible to understand one without reference to the other. Her point is not simply that race and sexuality are analogous dispositifs of biopower – though she did make this argument in an earlier article (McWhorter 2004). Instead, revising and deepening her analysis, she argues that race and sexuality are 'historically codependent and mutually determinative' (McWhorter 2009, 35), such that approaching them separately renders them opaque. Given this, it is now time to turn to a discussion of sexuality and sex within biopolitics.
Sex/Gender
Much to his credit, Foucault saw the deployment of sexuality as central to the emergence and operation of biopower, in that it drew together the two poles of disciplinary power and biopolitics of population. Within this argument, he also claimed that rather than constituting the material anchor point of the deployment of sexuality, 'biological' sex was in fact 'the most speculative, most ideal, and most internal element' (Foucault 1990, 155) in the deployment of sexuality in biopower. Interestingly, the various implications of this provocative claim have not been explored fully in biopolitics debates, and in particular, the implications for thinking about sex and sexual difference have been particularly effaced. In this section, then, I explore this aspect of biopolitics in more detail. I have already discussed the failure of biopolitical theorists such as Agamben and Esposito to adequately address sexual difference in their conceptions of biopolitics; here, I build on this discussion to take up work that has made inroads into thinking about sexual difference and gender in a biopolitical frame. First, I pick up on arguments made by Adriana Cavarero about the role of sexual difference in the origins of the formation of the Western state, and by way of contrast, consider a recent genealogy of gender as a biopolitical dispositif. My focus in this section, though, will be on the work of Elizabeth Grosz, who offers a feminist analysis of Charles Darwin's theory of evolution that emphasizes it productive aspects for thinking about sexual difference.
In feminist philosophy, the notion of sexual difference is often associated with the work of French feminist, Luce Irigaray. Her influence on feminist philosophy has been profound, though not without controversy. In taking up some of Irigaray's ideas, the Italian feminist philosopher and classicist, Adriana Cavarero, has also sought to extend them into a deconstruction of the tradition of Western philosophy that identifies the phallogocentric metaphysics at work in Ancient thinkers, as well as questions the relationship established between sexual difference and the State. Also influenced by Hannah Arendt, Cavarero has additionally advanced a more positive agenda, which is to bring forth birth or natality as the key political principle rather than death. Her book most relevant to biopolitical debates is that of Stately Bodies, in which she argues that sexual difference has an originary function in Western politics. As I mentioned in an earlier chapter, Cavarero argues in this book that the body excluded from the polis is the specifically female body and its generative capacity, while the body reincorporated into the polis is male. This claim is particularly developed through an analysis of the organological metaphor of the state across texts including Plato's Timaeus, Sophocles' Antigone, Thomas Hobbes' Leviathan and Shakespeare's Hamlet. Cavarero writes at one point in the book that,
[w]e are all born of a woman, and every body comes from a woman's body, as Plato, despite his logocentric fantasies, well knew. The entire culture of the West confirms this... It comes as no surprise, then, that the female is expelled from an idea of politics created in order to keep at bay birth, which instead assumes death as its training ground.
(Cavarero 2002, 159)
This line of analysis has significant implications for conceptions of biopolitics, especially those that see its history stretching back to the origins of Western politics. However, the implications of sexual difference for biopolitics as a rationality of power have not been explicitly analyzed. In keeping with a more genealogical approach, one might ask, for instance, whether there are significant historical transformations in the ways in which sexual difference is manifest as an aspect of biopolitics. Or, in other words, how is sexual difference itself implicated in the operation of biopower? To put the point more strongly, one might ask, is sexual difference an inherent feature of life that becomes entangled with biopolitics, or does biopolitics itself produce sexual difference?
Questions similar to these have recently been addressed by Jemima Repo (2016) in her genealogy of gender. Repo traces the emergence of the notion of gender in the work of controversial psychiatrist, John Money, and its subsequent deployment in feminist theory and in neoliberal forms of governance to argue that gender was a key term 'around which the psychiatric, economic, and demographic discourses regulating sexual difference converged over the past decades, resulting in the deployment of gender in disciplinary, biopolitical and neoliberal discourses, practices and contexts' (Repo 2016, 147). However, as this suggests, Repo takes it as given that sexual difference is something to be regulated, rather than a regulatory device in itself. Following Foucault's claim, though, that sex is the most ideal element in the deployment of sexuality, might it be that sexual difference is itself the most ideal element of the deployment of gender? This would suggest that sexual difference is itself a biopolitical apparatus, in need of critical genealogical analysis. While this is not the place to follow this line of thinking much further, I want to consider one recent feminist theoretic intervention that places sexual difference within a biopolitical framework, even though it ultimately misses the critical point regarding sexual difference. This is the recent work of Elizabeth Grosz on Charles Darwin and the evolutionary emergence of sex.
Grosz began publishing on Darwin in her 2004 book, The Nick of Time; since then, she has reworked her analysis in Time Travels (2005) and Becoming Undone (2011). In general, Grosz's interest in Darwin stems from the conviction that greater attention to biology is in the interest of contemporary feminist theory, and further, that Darwin's account of nature is especially valuable for the way in which it makes difference an irreducible and productive element of biology and biological processes such as natural selection. Most interesting for her, though, is Darwin's account of sexual selection, of which she claims that the Darwinian model of sexual selection anticipates contemporary feminist concerns with sexual difference by outlining a 'nonessentialist understanding of the (historical) necessity of sexual dimorphism' (Grosz 2004, 67). Moreover, of sexual selection, she writes that,
[s]exual selection ensures that sexual difference remains, at least into the foreseeable future of the human, and beyond – irreducible and impossible to generalize into a neutral or inclusive humanity. Sexual difference entails that, from the "moment" there is the human – and even long before – the human exists in only two nonreducible forms.
(Grosz 2004, 67)
However, Grosz's enthusiasm for Darwin's theory of sexual selection also runs into a problem, insofar as Darwin argued for a form of sexual complementarity in which males were active, aggressive and intellectually superior, while females were typically passive, non-aggressive and intellectually on a par with 'the lower races, and therefore of a past and lower status of civilization' (Darwin cited in Grosz 2004, 76). In The Nick of Time, Grosz's response to this problem is somewhat unsatisfactory and boils down to the idea that it is not surprising that Darwin adopted and reiterated ideas from his own culture and time. She takes up this problem again in more depth in Becoming Undone, where she makes the surprising claim that (apart from his ignorance of the gene and of sex-specific hormones) 'Darwin's work is not in need of revision, updating or scientific modification' (Grosz 2011, 120). She then goes on to develop a reading that emphasizes the transformability of sexual complementarity and its contingency and variability among species.
This later interpretation and defence of Darwin may allow more movement in considering the question of sexual complementarity, but it nevertheless fails to address the underlying problem here. For, on the one hand, sexual difference is cast as a universal feature of humanity, appearing at the origins of humanity and continuing to exist for the foreseeable future. At the same time, though, sexual difference takes variable forms, such that it is both trans-historical, that is, operating at the level of evolution, and historically contingent; in other words, sexual difference is both natural and cultural. The question to ask, then, is how certain cultural and historically specific manifestations of sexual difference become reified as manifestations of nature, and, further, how these reifications are tied to particular configurations of power. Specifically in relation to biopolitics, it would be salient to ask how the configuration of sexual difference in terms of complementarity contributes to the management and regulation of individual bodies and populations. And, in this regard, it can hardly be controversial to claim that ideas of reproductive complementarity have themselves been central to the normalization of sexed bodies. As such, sexual difference itself has, it seems, operated as a mechanism for the biopolitical regulation of bodies, and is not simply something regulated by biopower.
This brief discussion suggests that in order to understand (and, if necessary, free ourselves from the grasp of) modern biopolitics, an account of the ways in which gender and sexual difference is intimately related to the political management of life is required. In broad strokes, I suggest that such an account would need to do (at least) two things. First, it would need to reconsider the emergence of the biopolitical state in light of feminist critiques of the patriarchal foundations of Western politics. This would entail returning to feminist critiques of the state, of law and of political economy that see these as fundamentally patriarchal, and undertaking the work of bringing these into conversation with theories of biopolitics. To my knowledge, this task remains almost wholly to be done – in short, theorists of biopolitics have not engaged with feminist critiques of Western politics, and, to a large extent at least, the reverse is also true. Second, literature on biopolitics must attempt to approach the reconceptualization of life that many commentators in these debates see as required through the matrix of sexual difference, which itself must be understood as both a fundamental feature of life, and an effect of the biopolitical constitution and management of it. By this, I mean that an analysis of sexual difference and its role in the biopolitical management of life ought not ignore the history of concepts of reproduction and notions of sexual opposition and complementarity, and sexual selection and so on, that arose along with it. As naturalized as the notion of reproduction is, it is easy to forget that it too has a history, one that is deeply embedded in the emergence of the modern life sciences in the eighteenth and nineteenth centuries that Foucault saw as so central to the emergence of biopower. This suggests that it would be a mistake to assume from the start that sexual difference will provide a key to unlock the patriarchal foundations of Western biopolitics; we may find instead that it is one of its central motors.
Before concluding this chapter, I would like to comment briefly on a third axis of subjectivation, which has arguably also been central to the operation of biopower: that of disability. As we saw earlier, at one point Foucault offers a definition of racism in which it need not refer explicitly to race but acts as a general principle or apparatus of exclusion. He writes,
racism against the abnormal, against individuals who, as carriers of a condition, stigmata, or any defect whatsoever, may more or less randomly transmit to their heirs the unpredictable consequences of the evil, or rather of the non-normal that they carry within them... It is an internal racism that permits the screening of every individual within a given society.
(Foucault 2003a, 316–17)
In its referencing of the biopolitical centrality of the normal, and of normalization, what is ultimately at issue in this proposal is the ways in which the abnormal is identified and managed, either through rehabilitation or elimination, in biopolitical apparatuses. If this is so, then the notion of racism that Foucault here proposes may also yield critical leverage for elucidating the ways in which disability has been conceptualized, treated and otherwise made subject to the operations of biopolitical normalization.
As Shelley Tremain (2005, 5) argues, the biopolitical apparatuses 'erected to secure the well-being of the general population' have brought the disabled subject into 'discourse and social existence'. This is because the 'practices, procedures, and policies [of biopower] have created, classified, codified, managed, and controlled social anomalies through which some people have been divided from others and objectivized as (for instance) physically impaired, insane, handicapped, mentally ill, retarded, and deaf' (Tremain 2005, 5–6). Tremain's point it that as an axis of subjectivity, disability as an identity category is constitutively related to biopolitical apparatuses of normalization and control. Further, true to Foucault's comment that sex may be the most ideal element in the deployment of sexuality in biopower, Tremain argues that we ought not think that disability is simply a discursive or cultural construct anchored in the biological reality of impairment. Rather, impairment is itself constituted through technologies such as prenatal testing, as the most ideal element in the discursive and institutional apparatuses of disability. As Tremain puts it, impairment is 'materialized' through the reiteration of culturally specific norms of human morphological function and structure, and then 'naturalized as an interior identity or essence on which culture acts' (Tremain 2006, 39). The upshot of Tremain's perspective is that the designation of some bodies as disabled or impaired is so imbued with the precepts of biopolitical apparatuses of normalization that disability is virtually unthinkable apart from these.
Two recent interventions by leading scholars in disability studies seek to further elucidate and critique the position of disability in biopolitics, and apparatuses of normalization in particular. First, Lennard Davis (2013, 1) has argued that the 'mythos of the normal body' that 'created the conditions for the emergence and subjection of the disabled body, the raced body, the gendered body, the classed body, the geriatric body – and so on' is today being largely outpaced by a neoliberal apparatus of diversity. Even so, he goes on, diversity is nevertheless reliant on the exclusion of groups of people construed as 'abject or hypermarginalized' (Davis 2013, 4). For him, disabled subjectivity is excluded from diversity on this basis, and the pertinent question to ask, then, is whether 'diversity can ever encompass disability' (Davis 2013, 5). Davis ultimately concludes that as disability operates as a kind of constitutive outside to diversity, it cannot be encompassed within its terms. Somewhat contrary to this, David Mitchell and Sharon Snyder (2015) have argued that disabled subjectivities are increasingly integrated within neoliberal biopolitics insofar as formerly stigmatized groups come to 'approximate historically specific expectations of normality' (Mitchell and Snyder 2015, 2). Yet, this neoliberal inclusion comes at a cost since it tends to 'reify the value of normative modes of being', particularly in terms of ablebodiedness and heteronormativity. For them, the critical task of disability studies is not simply to oppose the imposition of barriers and devices of exclusion, but to bring forth the ways that 'disability subjectivities are not just characterized by socially imposed restrictions, but, in fact, productively create new forms of embodied knowledge and collective consciousness' (Mitchell and Snyder 2015, 2). This, they argue, allows for a more radical challenge to the contemporary neoliberal and biopolitical order of 'able-nationalism'.
Conclusion
In this chapter, I have explored the thematic of subjectivity, particularly through the axes of race and sex/gender, and to a lesser extent, disability. The broad argument that I advance in this chapter is that processes of subjectivation are central to the biopolitical management of life and its differential valuation; indeed, it might be that processes of subjectivation act as a principle means by which life is brought into politics. In examining the ways in which subjectivity has been thought about within the work of the key biopolitical theorists of Agamben and Foucault, I show that the former of these addresses subjectivity through the question of the accession to language. One implication of this is that Agamben offers very few conceptual tools for the analysis of how subjectivities are produced and governed within liberal and neoliberal biopolitics. Foucault's account of subjectivity is well known, and I suggest here that it remains productive for considering the ways in which axes of subjectivation such as race, sex/gender and disability operate in biopolitics, even if Foucault did not address these topics himself or did so only schematically. This productivity is demonstrated by the numerous theorists who have subsequently used his account of subjectivity and its embeddedness in power/knowledge – which does not mean its determination by power/knowledge – in thinking about race, sex/gender and disability.
The bulk of the chapter is spent considering these axes of subjectivation. Of race, I look to recent work by Weheliye and others, while in the section on sex/gender I look in most detail at the recent Darwinian explorations of Elizabeth Grosz. I close this section with brief comments on the biopolitics of disability. One point that I have made but wish to emphasize here is that race, sex/gender and disability are not separate, but are intermingled and co-constitutive in various ways. In this, we can speculate that as well as being part of the apparatus for constructing sexual difference, the notion of sexual complementarity is also racialized. Similarly, while I have not discussed sexuality in this chapter, we must also recognize the ways this is also imbricated with race, sex/gender and disability. This is made clear in the exemplary studies of Jasbir K. Puar (2007) and Robert McRuer (2006) for instance. The broad thesis underlying this chapter is that subjectivity is the battleground of biopolitics. If this thesis is correct, then it implies that what is often at issue in biopolitics is not the reduction of the subject to biological life, but the suspension of biological existence in pursuit of specific forms of subjectivity. Further, it suggests that the possibilities for living available in contemporary biopolitical formations might also operate as central points of resistance to the biopolitical management and regulation of individuals and populations.
Notes
In the deeply problematic sentence, 'when crushed into its racial dimension, the body has been the object of an exclusion taken to the extreme of annihilation' (Esposito 2015, 13).
In commenting on the view that husbands could appropriate their wives as objects by virtue of their sexual organs (Esposito 2015, 47).
In addition, there are a number of insightful analyses available of racialized political phenomena such as slavery, immigration and statelessness that utilize and develop a biopolitical framework (e.g. Yeng 2015). While I am unable to discuss it here, this literature makes it clear that race is central, in multiple ways, to biopolitics.
Grosz clarifies later on the same page that this 'irreducible binarism... itself generates endless variety on either side of its bifurcation, and indeed produces variations – the intersexes – that lie between bifurcated categories' (Grosz 2004, 67).
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Heyes, C. J. (2007). Self-Transformations: Foucault, Ethics, and Normalized Bodies. Oxford, Oxford University Press.
Lettow, S. (2014). Generation, Genealogy and Time: The Concept of Reproduction From Histoire Naturelle to Naturephilosophie. Reproduction, Race and Gender in Philosophy and the Early Life Sciences. Ed. Lettow, S. Albany, State University of New York: 21–44.
McRuer, R. (2006). Crip Theory: Cultural Signs of Queerness and Disability. New York, New York University Press.
McWhorter, L. (2004). 'Sex, Race and Biopower: A Foucauldian Genealogy'. Hypatia 19(3): 38–62.
McWhorter, L. (2009). Racism and Sexual Oppression in Anglo America: A Genealogy. Bloomington, Indiana University Press.
Mbembe, A. (2003). 'Necropolitics'. Public Culture 15(1): 11–40.
Mills, C. (2011). 'The Political Economy of Personhood'. Available at: <http://nationalhumanitiescenter.org/on-the-human/2011/04/political-economy-of-personhood/>. Accessed 24 April 2017.
Mitchell, D. T. and Snyder, S. L. (2015). The Biopolitics of Disability: Neoliberalism, Ablenationalism and Peripheral Embodiment. Ann Arbor, University of Michigan Press.
Motha, S. (2012). Colonial Sovereignty, Forms of Life and Liminal Beings in South Africa. Agamben and Colonialism. Eds. Svirsky, M. and Bignall, S. Edinburgh, Edinburgh University Press: 128–151.
Puar, J. K. (2007). Terrorist Assemblages: Homonationalism in Queer Times. Durham, Duke University Press.
Repo, J. (2016). The Biopolitics of Gender. Oxford, Oxford University Press.
Stoler, A. L. (1995). Race and the Education of Desire: Foucault's History of Sexuality and the Colonial Order of Things. Durham and London, Duke University Press.
Tremain, S., Ed. (2005). Foucault and the Government of Disability. University of Michigan Press.
Tremain, S. (2006). 'Reproductive Freedom, Self-Regulation and the Government of Impairment in Utero'. Hypatia 21(1): 35–53.
Weheliye, A. G. (2014). Habeas Viscus: Racializing Assemblages, Biopolitics and Black Feminist Theories of the Human. Durham, Duke.
Yeng, S. (2015). The Biopolitics of Race: State Racism and U.S. Immigration. Lanham, Lexington Books.
Ziarek, E. (2012). Feminist Aesthetics and the Politics of Modernism. New York, Columbia University Press.
Concluding remarks
In this book, I have provided an overview of key theoretical contributions and traced the broad contours of various current debates and topics for further consideration in biopolitical studies. The book proceeded in two parts; in the first, I looked closely at the theorizations of biopolitics provided by key figures, primarily Foucault, Agamben, Arendt and Esposito, and also included a brief discussion of Hardt and Negri. Today, it could almost be said that the key line of differentiation between scholars of biopolitics is whether their intellectual sympathies mesh with the genealogical approach to biopower initiated by Foucault, or the ontologically focused 'paradigmatic' approach of Agamben. As we saw, Agamben casts his reflections on biopolitics as an attempt to correct or complete the previous analyses of Foucault (Agamben 1998, 9), but in fact his prior philosophical commitments ensure that his own account significantly differs from Foucault's. By this point, it will probably be evident that my own sympathies lie more with Foucault's genealogy than with the Agambenian framework of sovereign exceptionalism and bare life. In my own work on reproductive technologies, I have found Foucault's texts productive, and his concepts sufficiently analytically flexible to yield new insights about phenomena that he could not have discussed himself (such as prenatal testing technologies). This is not to say that his work is beyond critique; far from it. But the central aphoristic characterization of biopower as a power that aims to 'foster life or disallow it to the point of death' (Foucault 1990, 138), and the analytic focus on the integration of knowledge as styles of thinking with the institutions and apparatuses of power seems to me to provide an unparalleled conceptual and methodological framework for the study of biopolitics.
In the second part of the book, I sought to address a number of what seem to me to be either key issues in current debates, and/or topics that require further examination if the interdisciplinary field of biopolitical studies is to continue to flourish. In brief, these topics roughly fall under the headings of 'Politics', 'Life' and 'Subjectivity'. The broad argument that I made across these sections is that subjectivity, or more accurately, subjectivation, is the mechanism for the integration of life and politics in the biopolitical nexus. This means that the constitution of styles of subjectivity, including the means for our own self-understanding, are significant mechanisms of biopolitical governance, though not the only ones. Within this broad frame, I considered topics such as the debate around the perceived necessity for a conception of life as immanent to itself, indicated in Agamben's notion of 'form-of-life' and Esposito's turn to vital normativity. I also discussed more empirically driven theses that the contemporary biosciences and biomedicine are themselves producing new concepts of life and making new forms of subjectivity possible, and indeed, mandatory. These ideas, developed most prominently by Nikolas Rose and Paul Rabinow, also bring to our attention the relative lack of discussion of technology within biopolitical studies. Yet, given the technological saturation of contemporary existence, at least in advanced capitalist societies, this occludes the multiple ways in which technology is itself integral to the formation of subjectivity and forms of life. Further, this leaves undisclosed the ways in which technologies have also been indispensably integrated with biopolitics, such that technological developments have at times made biopolitical configurations possible, and vice versa.
One area of contemporary life in which this integration comes to the fore is that of the medical, social and political management of human reproduction. In fact, while I addressed matters of reproduction explicitly in only one section of the book, this yields one driving argument of the book. This is that while virtually all of the contemporary theorists of biopolitics have failed to take much notice of questions about reproduction, it is in fact a key problem of biopolitical governance. It is possible to get some guidance on this from Foucault, especially in his considerations on the emergence of modern biology and the concept of heredity, as well as his later reflections on the deployment of sexuality. But, in Agamben the problem of reproduction and the related question of sexual difference is systematically occluded. Esposito is more cognizant of the biopolitical implications of reproduction, insofar as he sees the 'suppression of birth' as one of three key biopolitical dispositifs for instance, but his thinking remains tied to problematic tropes.
In arguing that reproduction is a key biopolitical problem, I do not only mean that contemporary configurations of the governance and economy of reproduction are themselves biopolitical. They may well be, but the point is deeper. For in my view, unless the theorization of biopolitics and its emergence can come to see how the long-standing exclusion of the generative bodies of women plays a constitutive role in the tradition of Western political and philosophical thinking, then it will fail to break from this phallic logic. Further, in remaining so tied, it will ensure that the 'crossroads of biopolitics and patriarchy', will remain undisclosed.
Fortunately, some scholars who engage with the concept of biopolitics are beginning to theorize more fully the central role of reproduction and the related alignment of women with procreative capacity. As we saw, Hannah Arendt's work can contribute to this insofar as it reorients political philosophy away from death and toward natality. Even so, Arendt was no feminist and her conception of natality remains significantly ambiguous in its obfuscation of the female body that gives birth. Beyond this, feminist scholars such as Penelope Deutscher (2017) are leading the way in re-interpreting Foucault's work with the aim of highlighting his thinking about procreation and reproduction. As this work advances, the onus increasingly lies with (bio)political theorists to seriously engage with it.
Importantly, putting emphasis on the role of reproduction in biopolitics does not mean that biopolitics is solely life-fostering. Rather, I have argued throughout that the life-fostering and life-denying aspects of biopolitics are deeply integrated; consequently, conceptions of biopolitics that focus only on one aspect of this dynamic end up being misleading. This double logic means that a focus on the generative female body also requires examination of the ways that women have been associated with a principle of life by virtue of their procreative body, which also positions them as bringers of death (Deutscher 2017). Even so, this imbrication of life and death does not necessarily give credence to Agamben's conception of bare life, understood most strictly as 'life exposed to death' either; rather, it requires examination of the ways in which any given situation the fostering of life operates in tension or tandem with the life-denying logics of thanatopolitical or necropolitical violence. Thus, if there is an exemplary body of biopolitics today (itself a contestable claim), it is not the Muselmann as Agamben suggests, or even the over-comatose (Agamben 1998, 186), but cases of posthumous pregnancy. In such cases, a pregnant woman who suffers severe brain damage or brain death may be ventilated for weeks to ensure that the foetus she carries reaches a gestational age at which they may be 'viable' after extraction from her body. In this, the woman is literally rendered as an incubator, with her body kept in a state of the functional simulation of life in order to yield up another life. Further, after foetal extraction by caesarean section, her organs may be extracted for transplant in order to save the life of another who may otherwise die. Thus, we have here a female body doubly mined for its generative capacity to give life, even in and after death. So, this focus on the generative female body would not be a celebration, but a much-needed critical examination. Further, the examination of the generative female body as a particularly intensive site of the intermingling of the principles of life and death may also yield critical space for dislodging their fixity and primacy in the political-philosophical imaginary.
This brings us to the question of whether biopolitics is something to be overcome, or whether we are already witnessing the demise of biopolitical configurations and the emergence of new forms of social and political existence that cannot easily be elucidated within this conceptual framework. Agamben, Esposito, and to some extent Hardt and Negri, have all argued for the more or less urgent need for the formation of new concepts of life and of new forms of living. In contrast to this, other scholars are already using the terminology of 'beyond biopolitics', with this giving title to books and conferences.
However, it is not especially clear just what is at stake in this phrase, and several inflections seem probable. First, it may be said to indicate the political and social need to overcome biopolitics in the manner claimed by the theorists mentioned above. To a large extent, this view relies on biopolitics being conceived as a unitary historical phenomena, with its constitutive logic underpinning social and political systems as a whole. In short, it is a totalizing view of biopolitics as driven by a single political-philosophical logic that is overwhelmingly violent or oppressive or exclusionary and so on – and hence the need to overcome it. Second, the formula of 'beyond biopolitics' might also mean that the intermeshing of biology and politics that the term 'biopolitics' highlights has itself been succeeded by new configurations, such that we are no longer living, or will soon no longer be living, in an era of biopolitics. This raises questions about the historicity and temporality of biopolitcs. As we saw, there is considerable dissension over the origins and duration of biopolitics as a political rationality, with some seeing biopolitics as a specifically modern phenomenon, insofar as it is tied to Enlightenment styles of thinking about the world and the place of the human in it. Others find its origin in ancient Greece and early Christianity. But, either way, as an historical phenomenon, it is possible that we are now witnessing its passing, either in part – where biopolitical phenomena might be mixed with those that are not obviously biopolitical – or insofar as it is succeeded by a new regime of power.
Finally, related to this, the formula might be meant to pick out the limits of the concept itself, and the intellectual or analytic need to find new concepts to understand the emerging political configurations of the contemporary world, or indeed, the longstanding ones. This inflection goes to the issue of the 'truth content' of the concept, as well as to its analytic perspicacity. To be sure, there are a number of critiques of the concept from this perspective. At the same time, though, there are an abundance of studies that continue to mobilize the concept, presumably because it reveals certain aspects of phenomena and circumstance that are not otherwise made evident. At the same time, the concept and theoretical frameworks of biopolitics are being stretched in different ways. For instance, there are important studies that extend the concept into non-Western contexts, as well as to non-human ones. In my view, this demonstrates that the concept remains valuable as an analytic tool. However, continued conceptual reflection is required in order to prevent the concept from submerging under the weight of its own ambiguity.
References
Agamben, G. (1998). Homo Sacer: Sovereign Power and Bare Life. Trans. Heller-Roazen, D. Stanford, Stanford University Press.
Deutscher, P. (2017). Foucault's Futures: A Critique of Reproductive Reason. New York, Columbia University Press.
Foucault, M. (1990). The History of Sexuality: An Introduction, Volume 1. Trans. Hurley, R. New York, Vintage Books.
Index
abandonment –, , , , , , , ,
Agamben, Giorgio: bare life , , , –, , , , ; exception , , –, , ; form-of-life –, , , , , , –, , ; glory –; homo sacer –, –, , ; oikonomia, , , , , , ; paradigms as a method , , , , , , , ; testimony –,
Arendt, Hannah: action –, , , , , ; bureaucracy ; labor –, ; natality , , , –, –, , , , , , , ; work –,
Aristotle , , , , , , , , ,
Benjamin, Walter , –, , ,
biocapitalism see capitalism
biology , , ; biocracy , , ; birth , , –, ; freedom and –; intersection with politics , , –, , , ; life as scientific object , , , , –, , ; meaning natural characteristic (e.g., 'zoē') , , , , , , , –, , , , , , , , , , ; population , –, –, ; scientific discipline of , –, , , , , –, , , , ; socio-biology , , ;
biopower (distinguished from biopolitics) , –,
bioscience (including biomedicine and biotechnology) , , , , , , –, ,
biosociality , ; see also Rabinow, Paul
birth see pregnancy and Arendt
Bisclavret –
Butler, Judith , , –
Canguilhem, Georges , , –, , –
capitalism , –, –,
Cavarero, Adriana , , , , –, –
choice ,
citizenship: bare life , ; biological citizenship , ; nationality ; rights , , –,
colonialism , , , , , , , , –
concentration camps: Agamben on –, –, , ; Arendt on –, –, , ; Esposito on , ; and ethical responsibility –, ; as an experiment –, –; as nomos , ,
Darwin, Charles , , , , , –,
death: as aspect of immunitary paradigm –, –, ; as central feature of politics , , , , , , , ; certificates ; in defining life ; as 'letting die' –, , , , , ; meaning of in camps –, –; and medicine –, , , ; ; as related to bare life , , , ; as related to racism –; , , , –, , , , ; sovereign right of death –, , , , ; in warfare –,
Deleuze, Gilles , , , , –, ,
Derrida, Jacques , , , ,
disability , , , , –
dispositifs: Foucault on –, ; of gender –; in immunitary paradigm , , –; of person –; of race
Esposito, Roberto: on auto-immunity –; immune tolerance –; immunitary paradigm –; maternal immunity –; personhood , –
ethics: Agamben on –, ; Foucault on –
eugenics , , ; Esposito on , ; flexible ; Foucault on –,
flesh –,
Foucault, Michel: archaeology , , , ; disciplinary power , –,, , –, , –, , , ; genealogy , –, , , , ; liberalism –, , ; normalization –, , , –, ; population , , –, , ; power, analytics of , –
freedom: Arendt on , , –, ; Foucault on , –, , , –; paradox of ,
gender see sex/gender
genetics , , , , , , , , , , ,
genocide , , ,
government: Agamben on , , –, ; Foucault on , , –, , , , ; political economy , ; political institution of , , , , ; replacing biopolitics , ; rights ; technology and ; utility ; violence ,
Grosz, Elizabeth , –
Guantanamo , ,
Hardt, Michael see Negri
Heidegger, Martin , , , , ,
heredity , , , , , , ,
Hobbes, Thomas , , , , , ,
Holocaust , , , ,
immanence: life –, –, ; multitude , ; norms –; oikonomia, , ; power
imperialism: Arendt on –, , , ; Hardt and Negri on , , ; see also Colonialism
Lemke, Thomas , ,
liberalism: biopolitics and , , , –, ; Chicago School , ; diversity ; neoliberalism , , , , , , ; Ordoliberalism , ; tolerance ; 'way of war' –
life: different meanings –; human life , , , , ; impersonal , , ; new concept of –, ; scientific object , –, , , , , , –, , ; valuation of ; work of art –; see also bare life, form-of-life
McWhorter, Ladelle ,
Mbembe, Achille ,
muselmann–, , ,
Nancy, Jean-Luc , , ,
natality: Arendt on , –, , –, ; political significance of , , , , , , , , , , ; Vatter on , –,
nation–state , , , ; Agamben on ; Arendt on , , –, , , , ; Esposito on ; Negri on , –, ,
Nazism ; Agamben on , , , , ; Arendt on –, , –, , , , , ; Esposito on , –, , , ; Foucault on –; Negri on
necropolitics –,
Negri, Antonio: biopolitical production , –, ; Empire , –; multitude , , –,
neoliberalism see liberalism
norms , , , , ; abnormality , , , ; heteronormativity , ; law and ; normalization , , –, , , , , , , , ; vital norms , , –, –, ,
personhood , –, , , , , , –
political economy , –, , , ,
power see Foucault, biopower, government, sovereignty
pregnancy: abortion , , ; maternal body , , , , , –, , ; placenta ; posthumous ; sterilization , , , ; surrogacy , ; technologies of , , ; see also birth
Rabinow, Paul , , , , , , , , , –, ,
race , –,
racism , , –; Arendt on , , , –, , ; Esposito on ; Foucault on , –, , , –, –,
rape , , ,
refugees , , , –
reproduction –; see also pregnancy
rights , , , –, , ; Agamben on , , –; Arendt on , –, , , , –; Butler on ; Esposito on ; Foucault on , , , , , , , ; human rights –, , , , –, ; right to die , ; Rose, Nikolas , , , –, , , , , –, ,
Schmitt, Carl , , –
security , , , , , , –,
sex/gender , , , , , , , , –
sexual difference , , , , , , –,
sexuality –, , , , , , , , ,
Simondon, Gilbert , ,
slavery , , , , ,
sovereignty –, , , ; Agamben on –, , , –, , –, , ; Arendt on ; Esposito on –, , , ; Foucault on , –, , , , , –, , , ; Negri on –
Stalinism , , , , ,
statelessness see refugees
statistics , , , , , , ,
Stoler, Ann , ,
subjectivity , , , , –; Agamben on , , , –; biopolitical importance of , , , –, –, –; Butler on ; Foucault on , –, –; Negri on , ,
technology , , , , –, ; biotechnology , , , –, , ; forms of life and subjectivity –, , –, ; life support –, ; transplant , ,
territory , ,
totalitarianism: Arendt on , , –, –, –, , , ; Negri on ,
truth: Foucault on , , , , , ; Rabinow and/or Rose on , , ,
Vatter, Miguel –,
violence , , , –, –, ; abandonment to , , , , ; care as opposite of ; contra power –; democratic state , , , ; divine , , ; horror of ; legal violence , –, , , , ; racism and racial , –; sovereign , , , , , , , , ; thanatopolitical , ; see also colonialism, rape, slavery, war
Waldby, Catherine ,
war , –, , ; Boer War , , ; Clausewitz ; First World War –, , , ; liberalism and –; metaphor of , ; race war ; Second World War , , , , , ; warfare –, ; War on Terror
Weheliye, Alexander
1. Cover
2. Half-Title
3. Title
4. Copyright
5. Contents
6. Acknowledgements
7. Introduction
1. PART I
1. 1 A new regime of power: Foucault
2. 2 Biopolitics as thanatopolitics: Agamben
3. 3 Totalitarianism and the political animal: Arendt
4. 4 Affirmative biopolitics: Negri and Esposito
2. PART II
1. 5 Politics: Sovereignty, violence, rights
2. 6 Life: Biology, technology, reproduction
3. 7 Subjectivity: Persons, race, gender
8. Concluding remarks
9. Index
|
{
"redpajama_set_name": "RedPajamaBook"
}
| 8,003
|
The paint on the once-revered MV Lady Rose is peeling and her hull is rusting as she sits along a private dock in Tofino in May 2015. SUSAN QUINN PHOTO
Alberni man wants to resurrect iconic MV Lady Rose
Sailor Mike Wright sees tourism potential in rusting ship
Elena Rardon
The iconic MV Lady Rose could be making a return to the Alberni Valley after six long years.
Boater Mike Wright presented a plan to the Port Alberni Maritime Heritage Society on Wednesday, Sept. 13 to bring the vessel back to Port Alberni as a floating museum and focal point for tourism.
Originally christened "Lady Sylvia," the MV Lady Rose was built in Scotland and first launched in 1937. She was the first single-propeller diesel vessel ever to traverse the Atlantic under her own power. The vessel spent 70 years ferrying passengers and supplies from Port Alberni to Bamfield and Ucluelet, but had her route taken over by the MV Frances Barkley in 2007.
The boat has been sitting tied to a dock in Tofino since its last voyage down the Alberni Inlet in March of 2011. Current owner Jamie Bray planned to turn her into a floating restaurant in front of Jamie's Whaling Station, but Bray instead purchased the Rainforest Inn in late 2011. Since that time, the Lady Rose has seen significant rusting and wear. It no longer looks like the ship that used to run the Barkley Sound.
"It's pretty disappointing," Wright admitted. "But when you do a good clean-up, things start to look better."
Wright is offering to pay for some of the costs to bring the Lady Rose back to the Valley, and he is committed to a two-year business plan, but he is looking for some indication of support for the community.
"Community support is needed," he said on Wednesday. "I don't believe in going to the city for money."
Wright suggested that a dry land berth might be the best option for the vessel, which is no longer in any condition to travel. "There's a little cove behind the Port Boat House," he said.
This is where the Port Alberni Port Authority has tentatively decided to install an oil spill-response base if Kinder Morgan's Trans Mountain pipeline expansion project goes through, but Wright said he has received a letter of interest from PAPA.
Federal and provincial funding could be available for the Lady Rose as a heritage piece. Volunteer support would also be key. A Lady Rose Foundation could be started to turn it into something that pays for itself, and guided tours, corporate meetings and weddings on board could generate revenue.
Overall, Wright believes that the Lady Rose is an iconic piece of history that deserves more than to rust in the waters in Tofino.
Wright said he saw the Lady Rose during a recent trip to the west coast, and asked Bray how people in Port Alberni could get a hold of it. Bray offered to donate it.
"He wants to see it come here, too," said Wright.
Wright's enthusiasm was commended, but members of the heritage society brought up a number of concerns regarding asbestos content on the ship and the cost of having the vessel restored.
Alberni Valley Museum manager Jamie Morton said, "It's a real problem getting volunteers involved. The McLean Mill Society is having this problem, too."
Members of the heritage society pitched other possibilities for the Lady Rose, including a plan to beach the boat on Hohm Island, and another idea to make the vessel seaworthy again and send it back to Scotland in a documented process. But Wright was firm in his belief that he wants the ship to be "accessible" to the public.
The society agreed to consider Wright's plan and get back to him with their decision. In the meantime, Wright will do more research into feasibility and put some work into cleaning the vessel.
Wright said he felt fairly optimistic following his conversation with the heritage society.
"I think it was intelligent, careful and positive," said Wright. "A lot of it is about attitude."
elena.rardon@albernivalleynews.com
Mike Wright speaks to members of the Marine Heritage Society on Wednesday, Sept. 13 about his plan to bring back the MV Lady Rose. ELENA RARDON PHOTO
THE MV Lady Rose's rusting hull is reflected in the water of Jamie's Whaling Station's docks in Tofino in May 2015. SUSAN QUINN PHOTO
After 70 days, B.C. state of emergency to end
B.C. doctor's fentanyl-laced pot warning may be false alarm
|
{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
| 7,750
|
<?php
namespace Magento\Sendfriend\Block;
use Magento\TestFramework\Helper\Bootstrap;
class SendTest extends \PHPUnit_Framework_TestCase
{
/**
* @var \Magento\Sendfriend\Block\Send
*/
protected $_block;
protected function setUp()
{
$this->_block = Bootstrap::getObjectManager()->create('Magento\Sendfriend\Block\Send');
}
/**
* @param string $field
* @param string $value
* @dataProvider formDataProvider
* @covers \Magento\Sendfriend\Block\Send::getUserName
* @covers \Magento\Sendfriend\Block\Send::getEmail
*/
public function testGetCustomerFieldFromFormData($field, $value)
{
$formData = ['sender' => [$field => $value]];
$this->_block->setFormData($formData);
$this->assertEquals(trim($value), $this->_callBlockMethod($field));
}
/**
* @return array
*/
public function formDataProvider()
{
return [
['name', 'Customer Form Name'],
['email', 'customer_form_email@example.com']
];
}
/**
* @param string $field
* @param string $value
* @dataProvider customerSessionDataProvider
* @covers \Magento\Sendfriend\Block\Send::getUserName
* @covers \Magento\Sendfriend\Block\Send::getEmail
* @magentoDataFixture Magento/Customer/_files/customer.php
*/
public function testGetCustomerFieldFromSession($field, $value)
{
$logger = $this->getMock('Magento\Framework\Logger', [], [], '', false);
/** @var $session \Magento\Customer\Model\Session */
$session = Bootstrap::getObjectManager()->create('Magento\Customer\Model\Session', [$logger]);
/** @var \Magento\Customer\Api\AccountManagementInterface $service */
$service = Bootstrap::getObjectManager()->create('Magento\Customer\Api\AccountManagementInterface');
$customer = $service->authenticate('customer@example.com', 'password');
$session->setCustomerDataAsLoggedIn($customer);
$this->assertEquals($value, $this->_callBlockMethod($field));
}
/**
* @return array
*/
public function customerSessionDataProvider()
{
return [
['name', 'John Smith'],
['email', 'customer@example.com']
];
}
/**
* Call block method based on form field
*
* @param string $field
* @return null|string
*/
protected function _callBlockMethod($field)
{
switch ($field) {
case 'name':
return $this->_block->getUserName();
case 'email':
return $this->_block->getEmail();
default:
return null;
}
}
}
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 1,684
|
\section{Introduction}
The regime of Low Earth Orbits (LEO), \ie altitudes below \SI{1500}{\kilo\meter} \cite{Montenbruck.2001}, offers great possibilities for Earth observation, thermospheric investigations, and the global connectivity market. This draws the attention of various actors and leads to a sharp rise in satellite numbers.
Growing numbers of satellites (and debris) do not only pose a threat to functional satellites. Every collision produces a huge amount of debris objects, which in turn threaten other satellites. Analyses have shown that this could end in an avalanche-like process, referred to as the Kessler syndrome, rendering the LEO regime useless for future generations \cite{Kessler.1980,Kessler.1991,Kessler.2000}. This highlights the need to minimize collision risk. Besides active debris removal, an obvious option for functional satellites is the implementation of collision avoidance manoeuvres (CAMs) in case of a predicted close encounter with another object.
Typically, such manoeuvres are performed with impulsive thrusters to deflect the satellite trajectory. Satellites without thrusting capabilities, on the other hand, need other strategies to evade potential collisions. In LEO, aerodynamic drag due to the remaining atmospheric particles represents significant natural perturbing forces. There has been research on how to use them to control and manoeuvre asymmetrically-shaped satellites, \eg for satellite formation flight \cite{Traub.2020}.
Although achievable forces are magnitudes smaller than what is possible with chemical thrusters, satellite orbits can be measurably altered given enough time, which allows for the implementation of CAMs. Satellites without thrusters can, therefore, greatly benefit from CAMs using aerodynamic drag.\par
One such satellite is the \emph{Flying Laptop}\xspace of the Institute of Space Systems, University of Stuttgart. It orbits the Earth in a nearly circular polar orbit at an altitude of $~\SI{600}{\kilo\meter}$.
From its launch into LEO in July 2017 until September 2022, the \emph{Flying Laptop}\xspace{}'s operators received over 5000 warnings for over 150 close encounters from the Joint Space Operations Center (JSpOC), with the lowest predicted miss distance reported as \SI{29}{\meter}. Since the \emph{Flying Laptop}\xspace{} does not possess any thrusters, it has no possibility to perform impulsive collision avoidance manoeuvres. Thus, so far the operators had to remain without action upon reception of a collision warning. As previously pointed out, though, the orbital altitude and the asymmetric shape of the satellite, in principle, allow control via aerodynamic drag. The \emph{Flying Laptop}\xspace will be used as an example satellite in this work.
At first, fundamental concepts are covered. The developed analysis tool is then introduced in \cref{ch:methodology}. Lastly, results of exemplary analyses for the \emph{Flying Laptop}\xspace are presented and discussed in \cref{sec:results_discussion}. The research presented here is based on the corresponding author's master thesis \cite{Turco.2022}.
\section{Fundamentals}
\subsection{Satellite aerodynamics}
\label{sec:satellite_aerodynamics}
Satellites in LEO are subject to a measurable perturbing force due to the remaining atmospheric particles, which impinge on the satellite surface. The aerodynamic forces acting on a satellite can be separated into lift and drag. Drag is defined as the component acting anti-parallel to the satellite velocity relative to the local atmosphere, whereas lift acts in perpendicular direction to that. The specific drag force $ \v{f}_D $ depends on the satellite's dimensionless aerodynamic drag coefficient $ C_D $, the reference area $ A_{ref} $, the satellite mass $ m $, the atmospheric density $ \rho $ and the satellite's velocity relative to the local atmosphere $ \v{v}_{rel} $ \cite{Vallado}:
\begin{equation}
\v{f}_D = -\frac{1}{2} \rho \frac{C_D A_{ref}}{m} v_{rel}^2 \frac{\v{v}_{rel}}{v_{rel}}
\label{eq:drag}
\end{equation}
Neglecting atmospheric winds, the relative velocity equals the satellite's orbital velocity.
Lift coefficients are usually considerably smaller than the respective drag coefficients. Although aerodynamic lift can be used for satellite control under the right conditions \cite{Traub.2020, Traub.2022, Omar.2013}, perturbing lift forces are not further pursued here.\par
The ballistic coefficient $ \beta $ is a dimensional coefficient measuring to which extent a satellite is affected by aerodynamic drag. It is defined as \cite{Vallado}
\begin{equation}
\beta = \frac{m}{C_D A_{ref}}.
\end{equation}
Here, the inverse ballistic coefficient $\beta^*$ is used, since it shows a direct proportionality to the drag coefficient and the reference area:
\begin{equation}
\beta^* = \frac{C_D A_{ref}}{m}
\label{eq:ballistic_coeff}
\end{equation}
Satellites with higher inverse ballistic coefficients experience higher drag forces than such with low inverse ballistic coefficients.\par
In \cref{eq:drag}, atmospheric density and satellite velocity are not controllable and the satellite mass is constant for thrusterless satellites. If a satellite is asymmetrically-shaped control authority lies in the variation of the ballistic coefficient. By changing its attitude and thereby changing the effective cross-section and the drag coefficient, the experienced drag force can be regulated.\par
The atmospheric density may be obtained by employing one of various atmospheric models of differing complexity. The NRLSMSISE-00 model \cite{Picone.2002} considers time, location and solar as well as geomagnetic activity via the solar radio flux $ F_{10.7} $ and the 3-hourly planetary equivalent amplitude values $ a_p $ to deliver estimations for the atmospheric constitution, temperature and total mass density \cite{Vallado,Doornbos.2011}. The JB-2008 model \cite{Bowman.2008} utilizes further proxies for solar activity. Besides the observed solar radio flux at \SI{10.7}{\centi\meter} ($ F_{10.7} $), proxies have to be provided for \SIrange{26}{34}{\nano\meter} solar extreme ultraviolet (EUV) emission ($ S_{10} $), solar middle ultraviolet (MUV) radiation at \SI{280}{\nano\meter} ($ M_{10} $) and X-ray emission of the sun ($ Y_{10} $). The disturbance storm time ($ Dst $) is used as an index for geomagnetic activity.\par
The \citet{ISO1422} has defined different levels of solar and geomagnetic activity by specifying values for the various parameters, which can be useful to compare different scenarios of solar and geomagnetic activity. The respective indices can be found in \cref{tab:activity_levels}.
\subsection{Analytic estimation of achievable separation distance}
\label{sec:sep_distance}
\citet{Omar.2020} derived an analytic expression for the evolution of the difference in mean anomaly between two satellites with the same initial conditions but experiencing different drag forces. This allows for the computation of the influence of a change in ballistic coefficient by a potential manoeuvre on a satellite orbit.\par
The necessary assumptions include a circular orbit over a spherical Earth, a non-rotating atmosphere and negligible change in the semi-major axis during the manoeuvre.
The index $ r $ denotes an original reference orbit around Earth. A new perturbed orbit, indexed with $ n $, can be achieved by altering the ballistic coefficient of the satellite, \eg through a change in attitude. $ M_r $ and $ M_n $ further denote the respective mean anomalies of the orbits and
\begin{equation}
\phi(t) = M_r(t) - M_n(t)
\end{equation}
their difference.
The mean motion $ n $ of a satellite is the time derivative of its mean anomaly, which results from Earth's gravitational parameter $ \mu_E $ and the orbit semi-major axis $ a $, it follows that
\begin{equation}
\dot \phi_{rn} = n_r - n_n = \sqrt{\frac{\mu_E}{a_r^3}} - \sqrt{\frac{\mu_E}{a_n^3}}.
\end{equation}
Under the assumption that the change in semi-major axis during the manoeuvre is small compared to the original semi-major axis, \citet{Omar.2020} show that the first and second derivatives of the change in mean anomaly can be expressed as
\begin{align}
\dot \phi_{rn} &= \frac{3\delta \sqrt{\mu_E} }{2 a_r^\frac{5}{2}}\\
\ddot \phi_{rn} &= \frac{3 \sqrt{\mu_E} }{2 a_r^\frac{5}{2}} \dot{\Delta a}
\end{align}
where the change in semi-major axis is
\begin{equation}
\Delta a = a_r-a_n.
\end{equation}
The time derivative of the orbital energy $ \dot E $ is directly linked to the magnitude of aerodynamic drag $ f_{D} $ and the satellite velocity $ v $ via the work-energy theorem:
\begin{align}
E &= - \frac{\mu_E}{2a}\\
\dot E &= \frac{\mu_E}{2a^2} \dot a = f_{D} v = -\frac{1}{2} \beta^* \rho v^3
\label{eq:e_deriv}
\end{align}
Since the atmosphere is assumed to be non-rotating, the relative velocity equals the orbital velocity given by
\begin{equation}
v = \sqrt{\frac{\mu_E}{a}}.
\label{eq:orbital_velocity}
\end{equation}
Combining \cref{eq:e_deriv,eq:orbital_velocity} gives the time derivative of the semi-major axis as
\begin{equation}
\dot a = -\beta^* \rho \sqrt{\mu_E a}.
\end{equation}
Finally, this leads to the time derivative of the change in semi-major axis:
\begin{equation}
\Delta \dot a = \dot a_n - \dot a_r = - \rho \sqrt{\mu_E a} \left( \beta^*_n - \beta^*_r \right)
\end{equation}
Since $ \Delta a $ is small compared to the original semi-major axis, the semi-major axes can be assumed to be approximately equal $ a_r \approx a_n \approx a_0 $. Further, the density is assumed constant.
This yields an analytic expression for the second temporal derivative of $ \phi_{rn} $:
\begin{equation}
\ddot \phi_{rn} = -\frac{3\rho \mu_E}{2a_0^2} \left( \beta^*_n - \beta^*_r \right) = -\frac{3\rho \mu_E}{2a_0^2} \Delta \beta^*
\end{equation}
This can be expressed from the perspective of the reference trajectory as
\begin{equation}
\ddot \phi= \frac{3\rho \mu_E}{2a_0^2} \left( \beta^*_n - \beta^*_r \right).
\label{eq:omar_diff}
\end{equation}
By integration over time, one can deduce the built-up angular difference $ \phi$ between both trajectories. This angular separation can be converted into an in-track separation distance $ \Delta x $:
\begin{equation}
\Delta x = \phi a_0
\label{eq:omar_dx}
\end{equation}\par
An increase in ballistic coefficient ($\beta^*_n > \beta^*_r$) for a certain time leads to the satellite advancing in direction of flight from its original trajectory. This is schematically shown in \cref{fig:manoeuvre_schematic}. The satellite in off-nominal attitude has a higher ballistic coefficient, leading to a positive in-track separation distance compared to a flight in reference attitude at a later point in time $t_c$. Vice-versa, for a manoeuvre involving an attitude with a smaller-than-nominal ballistic coefficient the separation distance is negative.
\begin{figure*}[h]
\centering
\includegraphics{manoeuvre_schematic.pdf}
\caption{Concept of the collision avoidance manoeuvre using aerodynamic drag. The off-nominal attitude has an increased ballistic coefficient, leading to a positive in-track separation distance to the reference trajectory. It is to note, that the satellites started from equal initial conditions and the separation distance builds up over time.}
\label{fig:manoeuvre_schematic}
\end{figure*}
\subsection{Conjunctions}
\label{sec:conjunctions}
A conjunction is a close encounter between a satellite and another object, posing an inherent threat because of the risk of a potential collision. This section describes how conjunction warnings are issued and how collision probabilities can be calculated.\par
\subsubsection{Conjunction Data Messages}
\label{sec:CDMs}
JSpOC provides services for satellite operators regarding conjunction events \cite{NASA.2020,SafetyHandbook.2020}. A proprietary catalogue of satellites and debris objects is continuously screened for close encounters with a miss distance below a defined threshold. When such a close encounter is identified, more thorough orbit and uncertainty analyses are performed and a standardized Conjunction Data Message (CDM) is sent to the satellite operators \cite{CCSDSCDM.2013}.
Each CDM contains extensive information about the encounter, \ie meta and orbital data of the objects, the predicted time of closest approach, predicted state vectors of both objects as well as estimated uncertainties and the models used for orbit propagation and respective parameters. The stated ballistic coefficient is of importance because it will be used as a reference value later on.
\subsubsection{Calculation of the collision probability}
\label{sec:pc_calculation}
Predictions about conjunctions are fraught with uncertainties. Therefore, a probability of collision associated with a close encounter is used as a main metric for risk assessment. \citet{Foster.1992} developed a method to calculate a collision probability between two objects for a given conjunction geometry by reducing the three-dimensional problem to a two-dimensional one. Many different implementations based on this original method have been derived since then \cite{NASA.2020, Foster.1992, Klinkrad.2006, Alfano.2005b} and JSpOC uses the method as well \cite{JSPOC.2016,CovarianceRealism.2016}. The FOSTER-1992 method makes the following assumptions:
\begin{itemize}
\item The objects' relative velocity is so high that the encounter duration can be considered short and their motion can be assumed rectilinear, \ie with constant velocity and along a straight line, during the encounter.
\item The uncertainties in the position components each follow a Gaussian distribution.
\item Velocity uncertainties are neglected.
\item The uncertainties are constant during the encounter, their values are the predicted values at the time of closest approach.
\item The uncertainties in the primary and secondary object positions are uncorrelated and can thus be joined into the combined covariance by simple addition.
\item Each object is represented by a sphere fully enclosing it. Its radius is called the hard body radius.
\end{itemize}
\cref{fig:conjunction_geometry} depicts the geometry during a close encounter at the time of closest approach (TCA). The norm of the relative position vector between both objects at TCA $ \Delta\v{r}_{tca} = \v{r}_2 - \v{r}_1 $ is referred to as the miss distance. The plane normal to the relative velocity vector, which contains the relative position vector, is called the conjunction plane, or $ B $-plane. The relative position vector further defines the $ x $-axis of the conjunction frame.
\begin{figure}[h]
\centering
\includegraphics[width=\linewidth]{conjunction_geometry.png}
\caption{Illustration of the conjunction geometry at the time of closest approach with the conjunction plane in grey (adapted from \cite{Chen.2017}).}
\label{fig:conjunction_geometry}
\end{figure}
Uncertainties in the satellite's and secondary object's positions $ \m{C}_{i,RTN} $ are specified as covariance matrices in the respective RTN-frames (radial, transversal, normal) \cite{Vallado}. Such a covariance matrix is symmetric and describes a three-dimensional ellipsoid indicating the $ 1\sigma $-environment of the position. The elements on the main diagonal represent the covariances $ \sigma_R^2,\sigma_T^2, \sigma_N^2 $, \ie the respective radial, transversal and out-of-plane components of the position uncertainty. The principle axes of the $ 1\sigma $-ellipsoid are defined by the eigenvectors of the covariance matrix and might deviate from the RTN-axes due to off-diagonal elements not being equal to zero.\par
\citet{Foster.1992} transform the two objects' covariances to the ECEF-frame (Earth-centred Earth-fixed) and add them to form one combined covariance $ \m{C}_{tot,ECEF} $:
\begin{equation}
\m{C}_{tot,ECEF} = \m{C}_{1,ECEF} + \m{C}_{2,ECEF}
\end{equation}
The combined covariance is then assigned to the secondary object. On the other hand, the two objects' hard body radii are summed as well and assigned to the primary object, yielding the collision sphere with radius $ R_C $. Therefore, a collision occurs if the distance between both objects at $ t_{ca}\xspace $ is smaller than the radius of the collision sphere, \ie the collision sphere is intersected. Due to the assumption of rectilinear motion, it is possible to convert the three-dimensional problem to a two-dimensional one by projecting the combined covariance ellipsoid and the collision sphere onto the conjunction plane. The resulting covariance ellipse on the conjunction plane is characterized by the projected combined covariance matrix:
\begin{equation}
\m{C}_{tot,B} = \m{R}_{B,ECEF} \m{C}_{tot,ECEF} \transp{\m{R}_{B,ECEF}}
\end{equation}
where $\m{R}_{B,ECEF}$ is the transformation matrix from ECEF-frame to the conjunction plane.
The projected covariance $ \m{C}_{tot,B} $ represents a Gaussian probability distribution in the $ B $-plane. By integrating the probability density function over the collision area $ A_C $ (\ie the projected collision sphere), a probability of collision $ P_c $ can be estimated:
\begin{equation}
P_c = \frac{1}{2\pi \sqrt{\det(\m{C}_{tot,B})}} \int_{-R_C}^{R_C} \int_{-\sqrt{R_C^2-x_B^2}}^{\sqrt{R_C^2-x_B^2}} \exp(-A_B)\, \mathrm{d} y_B \, \mathrm{d} x_B
\label{eq:pc_foster}
\end{equation}
\begin{equation*}
\text{with:\quad} A_B = \frac12 \transp{\Delta \v{r}_B} \m{C}_{tot,B}^{-1} \Delta \v{r}_B
\end{equation*}
The collision probability is significantly influenced by shape and size of the covariance ellipsoid. \citet{Alfano.2005} developed a method to determine the maximum collision probability based on the aspect ratio $ \lambda = \frac{\sigma_{B,1}}{\sigma_{B,2}}$ between the semi-major and semi-minor axes of the projected covariance ellipse \cite{Alfano.2005,Mishne.2017}.
The maximum collision probability results from
\begin{equation}
P_{c,max,alf} = \frac{R_C^2}{2\lambda\sigma^2} \exp{\left( -\frac12 \left( \frac{\Delta r_{tca}}{\lambda\sigma} \right)^2 \right)}
\label{eq:pc_max_alfano}
\end{equation}
\begin{equation*}
\text{with\quad} \sigma=\frac{\Delta r_{tca}}{\sqrt{2}\lambda}.
\end{equation*}
This metric includes information about the eccentricity of the combined covariance matrix.
\section{Analysis tool}
\label{ch:methodology}
In the following section, the methodology behind the analysis tool for collision avoidance using aerodynamic drag is explained.
\subsection{The manoeuvre}
\label{sec:manoeuvre}
In \cref{sec:sep_distance}, an analytic equation was presented which allows the calculation of a relative in-track separation distance by the variation of a satellite's ballistic coefficient relative to a reference value. During conjunction screening and the orbit determination process, the JSpOC determines a mean ballistic coefficient from observations, which is used for propagation as well as conjunction analysis and will be considered the reference value. By deliberately taking an (off-nominal) attitude with a ballistic coefficient different to the reference value until the predicted TCA, a separation distance can be built up relative to JSpOC's predicted reference trajectory.
For satellites changing their ballistic coefficient via changes in attitude, manoeuvring constraints might be necessary. A common example are attitudes which cause the solar arrays to point away from the Sun and thus decrease the power generated by them. Such attitudes can only be held as long as battery power is sufficient. Other constraints might include pointing requirements imposed by mission objectives etc.
To be able to consider these constraints in the manoeuvre analysis, the manoeuvring time is divided into sections. Each section consists of two parts of specified duration $ t_1 $ and $ t_2 $, respectively. During the first part, the commanded manoeuvring attitude is attained. During the second part, a predefined attitude is implemented to meet the constraints.\par
\cref{fig:manoeuvre_timeline2} shows an exemplary timeline of such a manoeuvre and the respective courses of the angular separation $ \phi $ as well as its first and second derivatives. While the second derivative is directly proportional to the change in inverse ballistic coefficient and therefore constant in phases with constant attitude, the first derivative is a piece-wise linear function representing an angular motion relative to the reference trajectory.
In this example, the inverse ballistic coefficient in the commanded manoeuvring attitude is higher than the reference value. Therefore, the angular separation $ \phi $ grows quadratically while the satellite is in manoeuvring attitude. During the constrained charging phases, the inverse ballistic coefficient is $ \beta^*_{constr}<\beta^*_{ref} $, here. The negative value of the change in inverse ballistic coefficient and hence $ \ddot \phi $ causes the first derivative of the angular separation to decrease. $ \phi $ itself still increases, as long as the relative angular motion $ \dot \phi > \SI{0}{\per \second} $. The separation distance is directly proportional to $ \phi $ and therefore shows the same behaviour. The respective courses vary depending on the actual ballistic coefficients in the different phases.
This already shows that a manoeuvre is more effective the earlier it is implemented. Due to the build-up of relative angular motion $ \dot \phi $, the angular separation can be increased despite negative values of $ \ddot \phi $ in later phases. These might be caused by phases with an inverse ballistic coefficient lower than the reference value. In case of a return to the reference attitude, in which $ \ddot \phi = \SI{0}{\per \square\second} $, $ \phi $ will still grow linearly over time because of a constant relative angular motion.
\begin{figure*}[htb!]
\centering
\includegraphics{manoeuvre_timeline.pdf}
\caption{Exemplary consideration of operational constraints. Considered is a manoeuvre with a commanded inverse ballistic coefficient $ >\beta^*_{ref} $ and additional phases (grey), in which constraints dictate a flight with $ <\beta^*_{ref} $. The respective evolution of the angular separation $ \phi $ between the reference and the perturbed trajectory as well as its derivatives are shown.}
\label{fig:manoeuvre_timeline2}
\end{figure*}
\subsection{The tool}
\label{sec:tool}
The tool is based upon \cref{eq:omar_diff} by \citet{Omar.2020}. \cref{fig:tool_flowchart} shows its basic concept. Using only a CDM and pointing constraints as user inputs, the influence of a potential manoeuvre on the collision probability can be analysed.\par
The tool is programmed in \code{python} and can be operated via a Jupyter Notebook. The tool makes use of the \code{pyatmos} package for atmosphere modelling \cite{pyatmos.2021}, which has been slightly adapted. Sgp4-propagation algorithms are implemented with a package developed at IRS, which itself is based on the \code{skyfield} package \cite{skyfield.2022}.
\begin{figure}[H]
\centering
\includegraphics{tool_flowchart.pdf}
\caption{Flowchart of the developed analysis tool.}
\label{fig:tool_flowchart}
\end{figure}
\subsubsection{Loading input data}
A CDM is issued by JSpOC when a close encounter is detected and contains all the necessary information about a conjunction event. It serves as primary input for the tool. Besides the predicted conjunction geometry and the associated uncertainties, especially the satellite's reference inverse ballistic coefficient $ \beta^*_{ref} $ is of interest to examine a potential avoidance manoeuvre.
For the propagation of the satellite position in the next step, a current two-line element set (TLE) is loaded. This is also used to provide the satellite's semi-major axis $ a_0 $. The TLE can be loaded from \url{space-track.org} \cite{space-track}, where the 18th Space Defense Squadron publishes TLEs obtained through own observations.
\subsubsection{Density estimation}
The achievable separation distance during a specified interval depends on the average density the satellite will encounter. To estimate the average atmospheric density during the manoeuvre, an atmospheric model considering is evaluated at a number of points along the reference trajectory. The previously obtained TLE is propagated for this using the sgp4-algorithm.
The deviation in position due to the manoeuvre is considered negligible here, since it is small and only occurs in the horizontal plane and thus only minimally affects the density output.
The averaged density $ \bar \rho $ then results from the evaluation of the chosen atmospheric model at $ n $ evenly-distributed time steps $ t_i $:
\begin{equation}
\bar \rho = \frac1n \sum_{i=1}^{n} \rho(\v{r}_i, t_i, \v{K}_i)
\end{equation}
where $ \v{r}_i $ denotes the satellite's position and $ \v{K}_i $ the space-weather indices for the atmosphere model at time step $ t_i $.
The NRLMSISE-00 and JB-2008 density models are implemented and can be chosen for analysis.
The \code{pyatmos} package retrieves recent space-weather data from \citet{SET}'s website. The indices for solar activity are updated on a daily basis, whereas the geomagnetic index is updated every three hours.
\subsubsection{Calculation of achievable separation distance}
In principle, a manoeuvre can be performed by taking any specified attitude. The value for the respective inverse ballistic coefficient has to be provided.
For clarity, the manoeuvring inverse ballistic coefficient will be referred to as $ \beta^* $ in the following, the reference value as $ \beta^*_{ref} $ and the value for the constraint phases as $ \beta^*_{constr} $.
At the start of a manoeuvre, the separation distance as well as its derivative equal zero: $ \dot \phi(\SI{0}{\second})=\phi(\SI{0}{\second})=\SI{0}{\per \second} $. For section $ i $, the separation distance at the end of the phase can then be calculated as follows:
\begin{align}
\ddot \phi_{1,i} = &\frac{3 \bar \rho \mu_E}{2a_0^2} \left( \beta^* - \beta^*_{ref} \right)\\
\ddot \phi_{2,i} = &\frac{3\bar \rho \mu_E}{2a_0^2} \left( \beta^*_{constr} - \beta^*_{ref} \right)
\end{align}
The second derivative $ \ddot \phi $ of the angular difference between the mean anomaly of the reference and manoeuvring trajectory is constant within each of the two parts of the section.
The first derivative $ \dot \phi $ is a piece-wise linear function of gradient $ \ddot \phi $. It follows through integration over time:
\begin{align}
\begin{split}
\dot \phi_{1,i}(t) = &\ddot \phi_{1,i} \left[t-t_{end,i-1}\right]\\
&+ \dot \phi_{2,i-1}(t_{end,i-1})
\end{split}\\[2ex]
\begin{split}
\dot \phi_{2,i}(t) = &\ddot \phi_{2,i} \left[t-(t_{end,i-1}+t_1)\right]\\
&+ \dot \phi_{1,i}(t_{end,i-1}+t_1)
\end{split}
\end{align}
Finally, the angular separation of the mean anomalies can be obtained via another integration over time, yielding
\begin{align}
\begin{split}
\phi_{1,i}(t) = &\frac12 \ddot \phi_{1,i} \left[t-t_{end,i-1}\right]^2\\
&+ \dot \phi_{2,i-1}(t_{end,i-1}) \left[t-t_{end,i-1}\right]\\
&+ \phi_{2,i-1}(t_{end,i-1})
\end{split}\\[2ex]
\begin{split}
\phi_{2,i}(t) = &\frac12 \ddot \phi_{2,i} \left[t-(t_{end,i-1}+t_1)\right]^2\\
&+ \dot \phi_{1,i}(t_{end,i-1}+t_1) \left[t-(t_{end,i-1}+t_1)\right]\\
&+ \phi_{1,i}(t_{end,i-1}+t_1).
\end{split}
\end{align}
$ \phi_{2}(t_m) $ at the end of the last section is the desired separation, which can be translated into a separation distance via \cref{eq:omar_dx}.
If no constraint phases are considered, the resulting separation distance follows as:
\begin{align}
\Delta x = -\frac12 \ddot\phi t_m^2 = \frac{3\bar \rho \mu_E}{4a_0} \Delta \beta^* t_m^2
\label{eq:deltax_min}\\
\text{with\quad} \Delta \beta^* = \left( \beta^* - \beta^*_{ref} \right)
\end{align}
$ \Delta \beta^* $ is the change in inverse ballistic coefficient in the commanded attitude compared to the reference value.
\subsubsection{Error propagation}
The influence of uncertainties in the parameters of \cref{eq:deltax_min} may in a first step be evaluated by applying a Gaussian error propagation \cite{Papula.2011}. The standard deviations of the averaged density, the semi-major axis, the difference in inverse ballistic coefficient and the manoeuvring time are denoted as $ \sigma_{\bar \rho} $, $ \sigma_{a_0} $, $ \sigma_{\Delta \beta^*} $, and $ \sigma_{t_m} $, respectively. The resulting standard deviation of the separation distance can be calculated by
\begin{equation}
\begin{split}
\sigma_{\Delta x} =
&\sqrt{ \left(\frac{\partial \Delta x}{\partial \bar \rho}\sigma_{\bar \rho}\right)^2
+ \left(\frac{\partial \Delta x}{\partial a_0}\sigma_{a_0}\right)^2}\\
&\overline{+ \left(\frac{\partial \Delta x}{\partial \Delta \beta^*}\sigma_{\Delta \beta^*}\right)^2
+ \left(\frac{\partial \Delta x}{\partial t_m}\sigma_{t_m}\right)^2}.
\label{eq:error_propagation}
\end{split}
\end{equation}%
The partial derivatives with respect to one parameter are to be evaluated with the other parameters set to their nominal value. They are given by:
\begin{align}
\frac{\partial \Delta x}{\partial \bar \rho} &= \frac{3 \mu_E}{4a_0} \Delta \beta^* t_m^2 \label{eq:error_propagation_rho}\\
\frac{\partial \Delta x}{\partial a_0} &= -\frac{3\bar \rho \mu_E}{4a_0^2} \Delta \beta^* t_m^2 \label{eq:error_propagation_sma}\\
\frac{\partial \Delta x}{\partial \Delta \beta^*} &= \frac{3 \bar \rho \mu_E}{4a_0} t_m^2 \label{eq:error_propagation_CB}\\
\frac{\partial \Delta x}{\partial t_m} &= \frac{3 \bar \rho \mu_E}{2a_0} \Delta \beta^* t_m \label{eq:error_propagation_tc}
\end{align}
The uncertainties in the parameters may be described by an uncertainty level $ s \ge 0 $, so that $ \sigma_p=s_p p $ is the respective standard deviation of any parameter $ p $. It follows that uncertainties of a given uncertainty level in any of the first three parameters affect the standard deviation of the achievable separation distance in the same way. The manoeuvring time, however, has an impact that is higher by a factor of $ 2 $.
The uncertainty in the separation distance leads to an additional uncertainty $ \sigma_{\Delta x} $ of the satellite position in the direction of flight. With the assumption that the additional uncertainty is uncorrelated to the uncertainty stated in the CDM $ \sigma_T $, these two can be added:
\begin{equation}
\sigma_T' = \sigma_{T} + \sigma_{\Delta x}
\end{equation}
$ \sigma_T' $ is the updated transverse position uncertainty, which will be used to calculate the collision probability in the next step. The assumption of the uncertainties being uncorrelated is a first approach and needs to be further investigated in future work.
\subsubsection{Calculation of collision probability}
An avoidance manoeuvre changes the conjunction geometry. Therefore, a new point of closest approach has to be determined considering the perturbed trajectory of the satellite. Afterwards, the collision probability is calculated with the Foster-1992 method, as described in \cref{sec:pc_calculation}. All assumptions mentioned there apply.\par
The position of the manoeuvring satellite at the original TCA $ t_{tca} $ can be determined by regarding the built-up separation distance:
\begin{equation}
\v{r}_{1}' \left(t_{tca}\right) = \v{r}_{1,tca} + \Delta x \frac{\v{v}_{1,tca}}{\left| \v{v}_{1,tca} \right|}
\end{equation}
Based on this, a new time of closest approach $ t_{tca}' $ may be determined by finding the minimum distance between the objects over time:
\begin{equation}
\begin{split}
t_{tca}' = t_{tca}+\Delta t \\
\Rightarrow \left| \Delta \v{r} \right|
&= \left| \v{r}_2 - \v{r}_1' \right|\\
&= \left| \v{r}_{2,tca} + \Delta t \v{v}_{2,tca} - \v{r}_{1,tca}' - \Delta t \v{v}_{1,tca} \right|\\ &\text{\quad is minimum}
\end{split}
\end{equation}
The elements of the covariance matrix describing the position uncertainty of the satellite are updated according to
\begin{equation}
C_{RTN,1,{i,j}}' =
\begin{cases}
\sigma_T' = \left(\sigma_{T} + \sigma_{\Delta x}\right)^2 &, \ i=j=1\\
C_{RTN,1,{i,j}} &, \ \text{else}
\end{cases}.
\end{equation}
Together with the updated time and point of closest approach, a probability of collision may be calculated employing \cref{eq:pc_foster} for the different manoeuvring attitudes and depending on the manoeuvring time.
The tool's computed collision probabilities were verified using exemplary conjunctions from literature \cite{Klinkrad.2006,Chen.2017,CARA}. Notably, this is not the value determined by JSpOC, since they perform further scaling of the covariance matrices before determining the collision probability, resulting in different collision probabilities stated in the CDM \cite{JSPOC.2016, CovarianceRealism.2016}.
\section{Results}
\label{sec:results_discussion}
This chapter presents the results of analyses performed with the tool on exemplary close encounters of the \emph{Flying Laptop}\xspace. Achievable separation distances are investigated before the influence of parameter uncertainties on the collision probability is analysed. Solar and geomagnetic data for the different activity levels are defined as in \cref{tab:activity_levels}, the used atmosphere model is NRLMSISE-00. The aerodynamic data of the \emph{Flying Laptop}\xspace are defined in \cref{tab:aero_results} and determined via interpolation depending on the space weather data. Only the minimum and maximum drag attitudes of the \emph{Flying Laptop}\xspace are taken into account, since they maximize the difference in ballistic coefficient compared to the reference value thus also maximizing the achievable separation distance. For the constraint phases, the Nadir-pointing attitude is assumed.
\subsection{Achievable separation distance}
\label{sec:ana_achievable_separation_distance}
In this section, the maximum separation distance of the \emph{Flying Laptop}\xspace achievable via a change in ballistic coefficient is examined.
Its reference trajectory is based on a CDM received for a close encounter on April 7, 2022, while different levels of solar and geomagnetic activity are assumed. The reference inverse ballistic coefficient is $ \beta^*_{ref} = \SI{0.01794}{\square\meter\per\kilogram}$, which is the average of recent CDMs.
The density is evaluated and averaged over one orbital period of the reference orbit. The resulting values are
$ \bar \rho = \SI{1.158e-14}{\kilogram\per\cubic\meter} $ for low,
$ \bar \rho = \SI{1.650e-13}{\kilogram\per\cubic\meter} $ for moderate and
$ \bar \rho = \SI{1.020e-12}{\kilogram\per\cubic\meter} $ for high activity.
The tool is used to analytically estimate the achievable separation distance for a given manoeuvring time. The results are presented in \cref{fig:analysis_sep_dist}. The positive separation distances are established by a flight with increased inverse ballistic coefficient. Consequently, a decreased inverse ballistic coefficient allows for the build-up of negative separation distances. Further, the achievable separation distance grows linearly with increasing atmospheric density, superimposed on it is the smaller influence of the density on the ballistic coefficient (\cf \cref{eq:omar_dx}). For low solar and geomagnetic activity, the resulting low density causes the separation distance to turn out negligibly small at only $ \SI{1.465}{\kilo\meter} $ for a flight in maximum drag attitude and even smaller for a flight with minimum drag. At moderate activity, the separation distances are noticeably higher at $ \SI{19.35}{\kilo\meter} $ and $ \SI{-7.647}{\kilo\meter} $, respectively. For a high level of activity, the separation distances grow up to $ \SI{119.4}{\kilo\meter} $ and $ \SI{-46.81}{\kilo\meter} $.\par
\begin{figure*}[p!]
\centering
\includegraphics{sep_dist2.pdf}
\caption{Achievable separation distance of the \emph{Flying Laptop}\xspace for different levels of solar and geomagnetic activity.}
\label{fig:analysis_sep_dist}
\end{figure*}%
\subsection{Influence of manoeuvring constraints}
Next, the effect of additional constraints on the achievable separation distance shall be evaluated. The scenario is the flight in maximum drag attitude for the manoeuvre mentioned before at moderate solar and geomagnetic activity. Additionally, phases in which the \emph{Flying Laptop}\xspace points its solar panels to the Sun to re-charge is batteries are now taken into account. Therefore, the manoeuvring time is divided into individual sections of $ \SI{4}{\hour} $ duration consisting of two parts. In the first phase, the satellite takes the commanded attitude, while in the second an offset Nadir-pointing is performed, which results in an attitude suitable for charging. It results from a $ \SI{90}{\degree} $ rotation of the satellite around the direction of flight starting from the Nadir-pointing attitude. The duration of the two phases is varied while adding up to $ \SI{4}{\hour} $. \cref{fig:analysis_sep_dist_constrained} shows the resulting achievable separation distances depending on the available time. If no charging phase is carried out ($ \SI{4}{\hour} $/$ \SI{0}{\hour} $), the separation distance is equal to the one for moderate activity and maximum drag in \cref{fig:analysis_sep_dist}. For longer charging phases, the achievable separation distance decreases. It does, however, still develop monotonously. For a manoeuvre involving sections of $ \SI{1}{\hour} $ flight in maximum drag attitude followed by $ \SI{3}{\hour} $ of charging, the effects of both phases almost cancel out. If the charging phase is commanded even longer in relation to the first phase, the separation distance becomes negative, due to the ballistic coefficient in Nadir-pointing being smaller than the reference ballistic coefficient. Within the first $ \SI{12}{\hour} $, all manoeuvres' separation distances are below $ \SI{300}{\meter} $. After $ \SI{24}{\hour} $, the $ \SI{2}{\hour} $/$ \SI{2}{\hour} $ split manoeuvre achieves a separation distance of $ \sim \SI{300}{\meter} $. The quadratic course of the separation distance then leads to increasingly higher values.
\begin{figure*}[p!]
\centering
\includegraphics{sep_dist_constrained_zoomin.pdf}
\caption{Achievable separation distance of the \emph{Flying Laptop}\xspace for different constraints and moderate solar and geomagnetic activity. The constraints consider a manoeuvring and charging phase of variable duration, \eg $ \SI{3}{\hour} $/$ \SI{1}{\hour} $ is a flight in maximum drag attitude for $ \SI{3}{\hour} $, followed by a charging phase of $ \SI{1}{\hour} $.}
\label{fig:analysis_sep_dist_constrained}
\end{figure*}
\subsection{Influence of relative in-track position at TCA}
\label{sec:ana_conjunction_geometry}
To study the effects of the predicted conjunction geometry on possible manoeuvres, two past close encounters are compared in the following section. The first is the previous close encounter on April 7, 2022, (encounter A), while the second is a close encounter on March 30, 2022, (encounter B). The two conjunctions differ with regards to the in-track component of the predicted relative position, which is negative for encounter A and positive for encounter B. This is visible in \cref{fig:conjunction_geometries}.
A maximum manoeuvring time of five days is assumed for both scenarios. The resulting achievable separation distances are presented in \cref{fig:sep_distances}. For both encounters, the achievable separation distance in maximum drag attitude is higher.
\begin{figure}[h!]
\centering
\begin{subfigure}{\linewidth}
\centering
\includegraphics{conjunction_geometry_a.pdf}
\caption{Encounter A. The radial component of the relative position is $ \SI{0.1}{\meter} $.}
\label{fig:conjunction_geometries_a}
\end{subfigure}\\
\begin{subfigure}{\linewidth}
\centering
\includegraphics{conjunction_geometry_b.pdf}
\caption{Encounter B. The radial component of the relative position is $ \SI{-61}{\meter} $.}
\label{fig:conjunction_geometries_b}
\end{subfigure}
\caption{Predicted conjunction geometries at TCA projected onto the local horizontal plane of the \emph{Flying Laptop}\xspace. Velocities are scaled by a factor of $ 0.1 $.}
\label{fig:conjunction_geometries}
\end{figure}
\begin{figure}[h]
\centering
\begin{subfigure}{\linewidth}
\centering
\resizebox{\linewidth}{!}{\includegraphics{sep_dist_a.pdf}}
\caption{Encounter A.}
\label{fig:sep_distances_a}
\end{subfigure}\\
\begin{subfigure}{\linewidth}
\centering
\resizebox{\linewidth}{!}{\includegraphics{sep_dist_b.pdf}}
\caption{Encounter B.}
\label{fig:sep_distances_b}
\end{subfigure}
\caption{Achievable separation distances for both encounters depending on manoeuvring time.}
\label{fig:sep_distances}
\end{figure}
The miss distances at TCA shall be evaluated more carefully and are visualized in \cref{fig:miss_distances}. For the first encounter a flight in maximum drag attitude increases the miss distance monotonously up to $ \num{13.30} $ times the predicted miss distance of $ \SI{761.0}{\meter} $. A minimum drag manoeuvre, however, leads to a shrinking miss distance for shorter manoeuvre times. Flying in minimum drag attitude for $ \SI{50.69}{\hour} $ before TCA minimizes the miss distance to yield only $ \SI{1.964}{\meter} $. This is the result of an achieved separation distance which corresponds to the in-track relative position component in the predicted conjunction geometry (\cf \cref{fig:conjunction_geometries}). For longer manoeuvre duration and therefore higher absolute separation distances the minimum drag manoeuvre is able to increase the miss distance as well. For a minimum drag manoeuvre lasting the whole 5 days a miss distance of $ \SI{2.924}{\kilo\meter} $ can be achieved, which is an increase by $ \SI{284.3}{\percent}$ compared to the original prediction. Looking at scenario B, the situation is different. Here a minimum drag manoeuvre leads to a monotonously increasing miss distance of up to $ \SI{2.868}{\kilo\meter} $ compared to the $ \SI{554.1}{\meter} $ that were originally predicted. On the other hand, a flight in maximum drag attitude for $ \SI{42.64}{\hour} $ minimizes the miss distance to only $ \SI{61.72}{\meter} $, \ie $ \SI{11.14}{\percent} $ of the miss distance for no manoeuvre. However, for longer manoeuvre duration in maximum drag the miss distance increases and finally equals the achievable miss distance via minimum drag for a manoeuvring duration of $ \SI{71.71}{\hour} $. For even longer manoeuvres the miss distance is greater for a flight in maximum drag attitude.
To compare the risk associated with the manoeuvres, the maximum collision probability is calculated according to Alfano's method (\cf \cref{eq:pc_max_alfano}), as visible in \cref{fig:pc_alfano}. The aforementioned findings settle down in the collision probability as well. The increasing miss distance for a maximum manoeuvre in case A leads to a decreasing collision probability, whereas the minimum drag manoeuvre causes $ P_{c,max,alf} $ to show a maximum at exactly the manoeuvring time minimising the miss distance. For encounter B it is exactly the other way round. A maximum drag manoeuvre shorter than $ \SI{57.50}{\hour} $ leads to a maximum collision probability higher than if no manoeuvre were to be performed, thus worsening the situation. However, the growing achievable miss distances eventually lead to the fact that a manoeuvre in maximum drag attitude causes a lower maximum collision probability than a minimum drag manoeuvre.
\begin{figure}[h]
\centering
\begin{subfigure}{\linewidth}
\centering
\resizebox{\linewidth}{!}{\includegraphics{miss_dist_a.pdf}}
\caption{Encounter A.}
\label{fig:miss_distances_a}
\end{subfigure}\\
\begin{subfigure}{\linewidth}
\centering
\resizebox{\linewidth}{!}{\includegraphics{miss_dist_b.pdf}}
\caption{Encounter B.}
\label{fig:miss_distances_b}
\end{subfigure}
\caption{Miss distance depending on manoeuvring time.}
\label{fig:miss_distances}
\end{figure}
\begin{figure}[h]
\centering
\begin{subfigure}{\linewidth}
\centering
\resizebox{\linewidth}{!}{\includegraphics{pc_alfano_a_ratio.pdf}}
\caption{Encounter A.}
\label{fig:pc_alfano_a}
\end{subfigure}\\
\begin{subfigure}{\linewidth}
\centering
\resizebox{\linewidth}{!}{\includegraphics{pc_alfano_b_ratio.pdf}}
\caption{Encounter B.}
\label{fig:pc_alfano_b}
\end{subfigure}
\caption{Maximum probability of collision depending on manoeuvring time.}
\label{fig:pc_alfano}
\end{figure}
\subsection{Influence of parameter uncertainties}
Uncertainties in the parameters of the analytic equation for the achievable separation distance lead to additional in-track position uncertainties at the time of closest approach. This additional uncertainty $ \sigma_{\Delta x} $ can be expressed as a scaling factor for the respective element of the covariance matrix of the manoeuvring satellite at TCA. The updated covariance matrix follows as
\begin{equation}
\m{C}_{1,tca}' = \mat{1 &0 &0\\
0 &k^2 &0\\
0 &0 &1} \m{C}_{1,tca}
\end{equation}
\begin{equation}
\text{with\quad} k = \frac{\sigma_{T,1} + \sigma_{\Delta x}}{\sigma_{T,1}}
\end{equation}
where $ \m{C}_{1,tca} $ is the position covariance matrix and $ \sigma_{T,1} $ its in-track component of the satellite as defined in the CDM.
The effect of these uncertainties on the collision probability will be examined based on the exemplary close encounter of the \emph{Flying Laptop}\xspace on April 7, 2022. The reasonable CAM is a flight in maximum drag attitude, for which the manoeuvring time will be varied in the following analyses ($ \SI{10}{\hour} $, $ \SI{20}{\hour} $, $ \SI{30}{\hour} $, $ \SI{40}{\hour} $, and $ \SI{50}{\hour} $). For simplicity, no further constraints on the manoeuvre will be considered. At first, the influence of the scaling factor $ k $ on the collision probability will be quantified. Afterwards, it will be examined how uncertainties in the different parameters translate into $ k $ and thus affect $ P_c $.
In \cref{fig:scaling}, the resulting collision probability is plotted over $ k $ for the different manoeuvring times. If no parameter uncertainties are considered, $ k=1 $ and the collision probability can be decreased compared to the predicted value for any manoeuvring time $ >\SI{0}{\second} $. The collision probability tends to decrease monotonously with growing scaling factor for the manoeuvring times up to $ \SI{40}{\hour} $. The longest manoeuvre duration of $ \SI{50}{\hour} $, however, shows a maximum collision probability, which is reached for $ k_{max}=1.455 $. This means that parameter uncertainties in the analytic equation, which cause the in-track covariance to grow by $ \SI{45.5}{\percent} $, in fact lead to an increased collision probability compared to when neglecting the parameter uncertainties. It becomes evident as well, though, that the resulting collision probability is still below the one without manoeuvre.
\begin{figure*}[p!]
\centering
\includegraphics{uncertainty_scaling_intrack.pdf}
\caption{Collision probability over the in-track covariance scaling factor. In red and dashed is the collision probability for no manoeuvre.}
\label{fig:scaling}
\end{figure*}
\cref{fig:k} depicts the scaling factors depending on the uncertainty level $ s $ in the parameters $ \bar \rho $, $ a_0 $, $ \Delta \beta^* $ or $ t_m $ for a manoeuvring time of $ t_m = \SI{50}{\hour} $. Due to the linear error propagation, the first three parameters show the same effect on the uncertainty in the achievable separation distance and thus on the scaling factor, whereas the impact of an uncertainty in the manoeuvring time is twice as high (\cf \cref{eq:error_propagation_rho,eq:error_propagation_sma,eq:error_propagation_CB,eq:error_propagation_tc}).
An uncertainty in any one of atmospheric density, semi-major axis, or inverse ballistic coefficient difference of $ \SI{14.14}{\percent} $ results in a scaling factor of $ k_{max} $ leading to the maximum in collision probability. For the uncertainty in the manoeuvring time $ t_m $, this is the case at $ \SI{7.071}{\percent} $.
\begin{figure*}[p!]
\centering
\includegraphics{uncertainty_k.pdf}
\caption{In-track covariance scaling factor depending on an uncertainty level $ s $ in the individual parameters of the analytic equation. The manoeuvring time is $ t_m=\SI{50}{\hour} $. The scaling factor $ k_{max} $, for which $ P_c $ becomes maximum, is marked in red and dashed.}
\label{fig:k}
\end{figure*}
\section{Discussion}
\label{ch:discussion}
In the following chapter the results of the previous analyses are discussed. At first, the feasibility of aerodynamic collision avoidance manoeuvres for the \emph{Flying Laptop}\xspace is evaluated. After that, the effect of uncertainties is dealt with before possible manoeuvre strategies are discussed.
\subsubsection{Feasibility}
Atmospheric density is highly dependent on the activity of the Sun and Earth's geomagnetic activity. This already highlights the importance of considering current activity data when estimating effects of aerodynamic manoeuvres. By employing complex atmospheric models, recent and predicted values for the indices and proxies can be used to estimate the density on a given trajectory. In a first step, the \emph{Flying Laptop}\xspace's ability to perform CAMs depending on solar and geomagnetic activity was studied. As expected, the achievable separation distance in a given time is strongly influenced by the activity level, leading to separation distances which vary across two orders of magnitude.
\citet{Mishne.2017} argued that for a reasonable collision avoidance manoeuvre, the achievable separation distance must be in the range of $ \SI{900}{\meter} $ per three days in order for it to be greater than typical propagation uncertainties and thus useful. It can be concluded that the \emph{Flying Laptop}\xspace can achieve this separation distance for moderate and high levels of solar and geomagnetic activity. Only for low activity, the resulting forces due to aerodynamic drag appear to be too low for creating significant separations. Solar activity follows approximately an 11-year cycle and the next maximum is expected for 2025 \cite{Hathaway.2015, NOAA.2019}. \cref{fig:space_weather} presents the course of the solar flux at $ \SI{10.7}{\centi\meter} $ as proxy for the Sun's activity. This allows to conclude that the feasibility of aerodynamic CAMs will be given in the upcoming period until the next minimum in the solar cycle. During minimum phases, CAMs using aerodynamic drag might temporarily not be feasible for the \emph{Flying Laptop}\xspace, as long as solar activity and consequently atmospheric density is low.
It is to note, that these results strongly depend on the ballistic coefficient. The used ballistic coefficients are considered lower limits. hence, the achievable separation distances are lower limits as well and the absolute values might be lower for a minimum drag maneouvre. Here, further analysis of the error in the determined ballistic coefficient is necessary.
\begin{figure}[h]
\centering
\resizebox{\linewidth}{!}{\includegraphics{space_weather.pdf}}
\caption{Solar activity proxy $ F_{10.7} $, its 81-day average $ \bar F_{10.7} $ and geomagnetic activity index $ A_p $ over time. Historic and predicted data are obtained from \citet{SET}.}
\label{fig:space_weather}
\end{figure}
Considering constraints for constraint phases, \ie charging phases here, shows an influence on the achievable separation distance. It was observed, though, that even for a manoeuvre in which the \emph{Flying Laptop}\xspace holds maximum drag attitude for $ \SI{1.5}{\hour} $ followed by a charging phase of $ \SI{2.5}{\hour} $ the resulting separation distance fulfills the criterion mentioned before and can therefore be considered a useful manoeuvre. Considering very short manoeuvres under $ \SI{24}{\hour} $, a split of $ \SI{2}{\hour} $ manoeuvring and $ \SI{2}{\hour} $ charging is necessary for an achieved separation distance of $ \SI{300}{\meter} $ within the first day. Such constraints, however, are far beyond what is typically required for charging phases.
In the context of the \emph{Flying Laptop}\xspace's orbital evolution it is to note that the angle between the orbital plane and the Sun vector is continuously changing. The recent state demands charging phases as mentioned before when performing maximum drag manoeuvres. Minimum drag manoeuvres can be implemented without having to consider charging phases, since the satellite can be freely moved around the axis pointing to the direction of flight, such that the solar arrays can point to the Sun constantly. For a progressively decreasing local time of ascending node, this situation will change. For a LTAN around midnight (or noon, respectively), the orbital plane is oriented in a way allowing to point the solar arrays to the Sun during a maximum drag manoeuvre. In this case, charging phases were to be implemented during flights in minimum drag attitude. Looking at orbital plane orientations between these extreme cases, further tests will be necessary to determine whether enough power can be generated in the respective attitudes or whether charging phases are needed. Furthermore, tests should be performed in order to study the effects of uneven heating of the satellite structure, which may, in fact, deteriorate battery performance as well or lead to other negative effects. Still, this can be mitigated by making use of the remaining degree of freedom and commanding rotations around the direction of flight without worsening the manoeuvring performance.
Power could be saved by minimizing downlinks of the satellite, \eg by not establishing connection to the ground station during a pass at all or by downlinking fewer data.
However, monitoring the satellite's telemetry data is an important point when performing collision avoidance manoeuvres. Problems with the GPS receivers or the star trackers, for example, could potentially take the satellite its ability to attain commanded attitudes. Since this might worsen the situation at TCA, it is recommended to constantly monitor both telemetry as well as housekeeping data, like battery voltages, to ensure that the satellite performs the manoeuvre as desired.
\subsection{Manoeuvre suggestions}
When facing a predicted close encounter, the operators of the \emph{Flying Laptop}\xspace so far have three options. They can either command a flight in maximum or in minimum drag attitude or do not implement any CAM at all. Depending on the predicted collision probability, performing a CAM and thus reducing the risk is desirable, however. In \cref{sec:ana_conjunction_geometry} it was found that a positive separation distance (corresponding to a flight in maximum drag attitude) is desirable if the predicted in-track component of the relative position at TCA is negative. Vice versa, a negative separation distance is to be induced for a positive predicted in-track distance at TCA. For longer manoeuvre durations, though, it may be favourable to perform a contrary manoeuvre. Depending on the reference ballistic coefficient (as well as solar and geomagnetic activity) it might be possible to implement a manoeuvre which would lead to a shrinking miss distance if performed shortly but to drastically growing miss distances for longer manoeuvring times, as was the case in encounter B in \cref{sec:ana_conjunction_geometry}.
From this, possible manoeuvring strategies might be deduced. For short-term manoeuvres with little time left until predicted TCA the attitude which monotonously increases the miss distance is to be commanded, representing a conservative. This results in a maximum drag attitude if the in-track component of the relative position of the secondary object with respect to the manoeuvring satellite is negative and a minimum drag attitude if it is positive. For long-term manoeuvres, the strategy might be inverted if an analysis of the achievable separation and miss distances indicates so. At this point in time it can, however, not be recommended to perform a manoeuvre which leads to decreasing miss distances at first. Further analyses on the accuracy of the predicted separation distances and evaluation of actually performed CAMs are necessary before such a manoeuvre might be reliably performed.
In general, it can be concluded that an early implementation of a manoeuvre is more effective. In \cref{sec:manoeuvre}, it was shown that the early build-up of angular motion relative to the reference trajectory leads to an increasing separation distance even though the satellite might return to its reference attitude, which would be the case if the CAM were, for example, ended and the satellite returned to nominal operation. From another point of view, an early manoeuvre implementation leads to a shorter overall time spent in the manoeuvring attitude to achieve a specific separation distance compared to if the manoeuvre was to be performed at a later point.
For charging phases, this is already considered in the analysis tool, since they follow manoeuvring phases.
Another point was not addressed so far. For a secondary object in an orbit with a similar period to that of the satellite, it is realistically possible for further encounters to happen several orbits before and after the predicted close encounters. It was so far not analysed, how avoidance manoeuvres using aerodynamic drag affect previous or later encounters. This is an important aspect of future research. Whether sgp4-propagation of TLEs is sufficient for a reliable simulation is difficult to assess. High-fidelity orbit propagation might lead to further insights.
\subsection{Uncertainties in parameters}
The effects of uncertainties in the parameters required for the equation to estimate the achievable separation distance were investigated on an exemplary close encounter. Uncertainties in the average density, the semi-major axis and the change in inverse ballistic coefficient were covered. Using a linear error progression, it was calculated to which extent they affect the uncertainty in the separation distance. This uncertainty adds to the in-track component of the position covariance as given in CDMs.
For the exemplary encounter it was found that, depending on the manoeuvring time, uncertainties in the parameters affect the resulting collision probability differently. For shorter manoeuvres the collision probability strictly decreases for growing parameter uncertainties. On the other hand, the collision probability shows a maximum for a certain scaling factor of the in-track position covariance before strictly decreasing as well. The scaling factor to reach maximum collision probability translates into uncertainties of about $ \SI{10}{\percent} $ for either averaged density, semi-major axis, or change in inverse ballistic coefficient. Alternatively, a manoeuvring time with an uncertainty of about $ \SI{5}{\percent} $ has the same effect. Taking into account, that these parameters are all fraught with uncertainties and that these uncertainties add up, the individual uncertainties must even be lower to maximise $ P_c $. Still, it becomes clear that, in the case at hand, for any additional uncertainty the collision probability is clearly lower than the one which was previously calculated in the case of not manoeuvring due to the positive impact of the increased miss distance.
\subsection{Uncertainty quantification}
Looking at the individual parameters, the uncertainty levels may be assessed. All the findings are summarized in \cref{tab:uncertainties}.
Atmospheric density varies over two orders of magnitude at the altitude regime of the \emph{Flying Laptop}\xspace, when comparing low to high solar and geomagnetic activity. The uncertainty in density is, thus, a major factor and caused by different aspects. First of all, the atmospheric model in use provides only estimations of the density. While no model provides overall better estimations than another, an average uncertainty of $ \SIrange{10}{15}{\percent} $ needs to be assumed according to literature \cite{Sagnieres.2017, Vallado.2014b}. For short-term predictions uncertainties might even be significantly higher, which leads to the next aspect. The implemented atmospheric models need indices and proxies for solar and geomagnetic activity as input, which have to be predicted in order to estimate atmospheric densities at future points in time. \citet{Vallado.2014b} analysed solar flux predictions and found that they can show significant error when compared to the actual observations, especially during solar maxima. Conclusively, they stated a standard deviation of $ \SIrange{20}{40}{} $ solar flux units for 45-day forecasts, leading to density variations of $ \SIrange{150}{300}{\percent} $. The geomagnetic activity was found to be accurate within $ \SI{20}{} $ units, resulting in a density error of up to $ \SI{50}{\percent} $. Regarding the error in atmospheric density induced through positional errors due to sgp4-propagation, a first estimation has been established. A \emph{Flying Laptop}\xspace trajectory was propagated for five days using sgp4 and a high-fidelity orbit propagation and the encountered densities were each averaged. The difference was found to be $ <\SI{1}{\percent} $ \cite{Turco.2022}.
The semi-major axis is obtained from TLEs. Looking at recent TLEs for the \emph{Flying Laptop}\xspace, the obtained semi-major axis showed variations of up to $ \SI{50}{\meter} $.
Compared to the JSpOC observations, though, differences of up to $ \SI{10}{\kilo\meter} $, \ie $ \SI{0.1435}{\percent} $, were found.
The accuracy of the change in inverse ballistic coefficient is dependent on the accuracy of the reference coefficient in the CDM and the accuracy of the inverse ballistic coefficient in manoeuvring attitude. Currently, no information on potential errors of the reference inverse ballistic coefficient is available. Regarding the manoeuvring attitude, two aspects influence the accuracy of the ballistic coefficient. First, the aerodynamic analysis of the \emph{Flying Laptop}\xspace, especially with the assumptions on gas-surface interaction introduces error. Its magnitude can so far not be assessed but the determined inverse ballistic coefficients are expected to be lower bounds, leading to the resulting separation distances being lower bounds with regards to $ \beta^* $ as well. The second aspect is the pointing accuracy of the \emph{Flying Laptop}\xspace. Deviations from the assumed attitude affect the surface exposed to the flow, the angle of incidence of the atmospheric particles and thus the inverse ballistic coefficient. The pointing error is stated to be within $ \SI{0.042}{\degree} $ \cite{Eickhoff.2016}, which translates to an error in the inverse ballistic coefficient of $ \le \SI{5}{\percent} $.
Uncertainties in the manoeuvring time are not quantified at the moment. With a rotation speed of $ \SI{1.5}{\degree\per\second} $, the \emph{Flying Laptop}\xspace is able to turn into any specified attitude within $ \SI{120}{\second} $ and this can be considered in the manoeuvring time itself. Losses in manoeuvring time can further happen due to the \emph{Flying Laptop}\xspace returning to a safe mode losing star tracker functionality or similar events, which cause an attitude potentially different from the commanded one. To represent such events statistically, further research needs to be conducted on their frequency.
The results indicate the importance of estimating uncertainties in the parameters as realistically as possible to generate a trustworthy probability of collision. Under or over-estimating the uncertainties may lead to increased or decreased collision probabilities. This does not only include the parameter uncertainties but also the position covariance given in CDMs and further scaling by JSpOC, which is another source for a variation of the collision probability. Furthermore, correlations between the CDM covariance and the uncertainties in the parameters, which are so far assumed to be zero, should be investigated more deeply to obtain a realistic position covariance after a performed manoeuvre. This directly increases the fidelity of the determined collision probability. So far, the most reliable possibility for assessing the overall risk of a close encounter is the calculation of a maximum collision probability, which is detached from the objects' covariances in its worst-case formulation. While the collision probability itself might be mismodelled, the maximum collision probability shows a positive influence of any increase in miss distance through an aerodynamic manoeuvre, which only depends on the achievable separation distance.
\begin{table*}[h!]
\centering
\caption{Causes for uncertainties in the parameters of the analytic equation for the achievable separation distance.}
\label{tab:uncertainties}
\begin{tabular}{@{}llcc@{}}
\toprule
Parameter &Cause of uncertainty &Expected uncertainty &\begin{tabular}[c]{@{}l@{}}Expected uncertainty\\in the parameter\end{tabular} \\
\midrule
\multirow{4}{*}{Density $ \bar \rho $} &Atmospheric model accuracy &$ \SIrange{10}{15}{\percent} $ \cite{Vallado.2014b} & $ \SIrange{10}{15}{\percent} $\\
&Solar flux activity prediction &$\le \SIrange{20}{40}{units} $ \cite{Vallado.2014b} & $ \le \SIrange{150}{300}{\percent} $\\
&Geomagnetic activity prediction &$ \le \SI{20}{units} $ \cite{Vallado.2014b} &$ \le \SI{50}{\percent} $\\
&sgp4-propagation and density averaging &$ \le\SI{1}{\percent} $\textsuperscript{1} &$ \le\SI{1}{\percent} $\\
\midrule[0.25pt]
Semi-major axis $ a_0 $ &TLE accuracy &$ \le \SI{10}{\kilo\meter} $\textsuperscript{1} &$ \le \SI{1}{\percent} $\\
\midrule[0.25pt]
\multirow{3}{*}{\begin{tabular}[c]{@{}l@{}}Change in inverse ballistic\\coefficient $ \Delta \beta^* $\end{tabular}} &Reference inverse ballistic coefficient $ \beta^*_{ref} $ &? &?\\
&Aerodynamic analysis &? &?\\
&Pointing accuracy &$ \SI{0.042}{\degree} $ \cite{Eickhoff.2016} &FLP: $ \le \SI{5}{\percent} $ \\
\midrule[0.25pt]
Manoeuvring time $ t_m $ &Safe mode, loss of star trackers, etc. & - & -\\
\bottomrule
\end{tabular}
\textsuperscript{1} This is an estimate based on first analyses with the \emph{Flying Laptop}\xspace and requires further research.
\end{table*}
\section{Summary}
\label{sec:summary}
In this work, a tool was developed to analyse aerodynamic manoeuvres for satellite collision avoidance. Different manoeuvres can be compared with respect to their resulting miss distance and collision probability, considering solar and geomagnetic activity conditions.
Further constraints regarding charging phases during the manoeuvre can be considered and uncertainties in the various parameters that are used as input to the tool can be accounted for, as well. At this point, the tool relies solely on CDM data and (publicly) available TLEs, making it simple to use. No further orbit determination and propagation processes are required to estimate the effect of a manoeuvre.
Close encounters of the university satellite \emph{Flying Laptop}\xspace were used as exemplary cases. The use of aerodynamic manoeuvres for collision avoidance was found to be feasible for the \emph{Flying Laptop}\xspace for moderate and high solar and geomagnetic activity levels. Implementing charging phases of reasonable duration were shown to not hinder the applicability of aforementioned manoeuvres. First manoeuvre suggestions based on the conjunction geometry were deducted. Additionally, an exemplary conjunction was studied to evaluate the influence of parameter uncertainties. For this example, it was concluded that while the uncertainties may lead to slightly higher collision probabilities than when not accounting for them, the overall collision probability is decreased with a manoeuvre.
\section{Outlook}
\label{sec:outlook}
More research is necessary for a realistic approximation of the uncertainties in the parameters, their correlation to the position uncertainty defined in a CDM to obtain a reliable and less conservative collision probability. The uncertainty sources need to be addressed individually and their effect on the manoeuvre uncertainty studied.
At the Institute of Space Systems, University of Stuttgart, the \emph{Flying Laptop}\xspace further offers the possibility to verify the effect of aerodynamic manoeuvres. In future work, flight tests can be performed and analysed. Upcoming other missions can benefit from insights and the use of the analysis tool as well.
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
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LARRY BOHANNAN
Ask Larry: LPGA wage disparity
Q: Lexi Thompson earns $300,000 for winning the Kraft Nabisco and Matt Jones earns $1,152,000 for winning the Shell Houston Open. This is a good example of the wage disparity between men and women.
A: What it really shows is the disparity in the marketplace for men's and women's golf. The difference in money for the tours is based in harsh economics. It is based on television ratings and what television networks and sponsors are willing to pay to be associated with the different tours. Golf Channel was happy to report its biggest ratings ever for the Kraft Nabisco, a Sunday of 670,000 viewers. But the final round of the Shell Houston Open, shown opposite the Kraft Nabisco, drew just under 2 million viewers for NBC.
So, NBC is more interested in putting the Houston event on its airwaves than the Kraft, because it draws more ratings. Higher ratings draw more advertising interest, and more money for NBC. And the higher ratings mean more companies want to be involved with the men's tour than the women's tour. All of that translates to more money for purses. And that is why the Kraft Nabisco has a $2 million purse, while the Houston event's is $6.4 million.
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{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
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Mathieu van der Poel wins Dutch Road Championship
Jul 02 2018 02:52 am CET
Photo of Mathieu van der Poel by Beobank-Corendon
Mathieu van der Poel of Corendon-Circus is the new Dutch road champion after beating Danny van Poppel and Ramon Sinkeldam in an impressive sprint in Hoogerheide.
The parcours of the road race was 221 kilometers long, on which Groupama-FDJ's Sinkeldam was hoping to defend his title earned in 2017.
LottoNL-Jumbo's Danny van Poppel was one of the more active riders on the day, first disrupting things in the lead group and later going solo, to be caught by a chase group with Van der Poel and then the reduced peloton.
Koen de Kort of Trek-Segafredo then attacked at just under two kilometers but he two was soon reeled back in.
In the final sprint, which went slightly up-hill, Sinkeldam threw his cards onto the table first but was surpassed by Van der Poel, who as such won the Dutch title 31 years after his father Adrie.
"This is a childhood dream coming true," said Van der Poel shortly after taking the win. "Despite a tough final, I kept believing that I could finish it in the sprint. And I did finish it.
"Winning here in Hoogerheide makes it all even more special. I will definitely celebrate this. But I won't go too crazy, because the Val di Sole is on its way," he concluded.
MATHIEU VAN DER POEL
|
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"redpajama_set_name": "RedPajamaCommonCrawl"
}
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@interface ADUserListItemTableViewCell ()
@property (weak, nonatomic) IBOutlet UIImageView *avatarImageView;
@property (weak, nonatomic) IBOutlet UILabel *loginLabel;
@property (weak, nonatomic) IBOutlet UILabel *detailLabel;
@property (strong, nonatomic) ADUserListItemViewModel *viewModel;
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@implementation ADUserListItemTableViewCell
- (void)awakeFromNib {
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self.viewModel = viewModel;
[self.avatarImageView sd_setImageWithURL:viewModel.user.avatarURL placeholderImage:[UIImage imageNamed:@"default_avatar"]];
self.loginLabel.text = viewModel.login;
self.detailLabel.text = viewModel.user.HTMLURL.absoluteString;
}
- (IBAction)onOperationButtonClicked:(UIButton *)sender {
sender.enabled = NO;
[[[self.viewModel.operationCommand execute:self.viewModel]deliverOnMainThread]subscribeCompleted:^{
sender.enabled = YES;
}];
}
@end
|
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"redpajama_set_name": "RedPajamaGithub"
}
| 461
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Special Alert: OCC issues CRA final rule
Buckley Special Alert
On May 20, the Office of the Comptroller of the Currency announced a final rule to modernize the regulatory framework implementing the Community Reinvestment Act. The final rule marks the culmination of a three-year effort led by the Treasury Department to revamp the CRA and arrives exactly six weeks after the comment period on the notice of proposed rulemaking (NPR) closed on April 8, 2020.
Significantly, while the Federal Deposit Insurance Corporation joined the OCC in issuing the NPR, the FDIC did not join in promulgating the final rule. The Federal Reserve Board was not party to the NPR or the final rule. Accordingly, banks whose prudential regulator is the FDIC or the Federal Reserve will continue to be subject to the existing CRA regulations.
The OCC's rule, while technically effective October 1, 2020, provides for at least a 27-month transition period for compliance based on a bank's size and business model. Large banks and wholesale and limited purpose banks will have until January 1, 2023 to comply, and small and intermediate banks that opt-in to the final rule's performance standards will have until January 1, 2024. In the interim, a performance evaluation conducted after October 1, 2020, and before January 1, 2023 or 2024, as applicable, would permit banks to rely on the current performance standards and tests or on the final rule.
The following are key takeaways from the final rule:
Metrics: Like the NPR, the final rule establishes a series of metrics as the primary basis for evaluating a bank's CRA performance. These metrics, applied to each assessment area and bankwide, will use a bank's call report data to determine the amount of its qualifying activities — specifically, the volume of mortgage, consumer, small business and small farm loans, and community development lending and investments. The value of certain activities, such as investments in community development financial institutions and in CRA deserts, would be upwardly adjusted through the use of multipliers.
Metric-Based Benchmarks: The final rule states that there will be quantitative benchmarks established as part of the metrics for determining a bank's CRA rating. However, unlike the NPR, the OCC did not provide the specific rating thresholds for determining sufficient CRA activity based on existing data. Instead, banks will submit their lending data according to the contours of the final rule, which the OCC will then analyze to set thresholds for outstanding, satisfactory, needs to improve, and substantial noncompliance ratings.
Deposit-Based Assessment Areas: The final rule preserves and expands upon the current requirements tying assessment area delineation to locations where a bank has a physical presence. Under the final rule and as provided in the NPR, banks sourcing 50 percent or more of their retail domestic deposits from outside their facility-based assessment areas must also designate assessment areas wherever they receive five percent or more of those deposits. The final rule permits banks to delineate those assessment areas as broadly as statewide, which is more flexible than allowed by the NPR.
Consumer Lending: The final rule provides for the mandatory inclusion of consumer loans, other than credit cards and overdraft products, in a bank's CRA evaluation. Credit card and overdraft products were included in the NPR. Note that inclusion of consumer loans is essentially optional under the current CRA framework.
List of Pre-Approved CRA Activities: Accompanying the final rule, and consistent with the NPR, the OCC published a nonexhaustive illustrative list of activities that qualify for CRA consideration. The final rule also establishes a process for stakeholders to submit additional items for inclusion on the list.
Small and Intermediate Banks: The final rule provides that small banks, defined as those with assets of $600 million or less (up from $326 million) and intermediate banks, defined as those with assets of $2.5 billion or less (up from $1.305 billion), will have the option to decide whether to be evaluated under existing CRA criteria or the new framework.
Retention of Wholesale and Limited Purpose Designations: The final rule preserves wholesale and limited purpose designations and the applicable community development test. The NPR had eliminated those designations and contemplated subjecting such institutions to the same evaluation standards as large banks.
Mortgage Lending: As with the NPR, the final rule provides that the OCC will no longer qualify mortgages made to middle- and upper-income borrowers in low- or moderate-income census tracts for CRA consideration.
The final rule is nearly 400 pages, and contains significantly more information than summarized above. Buckley will offer a webcast in the near future to discuss the final rule in greater detail.
If you have any questions about the CRA or other related issues, please visit our Fair Lending practice page or contact a Buckley attorney with whom you have worked in the past.
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{
"redpajama_set_name": "RedPajamaCommonCrawl"
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\section{Introduction}
We work over the complex number field $\mathbb {C}$.
In \cite{KL},
A.~K\"{u}ronya and V.~Lozovanu give the following Reider-type theorem for higher syzygies on abelian surfaces:
\begin{thm}[{\cite[Theorem 1.1]{KL}}]\label{thm_KL}
Let $p$ be a non-negative integer,
$X$ be an abelian surface,
and $L$ be an ample line bundle on $X$ with $(L^2) \geq 5(p+2)^2$.
Then the following are equivalent.
\begin{enumerate}
\item $X$ does not contain an elliptic curve $C$ with $(L.C) \leq p+2$,
\item Property $(N_p)$ holds for $L$.
\end{enumerate}
\end{thm}
We refer the readers to \cite{Ei}, \cite[Chapter 1.8.D]{La} for the definition of property ($N_p$).
We just note here that ($N_p$)'s consist an increasing sequence of positivity properties.
For example,
($N_0$) holds for $L$ if and only if $L$ defines a projectively normal embedding,
and ($N_1$) holds if and only if ($N_0$) holds and the homogeneous ideal of the embedding is generated by quadrics.
In the note,
we show a slight improvement of \autoref{thm_KL} as follows.
\begin{thm}\label{main_rem}
In \autoref{thm_KL},
it is enough to assume $(L^2) > 4(p+2)^2$ instead of $(L^2) \geq 5(p+2)^2$.
\end{thm}
\begin{rem}
In \cite{AKL},
the equivalence of (1) and (2) in
\autoref{thm_KL} is proved for $K3$ surfaces under the assumption $(L^2) > \frac12 (p+4)^2$.
\end{rem}
In \cite{KL}, $(2) \Rightarrow (1)$ in \autoref{thm_KL} is proved under the assumption
$(L^2) \geq 4p+5$.
Hence to prove \autoref{main_rem},
it suffices to show the converse $(1) \Rightarrow (2)$ under the assumption $(L^2) > 4(p+2)^2$.
To show $(1) \Rightarrow (2)$,
we use the following result in \cite{LPP} (see also \cite[Section 3]{KL}):
\begin{thm}[\cite{LPP}]\label{thm_LPP}
Let $p$ be a non-negative integer,
$X$ be an abelian variety,
and $L$ be an ample line bundle on $X$
such that there exists an effective
$\mathbb {Q}$-divisor $F_0$ on $X$ satisfying
\begin{itemize}
\item[(i)] $\frac{1}{p+2}L-F_0$ is ample,
\item[(ii)] the multiplier ideal $\mathcal {J}(X,F_0)$ coincides with the maximal ideal
$\mathcal {I}_o$ of the origin $o \in X$.
\end{itemize}
Then $(N_p)$ holds for $L$.
\end{thm}
To construct such a divisor $F_0$,
K\"{uronya} and Lozovanu use
Okounkov bodies as a key tool.
Instead, we use a standard technique developed in the study of Fujita's base point freeness conjecture \cite{EL}, \cite{He}, \cite{Ka} \cite{Ko}, etc.
We also note that G.~Pareschi \cite{Pa} proved a conjecture of R.~Lazarsfeld
which claims that for an ample line bundle $L$ on an abelian variety,
$(N_p)$ holds for $L^{\otimes p+3}$.
On the other hand, results in \cite{KL}, \cite{LPP} can prove ($N_p$) for $L$ which is not necessarily a multiple of a line bundle.
\subsection*{Acknowledgments}
The author was supported by Grant-in-Aid
for Scientific Research (No.\ 26--1881, 17K14162).
\section{Preliminaries}
We recall some notations. We refer the reader to \cite{KM}, \cite[Chapter 9]{La2} for detail.
Let $X$ be a smooth variety
and $D=\sum_i d_i D_i$ be an effective $\mathbb {Q}$-divisor on $X$.
Let $f: Y \rightarrow X $ be the log resolution of $(X,D)$
and write
\[
K_Y = f^* (K_X +D) + F
\]
with $F=\sum_j b_j F_j$.
Here we assume that if $F_j$ is the strict transform of $D_i$,
we take $b_j=-d_i$,
and all the other $F_j$'s are exceptional divisors of $f$.
The pair $(X,D)$ is called \emph{log canonical} (resp.\
\emph{klt}) at $x \in X$
if $b_j \geq -1$ (resp.\ $b_j > -1$) for any log resolution $f$ and any $j$ with $x \in f(F_j)$.
If $(X,D)$ is log canonical at $x$ and $b_j > -1$ for any log resolution $f$ and any \emph{exceptional} $F_j$ with $x \in f(F_j)$,
$(X,D)$ is called
\emph{plt} at $x$.
The pair $(X,D)$ is called log canonical (resp.\ klt, plt) if $(X,D)$ is log canonical (resp.\ klt, plt) at any $x \in X$.
The \emph{multiplier ideal} $\mathcal {J}(X,D)$ of $(X,D)$ is defined as
\[
\mathcal {J}(X,D) := f_* \mathcal {O}_Y(\lceil F \rceil),
\]
which does not depend on the choice of the log resolution $f$.
The \emph{log canonical threshold} of $D$ is defined to be
\[
\lct(D)= \max\{ s \geq 0 \, | \, (X,s D) \text{ is log canonical}\}.
\]
Similarly,
the log canonical threshold of $(X,D)$ at $x \in X$ is
\[
\lct_x(D)= \max\{ s \geq 0 \, | \, (X,s D) \text{ is log canonical at } x\}.
\]
A subvariety $Z$ of $X$ is called a \emph{log canonical center} of $(X,D)$
if there exists a log resolution $f$ of $(X,D) $ and some $j$ such that $b_j \leq -1$ and $f(F_j)=Z$.
We note that a log canonical pair $(X,D)$ is plt if and only if $(X,D)$ has no log canonical center of codimension $\geq 2$.
\begin{rem}\label{rem_max_ideal}
If $\{x\}$ is the \emph{unique} log canonical center of $(X,D)$ containing $x$,
we have $\Supp \mathcal {O}_X / \mathcal {J}(X,D) =\{x\}$ around $x$.
Furthermore,
if $(X,D) $ is log canonical at $x \in X$,
every negative coefficient of $\lceil F \rceil$ over $x$ is $-1$.
Hence $\mathcal {J}(X,D)$ is reduced at $x$,
that is,
$\mathcal {J}(X,D) = \mathcal {I}_x$ holds around $x$ in such a case.
\end{rem}
The following theorem about the existence of minimal log canonical centers is known
(see \cite{EL},\cite{He},\cite{Ka}).
We note that for surface this theorem can be easily proved.
\begin{thm}\label{thm_minimal _center}
Let $X$ be a smooth variety
and $D$ be an effective $\mathbb {Q}$-divisor on $X$ such that $(X,D) $ is log canonical.
Then every irreducible component of the intersection of two log canonical centers of $(X,D)$
is also a log canonical center of $(X,D)$.
In particular, if $(X,D) $ is log canonical and not klt at $x \in X$,
there exists the unique minimal log canonical center $Z$ of $(X,D)$ containing $x$.
Furthermore, $Z$ is normal at $x$.
\end{thm}
\section{Proof}
\subsection{Proof of \autoref{main_rem}}
As stated in Introduction,
we will construct a divisor $F_0$ in \autoref{thm_LPP}.
In fact, we can construct such a divisor easily by a standard technique developed in the study of Fujita's base point freeness conjecture.
Roughly speaking,
we cut down minimal log canonical centers to obtain a $0$-dimensional log canonical center.
In higher dimensional case,
the arguments and suitable explicit estimation are difficult and complicated,
but they are relatively easy for surfaces (see \cite[Section 2]{EL}).
\begin{comment}
\begin{thm}\label{main_lem}
Let
$B$ be an ample $\mathbb {Q}$-divisor on a smooth projective surface $S$.
Let $x \in S$.
If $(B^2) >2 $ and $(B.C) \geq 2$ hold for all curve $C \subset X$ through $x$,
there exists an effective $\mathbb {Q}$-divisor $F_0$ on $S$ such that $B- F_0$ is ample and $\mathcal {J}(X,F_0)=\mathcal {I}_x$ around $x$.
\end{thm}
\end{comment}
Throughout this subsection,
$X$ is an abelian surface
and $\pi : X' \rightarrow X $ is the blow-up at the origin $o \in X$.
We denote by $E \subset X'$ the exceptional divisor.
\vspace{2mm}
\begin{prop}\label{main_lem}
Let
$B$ be an ample $\mathbb {Q}$-divisor on $X$.
If $\pi^* B -2E $ is big and $(B.C) > 1$ holds for any elliptic curve $C \subset X$,
there exists an effective $\mathbb {Q}$-divisor $F_0$ on $X$ such that $B- F_0$ is ample and $\mathcal {J}(X,F_0)=\mathcal {I}_o$.
\end{prop}
\begin{rem}
If $X$ is a general surface,
we usually assume $(B.C) \geq 2$ for a curve $C$
to construct a divisor $F_0$ as in \autoref{main_lem} (see \cite[1.10 Variant]{EL}, for example).
It is not so surprising that we can relax the numerical condition for abelian surfaces.
\end{rem}
By \autoref{rem_max_ideal},
we want $F_0$ such that $\{o\}$ is the unique log canonical center of a log canonical pair $(X,F_0)$.
Considering a small perturbation,
it suffices to find an effective $\mathbb {Q}$-divisor with a $0$-dimensional minimal center:
\begin{lem}\label{lem_minimal_center}
To prove \autoref{main_lem},
it suffices to construct an effective $\mathbb {Q}$-divisor $F$ on $X$ such that $B-F$ is ample,
$(X,F)$ is log canonical and has a $0$-dimensional minimal center.
\end{lem}
\begin{proof}
Let $\{x\} \subset X$ be a $0$-dimensional minimal center of $(X,F)$.
Choose a general effective divisor $F'$ on $X$ which contains $x$.
By replacing $F$ with $c_{\varepsilon} F + \varepsilon F'$ for $0 < \varepsilon \ll 1$,
where $c_{\varepsilon}= \max\{ s \geq 0 \, | \, (X,s F+\varepsilon F') \text{ is log canonical}\}$,
we may assume that $\{x\} \subset X$ is the unique minimal center of $(X,F)$
as in the proof of \cite[Proposition 2.4]{He}.
We note that the ampleness of $B - F$ is preserved if $\varepsilon$ is sufficiently small.
Then we have $\mathcal {J}(X,F)=\mathcal {I}_x$ by \autoref{rem_max_ideal}.
Set $F_0=t_x^* F$ to be the pull back of $F$ by the automorphism
\[
t_x :X \rightarrow X \quad : \quad y \mapsto y+x .
\]
Then $B - F_0 \equiv B - F $ is ample and $\mathcal {J}(X,F_0)=\mathcal {I}_o$ holds.
\end{proof}
\begin{proof}[Proof of \autoref{main_lem}]
We will construct $F$ as in \autoref{lem_minimal_center}.
Since bigness is an open condition,
$\pi^* B - t E $ is big for a rational number $t $ if $0 < t-2 \ll 1$.
Fix such $t$
and let $\pi^* B - t E =P_t+N_t$ be the Zariski decomposition.
Since $P_t$ is nef and big,
we can find an effective divisor $N'$ such that $P_t-\frac{1}{k} N'$ is ample for $k \gg 0$ by \cite[Theorem 2.3.9]{La}.
For $k \gg 0$,
we choose a general effective $\mathbb {Q}$-divisor $A \equiv P_t-\frac{1}{k} N'$ and set
\begin{align}\label{eq_def_D}
D' := A + \frac{1}{k} N' +N_t \equiv \pi^* B - t E , \quad D:=\pi_* D' \equiv B.
\end{align}
Then $D$ is an effective $\mathbb {Q}$-divisor on $X$ with the multiplicity $\mult_o (D) \geq t$.
Hence we have
\[
c:= \lct(X,D) \leq \lct_o (X,D)
\leq \frac{2}{t} <1.
\]
If there exists a $0$-dimensional minimal center $\{x\}$ of $(X,cD)$,
we can take $F:= cD$, which satisfies the condition in \autoref{lem_minimal_center}.
Hence we may assume that there exists no $0$-dimensional minimal center of $(X,cD)$.
Then there exists a $1$-dimensional minimal center $C \subset X$ of $(X,cD)$.
\begin{claim}\label{clm1}
The minimal center $C$ is an elliptic curve containing the origin $o$.
\end{claim}
\begin{proof}
Since $ C$ is a log canonical center of $(X,cD)$,
the coefficient
of $C$ in $D$ is $\frac{1}{c} >1$.
Since $A$ is general and $k$ is sufficiently large in \ref{eq_def_D},
the strict transform $C' \subset X'$ of $C$ is contained in $\Supp N_t$.
Hence $C' $ is a negative curve
and we have
\begin{align}\label{eq_ineq}
0 > (C'^2)= ((\pi^*C - m E)^2)=(C^2) -m^2,
\end{align}
where $m := \mult_o(C) \geq 0$.
Since $C $ is a curve on an abelian surface,
$(C^2) \geq 0$ holds.
On the other hand,
$m=0$ or $1$
since a minimal center is normal by \autoref{thm_minimal _center}
and hence $C$ is smooth.
Thus $(C^2)=0 $ and $m=1$ hold by \ref{eq_ineq}.
Since $X$ is an abelian surface,
$(C^2)=0$ implies that $C$ is an elliptic curve.
\end{proof}
We cut down the minimal log canonical center $C$.
We refer the reader to \cite{He}, \cite{Ka}, \cite[Section 10.4]{La2} for detail.
By assumption and \autoref{clm1},
we have
\[
(B-C.C) =(B.C) -(C^2) = (B.C) >1.
\]
Thus there exists an effective $\mathbb {Q}$-divisor $\overline{D}_1 \equiv (B-C)|_C $ on $C$
with $\mult_o (\overline{D}_1) >1$.
We take such $\overline{D}_1 $
so that $\mult_p (\overline{D}_1) < 1 $ for any $p \in C \setminus \{o\}$.
Since the coefficient of $C$ in $cD$ is $1$,
$cD - C$ is effective hence nef because $X$ is an abelian surface.
Thus
\[
B-C =(1-c) B +(cB-C)
\equiv (1-c)B+(cD-C)
\]
is ample by $c <1$.
Hence
we can take an effective $\mathbb {Q}$-divisor $D_1 \equiv B-C$ on $X$
such that $C \not \subset \Supp D_1$ and $D_1|_C = \overline{D}_1$ as in Step 2 in the proof of \cite[Proposition 3.2]{He}.
We take general such $D_1$,
then $(X,C+D_1)$ is klt outside $C$.
Since $\mult_o (D_1|_C) = \mult_o (\overline{D}_1) >1$,
$(X,C+D_1)$ is not log canonical at $o \in X$ by adjunction \cite[Theorem 5.50]{KM}.
On the other hand, since $\mult_p (D_1|_C) =\mult_p (\overline{D}_1) < 1 $ for any $p \in C \setminus \{o\}$,
$(X,C+D_1) $ is plt in a neighborhood of $C \setminus \{o\}$ by inversion of adjunction \cite[Theorem 5.50]{KM}.
Set
\[
F=C+c_1 D_1 ,
\]
where $c_1 :=\max \{ s \geq 0 \, | \, (X,C+s D_1) \text{ is log canonical}\} <1$.
Then $\{o\} \subset X$ is the minimal log canonical center of $(X,F)$.
Since
\[
B-F= B- (C+c_1 D_1) \equiv (1-c_1)(B-C)
\]
is ample by $c_1 <1$,
this $F$ satisfies the condition in \autoref{lem_minimal_center}.
Hence
\autoref{main_lem} is proved.
\end{proof}
\begin{cor}\label{thm_suff}
Let $p$ be a non-negative integer
and $L$ be an ample line bundle on $X$.
If $\pi^* L -2(p+2)E$ is big and $(L.C) > p+2$ holds for any elliptic curve $C \subset X$,
then $(N_p)$ holds for $L$.
\end{cor}
\begin{proof}
We can apply \autoref{main_lem} to $B=\frac{1}{p+2} L$.
Then Property ($N_p$) holds for $L$ by \autoref{thm_LPP} and \autoref{main_lem}.
\end{proof}
\begin{proof}[Proof of \autoref{main_rem}]
As stated in Introduction,
it suffices to show $(1) \Rightarrow (2)$ under $(L^2) > 4(p+2)^2$.
In this case,
$\pi^* L -2 (p+2)E$ is big.
Hence $(1) \Rightarrow (2)$ follows from \autoref{thm_suff}
\end{proof}
\subsection{Remark}
By a similar argument as in the proof of \autoref{main_lem},
we can show the following lemma,
which recovers \cite[Theorem 2.5 (1)]{KL}.
We denote by $\coeff_C(D)$ the coefficient of a prime divisor $C$ in a $\mathbb {Q}$-divisor $D$.
\begin{lem}
Let $X$ be a smooth projective surface and $B$ be an ample $\mathbb {Q}$-divisor on $X$.
Let $\pi : X' \rightarrow X $ be the blow-up at a point $x \in X$ and $E \subset X'$ be the exceptional divisor.
Assume that $\pi^* B -2E $ is big
and let $ \pi^* B -2E =P+N$ be the Zariski decomposition.
If
\begin{align}\label{eq_condition}
\coeff_{C'} (N) < 1 \quad \text{ for any prime divisor } C' \text{ on } X' \text{ with } (C'.E)=1,
\end{align}
there exists an effective $\mathbb {Q}$-divisor
$F_0 \equiv c_0 B$ for some $0 < c_0 < 1$ such that $\mathcal {J}(X,F_0 ) = \mathcal {I}_x$ holds in a neighborhood of the point $x$.
\end{lem}
\begin{proof}
As in the proof of \autoref{main_lem},
we take $t$ with $0 < t-2 \ll 1$ and let $\pi^* B -t E =P_t +N_t$ be the Zariski decomposition.
We note that $N_t$ also satisfies \ref{eq_condition} by \cite[Proposition 1.16]{BKS},
i.e.\
$\coeff_{C'} (N_t) < 1$ holds if $(C'.E) = 1$.
We set $D'$ on $X'$ and $D=\pi_* D'$ on $X$ as \ref{eq_def_D}.
We can take $D'$ so that $\coeff_{C'} (D') < 1$ holds if $(C'.E) = 1$.
Since $\mult_x (D) \geq t >2$, we have
$
c:= \lct_x(X,D)
\leq \frac{2}{t} <1.
$
Assume that the minimal center of $(X,cD)$ at $x$ is a curve $C$.
Then $C $ is smooth at $x$ and
hence we have $(C'.E) =1$ for the strict transform $C'$ of $C$.
Thus $\coeff_C (D)= \coeff_{C'} (D') < 1$ holds by the definition of $D'$.
Since $c <1$, we have $\coeff_C (c D) < 1$,
which contradicts the assumption that $C$ is the minimal center of $(X,cD)$.
Thus the minimal center of $(X,cD)$ at $x$ is $\{x\}$.
As in the proof of \autoref{lem_minimal_center},
we can take $F_0$ as a small perturbation of $cD \equiv cB$.
\begin{comment}
Set $F:=c D \equiv c B$.
We take a general effective $\mathbb {Q}$-divisor $F_1\equiv B$ containing $x$ on $X$
and set $F_0 := c_{\varepsilon} F + \varepsilon F_1 \equiv (c_{\varepsilon}c + \varepsilon) B$ for $0 < \varepsilon \ll 1$, where
$c_{\varepsilon}= \max\{ s >0 \, | \, (X,s F+\varepsilon F') \text{ is log canonical}\}$.
Then $\{x\} $ is the unique log canonical center of $(X,F_0)$ at $x$.
Since $\varepsilon $ is sufficiently small,
$c_0:= c_{\varepsilon}c + \varepsilon$ is sufficiently close to $c$
and hence we have $c_0 <1$.
\end{comment}
\end{proof}
\section{Naive Questions}
We end this note by questions in higher dimensions.
As a generalization of Fujita's base point freeness conjecture \cite{Fu},
the following conjecture is known:
\begin{conj}[{\cite[5.4 Conjecture]{Ko2}}]\label{conj_fujita}
Let $Y$ be a smooth projective variety, $y \in Y$ be a point and
$L$ be a nef and big line bundle on $Y$ . Assume that if $y \in Z \subsetneq Y$ is an irreducible
subvariety then
\[
(L^{\dim Z}.Z) \geq (\dim Y)^{\dim Z}, \quad \text{and} \quad (L^{\dim Y}) > (\dim Y)^{\dim Y}.
\]
Then $K_Y + L$ is base point free at $y$.
\end{conj}
The condition (1) in \autoref{thm_KL} and the condition $(L^2) > 4(p+2)^2$ in \autoref{main_rem} are equivalent to
\[
(B.C) >1=(\dim C)^{\dim C}, \quad \text{and} \quad (B^2) > 4=(\dim X)^{\dim X}
\]
respectively,
where $B=\frac{1}{p+2} L$ and $C \subset X$ is any elliptic curve.
Naively,
we have the following question as an analog of \autoref{conj_fujita}:
\begin{ques}\label{ques1}
Let $p$ be a non-negative integer,
$X$ be an abelian variety,
and $L$ be an ample line bundle on $X$.
Set $B=\frac{1}{p+2} L$.
Assume that $(B^{\dim Z} .Z) > (\dim Z)^{{\dim Z}}$ holds for any abelian subvariety $\{o\} \neq Z \subset X$.
Then does property ($N_p$) hold for $L$?
\end{ques}
We note that in this question we assume $(B^{\dim Z} .Z) > (\dim Z)^{{\dim Z}} $, which is weaker than $(B^{\dim Z} .Z) \geq (\dim X)^{\dim Z}$ for $Z \subsetneq X$.
By \autoref{thm_suff},
we also have a little stronger version:
\begin{ques}\label{ques2}
Let $p$ be a non-negative integer,
$X$ be an abelian variety,
and $L$ be an ample line bundle on $X$.
Let $\pi : X' \rightarrow X $ be the blow-up at the origin $o \in X$.
Set $B=\frac{1}{p+2} L$.
Assume that for any abelian subvariety $\{o\} \neq Z \subset X$, $\left( \pi^*B - (\dim Z)E \right) |_{Z'} $ is big, where $Z' \subset X'$ is the strict transform of $Z$.
Then does property ($N_p$) hold for $L$?
\end{ques}
\begin{rem}
If $X$ is simple, that is, there exists no abelian subvariety $\{o\} \neq Z \subsetneq X$,
then $(B^{g}) > g^g$ is the unique condition in \autoref{ques1} for $g=\dim X$.
The following are known in any dimension.
\begin{enumerate}
\item By \cite[Corollary B]{LPP},
($N_p$) holds for $L$
if $(X,L)$ is very general
and $(B^g) > (4g)^g /2$.
\item J.~Iyer studied the projective normality, i.e.\ ($N_0$), for simple abelian varieties.
By \cite[Theorem 1.2]{Iy},
if $X$ is simple and $(B^g) >( g{!} )^2$ for $B=\frac12 L$,
then $L$ satisfies ($N_0$).
We note that
\[
( g{!} )^2 = \prod_{i=1}^g i (g+1-i) \geq \prod_{i=1}^g g =g^g.
\]
\end{enumerate}
\end{rem}
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Q: Laravel 5.6 update foreign key in database after storing resource I have a resource 'imagerequest' and a resource 'events'. For each of these resources I have a Modal and the relationship is as following:
1 Imagerequest can have many Events:
Imagerequest model:
/**
* Get the events for an Image Request.
*/
public function events()
{
return $this->hasMany('App\Event');
}
Event model:
/**
* Get the Image Request that belongs to the Event
*/
public function imagerequest()
{
return $this->belongsTo('App\ImageRequest');
}
Now in my form when creating a new Imagerequest I have an other form to create multiple events but my problem is when I store a new Event there is no image_request_id because the new image_request_id doesn't exist yet since I haven't created the imagerequest yet.
So in my database image_request_id is always empty
id user_id image_request_id location trainnr created_at updated_at
1 53 NULL location NULL 2019-05-28 10:49:45 2019-05-28 10:49:45
Do I need to write an update query after saving the imagerequest to fill up this image_request_id?
How I create a new Event in my EventController:
$event = Auth::user()->events()->save(new Event($request->all()));
How I create a new Imagerequest in my ImagerequestController:
$imageRequest = new ImageRequest(
array_merge(
$request->all(),
['status' => self::STATUS_NEW]
)
);
$imageRequest = Auth::user()->imageRequests()->save($imageRequest);
EDIT my front end for adding an event
A: You need to add the relationship
have a look at the docs for more help with this
From my quick look you can do it something like this:
$event = Auth::user()->events()->save(new Event($request->all()));
$imageRequest = new ImageRequest(
array_merge(
$request->all(),
['status' => self::STATUS_NEW]
)
);
$event->imagerequest()->create($imageRequest);
I would also advise that you validate any request data before you update the database also.
EDIT base on more info
you can do something like this then:
$imageRequest->attach($request->eventIds);
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{
"redpajama_set_name": "RedPajamaStackExchange"
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Q: How to get the value of a checkbox in data template of a combobox in datagrid in silverlight4 I have a combobox inside a datagrid in silverlight4. I have placed a check box inside data template of combobox to make combobox multi-selectable. Now i want to get the values for selected items in combobox.So how i can do that ?
Here is my code :
<ComboBox Name="cbxitmes" Height="23" Width="255" IsSynchronizedWithCurrentItem="False"
ItemsSource="{Binding Path=GetItems,Mode=TwoWay}"
SelectedValue="{Binding Mode=TwoWay, Path=myname}" SelectedValuePath="myvalue">
<ComboBox.ItemTemplate>
<DataTemplate>
<CheckBox Content="{Binding myname}" ></CheckBox>
</DataTemplate>
</ComboBox.ItemTemplate>
</ComboBox>
Please Help me guys.
Thanks,
A: Take a look at this example, it might help http://www.codeproject.com/Tips/452756/Add-checkbox-inside-Combobox-in-Silverlight
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{"url":"https:\/\/electronics.stackexchange.com\/questions\/linked\/72582","text":"16k views\n\n### OR-ing power supplies (diode or mosfet)\n\nI have got two power supply options on my board: USB 5.0V DC Power Jack of 5.0V My intention is to design a or-ing power supply option, as either of them will be used at a time. Possible solutions ...\n12k views\n\n### What is killing my MOSFETs\n\nThis is my first post here on electronics stackexchange. I am a hobbyits in electronics, and a professional in programming. I am working on a inductor circuit to heat a workpiece. I have a working ...\n9k views\n\n### Why MOSFET source is indicated with arrow ?\n\nI know that , a basic MOSFET contains source and drain , and either it's a NMOS or PMOS ; it is indicated by an arrow at source . But let's look at a fabricated NMOS. Here we can easily see that ...\n9k views\n\n### What is MOSFET gate drive capability and why do I care about it?\n\nSomeone told me that this circuit has \"poor gate drive capability\": simulate this circuit \u2013 Schematic created using CircuitLab What exactly does that mean? I tested it with an LED as a load ...\n19k views\n\n### Why have a bulk terminal at all? (MOSFET)\n\nI must be overlooking something but I really don't see the need to have a bulk terminal. I am studying the body effect whereby it somehow interferes with the threshold voltage. I don't fully ...\n2k views\n\n### Why does CircuitLab have the MOSFET symbols it has?\n\nCircuitLab has [used to have -- they have changed their symbols since this question was asked] these goofy things: simulate this circuit \u2013 Schematic created using CircuitLab My eyes are much ...\n7k views\n\n### NMOS FET with a negative drain\n\nI'm trying to create a switch between a -15V and +15V supply using a gate signal of 0V to +3V. With the positive-going Gate voltage, I tried an NMOS FET. This works as expected with a drain of +10V ...\n4k views\n\n### What does \u201cno bulk\u201d mean in MOSFETs?\n\nThe 3rd and 4th column in the image below (from wikipedia) says \"enhancement, no bulk\". What does \"no bulk\" mean?\n4k views\n\n### MOSFET bulk connection to the source\n\nWhy is bulk always known to be associated with source and not the drain? What would be the result or outcome if I connect the bulk to the drain instead of the source?\n993 views\n\n### How do SRAM bit line \u201cgates\u201d work?\n\nI'm currently learning about the operation of 6 transistor static random access memory cells and I've hit a wall in understanding exactly how the read\/write operations work. More specifically I don't ...\n357 views\n\n### Does linearity of a N-MOSFET continue for $V_{DS}$ below $0V$?\n\nI operate a N-MOSFET (2N7000) in a simulator in the linear region very close to $V_{DS}=0$. In fact $V_{DS}$ is between $-50nV$ and $+50nV$. In the simulator the output seems linear, but is ...\n342 views\n\n### Why is there voltage out of the gate of my transitor\n\nI'm a big beginner in electricity :-D I just discovered the \"Circuit wizard\" tool and I'm drawing a circuit with a \"MOSFET n channel\" transistor. As I understand, when there is voltage on the gate, ...\n519 views\n\n### Would a four terminal mosfet work equally well if the source-drain polarity were switched?\n\nI noticed a question regarding mosfet polarity asked here and wondered whether a four terminal mosfet would behave the same way under reversed polarity. Since the four terminal mosfet has no internal ...","date":"2020-01-18 12:51:36","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\": 0.6202523112297058, \"perplexity\": 2694.2441896623527}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2020-05\/segments\/1579250592565.2\/warc\/CC-MAIN-20200118110141-20200118134141-00536.warc.gz\"}"}
| null | null |
{"url":"https:\/\/dmtcs.episciences.org\/7487","text":"## Darij Grinberg - The Elser nuclei sum revisited\n\ndmtcs:7012 - Discrete Mathematics & Theoretical Computer Science, June 3, 2021, vol. 23 no. 1 - https:\/\/doi.org\/10.46298\/dmtcs.7012\nThe Elser nuclei sum revisited\n\nAuthors: Darij Grinberg\n\nFix a finite undirected graph $\\Gamma$ and a vertex $v$ of $\\Gamma$. Let $E$ be the set of edges of $\\Gamma$. We call a subset $F$ of $E$ pandemic if each edge of $\\Gamma$ has at least one endpoint that can be connected to $v$ by an $F$-path (i.e., a path using edges from $F$ only). In 1984, Elser showed that the sum of $\\left(-1\\right)^{\\left| F\\right|}$ over all pandemic subsets $F$ of $E$ is $0$ if $E\\neq \\varnothing$. We give a simple proof of this result via a sign-reversing involution, and discuss variants, generalizations and refinements, revealing connections to abstract convexity (the notion of an antimatroid) and discrete Morse theory.\n\nVolume: vol. 23 no. 1\nSection: Combinatorics\nPublished on: June 3, 2021\nSubmitted on: December 22, 2020\nKeywords: Mathematics - Combinatorics,05C30, 18G85, 57Q70","date":"2021-09-19 05:14:04","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.582746148109436, \"perplexity\": 659.5645359176339}, \"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-2021-39\/segments\/1631780056711.62\/warc\/CC-MAIN-20210919035453-20210919065453-00155.warc.gz\"}"}
| null | null |
Q: I'd like some clarification in this theorem proof. Let $(P,Sc,1)$ a Peano's system, then $P=\{1\}\cup Sc\{P\}$
They use the third Peano's axiom, in which if $A\subseteq P, 1\in A$ and $Sc(a)\subseteq A\Rightarrow A=P$.
But their proof says in the begining: It's enough to prove that $Sc(A)\subseteq A$
How do they know that $A\subseteq P$?
A: By definition, $P$ is closed under sucessors, i.e., $\operatorname{Sc}(P)\subseteq P$. Also, by definition $1\in P$, so $\{1\}\subseteq P$.
So if we define $A=\{1\}\cup\operatorname{Sc}(P)$, it is the union of two subsets of $P$, hence a subset of $P$ itself. The proof goes on to showing that $1\in A$ and $\operatorname{Sc}(A)\subseteq A$, from which the third Peano axiom (induction) implies that $P=A=\{1\}\cup\operatorname{Sc}(P)$.
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{
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}
| 7,981
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Ectodermal dysplasia with corkscrew hairs is a skin condition with salient features including exaggerated pili torti, scalp keloids, follicular plugging, keratosis pilaris, xerosis, eczema, palmoplantar keratoderma, syndactyly, onychodysplasia and conjunctival neovascularization.
See also
Skin lesion
References
Further reading
Genodermatoses
Ectoderm
|
{
"redpajama_set_name": "RedPajamaWikipedia"
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Home Current-Affairs-gk-Quiz DAILY-GK-QUIZ gk-Quiz QUIZ Daily Current Affairs Quiz: 01 June 2018
Daily Current Affairs Quiz: 01 June 2018
IBTSINDIA.COM - Friday, June 01, 2018 Current-Affairs-gk-Quiz, DAILY-GK-QUIZ, gk-Quiz, QUIZ,
Q1. Every year, on which of the following day, the World Health Organization and partners mark World No Tobacco Day (WNTD)?
(a) 16 May
(b) 19 May
(c) 31 May
(d) 27 May
(e) 24 May
S1. Ans.(c)
Sol. Every year, on 31 May, World Health Organization and partners mark World No Tobacco Day (WNTD). The theme of World No Tobacco Day 2018 is "Tobacco and heart disease."
Q2. The Third Home Affairs' Dialogue between India and U.K. was held recently in ______________.
(a) Bengaluru
(b) New Delhi
(c) Chandigarh
(d) Guwahati
(e) Pune
S2. Ans.(b)
Sol. The Third Home Affairs' Dialogue between India and U.K. was held in New Delhi. The Government of India Delegation was led by Mr Rajiv Gauba, Union Home Secretary and the Delegation from Home Office of the U.K. was led by its Second Permanent Secretary, Ms. Patsy Wilkinson.
Q3. As many as 26 countries, including India, will participate in June 2018 in the world's largest international maritime biennial exercise, named ___________.
(a) RAMPAGE
(b) PACIRIM
(c) RIMPACIF
(d) RIMPAC
(e) None of the given options is true
S3. Ans.(d)
Sol. As many as 26 countries, including India, will participate in the biennial Rim of the Pacific, RIMPAC military exercise from June 27 to August 2, in and around the Hawaiian Islands and Southern California. The announcement was made by the Pentagon. It is the world's largest international maritime exercise.
Q4. The joint military exercise SURYA KIRAN between India and Nepal has begun at Pithoragarh in Uttarakhand. This is the __________ edition of this exercise.
(a) 9th
(b) 10th
(c) 11th
(d) 12th
(e) 13th
S4. Ans.(e)
Sol. The joint military exercise SURYA KIRAN-13 between India and Nepal began at Pithoragarh in Uttarakhand. The military exercise is a biannual event which is conducted alternatively in Nepal and India every six months.
Q5. India has released 33.10 Crore rupees to Nepal towards the cost of two road packages of Birgunj-Thori Road being implemented under which of the following projects in Nepal with Government of India's grant assistance?
(a) Postal Highway Projects
(b) International Highways Development Project
(c) Bharatmala Project
(d) Golden Quadrilateral Project
(e) Project Sadak Sahayata
S5. Ans.(a)
Sol. India has released 33.10 Crore rupees to Nepal towards the cost of two road packages of Birgunj-Thori Road being implemented under Postal Highway Projects in Nepal with Government of India's grant assistance. The amount has been released towards 25% of the tendered cost (including 10% mobilization advance) of the Postal Highway Projects.
Q6. Yoga guru Baba Ramdev's Patanjali launched a new swadeshi messaging application called ________ to compete with WhatsApp.
(a) Shankh
(b) Kimbho
(c) Samvad
(d) Parichay
(e) Doorbhash
Sol. Yoga guru Baba Ramdev's Patanjali launched a new swadeshi messaging application called 'Kimbho' to compete with WhatsApp. Kimbho caters to private and group chats with free phone and video calling. The tagline of Kimbho is "Ab Bharat Bolega".
Q7. India's biggest lender State Bank of India (SBI) has revised its interest higher on retail fixed deposits or FDs below ____________.
(a) Rs. 50 lakh
(b) Rs. 75 lakh
(c) Rs. 1 crore
(d) Rs. 1.25 crore
(e) Rs. 1.50 crore
Sol. India's biggest lender State Bank of India (SBI) has revised its interest higher on retail fixed deposits or FDs below Rs. 1 crore. SBI has revision interest rates by up to 25 basis points in select maturities. The interest rate on SBI FDs with a maturity of one year to less than two years has been increased to 6.65% for the public, from 6.4% earlier.
Q8. Name the Indian botanist who has received the prestigious Linnean Medal in Botany from the Linnean Society of London and has become the first Indian to win the award.
(a) K. S. Manilal
(b) G. S. Venkataraman
(c) Janaki Ammal
(d) Birbal Sahni
(e) Kamaljit S. Bawa
Sol. Indian botanist Kamaljit S. Bawa (president of Bengaluru-based non-profit Ashoka Trust for Research in Ecology and the Environment) received the prestigious Linnean Medal in Botany from the Linnean Society of London. Dr. Bawa is the first Indian to win the award ever since it was first constituted in 1888.
Q9. Name the Kerala cartoonist who has won an international award, in the best caricature category at the 13th edition of the World Press Cartoon awards, instituted by an organization based in Lisbon, Portugal.
(a) O. V. Vijayan
(b) Thomas Antony
(c) Ravi Shankar
(d) K. Shankar Pillai
(e) P. K. Manthri
Sol. Kerala cartoonist Thomas Antony has won an international award, in the best caricature category. Antony is among the nine winners at the 13th edition of the World Press Cartoon awards, instituted by an organisation based in Lisbon, Portugal.
Q10. What is the theme of World No Tobacco Day 2018?
(a) Tobacco – a threat to development
(b) Get Ready for Plain Packaging
(c) Tobacco use and advocating for effective policies to reduce tobacco consumption
(d) Tobacco and heart disease
(e) Raise taxes on tobacco
S10. Ans.(d)
Q11. Name the famous Indian personality who has been awarded the 'most inspiring icon of the year for Social Welfare' award by the Dadasaheb Phalke International Film Festival (DPIFF) for his exemplary contribution in the field of social welfare.
(a) Salman Khan
(b) Shabana Azmi
(c) Nandita Das
(d) Yuvraj Singh
(e) Nafisa Ali
Sol. Indian star cricketer Yuvraj Singh has been awarded the 'most inspiring icon of the year for Social Welfare' award by the Dadasaheb Phalke International Film Festival (DPIFF) for his exemplary contribution in the field of social welfare.
Q12. Name the Former Himachal Pradesh governor and Congress leader who had passed away recently in Indore.
(a) Urmila Singh
(b) V. S. Ramad
(c) Vishnu Sadashiv Kokje
(d) Hokishe Sema
(e) Mahaveer Prasad
S12. Ans.(a)
Sol. Former Himachal Pradesh governor and Congress leader Urmila Singh died due to prolonged illness at a private hospital in Indore. She was 71.
Q13. Name the tech company that has completed the acquisition of WongDoody Holding Company, a US-based digital creative and consumer insights agency, for a total consideration of upto $75 million.
(a) TCS
(b) Reliance Jio
(c) Infosys
(d) WIPRO
(e) Microsoft
S13. Ans.(c)
Sol. Infosys has completed the acquisition of WongDoody Holding Company, a US-based digital creative and consumer insights agency, for a total consideration of upto $75 million.
Q14. The Government of India and the World Bank signed a loan agreement to provide additional financing for the Pradhan Mantri Gram Sadak Yojana (PMGSY) Rural Roads Project, implemented by Ministry of Rural Development, Govt. of India. What is the amount of this loan agreement?
(a) $100 million
(b) $200 million
(c) $300 million
(d) $400 million
(e) $500 million
S14. Ans.(e)
Sol. The Government of India and the World Bank signed a $500 million loan agreement to provide additional financing for the Pradhan Mantri Gram Sadak Yojana (PMGSY) Rural Roads Project, implemented by Ministry of Rural Development, Govt. of India, which will build 7,000 km of climate resilient roads, out of which 3,500 km will be constructed using green technologies.
Q15. Name the PSU which has emerged as India's most profitable state-owned company for the second consecutive year.
(a) HPCL
(b) GAIL
(c) BHEL
(d) Indian Oil Corporation
(e) ONGC
Sol. Indian Oil Corporation has emerged as India's most profitable state-owned company for the second consecutive year. Indian Oil posted a record profit of Rs21,346 crore in 2017-18, followed by ONGC, whose profit stood at Rs19,945 crore.
Current-Affairs-gk-Quiz, DAILY-GK-QUIZ, gk-Quiz, QUIZ
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\section{INTRODUCTION}
First revealed by the {\em Einstein} observatory, the spectroscopic
temperature of the intracluster medium (ICM)
in many cD clusters of galaxies decreases toward the center by a factor of
$2-3$ (e.g., Canizares et al. 1988; Mushotzky \& Szymkowiak 1988; see also Fabian 1994
for a review). Generally, the appearance of such a central cool
component (hereafter CCC) has been described in two
possible but competing ways. One is a single-phase (hereafter 1P) scenario, in which
a mono-nature plasma showing an inward temperature decrease is assumed to permeate the whole
cluster (e.g., Allen et al. 2001; Vikhlinin et al. 2006). The other is a two-phase (hereafter 2P)
scenario, in which the cluster's central region is assumed to be
occupied by a mixture of two gas components characterized by
discrete temperatures (i.e., hot and cool), of which the relative
volume filling factors slowly vary with radius (e.g.,
Fukazawa et al. 1994, 1998; Takahashi et al. 2009; see Makishima et al. 2001 for a review). Although
in many cases it is difficult to distinguish between the 1P and 2P models due to insufficient
data quality, the implications of these models on the formation
and evolution of the CCC are intrinsically
different. In the 1P scenario the CCC is simply interpreted as the dense cluster core, where radiative
losses have significantly lowered the gas temperature (see, e.g.,
Peterson \& Fabian 2006 for a review), while
the 2P scenario favors an interpretation that the CCC is caused
by the cD galaxy, rather than being a cooling portion
of the ICM (Makishima et al. 2001 and references therein). Furthermore, the choice
between the two scenarios has considerable impact on the accuracies of the measurements
of both gas metallicities (e.g., the ``Fe-bias'' discussed in Buote
2000) and, in some extreme cases, gravitating mass distributions
in clusters. For the latter one, the systematic mass bias induced by different ICM models can reach
$\sim 10$\% for the cluster's central region,
which is comparable to the typical deviation between the results
obtained in X-ray and lensing studies
(e.g., Nagai et al. 2007; Mahdavi et al. 2008). Hence, it is of fundamental and urgent importance
to determine which of the two scenarios is valid, based on the analysis of existing
high quality X-ray data of nearby, bright clusters.
So far, only a few works have been done to examine the preference
between the 1P and 2P ICM models of
cD groups and clusters. Of these, Fukazawa et al. (1994) and Ikebe et al.
(1999) used the {\em ASCA} data to show that a 2P model with temperatures of 1.4 keV and 4.4 keV for the two components is required to describe the CCC of
the Centaurus cluster. Their results were recently confirmed by the analysis of the high quality data acquired
with the European Photon Imaging Camera (EPIC) and Reflection Grating Spectrometer (RGS) onboard {\em XMM-Newton}\/\ (Takahashi
et al. 2009). The preference for the 2P scenario was also reported in the works on the Fornax cluster
(Ikebe et al. 1996; Matsushita et al. 2007) and the NGC 5044 group (Buote \& Fabian 1998; Buote et al. 2003;
Tamura et al. 2003) with {\em ASCA}, {\em Chandra}, {\em XMM-Newton},
and {\em Suzaku} observations. However, due to the lack of such studies on more representative relaxed clusters
or groups that exhibit strong CCCs, it is not yet clear whether or not the 2P scenario do surpass
the 1P scenario in general, and if yes, to what extent the 2P model can be applied to constrain the origin of
the CCC.
To address these issues, we study in this work the rich, bright cD
cluster of galaxies Abell 1795 (A1795 hereafter), by jointly analyzing
the archival {\em Chandra}, {\em XMM-Newton}, and {\em Suzaku}\/\ data. A1795 is suited to our
purpose as it is a relaxed, nearby (at a redshift of z $=0.0625$) cluster of galaxies
with a luminous X-ray emission
(with a $2.0-10.0$ keV luminosity of $L_{X}\approx 1.0\times10^{45}$ ergs $\rm s^{-1}$; Xu et al. 1998), which is found
peaked at the giant cD galaxy PGC 049005 (Jones \& Forman 1984; Briel \&
Henry 1996). Under the 2P assumption,
Xu et al. (1998) resolved a cool ($T_{X} \approx 1.7$ keV) and a hot
($T_{X} \approx 6.5$ keV) component in
the central 220$h_{71}^{-1}$ kpc region of this cluster with {\em
ASCA}, although Ettori et al. (2002) employed the
1P model instead to fit {\em Chandra} spectra extracted from
the same region, and found an inward temperature decrease
to $\approx 2.5$ keV. In the past 11 years, X-ray observations of this
cluster have been accumulated to $\approx 300$ ks, 140 ks, and 270 ks
with {\it Chandra}, {\it XMM-Newton}, and {\it Suzaku}, respectively. Clearly this
is an ideal target for utilizing as much of existing high-quality
X-ray data as possible to judge which spectral model is more correct.
The layout of this paper is as follows. Section 2 gives a brief description
of our data reduction
procedure. The data analysis and results are described in \S 3. We
discuss the physical implication of our analysis in \S 4, and
summarize our work in \S 5. Throughout the
paper we assume a Hubble constant of $H_0=71$$h_{71}$ km s$^{-1}$
Mpc$^{-1}$, a flat universe with the cosmological parameters of
$\Omega_M=0.27$ and $\Omega_\Lambda=0.73$, and quote errors by the
90\% confidence level unless stated otherwise. At the redshift of this
cluster, 1$^{\prime}$\ corresponds to about 72$h_{71}^{-1}$ kpc. To compare with previous
results, we adopt the solar abundance standards of Anders \& Grevesse (1989).
\section{OBSERVATION AND DATA REDUCTION}
\subsection{X-ray Observation}
\subsubsection{{\it Chandra}}
In Table 1, we list 11 {\em Chandra}\/\ datasets of A1795 obtained with
the advanced CCD imaging spectrometer (ACIS),
which are all used in the analysis. Among the 11 observations, five
ACIS-S3 exposures focused on the cluster center, and six ACIS-I
exposures covered off-center regions up to $r \sim 1.5$ Mpc. All data
were telemetered in VFAINT mode, except for the one with ObsID 494 in FAINT
mode. Using CIAO v4.4 and CALDB v4.4.7, we removed bad pixels and
columns, as well as events with {\em ASCA}\/\ grades 1, 5, and 7. We then
executed the gain, CTI, and astrometry corrections. The cumulative exposure
time was reduced from $\approx 170$ ks to $\approx 163$ ks after removing
intervals contaminated by occasional background flares with count rate
$>$$20$\% of mean value, which were detected by
examining $0.3-10.0$ keV lightcurves extracted from source free
regions near the CCD edges (e.g., Gu et al. 2009). When available, ACIS-S1
data were also used to crosscheck the determination of the
contaminated intervals. The obtained clean exposure time for each observation is listed
in Table 1. In Figure 1{\it a} we show the combined ACIS
image, which has been corrected for exposure but not for background.
\subsubsection{{\it XMM-Newton}}
{\em XMM-Newton}\/\ was used to observe the central and the south periphery
regions of A1795 on 2000 June 26
and 2003 January 13, respectively, both with the EPIC operated in the full window mode with
thin filter, and with the RGS in the
spectroscopy mode (Table 1). Data reduction and calibration were carried out
with SAS v11.0.1. In the screening process we set $FLAG=0$,
and kept events with $PATTERNs$ 0--12 for MOS cameras and events with $PATTERNs$ 0--4
for pn camera. By examining lightcurves extracted in
$10.0-14.0$ keV and $1.0-5.0$ keV from source free regions, we rejected time intervals
affected by hard- and soft-band flares, respectively, in
which the count rate exceeds a 2$\sigma$ limit above the quiescent
mean value (e.g., Katayama et al. 2004; Nevalainen et al. 2005).
The cleaned MOS1, MOS2, and pn datasets of the central pointing have exposure times of 43.6
ks, 40.3 ks, and 30.2 ks, respectively (Table
1). The RGS data were screened using the method described in Tamura et al. (2001).
\subsubsection{{\it Suzaku}}
As listed in Table 1, A1795 was also observed by {\em Suzaku}\/\ on 2005
December 10, with five pointings aimed at its
central region, near south and north regions (offset $\approx 12^{\prime}$), and
far south and north regions (offset $\approx 24^{\prime}$). The onboard
X-ray Imaging Spectrometer (XIS; Koyama et al. 2007)
and the Hard X-ray Detector (HXD; Takahashi et al. 2007) were both operated in
normal modes. The same XIS datasets were already utilized by
Bautz et al. (2009). In analyzing the obtained data, we used the software
HEASoft 6.11.1 and latest CALDB (20111109 for the XIS and 20110913 for
the HXD). We started with version 2.0
processing data, and removed the data obtained either near South Atlantic
Anomaly or at low elevation angles from the Earth rim
($<5^{\circ}$ and $<20^{\circ}$ for night and day,
respectively). Cut-off rigidity criteria of $> 8$ GV for HXD-PIN data
were also applied. For the XIS instrument, we further examined $0.3-10.0$ keV
lightcurves of a source free region in each CCD, and
filtered off anomalous time bins with count rate above the 2$\sigma$
limit of the quiescent mean value. The combined XIS image is shown in Figure 1{\it b}.
The data obtained in the far north field are contaminated seriously by solar wind charge exchange
emission (Fujimoto et al. 2007), and thus are not included in the
following data analysis. The obtained clean exposure times are listed
in Table 1. We do not consider HXD-GSO data in this paper.
\subsection{Point Sources, Background, and Systematic Uncertainties}
We excluded point sources detected beyond a $3\sigma$ threshold on the ACIS images
with the CIAO tool {\tt celldetect}, and masked the corresponding regions on the
EPIC and XIS images after considering the differences in the Point Spread Functions (PSFs) of
the instruments. The mask regions have radii of $12^{\prime\prime}$
and $70^{\prime\prime}$ for the EPIC and XIS, respectively. The point sources outside the ACIS sky
coverage were detected and masked using the EPIC images. The ancillary
response files (ARFs) and the redistribution matrix
files (RMFs) were calculated with the CIAO tools {\tt mkwarf} and {\tt
mkacisrmf} for the ACIS, and
with the SAS tools {\tt arfgen} and {\tt rmfgen} for the EPIC. We calculated the RMFs for the RGS
using the SAS tool {\tt rgsrmfgen}, after blurring the line spread function by convolving the
raw RMFs with the surface brightness profile (\S 3.3.1) extracted from
the $0.5-8.0$ keV ACIS image along the dispersion direction. As for the XIS, we
generated the response files using {\tt xissimarfgen} (version 2008-04-05) and
{\tt xisrmfgen} (version 2007-05-14). To enhance the precision of the
Monte-Carlo simulation carried out by {\tt xissimarfgen}, we created the initial
feed photon list by combining all the exposure-corrected and background-subtracted
ACIS-I images. The method to
build the response of HXD-PIN is described in \S 3.4.
We estimated the background as a combination of three independent
components, i.e., non X-ray background (NXB), cosmic X-ray background
(CXB), and Galactic emission. For all the XIS data, we created the NXB spectra
using a dark earth observation database. The CXB and
Galactic emission components in the XIS data were estimated by analyzing a spectrum extracted from
a region about $26^{\prime}-30$$^{\prime}$\ ($\approx
1.9-2.2$$h_{71}^{-1}$ Mpc at the distance of
A1795) south of the cluster center, which is covered by the dataset
with ObsID = 800012050 (Table 1); a similar background region was used in Bautz et al. (2009).
Since this region is outside the virial
radius $r_{\rm 200} = 1.9$$h_{71}^{-1}$ Mpc, where the brightness of cluster emission is expected to be $\sim 10^{-13}$ $\rm ergs$ $\rm cm^{-2}$ $\rm
s^{-1}$ $\rm deg^{-2}$ in $0.5-2.0$ keV (Roncarelli et al. 2006),
lower than the detection limit of the XIS, i.e., $\sim 10^{-12}$ $\rm ergs$ $\rm cm^{-2}$ $\rm
s^{-1}$ $\rm deg^{-2}$ (Bautz et al. 2009), the ICM component can be ignored in the background fitting. After
subtracting the NXB from the extracted spectrum, we fitted
the resulting spectrum with an absorbed power law model (photon index
$\Gamma = 1.4$) describing the CXB, and two unabsorbed optically thin thermal models
(abundance = 1 $Z_\odot$, temperatures = 0.08 keV
and 0.2 keV; Snowden et al. 1998) describing the Galactic emission. The $2.0-10.0$ keV CXB flux was
estimated as $6.6\times10^{-8}$ $\rm ergs$ $\rm cm^{-2}$ $\rm s^{-1}$
$\rm sr^{-1}$, which is consistent with the results of Kushino et
al. (2002). The obtained CXB + Galactic
background template was applied to all the subsequent
XIS spectral analysis assuming the same uniform CXB + Galactic emission distribution over
the field of view (e.g., Sato et al. 2008).
In order to achieve an accordant background model for the {\em Chandra} ACIS, we utilized
the above CXB + Galactic background template and calculated the ACIS
NXB spectra in an adaptive way as follows. For all ACIS-I
observations, background spectra were
extracted from regions $17^{\prime}-21$$^{\prime}$\ ($\sim 1.3-1.5$$h_{71}^{-1}$ Mpc) away
from the cluster center, where the ICM brightness was measured to be
weak but cannot be neglected, on a level of 10\% to 40\% of the CXB + Galactic
components (Bautz et al. 2009). We subtracted the CXB + Galactic
components from the extracted spectra, whose brightness was fixed at the
XIS result obtained above, and fitted the resulting spectra
with an empirical NXB model that consists of a broken power law
and five narrow Gaussian lines (e.g., Humphrey \& Buote 2006), plus a thermal APEC
model for the cluster emission. Since the
temperature and metal abundance of this ICM component cannot be well
constrained, due to the relatively high
NXB level of the ACIS, we fixed them to previous {\em Suzaku} results (temperature =
2.5 keV, abundance = 0.1 $Z_\odot$) reported in Bautz et
al. (2009). In the $r=17^{\prime}-21$$^{\prime}$\ region, average surface brightness of the ICM
component was obtained as $1.7\pm0.8 \times 10^{-12}$ $\rm ergs$ $\rm cm^{-2}$ $\rm
s^{-1}$ $\rm deg^{-2}$ in $0.5-2.0$ keV, which is about $25$\% of the
CXB + Galactic components, in rough consistent with Bautz et al. (2009).
The ACIS-I NXB model was thus determined. The same procedure was applied to all ACIS-S data, but
source free regions on the S1 chip were used as the background regions
instead. To crosscheck above background model with the blank sky
templates, we applied both backgrounds to the spectral analysis in
\S3.1.1. By fitting the background-subtracted ACIS spectra extracted from
$9^{\prime}.8-17^{\prime}.0$ with a single-phase thermal model, we
found that the ICM temperatures with the two background models differ
by 0.35 keV, which is less than the 1$\sigma$ uncertainties of 0.5 keV. The
error caused by background model is much smaller for the inner
regions, where the ICM emission contributes to
$>80$\% of the total counts in $0.5-10.0$ keV (Table 2).
The background model for the {\em XMM-Newton} EPIC data was determined
with a method similar to that used for the {\em Chandra} ACIS
data. The EPIC spectra extracted from $19-25$$^{\prime}$\ ($\sim
1.4-1.8$$h_{71}^{-1}$ Mpc) away from
the cluster center, covered by the data with ObsID = 0109070201, were
fit with a model consisting of the CXB + Galactic components, the EPIC NXB model
including a broken power law and six Gaussian lines (e.g., Gastaldello et al. 2007; Zhang et al. 2009),
and a cluster component approximated by an APEC model. The temperature and
abundance of the cluster component were fixed at 2.4 keV and 0.1
$Z_\odot$ (Bautz et al. 2009), respectively, and the $0.5-2.0$ keV surface
brightness was obtained as $1.4\pm1.1 \times 10^{-12}$ $\rm ergs$ $\rm cm^{-2}$ $\rm
s^{-1}$ $\rm deg^{-2}$ ($\approx 15$\% of the CXB + Galactic components), in
agreement with Bautz et al. (2009) and our
{\em Chandra} estimate.
The best-fit CXB + NXB + Galactic models were used to create background spectra.
In Figure 2{\it a}, we plot the NXB-subtracted background spectra
against the best-fit CXB and Galactic components for the ACIS, EPIC,
and XIS, as well as the cluster emission component for the former two instruments. For the RGS data, we employed the blank sky template (e.g.,
Takahashi et al. 2009) in the following spectral analysis.
Errors quoted below in the spectral fittings were estimated by
taking into account both statistical and
systematic uncertainties. The former was calculated by scanning
over the parameter space with
the {\tt XSPEC} command {\em steppar}, as the fitting was repeated for a few iterations at each step to
ensure that the actual minimum $\chi^2$ is found. For the latter,
we altered the normalization of the CXB spectra by 10\% (Kushino et
al. 2002) to approximate its field-to-field variation, and
similarly, re-normalized the NXB components by 2\%, 2\%, and 1\% for the ACIS, EPIC,
and XIS data (e.g., Hickox \& Markevitch 2006; De Luca \& Molendi 2004; Tawa et al. 2008),
respectively, to assess the error ranges. We
have also tried assigning a systematic error of 2\% to approximate
the unassessed calibration uncertainties of the EPIC data (e.g., Takahashi et al. 2009), which turned out to have negligible
impacts on the results and thus was not included in the actual fittings.
\section{ANALYSIS AND RESULTS}
\subsection{Azimuthally Averaged Spectral Analysis}
\subsubsection{Single-phase ICM Model}
We extracted the ACIS and EPIC spectra from eight thin annuli,
i.e.,
$0-30$$h_{71}^{-1}$ kpc ($0^{\prime}-0^{\prime}.4$),
$30-51$$h_{71}^{-1}$ kpc ($0^{\prime}.4-0^{\prime}.7$),
$51-80$$h_{71}^{-1}$ kpc ($0^{\prime}.7-1^{\prime}.1$),
$80-116$$h_{71}^{-1}$ kpc ($1^{\prime}.1-1^{\prime}.6$),
$116-238$$h_{71}^{-1}$ kpc ($1^{\prime}.6-3^{\prime}.3$),
$238-354$$h_{71}^{-1}$ kpc ($3^{\prime}.3-4^{\prime}.9$),
$354-707$$h_{71}^{-1}$ kpc ($4^{\prime}.9-9^{\prime}.8$) and
$707-1335$$h_{71}^{-1}$ kpc ($9^{\prime}.8-18^{\prime}.5$), and the XIS
spectra from four thick annuli, i.e., $0-144$$h_{71}^{-1}$ kpc ($0^{\prime}-2^{\prime}.0$),
$144-320$$h_{71}^{-1}$ kpc ($2^{\prime}.0-4^{\prime}.4$),
$320-700$$h_{71}^{-1}$ kpc ($4^{\prime}.4-9^{\prime}.7$) and
$700-1444$$h_{71}^{-1}$ kpc ($9^{\prime}.7-20^{\prime}.0$). When
fitting these spectra, the lower
energy cut was set at 0.7 keV, 0.7 keV, and
0.6 keV for the ACIS, EPIC, and XIS, respectively, while the
upper cut was fixed at 8.0 keV for all the three
detectors. Also, the Si K edge
($1.8-1.9$ keV) was excluded from all the XIS spectra. Independent
fittings were carried out for the {\em Chandra} (ACIS-S and ACIS-I), {\em XMM-Newton}
(MOS and pn), and {\em Suzaku}\/\ (BI and FI) spectrum sets, by applying
a common absorbed APEC model and linking all annuli with {\tt XSPEC}
model PROJCT, which performs projection of 3-D shells onto 2-D
annuli to evaluate the projected emission of outer shells on inner
ones. For each shell, the gas temperature, metal abundance, and column density of the
neutral absorber were set free. The best-fit model yielded $\chi^{2}/\nu = 1750/1496$, 1040/881,
and 845/775 for the ACIS, EPIC, and XIS spectra, respectively.
Figure 2{\it b} and 2{\it c} show the best-fit deprojected 1P fittings
to the spectra extracted from $320-700$$h_{71}^{-1}$ kpc and core
regions ($0-80$$h_{71}^{-1}$ kpc for the ACIS and EPIC,
and $0-144$$h_{71}^{-1}$ kpc for the XIS), respectively.
As shown in Figure 3{\it a}, the best-fit deprojected 1P temperature drops inwards
from $\approx 6.0$ keV at $\approx 100-350$$h_{71}^{-1}$ kpc to $\approx
3.0$ keV in the central 30$h_{71}^{-1}$ kpc, which shows apparent
diagnostic of a cool core (e.g., Sanderson et al. 2006). On the other
hand, in the cluster's outskirt ($\approx 350-1100$$h_{71}^{-1}$
kpc), the temperature declines outwards steeply down to $\approx
3.0$ keV. Generally, our 1P temperature profiles are consistent with
previous reports (e.g., Ettori et al. 2002; Vikhlinin et al.
2006; Snowden et al. 2008; Bautz et al. 2009). Incidentally, in
$\approx 150-300$$h_{71}^{-1}$ kpc, the temperature obtained with the
ACIS spectra is by about $1.0$ keV higher than those measured with the EPIC and XIS
spectra. This discrepancy can be ascribed to a local
temperature structure (see \S3.2.2), which is smoothed to some extent in the {\em XMM-Newton}\/\
and {\em Suzaku}\/\ data.
Although the best-fit 1P model can reproduce the three data groups
with reasonable goodness, i.e., $\chi^{2}/\nu \approx 1.09 \sim 1.18$, the
fitting residuals become drastically significant in the central 80$h_{71}^{-1}$ kpc
regions, where the averaged gas temperature is measured to be $\simeq
4.5$ keV (Fig. 3{\it a}). As shown in Figure 2{\it c}, the 1P model significantly underestimates the
Fe-L blend and the continuum in $>$ 5.0 keV. This indicates that these
annular spectra exhibit stronger multi-temperature nature than is
predicted by foreground and background contributions from different
radii that are accounted for by the PROJCT model.
To further examine the issues with the 1P ICM model for these regions,
following, e.g., Cavagnolo et al. (2008), we fitted the ACIS spectra in $0.7-4.0$ keV and $4.0-8.0$ keV with the
1P model, both extracted from the central 80$h_{71}^{-1}$ kpc
and corrected for projection effects. The
1P gas temperature obtained in $0.7-4.0$ keV ($T_{\rm 0.7-4.0 \ keV} = 3.5\pm0.2$ keV) became lower than its counterpart
in $4.0-8.0$ keV ($T_{\rm 4.0-8.0 \ keV} = 5.7\pm0.7$ keV) at the 99\% confidence
level. The bandpass dependence cannot be ascribed to the
calibration problem of {\em Chandra} at soft band, because a similar
dependence can be obtained using the {\em XMM-Newton} data, with
$T_{\rm 0.7-4.0 \ keV} = 3.3\pm0.1$ keV and $T_{\rm 4.0-8.0 \ keV} = 4.7\pm0.5$ keV. Thus, the
1P modeling of the ICM is considered inadequate, and a
2P ICM spectral model is hence invoked.
\subsubsection{Two-phase ICM Model}
Next we
investigate whether or not the 2P ICM model, as inferred in the above 1P
analysis, is applicable to the spectra
extracted in the inner 80$h_{71}^{-1}$ kpc region, after the projection
effects are taken into account. Here, the 2P ICM components were both
represented by thermal APEC models, which were constrained to have the same
metallicity and absorption. Both the cool- and hot-phase ICM
temperatures ($T_{\rm c}$ and $T_{\rm h}$, respectively) were tied
among all the thin shells in central 80$h_{71}^{-1}$ kpc, while the
1P ICM model (\S 3.1.1) was
retained for the outer parts. This fitting yielded overall
$\chi^{2}/\nu$ = 1608/1494 and 929/879 for the ACIS and
EPIC spectrum sets, respectively. The fitting goodness was compared to the 1P one (\S 3.1.1)
using an F-test, which yielded $F$-statistic of 66.0 and
52.5 for the ACIS and EPIC data, respectively, indicating that the 2P
model gives a better fit than the 1P model at $>$ 99\% confidence
level. As shown in Figure 2{\it
c}, the 2P ICM model apparently better reproduces the Fe-L
blend and the continuum in $>$ 5.0 keV than its 1P
counterpart. This shows that the 2P ICM model is
strongly and consistently required for the inner 80$h_{71}^{-1}$ kpc
region even after removing (via PROJCT) foreground and background
contributions from the outer shells.
The 2P temperatures
($T_{\rm c}$, $T_{\rm h}$) were determined
as ($2.2 \pm 0.2$, $5.7 \pm 0.4$) keV and ($2.0 \pm 0.2$, $5.0 \pm
0.3$) keV by using the ACIS and EPIC data, respectively. For the
regions with $r > 80$$h_{71}^{-1}$ kpc, the 2P model does not
significantly improve the fitting over the 1P model.
The same 2P ICM model was also applied to the thick shell spectra
obtained with the XIS. Compared to the 1P case (\S 3.1.1; $\chi^{2}/\nu
\approx 845/775$), the 2P ICM model again gave
significantly better fit to the XIS spectra within central
$320$$h_{71}^{-1}$ kpc region, with reduced chi-squared improved to
$\chi^{2}/\nu$ = 802/773. Since the PSF of the XIS is much wider than
those of the ACIS and EPIC, with the XIS alone we cannot constrain the
boundary between the 1P and 2P regions to a small radius, as we did
with the ACIS and EPIC data. To examine the PSF effect on the
spectral fitting of the innermost thick shell ($0-144$$h_{71}^{-1}$
kpc), we performed a ray tracing simulation (e.g., Ishisaki et al. 2007;
Reiprich et al. 2009) based on the ACIS image, and found that the
$0.5-8.0$ keV counts in this shell are mostly ($\approx 75$\%) from
the emission originated in $0-80$$h_{71}^{-1}$ kpc, only a small
fraction ($\approx 5$\%) come from $116-238$$h_{71}^{-1}$ kpc region
where a hotter component ($T_{\rm X}\approx 6.5$ keV; Figure
3{\it a}) is detected with the
ACIS data. Hence this hot component cannot bias the 2P result significantly. In fact, the best-fit 2P temperatures
for this region, $T_{\rm c} = 2.1
\pm 0.5$ keV and $T_{\rm h} = 5.5 \pm 0.4$ keV, are consistent with those obtained in
the thin shell analysis. Thus, the PSF of the XIS little affects the 2P
results obtained for the innermost thick shell.
As the most consistent form of our 2P analysis, we simultaneously
fitted the thin shell spectra (the ACIS and EPIC) and those from the
thick shell (the XIS), by allowing to vary independently the pair of 2P temperatures of
each inner shell, i.e., $\leq
80$$h_{71}^{-1}$ kpc and $\leq 320$$h_{71}^{-1}$ kpc for the ACIS/EPIC and XIS
cases, respectively. Nearly the same fit
goodness was achieved (Table 2), and the best-fit $T_{\rm c}$ and $T_{\rm h}$, as
shown in Figure 4{\it a} and Table 2, exhibit nearly insignificant spatial variations across the inner
shells. The values of $T_{\rm c}$ and $T_{\rm h}$ are consistent
among the three instruments, and broadly consistent with the {\em ASCA}\/\ result, ($T_{\rm c}$, $T_{\rm h}$) =
($1.7 \pm 0.3$, $6.5 \pm 0.6$) keV, reported in Xu et
al. (1998). In short, the ICM in the cool core of this cluster can be described by
two discrete temperatures, $T_{\rm c}\approx 2.0-2.2$ keV and $T_{\rm
h}\approx 5.0-5.7$ keV, which are both consistent with being spatially
constant. This makes the simple 2P ICM picture (Makishima et
al. 2001) a natural and reasonable description.
\subsubsection{A Weak 0.8 keV Spectral Component Within the Central
144$h_{71}^{-1}$ kpc}
Employing the 1P formalism, Fabian et al. (2001) reported a
filamentary structure in the inner 50$h_{71}^{-1}$ kpc region of
A1795, with a possible temperature of $\sim 1$ keV, apparently lower
than the value of $T_{\rm c}$ obtained in our 2P analysis. To look for such a
component, we added a third APEC component to the 2P
ICM model describing the XIS data from the central 144$h_{71}^{-1}$ kpc
region. Indeed, we obtained a significantly better fit ($\chi^{2}/\nu =
791/771$) by adding a third APEC component, whose temperature is
$kT_{\rm 3} = 0.8$$\pm 0.4$ keV, while its metal abundance,
absorption, and redshift were tied to those of the 2P ICM components.
An $F$-test indicates that the probability of this improvement being caused by chance is
$< 5 \times 10^{-3}$. The $0.3-10.0$ keV luminosity of this 0.8
keV component is $3.6^{+1.2}_{-2.6}\times10^{42}$ ergs
$\rm s^{-1}$, which is consistent with the
luminosity of the filamentary structure measured with the ACIS in
Fabian et al. (2001; $\sim 4\times10^{42}$ ergs $\rm s^{-1}$).
As shown in Table 2, the values of $T_{\rm c}$ and $T_{\rm h}$, as
well as the ICM abundances, remain
nearly the same by adding this 0.8 keV component, because it
contributes only $\approx 0.7$\% of the total $0.3-10.0$ keV
luminosity of the central 144$h_{71}^{-1}$ kpc region ($\approx 5.0\times10^{44}$ ergs $\rm s^{-1}$).
In order to crosscheck the XIS result, we
analyzed the {\em XMM-Newton}\/\ RGS and deprojected EPIC spectra extracted from the central
80$h_{71}^{-1}$ kpc region by fitting them with three component
spectral model, i.e., 2P ICM plus a
third APEC component. Since the RGS is not so sensitive to hot
gas components, we fixed $T_{\rm c}$ and $T_{\rm h}$ at the best-fit
EPIC results, i.e., 2.0 keV and 5.0 keV, respectively. By fitting the
RGS1 and RGS2 spectra simultaneously in $6-23$~\AA , a
weak 0.8 keV component was again detected, with a probability of $6
\times 10^{-2}$ for the detection to be caused by chance. The fitting
of the EPIC spectra was improved to $\chi^{2}/\nu =
923/878$ by including the 0.8 keV component, significantly better than
previous 2P fitting in terms of $F$-test ($>98$\% confidence level).
The $0.3-10.0$ keV luminosities
of this component determined with the RGS and the EPIC are $2.7 \pm
2.3\times10^{42}$ ergs $\rm s^{-1}$ and $2.6 \pm 1.7\times10^{42}$
ergs $\rm s^{-1}$, respectively, nicely consistent with the XIS value
and the ACIS result in Fabian et al. (2001).
\subsubsection{Filling Factor of the Cool Phase}
Using the high resolution {\it Chandra}, {\it XMM-Newton}, and {\it
Suzaku} data, we have demonstrated that the 2P ICM model gives a
significantly better description of the ICM thermal condition in the central
80$h_{71}^{-1}$ kpc of A1795 than the 1P counterpart. The cool and hot
phase ICM components have $0.3-10.0$ keV luminosities of about $1.4\times10^{44}$ ergs $\rm
s^{-1}$ and $2.5\times10^{44}$ ergs $\rm s^{-1}$, respectively. Also a
weak 0.8 keV component has been detected in the same region, which
has a $0.3-10.0$ keV luminosity of about $3.6\times10^{42}$ ergs $\rm s^{-1}$.
The 2P ICM picture implicitly assumes that these
phases with different temperatures coexist, each occupying a certain
fraction of the total volume under study in the cluster center, which can be
described as volume filling factor (e.g., Ikebe et al. 1999; Makishima et al. 2001;
Takahashi et al. 2009). To quantify this quantity, we followed the
method described in Ikebe et al. (1999) and
calculated the filling factor of the cool phase gas, $\eta_{\rm
c}(R)$, where $R$ denotes 3-D radius. With $\eta_{\rm c}$, the specific emission measures of the two
phases, $Q_{\rm c}(R)$ and $Q_{\rm h}(R)$, are described as
\begin{equation} \label{eq:em}
Q_{\rm c}(R) = n_{\rm c}(R)^{2}\eta_{\rm c}(R), \mbox{ } Q_{\rm h}(R) = n_{\rm h}(R)^{2}(1-\eta_{\rm c}(R)),
\end{equation}
where $n_{\rm c}(R)$ and $n_{\rm h}(R)$ are the density distributions
of the cool and hot
phases, respectively. Assuming a pressure balance between the two phases
\begin{equation} \label{eq:pb}
n_{\rm c}(R)T_{\rm c}(R) = n_{\rm h}(R)T_{\rm h}(R),
\end{equation}
we have
\begin{equation} \label{eq:vf}
\eta_{\rm c}(R) = \left[ 1+\left(\frac{T_{\rm h}(R)}{T_{\rm c}(R)}\right)^{2}
\left(\frac{Q_{\rm h}(R)}{Q_{\rm c}(R)}\right) \right] ^{-1}.
\end{equation}
Figure 4{\it b} shows radial profiles of $\eta_{\rm c}(R)$, calculated
with the best-fit deprojected ACIS, EPIC, and
XIS model parameters as listed in Table 2. Thus, the ACIS and EPIC datasets consistently
indicate that the hot component dominates in volume over its cool
counterpart not only in outer regions, but also in the core region
($<80$$h_{71}^{-1}$ kpc). The cool phase gas occupies up to $\eta_{\rm c} \approx 0.2$ of the volume
in the central $30$$h_{71}^{-1}$ kpc, whereas $\eta_{\rm c}$
declines steeply to below 0.05 outside the central $80$$h_{71}^{-1}$
kpc region. The XIS result for the inner $144$$h_{71}^{-1}$ kpc
is consistent with those of the ACIS and
EPIC, whereas the relatively large value of $\eta_{\rm c}$
indicated by the outer $144-320$$h_{71}^{-1}$ kpc XIS bin can be
attributed to the broad PSF of the XIS. In fact,
by performing a ray tracing
simulation (see \S3.1.2 for more details), the outer XIS
bin at
$144-320$$h_{71}^{-1}$ kpc was found to be contaminated by photons
scattered from the inner regions, in such a way that $\approx 37$\% of the emission
in this region actually comes from the central 144$h_{71}^{-1}$
kpc. In this case, a weighted mean of $\eta_{\rm c} \approx 0.08$
measured with the XIS over inner $144$$h_{71}^{-1}$ kpc and $\eta_{\rm
c} \approx 0.01$ measured with the other two missions at the
$144-320$$h_{71}^{-1}$ kpc region becomes $\approx 0.04$, in agreement
with the outer XIS point. Hence, after correcting for the PSF effect, all
data indicate the same 2P configuration in the central region.
\subsubsection{2P vs. Multi-phase ICM Model}
The ICM in the central $144$$h_{71}^{-1}$ kpc may alternatively in a
multi-phase condition (e.g., Kaastra et al. 2004), to be described by
a more complicated emission measure distribution in a wide temperature
range than that used in \S 3.1.2. To examine this possibility, we
analyzed the deprojected XIS spectra from the inner
144$h_{71}^{-1}$ kpc region with the same multi-temperature fitting
approach as used in Tamura et al. (2001) and Takahashi et al. (2009). That is, the spectra
were modeled as a cumulative contribution of seven APEC components, whose
temperatures are given as $T_{\rm 0}$, 1.5$T_{\rm 0}$,
(1.5$)^2$$T_{\rm 0}$,..., and (1.5$)^6$$T_{\rm 0}$, where $T_{\rm 0}$
is a base temperature and left free in the fitting. The seven components
were constrained to have the same metal
abundance, and suffer the same absorption. The fit has been
acceptable, and the goodness of fitting ($\chi^{2}/\nu=
786/769$) is as good as that of
the 2P ICM plus a 0.8 keV component model (Table 2). The base
temperature was constrained as $T_{\rm 0} = 0.7 \pm 0.2$ keV. As
shown in Figure 5, in the multi-phase model, only one
weak component at $\approx 0.7$ keV and
two strong components at 2.4 keV and 5.3 keV remain significant (68\%
confidence level), while the rest components cannot be
constrained. This is essentially identical to the 2P ICM plus
0.8 keV modeling. Our result is consistent with the multi-phase fitting
with the EPIC data in Kaastra et al. (2004), which also shows a two-temperature structure
(2.5 keV and 4.9 keV, see their Table 6) in the cluster center. Hence, the XIS data
prefers the discrete 2P model,
to a continuous temperature distribution, although the latter cannot
be ruled out based on available data.
\subsubsection{Metal Abundance and Absorption Distributions}
As shown in Figure 3{\it b}, the deprojected metal abundance profiles appear roughly consistent
between the 1P and 2P ICM modelings (\S3.1.1 and \S3.1.2,
respectively). Both profiles show a peak
in the shell of $30-51$$h_{71}^{-1}$ kpc, and a mild decline outwards.
This abundance profile, however, should be regarded as an average among
those of different elements, because we have so far assumed the
solar abundance ratios. To examine the ICM for possible deviation
from the solar ratios, we employed two VAPEC models and reran
the deprojected 2P fittings of the {\em Suzaku}\/\ XIS
spectra extracted from the central 320$h_{71}^{-1}$ kpc
region. Specifically, we left the abundances of O, Mg, Si and Fe free,
fixed the abundances of He, C and N at the solar value (Anders
\& Grevesse 1989), and tied the abundance of Ni to that of Fe and those of other
elements (Ne, Al, S, Ar and Ca) to that of Si. This model gave nearly
the same set of temperatures for the 2P ICM, and a fit
goodness ($\chi^{2}/\nu=787/761$) slightly better than that of the best-fit
APEC model ($\chi^{2}/\nu=802/773$). As shown in Table 3, the best-fit Fe and Si
abundances increase significantly towards the center, while the O and Mg abundances
are nearly constant throughout the cluster. This confirms the previous
ACIS result reported in Ettori et al. (2002; see also Matsushita et al. 2007).
We also added another APEC model ($A$ = 0.5
$Z_\odot$; see Table 2) in the central 144$h_{71}^{-1}$
kpc to account for the 0.8 keV component, while the model gave
nearly the same best-fit abundance profiles for the 2P ICM components.
To examine the possible existence of any intrinsic
absorption, we compared the absorption obtained in the spectral analysis
with the Galactic value. Regardless of the 1P or 2P modeling, the ACIS
and XIS fitting results revealed a significant excess absorption by
$1-2 \times 10^{20}$ $\rm cm^{-2}$ (Table 3)
beyond the Galactic value of $1.2 \times 10^{20}$ $\rm cm^{-2}$, at least
in the central 30$h_{71}^{-1}$ kpc region. The excess absorption could not be mitigated by
varying the abundances of specific elements (e.g., oxygen) whose emission lines couple with the
absorption feature. This result is in good agreement with the ACIS result of
Ettori et al. (2002; $\approx 2.5 \times 10^{20}$ $\rm cm^{-2}$). On
the contrary, our best-fit EPIC result ($\approx
1.1 \times 10^{20}$ $\rm cm^{-2}$), as shown in Table 3,
implies lack of excess absorption, in agreement with Nevalainen et
al. (2007) who used
the same {\em XMM-Newton}\/\ MOS data. The discordance among the three detectors
maybe caused by a temporal solar wind charge exchange emission, which
biased the EPIC absorption low. As
shown in Wargelin et al. (2004), the brightness of charge exchange can
reach $\sim 2\times10^{-6}$ photon $\rm s^{-1}$ $\rm arcmin^{-2}$
$\rm cm^{-2}$ in $0.5-0.9$ keV, which is sufficient to undermine the
absorption by $0.5-1.0 \times 10^{20}$ $\rm cm^{-2}$. The origin of
the central excess absorption detected with the ACIS and XIS, on the
other hand, still remains unclear.
\subsection{Projected 2-D Spectral Analysis}
As shown above, the azimuthally-averaged spectral analysis
prefers a view of the 2P ICM in the central
80$h_{71}^{-1}$ kpc of A1795 than the 1P counterpart. However,
we cannot exclude at present the possibility that the 2P preference is
artificially caused by an
anisotropic spatial distribution of 1P gas temperature that fluctuates
between the range from $T_{\rm c}$ to $T_{\rm h}$. To address this issue, we examine
two-dimensional (2-D) temperature distribution in the central region
of A1795. Below, 2-D ICM
abundance and $Q_{\rm c}$/$Q_{\rm h}$
distributions are also presented.
\subsubsection{Analysis Procedure}
Combining five {\em Chandra}\/\ ACIS-S pointings onto the central region
together, we have collected sufficient photons for a high resolution 2-D spectral
analysis. Following the procedure described in detail in Gu et
al. (2009), a set of $> 10000$ discrete points (${\bf r}_{\rm i}$, $i
= 1,2,3,...$), or ``knots'', were chosen
in the central
$240$$h_{71}^{-1}$ kpc, which are randomly distributed with a
separation of $<6$$h_{71}^{-1}$ kpc between any two adjacent knots. To
each knot we assigned a circular region, which is centered on the knot
and has an adaptive radius of $15-80$$h_{71}^{-1}$ kpc, ensuring that it
encloses $>10000$ photons in $0.5-8.0$ keV after all detected point
sources (\S2.2)
were excluded. The spectrum extracted from each
circular region was fitted with the 1P and 2P APEC models, both
subjected to an absorption that was set free. In the
2P spectral fitting, the temperatures of the cool and hot
components were fixed at 2.2 keV and 5.7 keV (i.e., best-fit ACIS
results; \S 3.1.2), respectively,
and the two components were assumed to have the same abundance and
absorption. For each knot ${\bf r}_{\rm i}$, we obtained the best-fit 1P gas
temperature, 1P/2P abundance,
and specific emission measure ratio between the cool and hot
components ($Q_{\rm c}$/$Q_{\rm h}$), as well as their $1\sigma$
errors.
Then, following Gu et al. (2009), we calculated continuous maps based
on the obtained knots. For any position ${\bf r}$ within the map
region, we defined a scale
$s({\bf r})$, so that there are $>10000$ net photons in a circular
region centered at ${\bf r}$, whose radius is $s({\bf r})$. The 1P
temperature at ${\bf r}$ was then calculated by a weighted mean of all
the knots ${\bf r}_{\rm i}$ in the circular region,
\begin{equation} \label{eq:tmap}
T({\bf r}) = \sum_{\bf r_i} (G_{\bf r_i}(R_{{\bf r},{\bf r_i}})T_{\rm c}({\bf r_i}))/\sum_{\bf r_i} G_{\bf r_i}(R_{{\bf r},{\bf r_i}}), \mbox{ } \rm when \mbox{ } \it R_{{\bf r},{\bf r_i}} < s({\bf r}),
\end{equation}
where $R_{{\bf r},{\bf r_i}}$ is the distance from ${\bf r}$ to ${\bf r_i}$, and $G_{\bf r_i}$ is the Gaussian kernel whose scale
parameter $\sigma$ is fixed at $s({\bf r_i})$. The use of compact
Gaussian kernel guarantees an angular resolution of $\sim
15$$h_{71}^{-1}$ kpc within central 100$h_{71}^{-1}$ kpc region. The
abundance and $Q_{\rm c}$/$Q_{\rm h}$ maps, along with their $1\sigma$ error maps, are calculated in the same
way. The resulting 1P temperature and abundance maps are shown in
Figure 6, and the 2P abundance (hereafter $A_{\rm 2P}$) and $Q_{\rm
c}$/$Q_{\rm h}$ maps are presented in Figure 7. The 0.8
keV component (\S 3.1.3) was not considered in the above 2-D analysis,
because it will introduce negligible effects, i.e., uncertainties of
about 0.05 keV and 0.02 $Z_\odot$, to the temperature and abundance
measurements, respectively.
\subsubsection{Relaxed Cool Core and SE High Temperature Arc}
The primary purpose of 2-D spectral analysis is to assess the validity
of spherically symmetric temperature distribution that we assumed in the radial 1P/2P analysis. To do
this, we examined the obtained 1P temperature map for any significant
anisotropic distribution. As shown in Figure 6{\it a}, the 1P temperature
distribution in the cluster central region exhibits an approximate elliptical
symmetry; the temperature variation in the azimuthal direction is
about 0.5 keV, 1.0 keV, and 1.0 keV at $r=20$$h_{71}^{-1}$ kpc,
$40$$h_{71}^{-1}$ kpc, and $80$$h_{71}^{-1}$ kpc, respectively, which
is insufficient, at least by a factor of three, to imitate the obtained 2P ICM condition. Similar morphology
is seen in the $Q_{\rm c}$/$Q_{\rm h}$ map of the core region as shown
in Figure 7{\it b}. This confirms that the 2P view is not an artifact caused
by a 1P ICM with large azimuthal asymmetry. It also indicates that the
two phases, in terms of the 2P view, must be separated on scales smaller than the
spatial resolution allowed by the present analysis ($\approx 15$$h_{71}^{-1}$ kpc).
The most prominent feature on the 1P temperature map (Fig. 6{\it a}) is a high-temperature arc
($\approx 7.5$ keV according to the 1P model) located at $\approx 100 - 180$$h_{71}^{-1}$ kpc
southeast (SE) of the cluster's center, with an open angle of $>$$120
^{\circ}$. Given the temperature difference of about 2 keV, the hot structure is significant
over the ambient on 90\% confidence level. This feature agrees
with a high temperature bin ($116-238$$h_{71}^{-1}$ kpc) on the deprojected ACIS temperature
profile shown in Figure 3{\it a}. A same high temperature structure
was found by Markevitch et al. (2001), who carried out
a 1P azimuthal spectral analysis for the southern part of the
cluster. Markevitch et al. (2001) also reported a density jump by a
factor of $1.3-1.5$ near the inner edge ($\approx 85$$h_{71}^{-1}$
kpc from the cluster center) of the high-temperature arc, and ascribed
the density jump to a cold front. Since both the high-temperature component
and the cold front locate outsides of the 2P region (i.e., $r < 80$$h_{71}^{-1}$ kpc),
they are unable to affect the obtained 2P result significantly, even for the
XIS thick shell as indicated by the ray tracing simulation shown in \S 3.1.2.
\subsubsection{A Possible Correlation Between Metal-rich and Cool-phase Gas}
A comparison of the $A_{\rm 2P}$ map (Fig. 7{\it a}) with the
$Q_{\rm c}$/$Q_{\rm h}$ map (Fig. 7{\it b}) suggests
a spatial correlation on a scale of $50-100$$h_{71}^{-1}$ kpc
between the two
quantities. Both maps reveal strongly inhomogeneous
distributions outside the relaxed core region, with substructures protruding
towards southwest, northeast, and
northwest of the cluster center by $\approx 80-100$$h_{71}^{-1}$ kpc.
Most of these substructures show both higher metal abundances,
typically by a
factor of $2-3$, and larger cool phase fractions,
than their neighborhoods. The best-fit $A_{\rm 2P}$ and $Q_{\rm
c}$/$Q_{\rm h}$ values for all the knot-centered circular regions in
$50-100$$h_{71}^{-1}$ kpc, obtained in our 2-D spectral analysis (\S
3.2.1), were compared directly in Figure 7{\it c}. Indeed, it reveals a positive
linear correlation, which is represented by
an analytic form as $A_{\rm 2P} = 0.38^{+0.11}_{-0.12} \times Q_{\rm
c}$/$Q_{\rm h}+ 0.48^{+0.08}_{-0.07}$ $Z_\odot$. The linear correlation
coefficient was obtained as 0.97.
To quantify the suggested correlation, following, e.g., Shibata et
al. (2001), we calculated a 2-D cross
correlation function as,
\begin{equation}\label{eq:tpcf}
\xi ({\rm R}) =
\left < \frac{\left \{ A_{\rm 2P}({\bf r}_{1}) - \overline{A_{\rm 2P}} \right \}
\left \{ Q_{\rm c}({\bf r}_{2})/Q_{\rm h}({\bf r}_{2}) - \overline{Q_{\rm c}/Q_{\rm h}}
\right \} }{\overline{A_{\rm 2P}}\overline{Q_{\rm c}/Q_{\rm h}}}
\right >_{\rm r_{12} = \rm R},
\end{equation}
where $\overline{A_{\rm 2P}}$ and $\overline{Q_{\rm c}/Q_{\rm h}}$ are averages
of $A_{\rm 2P}$ and $Q_{\rm c}/Q_{\rm h}$ maps, respectively, $\rm
r_{12}= |{\bf r}_{1} - {\bf r}_{2}|$ is the distance between ${\bf
r}_{1}$ and ${\bf r}_{2}$, and the bracket is ensemble
average. The obtained correlation function is shown in Figure 7{\it d}.
A reference profile ($\xi_{\rm ref} ({\rm R})$) was also calculated, based on
a series of random maps, $A_{\rm 2P, ran}$ and $Q_{\rm c,
ran}/Q_{\rm h, ran}$, which were obtained by randomizing the maps within
the same data range and smoothing to the same spatial resolution as the original $A_{\rm 2P}$ and $Q_{\rm
c}/Q_{\rm h}$ maps. The error bars shown in Figure 7{\it d} were calculated from the
variances of $\xi ({\rm R})$ when scattering the $A_{\rm 2P}$
and $Q_{\rm c}/Q_{\rm h}$ maps by their $1\sigma$ error
maps. Comparing to the $\xi_{\rm ref} ({\rm R})$, the $\xi ({\rm
R})$ profile shows a significant excess within 100$h_{71}^{-1}$ kpc,
reconfirming the result that cool-phase ICM is indeed more metal-rich.
Given the detected correlation, the assumption made in \S 3.1.2 and
\S3.1.6 that the two phases ICM have approximately the same
abundances may not stand now. Therefore, we refitted the
deprojected XIS spectra of the central $144$$h_{71}^{-1}$ kpc region
with the 2P ICM (VAPEC + VAPEC) + 0.8 keV component (APEC; 0.5
$Z_\odot$) model, same
as the one in \S3.1.6, except that the iron abundances of the
cool and hot phases were let float separately. The
absorption column densities of the three components were again tied together
in the fitting as a free parameter. We found that the abundance of
cool phase ICM ($A_{\rm Fe, c}$) does appear higher
than that of the hot one ($A_{\rm Fe, h}$); i.e., ($A_{\rm Fe, c}, A_{\rm Fe, h}$) =
($0.80 \pm 0.25$, $0.36 \pm 0.06$) $Z_\odot$. Compared to the
model presented in \S3.1.6 (Table 3), the current model improves fit goodness
to $\chi^{2}/\nu = 779/759$ from $787/761$, with a probability of $2
\times 10^{-2}$ for the improvement to be caused by chance.
Following,
e.g., Simionescu et al. (2008), the iron abundance to be obtained with
a 2P model assuming a single common metallicity can be estimated by
\begin{equation} \label{eq:aew}
A_{\rm Fe}^{\prime} \approx \frac{Q_{\rm c}A_{\rm Fe, c}+Q_{\rm h}A_{\rm Fe, h}}{Q_{\rm c}+Q_{\rm h}}.
\end{equation}
Eliminating $Q_{\rm c}$ and $Q_{\rm h}$ with Eq.(1),
we have
\begin{equation} \label{eq:avf}
A_{\rm Fe}^{\prime} \approx \frac{T_{\rm h}^2 \eta_{\rm c} A_{\rm Fe,
c} + T_{\rm c}^2 (1-\eta_{\rm c}) A_{\rm Fe, h}}{T_{\rm h}^2
\eta_{\rm c} + T_{\rm c}^2 (1-\eta_{\rm c})}.
\end{equation}
Adopting $\eta_{\rm c} = 0.08$ (Fig. 4{\it b}) and ($A_{\rm Fe, c}, A_{\rm Fe, h}$)
obtained above, $A_{\rm Fe}^{\prime}$ is then calculated as 0.51$Z_\odot$, which
agrees with that derived with the XIS spectra ($0.48\pm0.03$ $Z_\odot$, Table 3). Thus, all the ACIS
and XIS results obtained so far can be interpreted as evidence for the
relatively high metal abundance of the cool phase ICM.
\subsection{Hierarchical Gravitational Potential Structure}
\subsubsection{Central Excess in X-ray Surface Brightness}
In many cool-core clusters, the X-ray surface brightness exhibits an
central excess over a $\beta$ model that well fits the outer
region. Makishima et al. (2001) argued that the central excess is a combined
result of two major effects; the presence of a cool phase component,
and the existence of hierarchical potential structure. According to their
definition, the hierarchical potential is specified as a halo-in-halo structure, i.e.,
a smaller potential component is nested on a larger one, so that the
overall potential shows a central deepening relative to a simple King profile.
Here we examined the surface brightness profile of A1795 for such a central excess.
In order to measure the surface brightness profile precisely by taking
advantages of both high spatial resolution of the {\em Chandra}\/\ ACIS and
stable low background of the {\em Suzaku}\/\ XIS, we calculated
exposure-corrected surface brightness profiles from the inner 1000$h_{71}^{-1}$ kpc ACIS and
$1200-1800$$h_{71}^{-1}$ kpc XIS data, i.e., $S_{\rm ACIS}(r)$ and $S_{\rm
XIS}(r)$, respectively, where $r$ is projected radius. The method
described in, e.g., Markevitch et al. (1998), was employed to compensate
discrepancies on calibration and instrumental background between
the two instruments. First, we performed simulations using the {\tt xissim} tool (Ishisaki et
al. 2007) to smooth the {\em Chandra} images with the {\em Suzaku} PSF,
and extracted the simulated surface brightness profile in the central
1000$h_{71}^{-1}$ kpc, i.e., $S_{\rm ACIS}^{\prime}(r)$. Then, $S_{\rm
ACIS}^{\prime}(r)$ was scaled to $S_{\rm
XIS}(r)$ by solving $aS_{\rm ACIS}^{\prime}(r)-b=S_{\rm XIS}(r)$, where $a$ and
$b$ represent the differences in normalization and NXB, respectively. A modified
$S_{\rm ACIS}(r)$ with the original ACIS resolution was obtained by applying $a$ and $b$ to $S_{\rm ACIS}(r)$
in the same way as we normalized $S_{\rm ACIS}^{\prime}(r)$.
Then, the modified $S_{\rm ACIS}(r)$ and $S_{\rm XIS}(r)$, shown in Figure 8, were
fitted jointly with the $\beta$ model, $S(r)= S(0) \{1+(r/r_{\rm
c})^2\}^{-3\beta+1/2} + S_{\rm BG}$, where $S(0)$ is the central
brightness, $r_{\rm c}$ is the core radius, and $S_{\rm BG}$ is the background.
To represent the possible halo-in-halo structure,
a double-$\beta$ model was also tested as $S(r) = S_1(0) \{1+(r/r_{\rm c1})^2\}^{-3\beta_1 + 1/2} +
S_2(0) \{1+(r/r_{\rm c2})^2\}^{-3\beta_2 + 1/2}+ S_{\rm BG}$, where
subscripts 1 and 2 denote compact and extended components,
respectively. As shown in Table 4, the fit with the $\beta$ model can
be rejected on the 95\% confidence level, because it significantly
underestimates the surface brightness in central $\sim
100$$h_{71}^{-1}$ kpc, while the double-$\beta$ model gives an
acceptable fit to the data. The successful fits in Figure 8 are based
on a still more sophisticated model, a $\beta$+double-$\beta$ gas density model,
to be explained later. The properties of the central excess
brightness, represented by
the compact $\beta$ component ($\beta_1 \approx 0.75$, $r_{\rm c1} \approx 53$$h_{71}^{-1}$ kpc),
are consistent within errors between the
$0.5-3.0$ keV and $3.0-8.0$ keV bands. Such an energy-independent central
excess is likely to be caused by an intrinsic hierarchical potential shape,
rather than the presence of cool phase component. This result agrees with
the earlier {\em ASCA}\/\ result reported in Xu et al. (1998).
To further quantify the central emission excess, we corrected the surface brightness profile
for the projection effect using the best-fit deprojected 2P gas temperature and abundance profiles (Table 2; \S 3.1.2). The
observed surface brightness profile was modeled as
\begin{equation}
S(r)= \int^{\infty}_{\rm r}\Lambda(T_{\rm c},A_{\rm c})Q_{\rm c}(R)\frac{RdR}{\sqrt{R^{2}-r^{2}}}
+ \int^{\infty}_{\rm r}\Lambda(T_{\rm h},A_{\rm h})Q_{\rm h}(R)\frac{RdR}{\sqrt{R^{2}-r^{2}}} + S_{\rm BG},
\end{equation}
where $\Lambda$ is the cooling function, $Q_{\rm c}$ and $Q_{\rm h}$
are the same as in Eq.(1), and $S_{\rm BG}$ is the averaged background value.
Following Ikebe et al. (1999), we represented the specific emission measure
profiles by a 2P $\beta$+double-$\beta$ model, which consists of a
$\beta$ component for the cool phase ICM,
\begin{equation}
Q_{\rm c}(R) = \left \{
\begin {array} {c}
Q_{\rm 0,c} \left[ 1 + (\frac{R}{R_{\rm c,c}})^{2} \right] ^{-3\beta_{\rm c}}
\mbox{ } \mbox{ } \mbox{ } \mbox{ } R \leq 80h_{71}^{-1}\mbox{ } {\rm kpc}, \\
0 \mbox{ } \mbox{ } \mbox{ } \mbox{ } \mbox{ } \mbox{ } \mbox{ } \mbox{ } \mbox{ }
\mbox{ } \mbox{ } \mbox{ } \mbox{ } \mbox{ } \mbox{ } \mbox{ } \mbox{
} \mbox{ } \mbox{ } \mbox{ } \mbox{ } \mbox{ } \mbox{ } \mbox{ }
\mbox{ } \mbox{ } \mbox{ } \mbox{ } \mbox{ } \mbox{ }
R > 80h_{71}^{-1}\mbox{ } {\rm kpc},
\end{array}
\right .
\end{equation}
and a double-$\beta$ component for the hot phase ICM,
\begin{equation}
Q_{\rm h}(R) =
Q_{\rm 0,h1} \left[ 1 + \left(\frac{R}{R_{\rm c,h1}}\right)^{2} \right] ^{-3\beta_{\rm h1}}
+ Q_{\rm 0,h2} \left[ 1 + \left(\frac{R}{R_{\rm c,h2}}\right)^{2} \right] ^{-3\beta_{\rm h2}},
\end{equation}
where $Q_{\rm 0,c}$, $Q_{\rm 0,h1}$, and $Q_{\rm 0,h2}$ are the model
normalizations of the cool phase, the compact hot phase, and the
extended hot phase components, respectively. The filling factor
$\eta_{\rm c}$ is renormalized into these parameters.
With these preparations, we fitted the $0.5-8.0$ keV ACIS + XIS surface
brightness profiles using Eq.(8), where $\Lambda$ was calculated from
the best-fit 2P spectral parameters. The fit has
been successful with $\chi^{2}/\nu = 161/151$. The best-fit values of
$R_{\rm c,c}$, $R_{\rm c,h1}$, $R_{\rm c,h2}$, $\beta_{\rm c}$,
$\beta_{\rm h1}$, and $\beta_{\rm h2}$ are given in Table 4. The fittings with surface
brightness profiles extracted in different energies (i.e., $0.5-3.0$ keV and
$3.0-8.0$ keV bands) are shown in Figure 8. As presented in Table 4, all the fittings
yielded consistent parameters for the 2P $\beta$+double-$\beta$ model.
In some clusters, the central excess can be alternatively
explained by a cuspy dark matter distribution proposed in Navarro, Frenk, \& White (1996;
hereafter NFW model). The NFW model predicts
potential structure with a single spatial scale, instead of the dual structure assumed by
the $\beta$+double-$\beta$ model. Given the NFW dark matter density profile,
\begin{equation}
\rho_{\rm DM}(R) = \rho_{\rm c} \delta_{\rm c} \left( \frac{R}{R_{\rm s}}\right)^{-1} \left( 1 + \frac{R}{R_{\rm s}}\right)^{-2},
\end{equation}
where $\rho_{\rm c}$ is the critical density, $\delta_{\rm c}$ is characteristic density, and $R_{\rm s}$
is scale radius, the ICM density profile is expressed as
\begin{equation}
n_{\rm NFW} (R) = n_{\rm 0,NFW} \frac{\left(1+ \frac{R}{R_{\rm s}}\right)^{\frac{\alpha R_{\rm s}}{R}}-1}{e^{\frac{R}{R_{\rm s}}}-1},
\end{equation}
where $n_{\rm 0,NFW}$ is model normalization and $\alpha$ is a parameter related to the ICM temperature. Then
the projected ICM surface brightness profile was modeled as a single component,
\begin{equation}
S(r)= \int^{\infty}_{\rm r}\Lambda(T_{\rm 1P},A_{\rm 1P})n_{\rm NFW}^{2}(R)\frac{RdR}{\sqrt{R^{2}-r^{2}}} + S_{\rm BG}.
\end{equation}
We found this NFW model cannot give acceptable fits to the $0.5-8.0$ keV surface brightness profile for
the entire cluster, with a minimum $\chi^{2}/\nu = 307/155$. As shown in Figure 8, the central excess emission is
too strong to be explained by the NFW model that best-fits the $>100$$h_{71}^{-1}$ kpc region. Hence, we reconfirmed the
{\it ASCA} result reported in Xu et al. (1998) that the
halo-in-halo hierarchical model is preferred to the single-halo NFW model in A1795.
\subsubsection{Central Excess in Total Gravitating Mass}
Next we calculated the total gravitating mass profile $M(R)$ based on the best-fit
2P temperatures (\S3.1.2) and the emission measure profiles obtained with the $\beta$+double-$\beta$ model
(\S3.3.1). With the help of Eqs.(1)-(3), the best-fit emission measure profiles, $Q_{\rm
c}(R)$ and $Q_{\rm h}(R)$, were converted to gas density profiles of the cool
and hot phases, $n_{\rm c}(R)$ and $n_{\rm h}(R)$,
respectively. Then, under the assumptions of spherical symmetry
and hydrostatic equilibrium, the
gravitating mass profile $M(R)$ was calculated as
\begin{equation} \label{eq:totalmass}
M (R) = - \frac{R^2}{G \rho_{\rm g}(R)}
\frac{{\rm d}P (R)}{{\rm d}R},
\end{equation}
where $P(R)$ = $n_{\rm c}(R)k_{\rm B}T_{\rm c}(R)$ = $n_{\rm h}(R)k_{\rm B}T_{\rm h}(R)$ is
the gas pressure, $\rho_{\rm g}(R) = \mu m_{\rm p} \{ \eta_{\rm c}(R)
n_{\rm c}(R) + [1 - \eta_{\rm c}(R)] n_{\rm h}(R) \}$
is the averaged gas mass density, $\mu = 0.609$ is the approximated mean molecular weight, and $m_{\rm p}$ is the proton mass. As shown
in Figure 9, the obtained mass profile exhibits a significant shoulder-like feature at
$R_{\rm x} \approx 120$$h_{71}^{-1}$ kpc, enclosing an excess mass
($M_{\rm excess}$) of about $1.5 \times 10^{13} M_\odot$ above
a flat core mass distribution calculated with the extended hot phase
component alone. Our result agrees with the {\em ASCA}\/\ result, $R_{\rm x}
\approx 110$$h_{71}^{-1}$ kpc and $M_{\rm excess} \approx 2.1\times
10^{13} M_\odot$, as reported in Xu et al. (1998).
To examine systematic errors of the mass profile due to
different modelings of the ICM temperature distribution, we also calculated
the gravitating mass profile using the best-fit deprojected 1P spectral
parameters (\S 3.1.1). The projected X-ray surface brightness profile was
modeled with a single ICM component as
\begin{equation}
S(r)= \int^{\infty}_{\rm r}\Lambda(T_{\rm 1P},A_{\rm 1P})Q_{\rm
1P}(R)\frac{RdR}{\sqrt{R^{2}-r^{2}}} + S_{\rm BG},
\end{equation}
where the specific emission measure is given by a double-$\beta$ model,
\begin{equation}
Q_{\rm 1P}(R) = n^{2}_{\rm 1P}(R) =
Q_{\rm 0,1} \left[ 1 + \left(\frac{R}{R_{\rm c,1}}\right)^{2} \right] ^{-3\beta_{\rm 1}}
+ Q_{\rm 0,2} \left[ 1 + \left(\frac{R}{R_{\rm c,2}}\right)^{2} \right] ^{-3\beta_{\rm 2}}.
\end{equation}
The fit is as good as the 2P case, with $\chi^{2}/\nu = 165/154$. The
resulting mass profile, derived by substituting $P(R)$ = $n_{\rm 1P}(R)k_{\rm B}T_{\rm 1P}(R)$ and
$\rho_{\rm g}(R) = \mu m_{\rm p} n_{\rm 1P}(R)$ to Eq.(14), is shown in
Figure 9 by a red curve. Thus, the shoulder-like potential
structure is again found, although the 1P model systematically underestimates the
gravitating mass within 80$h_{71}^{-1}$ kpc by $\approx 30$\%. This
bias is negligible in $>80$ kpc.
Another concern is that, the assumption of hydrostatic equilibrium may
not hold for the ICM near the cold front located southeast of
the cluster center (\S 3.2.2), casting doubt on the validity of mass profile
obtained in the related region (Markevitch et al. 2001). To
investigate this, we excluded the southern half of the cluster and
recalculated mass profile for the northern half using the best-fit 2P
spectral results and surface brightness profile with the ACIS data. The 2P
temperatures, $T_{\rm c, north} = 2.0$ keV and $T_{\rm h, north} =
5.9$ keV, are consistent with previous ACIS results. The same is the
obtained $n_{\rm h, north}(R)$ profile, except for the annulus of the cold front (i.e., $R = 100-180$$h_{71}^{-1}$ kpc), where $n_{\rm h,
north}(R)$ is lower than $n_{\rm h}(R)$ by
$5-20$\% in $100-180$$h_{71}^{-1}$ kpc. After comparing $M_{\rm
north}(R)$ to $M(R)$ in Figure 9,
we found that the shoulder-like structure, though
diminished by $\approx 10$\%, still remains significant after
the high-temperature arc is excluded. Hence we conclude that the
existence of central
hierarchical structure is unambiguously confirmed in A1795.
\subsection{Hard X-ray Emission Component}
Here we examined the $12.0-50.0$ keV HXD-PIN data for possible
existence of any extra emission component. First, following, e.g.,
Nakazawa et al. (2009), we calculated the effective area of the
HXD-PIN considering the location and extension of the source. Based on the observed
best-fit gas temperature and gas density profiles
(Fig. 4{\it a} and Table 4), we calculated the ICM surface brightness distribution
map $S_{\rm hard}(x,y)$ in $12.0-50.0$ keV. Then we divided the
whole HXD field of view into
$1^{\prime}\times1^{\prime}$ blocks, calculated the point source
ARF for each block at $(x,y)$ using the ftool hxdarfgen, and convolved
all the monochromatic ARFs with the normalized $S_{\rm hard}(x,y)$. The PIN/XIS
cross normalization factor of 1.132 ({\em Suzaku}\/\ Memo
2007-11\footnote{http://www.astro.isas.ac.jp/suzaku/doc/suzakumemo/suzakumemo-2007-11.pdf})
was also adopted in the resulting ARF.
To assess the NXB level, we utilized the ``tuned'' NXB model, which
provides an optimized reproductivity by making use of HXD-GSO
information ({\em Suzaku}\/\ Memo
2008-03\footnote{http://www.astro.isas.ac.jp/suzaku/doc/suzakumemo/suzakumemo-2008-03.pdf}). The
CXB model was assumed as $N(E) = 8.69 \times 10^{-4} \times
(E/$1.0 keV$)^{-1.29} \times$ $\rm exp$$(-E/$40.0 keV) photons $\rm cm^{-2}$ $\rm s^{-1}$
$\rm keV^{-1}$, where the normalization was set to match the HXD-PIN opening
angle of 0.32 $\rm deg^{2}$ (e.g., Nakazawa et al. 2009).
As shown in Figure 10, the NXB-subtracted XIS and HXD-PIN spectra in
$4-50$ keV were
tentatively fitted with a CXB + ICM model. The CXB component was
modeled by a cutoff powerlaw with index of 1.29 (Boldt 1987), and
the ICM component was represented by the 2P model derived in \S3.1.2.
The 2P temperatures were set to ($T_{\rm c}$, $T_{\rm h}$) = ($2.1$, $5.5$) keV, and
their relative normalizations were fixed at the best-fit value in our
2P analysis with the XIS. The fitting is acceptable with $\chi^{2}/\nu
= 45/55$, and no additional hard X-ray component is required in
$12-50$ keV band. To obtain an upper limit on any
hard X-ray component, we fitted the combined XIS-HXD spectra with a 2P
ICM plus $\Gamma=2$ power-law model (Nakazawa et al. 2009). Taking
account of the NXB
systematic uncertainty of 2.0\% at 90\% confidence level (Fukazawa et
al. 2009), the upper
limit flux of power-law emission in
$12.0-50.0$ keV band was estimated to be $8.2 \times 10^{-12}$ ergs
$\rm cm^{-2}$ $\rm s^{-1}$. A similar upper limit on any thermal hard X-ray
excess, $6.2 \times 10^{-12}$ ergs $\rm cm^{-2}$ $\rm s^{-1}$, was obtained by
alternatively fitting the combined spectra with 2P ICM plus a 10 keV APEC
model.
\section{DISCUSSION}
We have analyzed {\em Chandra}, {\em XMM-Newton}, and {\em Suzaku}
data of the relaxed galaxy cluster A1795. The deprojected spectra extracted
from the central 80$h_{71}^{-1}$ kpc region have been successfully
reproduced by invoking two major ICM components with discrete
temperatures, indicating a 2P property. A third weak 0.8 keV component
is marginally detected in the core region. In \S3.2, we analyzed the
{\em Chandra} data, under 2-D (spatially) and 2P (spectroscopically)
formalism. The ICM metallicity was found to show a similar spatial
distribution to that of the cool phase
component, indicating that the cool phase ICM is more metal-enriched than
the hot phase one.
\subsection{Two-phase ICM Properties}
A joint fitting of the thin shell spectra from the ACIS and EPIC,
and thick shell spectra from the XIS, consistently indicates a clear
preference of the
two-phase ICM nature, over the single-phase one, in the central
80$h_{71}^{-1}$ kpc region of A1795. The ICM therein can be
characterized by two representative temperatures, $T_{\rm h} = 5.0 -
5.7$ keV and $T_{\rm c} = 2.0 -
2.2$ keV for the hot and cool phases, respectively, whose $0.3-10.0$
keV luminosities were estimated to be $L_{\rm h} = 2.5\times10^{44}$
ergs $\rm s^{-1}$ and $L_{\rm c} = 1.4\times10^{44}$ ergs $\rm
s^{-1}$.
Both components show
insignificant temperature gradients in the 80$h_{71}^{-1}$ kpc region. The hot phase ICM
dominates in volume over the cool phase one by a factor of $\geq 4$,
even in the innermost region. The 2P view is consistent with
previous {\em ASCA}\/\ results reported in Xu et al. (1998). Employing the
2P ICM model and fitting the cool phase and hot phase components with
$\beta$ and double-$\beta$ gas density models, respectively, we
quantified the shape of cluster potential. As shown in Figure 9, the
total gravitating mass profile exhibits a
central mass excess ($M_{\rm excess} \approx 1.5 \times 10^{13}
M_\odot$), which has a similar spatial scale to the region of
clear 2P property. This indicates a
potential hierarchy, with a 100$h_{71}^{-1}$ kpc level halo nested in
the center of a cluster-scale one.
As shown in Figure 3{\it b}, the ICM metallicity is also enhanced in the 2P region.
Furthermore, by comparing the 2-D metallicity and the emission measure ratio maps
created with the 2P ICM model, we found a possible spatial
correlation on $\geq 1\sigma$ level between the cool-phase fraction
(or $\eta_{\rm c}$) and the ICM metallicity in the $50-100$$h_{71}^{-1}$ kpc region. The 2-D maps
also revealed a filamentary distribution of the cool, metal-rich gas in
this region. The subsequent
2P fitting to the XIS spectra confirmed the metallicity enhancement
in the cool phase relative to the hot phase, giving the best-fit
iron abundances of the cool phase and hot phase ICM to be 0.80 $Z_\odot$
and 0.36 $Z_\odot$, respectively. Such a picture resembles the
previous findings of the metal-rich, multi-phase arms in M87
(Simionescu et al. 2008). In the innermost core of A1795, i.e., $r <
25$$h_{71}^{-1}$ kpc, however, a dip is seen in the abundance
distribution, which does not correlate with the cool phase
fraction. Such
a feature is also seen in the coolest spots of some other clusters,
while its origin still remains unclear (e.g., Sanders et al. 2006).
In \S 3.1.3 and \S 3.1.5, we showed that the XIS and RGS spectra of the
central 144$h_{71}^{-1}$
kpc region can be better reproduced by including an additional weak 0.8 keV
component, without significantly revising the properties of the cool and hot ICM
components. This 0.8 keV component has a
$0.3-10.0$ keV luminosity of $3.6^{+1.2}_{-2.6}\times10^{42}$ ergs
$\rm s^{-1}$, and is dimmer than the hot ICM component by two orders
of magnitude. Since the temperature and luminosity of this component
are consistent with those of an X-ray/H$\alpha$ filament in the core
of A1795 as reported in Fabian et al. (2001), the 0.8 keV emission might be
associated with a portion of ISM of the cD galaxy, which is presumably
confined within some filamentary magnetic structures.
The 2P formalism has also been successful for the central region of
the Centaurus cluster, as firstly reported by Fukazawa et al. (1994)
and Ikebe et al. (1999) with {\it ASCA} data, and recently reinforced
by Takahashi et al. (2009) using {\it
XMM-Newton}. According to Takahashi et al. (2009), the cool phase
and hot phase coexist and dominate in the
central 70$h_{71}^{-1}$ kpc of the Centaurus cluster, exhibiting a
temperature ratio $T_{\rm
c}/T_{\rm h} \approx 0.46$. These properties are analogous to those
of A1795. In addition, a similar central
excess in the total gravitating mass profile, with a radius of
50$h_{71}^{-1}$ kpc, has also been found in the Centaurus cluster
(Ikebe et al. 1999). Thus, the two clusters are very similar in the
ICM properties, as well as in the total mass distribution.
As shown in \S3.1.4 and Figure 4{\it b}, the cool ($\sim 2$ keV) phase component of A1795
occupies up to only 20\% volume even in the innermost region, and in
Figure 6 and 7, no apparent separation on the spatial distribution is seen
between the cool phase and hot phase ICM in the cluster
core. This indicates that the cool phase is substantially mixed
into the surrounding hot phase, instead of forming any
``cool-phase-only'' core on scale of $\geq 15$$h_{71}^{-1}$
kpc. Since such a
2P structure is seen predominately around a cD galaxy, with an
enhanced metallicity (particularly in the cool phase), the phenomenon
is very likely related to the cD galaxy.
\subsection{Bubble Uplifting Model}
One possible formation mechanism of the 2P structure could be an
AGN-driven gas transport. As indicated by numerical simulations, e.g., Churazov et al. (2001) and Guo
\& Mathews (2010), the buoyant bubbles created by AGN outbursts can drag a certain amount
of surrounding cool, metal-rich gas to larger radii. For example, the
ICM in central 30$h_{71}^{-1}$ kpc regions of the Hydra A cluster and
M87 are known to be in a 2P or multi-phase form (Ikebe et al. 1997; Molendi
2002). More recently, the distributions of cool phase ICM were
found to coincide with the powerful radio lobes and X-ray cavities, hence, the bubble
uplifting model has successfully
been invoked for both Hydra A cluster and M87 (Nulsen et al. 2002;
Simionescu et al. 2008). These properties may be useful to explain the origin
of the co-existing 2P ICM. As for A1795, by adopting the obtained
distributions of total gravitating mass and cool phase gas
mass ($M(R)$ and $M_{\rm c, gas}$, respectively; \S
3.3.2), the energy required to uplift the cool
component from cD galaxy center to current position (average $R
\approx 40$$h_{71}^{-1}$ kpc) is estimated as $\int_{0}^{R} G M_{\rm c, gas} M(R^{\prime})/{R^{\prime}}^{2} {\rm d}R^{\prime} \sim 10^{58}$ ergs, which
is consistent
with the amount of mechanical energy injected from its central AGN outburst (Rafferty et al. 2006).
Nevertheless, the faint X-ray cavity detected in A1795 is located
$\lesssim 20$$h_{71}^{-1}$ kpc from the center (Birzan et al.
2004), apparently much smaller than the radius of the cool phase ICM
as shown in
the obtained 2-D temperature and the $Q_{\rm c}$/$Q_{\rm h}$ maps (Fig. 6
\& 7). Hence the appearance of the 2P ICM cannot be
solely ascribed to bubble uplifting via recent AGN activity. Neither
can the cool phase ICM be bubble remnant, since the uplifted gas would
conductively mix and reach thermal equilibrium with the surrounding
hot phase ICM quickly, i.e., within $\sim 10^{6}$ yrs (e.g., Simionescu
et al. 2008), after the bubble is torn apart.
\subsection{cD Corona Model}
The most natural and effective way to sustain a stable 2P structure is
to separate the two ICM phases by magnetic fields. In such a case, the two
phases become thermally insulated with each
other, since the gyroradius of a thermal electron in a $1-20$ $\mu$G field
is smaller than its mean free path by about $10-11$ orders of magnitude
(Sarazin 1988). Actually, cluster central regions are often
threaded by rather strong ($\sim 10$ $\mu$G; Ge \& Owen 1993) magnetic
fields, which are possibly related to the cD galaxy. Employing a
general topological classification of such cD-galaxy-related magnetic
field lines into closed and open ones, Makishima et al. (2001)
proposed that the closed loops (with their both ends anchored to the
cD galaxy) are filled with the cool phase ICM, while the open-line
regions, connecting to outer regions, are permeated by the hot
phase. Thus, the cool phase component is expected to emerge as
numerous
filamentary substructures in the core region, while the hot phase distributes
throughout the vast cluster volume. In addition, the cool phase is
naturally more
metal-rich than the hot phase in this scenario, as the supernova yield
of the cD galaxy may be largely confined within the loops
(Takahashi et al. 2009), which are surrounded by intruding, less
contaminated hot phase ICM. This ``cD corona'' model is expected to
provide a natural account
for the appearance of metal-rich cool phase ICM in the center of
A1795.
As the cool phase is thermally insulated from the hot phase, and the
radiative transport between the two phases can be ignored (Sarazin
1988), a continuous, efficient heating source located in the central region is required to prevent it
from collapsing due to radiative cooling (see Aschwanden et al. 2001
for a review). As first pointed out by Rosner, Tucker \& Vaiana (1978;
hereafter RTV)
for solar corona, and applied successfully to the Centaurus cluster by
Takahashi et al. (2009), a loop structure as illustrated in Figure 11
has a built-in feedback mechanism to maintain thermal stability for
the cool phase plasma. In order to characterize the loop
heating mechanism in a quantitative way, we employed an updated
version of RTV model, an analytical
solution to the hydrostatic model introduced in Aschwanden \&
Schrijver (2002; AS02 hereafter) for solar corona and applied it to the 2P
ICM. This model considers a thin, arch-like loop
delineating magnetic field lines, for which the pressure between the
loop-interior and surrounding plasmas are in equilibrium. To constrain the
temperature and density structures of the loop, AS02 assumed mass and
momentum conservations along the loop, as well as a total energy
balance involving radiative loss rate, heating rate,
and thermal conductive flux. The heating rate function has been defined in
form of $E_{\rm H} = E_{\rm 0}$ ${\rm exp} (- h / h_{\rm H})$, where $E_{\rm
0}$ is heating rate at the bottom of loop, $h$ is
the height in the loop plane, and $h_{\rm H}$ is the scale height of
the heating source (Fig. 11). The analytic solution by AS02 gave a loop
temperature distribution increasing with $h$ in general, and the exact
temperature gradient was predicted to be sensitive to the heating scale
$h_{\rm H}$. As shown in Figure 9 of AS02, for a uniform heating,
i.e., $h_{\rm H} \geq H$, where $H$ is the loop height, the loop temperature
exhibits a mild decrease towards the loop bottom by $>60$\%, while for a small
scale heating at the bottom, i.e., $h_{\rm H} \ll H$, a large
part of the loop becomes near isothermal, with only a sharp, small-scale temperature
drop seen at the footpoints. Such a loop also obey a scaling law, as
defined in Eq.(29) of AS02 (also see Eq.19 below), which describes the
loop maximum temperature $T_{\rm max}$ as a function of $h_{\rm H}$,
$H$, and external
pressure at footpoint $P_{0}$. Another
important property of the loop-confined plasma is that, as shown in
Eq.(30) of AS02, $T_{\rm max}$ shows rather weak dependence on the heating
rate $E_{\rm 0}$, i.e., $T_{\rm max} \propto E_{\rm 0}^{2/7}$,
because a decrease (or increase) in $E_{\rm 0}$ will
make the loop thinner (or thicker), so as to keep the plasma
luminosity nearly equal to $E_{\rm 0}$.
Specifically, following AS02, the temperature distribution of the
loop-interior plasma is expected as
\begin{equation} \label{eq:ltp}
T(h)=T_{\rm max}\left\{1-\left[1-\left(\frac{2}{\pi}\right){\rm arcsin}\left(\frac{h}{H}\right)\right]^{a(H/h_{\rm H})}\right\}^{b(H/h_{\rm H})},
\end{equation}
and the density distribution can be calculated as
\begin{equation} \label{eq:ldp}
n(h)=\frac{P(h)}{k_{\rm B}T(h)}=\frac{P_{0}}{k_{\rm B}T(h)}{\rm
exp}\left[-\frac{\mu m_{\rm p}}{2 k_{\rm
B}}\int_{0}^{h}\frac{g(h^{\prime})}{T(h^{\prime})}d h^{\prime} \right],
\end{equation}
where
\begin{equation} \label{eq:sl1}
T_{\rm max}=\left(\frac{\pi H}{2}P_{0}\right)^{1/3}S_1^{AS}(H,h_{\rm H})
\end{equation}
is directly derived from the scaling law (Eq.29 of AS02), $g$ is the
gravitational
acceleration calculated from the obtained total mass profile (\S 3.3.2),
$a(H/h_{\rm H})$ and $b(H/h_{\rm H})$ are the best-fit analytic
functions, as defined in Eqs.(20) and (21) of AS02, respectively, to
reproduce the dependence of $T(h)$ on $H/h_{\rm H}$ derived in a
numerical way, and similarly,
$S_1^{AS}(H,h_{\rm H})$ is the best-fit empirical function of scaling law
factor given in Eq.(35) of AS02. We adopted the best-fit coefficients
of the hydrostatic solutions presented in Table 1 of AS02 to
characterize $a(H/h_{\rm H})$, $b(H/h_{\rm H})$, and
$S_1^{AS}(H,h_{\rm H})$, and then, to calculate $T_{\rm max}$,
$T(h)/T_{\rm max}$, and $n(h)$ based on the loop properties.
In short, given a set of ($P_{0},H,h_{\rm H}$), we may determine the
temperature and density distributions
of the loop-interior plasma.
Next we applied the AS02 model to the X-ray observations of A1795 and,
for a comparison, the Centaurus cluster, both harboring a prominent
cool phase in the central region. In the
calculation we determined the external pressures $P_{0}$ by the
temperatures and densities of the hot
phases measured at the innermost regions, and took the loop heights
$H\approx80$$h_{71}^{-1}$ kpc and 70$h_{71}^{-1}$ kpc for A1795 and the
Centaurus cluster, respectively,
as indicated by the radii of cool phases (\S 3.1.2; Takahashi et
al. 2009). Figure 12{\it a} shows the predictions of Eq.(19), i.e., $T_{\rm max}$ as a
function of $h_{\rm H}$, in comparison with the measured maximum cool
phase temperature, i.e., $T_{\rm max} \approx 2.4$ keV
and $2.2$ keV for A1795 and the Centaurus cluster (Fig. 4{\it a};
Takahashi et al. 2009), respectively. This comparison yields $h_{\rm H} \approx
11$$h_{71}^{-1}$ kpc for A1795 and 18$h_{71}^{-1}$ kpc for the
Centaurus cluster, which indicates
that the heating is concentrated at the bottoms of the loops.
Since the sizes of radio lobes of the central AGNs in A1795 and the
Centaurus cluster ($\approx 10$$h_{71}^{-1}$ kpc and
$15$$h_{71}^{-1}$ kpc, respectively; Ge \& Owen 1993; Taylor et
al. 2002) agree well with $h_{\rm H}$
estimated above, we speculate that the AGN feedback is a candidate
heating source for the coronal loops. The life time of the radio lobes can be estimated by
$t_{\rm sync} = \frac{9 m_{\rm e}^3 c^5}{4 e^4 \bar{B}^2 \gamma_{\rm
e}} \sim 10^{7} - 10^{8}$ yrs, roughly consistent with the
cooling time of the cool phase ICM. Furthermore, as shown in
Rafferty et al. (2006), the heating rates provided by the central AGN
outbursts in A1795 and the Centaurus cluster were estimated to be
$\simeq 1.6 \times
10^{44}$ ergs $\rm s^{-1}$ and $7.4 \times 10^{42}$ ergs $\rm
s^{-1}$, respectively, which agree well with their cool phase luminosities
($\simeq 1.4\times10^{44}$ ergs $\rm s^{-1}$ for A1795 and $1.0 \times 10^{43}$
ergs $\rm s^{-1}$ for the Centaurus cluster).
Having obtained the estimates of $H$ and $h_{\rm H}$ for the two
clusters, we then calculated, via Eq.(17), the $T(h)/T_{\rm max}$
profiles, and show the results in Figure 12{\it b}. Thus, a mild
inward temperature decrease is expected, especially for the Centaurus cluster. Actually, Takahashi et
al. (2009) showed (in their Fig. 4{\it a}) that the cool phase
temperature of the Centaurus cluster decreases mildly, from $\sim
2.2$ keV (as already used in Fig. 12{\it a}) at $\sim 25$$h_{71}^{-1}$ kpc
, to $\simeq 1.6$ keV = 0.72 $T_{\rm max}$ at $\lesssim
7$$h_{71}^{-1}$ kpc. This latter value, when plotted on Figure 12{\it
b} (with a dashed cross), agrees with the prediction. The same is
true in the A1795 case, which shows a more flat $T(h)/T_{\rm max}$
profile that is consistent with the measured value of $\simeq 0.94$ at
15$h_{71}^{-1}$ kpc (solid cross on Fig. 12{\it b}). As shown in
Figure 12{\it c}, we applied the obtained $T(h)$ to Eq.(18) and
calculated the normalized loop gas density profile $n(h)/n_{\rm 0}$ for
A1795, which again agrees well with the observed profile.
In summary, the cD corona view, combined with the AS02 modeling of the
loop-confined plasmas, can account for several important properties of
inner regions of A1795 and the Centaurus cluster. These properties
include the stable 2P structure, the higher metallicity of
the cool phase, the absolute values of $T_{\rm c}$, and the spatial
$T_{\rm c}$ and density distributions.
\section{CONCLUSION}
By analyzing the {\em Chandra}, {\em XMM-Newton}, and
{\em Suzaku} data of the X-ray bright galaxy cluster A1795, we report
clear preference for the 2P ICM model in the central 80$h_{71}^{-1}$
kpc, which consists of a cool phase
($2.0-2.2$ keV) and a hot phase ($5.0-5.7$ keV) component. This 2P
model provides significantly better fit to the deprojected spectra
than the 1P model with continuous temperature profile, while the
latter cannot be fully ruled out based on current data. Combining the
{\em Suzaku} XIS and the {\em XMM-Newton} EPIC \& RGS, we
have marginally detected a third weak 0.8 keV component in the inner
144$h_{71}^{-1}$ kpc region that can be ascribed to a portion of ISM
component of the cD galaxy. Based on a 2-D spectral
analysis with the ACIS data, we have revealed a
possible spatial correlation between the cool
phase and metal-rich gas in the $50-100$$h_{71}^{-1}$ kpc
region. A follow-up XIS analysis shows consistent result, that the
cool phase ICM does exhibit a higher metallicity ($\approx 0.80$
$Z_\odot$) than the hot phase one ($\approx 0.36$ $Z_\odot$). Hence, we have
successfully resolved a 2 keV, metal-rich component associated with the
cD galaxy, which is spatially mixed but thermally separated with
the surrounding 5 keV cluster component.
All these properties can be explained by the cD corona model incorporating with
the AS02 solution for quiescent coronal loops.
\section*{Acknowledgments}
This work was supported by the Grant-in-Aid for Scientific Research (S),
No. 18104004, titled "Study of Interactions between Galaxies and
Intra-Cluster Plasmas", and by the National Science Foundation of China
(Grant No. 10878001 and 10973010, and National Science Fund for
Distinguished Young Scholars) and the Ministry of Science and Technology
of China (Grant No. 2009CB824900 and 2009CB824904). L. G. was
supported by the Grand-in-Aid for JSPS fellows,
through the JSPS Postdoctoral Fellowship program for Foreign Researchers.
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Q: Medoo and Timezone I use "Medoo" as a database framework.
My problem is I'm in Brazil, but my database (which is MySQL) is in Los Angeles, meaning the timezone is different.
I use the NOW() function of the database a lot and everything is like Los Angeles time.
Eventually this causes me issues.
With pure PHP, I usually resolve this as follows, however, with Medoo I do not know how to resolve:
mysql_query ('SET time_zone = "America / Sao_Paulo"');
mysql_connect ("host", "user", "password") or
die ("Could not connect:". mysql_error ());
mysql_select_db ("database");
$ result = mysql_query ("myquery");
Did you realize that before the query, I made a SET in time_zone ? I do not know how to do this in Medoo or even leave it at the definitive setting.
Anyone have ideas on how to solve this?
A: Check out the documentation about initialization. There is a command option that will be executed after database connected. You can set the timezone there.
$database = new Medoo([
...
// [optional] Medoo will execute those commands after connected to the database for initialization
'command' => [
'SET SQL_MODE=ANSI_QUOTES'
]
]);
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Tim the Fox – Puzzle Tales – is a great game that will teach your child to assemble images from little pieces. Magical objects from the tales by European writers come alive after the completion of each level.
The game is aimed, first and foremost, at children aged 3 to 5, but will be fun for anyone aged 0 to 100.
Putting together bright pictures made up of little pieces with different shapes helps children develop attention and imagination, organizational skills and logic.
Your little one is sure to be entertained; after all putting together Tim the Fox puzzles is much more exciting than regular table-top puzzle!
• The game features pleasant background music.
Give a magical atmosphere to your kid in fairy puzzles «Tim the Fox — Puzzle Tales»!
← «Tim the Fox – Travel» is presented to your attention!
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The all familiar rebukes by our parents such as "eat your veggies" and "chew your food well" are an important part of our diet. Just ask the dinosaurs… Dinosaurs you say? I was scanning some interesting science journals which is customary for me since I am what you consider an arm-chair scientist and found this article. Most fat dinosaurs didn't chew.
I know it's a bit of a stretch so I won't draw parallels, but let's face it, in a world with time pressures; we have all been guilty of "inhaling" our lunch at some point. I keep coming back to the notion of eating what you need and not what you are wanting, but a little caveat should be added in that we should take and enjoy our food.
We take pride at Rice by offering different and sometimes exotic ingredients and for the most part students enjoy this. I congratulate those students who are willing to try new and exciting flavors. Those of you who do not veer away from the proverbial burger and fries diet are doing yourselves a disservice. The chefs are tremendous here and you can just taste the pride in their food along with amazing and complex flavors.
When you all become parents in the future, just remember those two phrases Mom & Dad taught us, "eat your veggies" and "chew your food well", because I guarantee you that you will need to say those words those often with your children. Very Often!
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{"url":"https:\/\/tex.stackexchange.com\/questions\/480472\/linear-comb-plot-choosing-origin-different-from-zero","text":"# Linear comb plot, choosing origin different from zero\n\nI'm plotting spectral data, sometimes with logarithmic axes and sometimes linear but with values in dB. The usual way for plotting these kind of diagrams (at least in my field) is to have all the lines start at the lower part of the graph, despite their values (negative, positive). For logarithmic comb plots, this can easily be achieved using log origin = infty. However, how can I use something like this for plots with negative values (in linear scale)? Currently, I'm shifting all points up by the smallest value and then shifting all yticklabels. However, this is cumbersome and not really an automatic solution. I have to find the smallest value by hand in order to find the optimal shift. Can this be improved? I've found this answer, but this is about logarithmic plots. I don't really understand the code, so I don't know if this can easily be modified.\n\nHere is a MWE with what I'm trying to achieve:\n\n\\documentclass{article}\n\n\\usepackage{pgfplots}\n\n\\begin{document}\n\\begin{tikzpicture}\n\\begin{axis}\n[\n% I'm using this:\nyticklabel = {\\pgfmathparse{\\tick-130}\\pgfmathprintnumber{\\pgfmathresult}},\ny filter\/.expression = {y + 130},\nymin = 0\n% I would like to have this:\n% comb origin = -infty % or something\n]\n(1, -6)\n(2, -80)\n(3, -85)\n(4, -120)\n(5, -120)\n(6, -120)\n(7, -120)\n(8, -120)\n(9, -120)\n};\n\\end{axis}\n\\end{tikzpicture}\n\\end{document}\n\n\nAnd the result looks like this:\n\nIf I don't do this shift, then the result looks like this (since all combs start at y = 0):\n\n\u2022 I read this question several times and still do not understand what you are asking. Could you please consider making your question clearer? E.g. by adding a sketch of the target output? (The way I read your question you already have achieved your target output.) \u2013\u00a0marmot Mar 23 at 1:13\n\u2022 Yes, what I am showing is the target output. I would like to have a key for linear comb plots to change the starting line (baseline?) for them, similar to log origin. I will add a picture of the unwanted output to the question. \u2013\u00a0pschulz Mar 23 at 9:17\n\nI think that you can use one \"very big number\" as replacement of infinity. Here I use 1000 for the value of \\verybignumber. Then I apply your code with \\verybignumber in place of 130 and I let pgfplots to set ymin.\n\n\\documentclass[border=7pt]{standalone}\n\\usepackage{pgfplots}\n\\def\\verybignumber{1000}\n\\begin{document}\n\\begin{tikzpicture}\n\\begin{axis}\n[\nyticklabel = {\\pgfmathparse{\\tick-\\verybignumber}\\pgfmathprintnumber{\\pgfmathresult}},\ny filter\/.expression = {\\verybignumber+y},\n]\n(1, -6)\n(2, -80)\n(3, -85)\n(4, -120)\n(5, -120)\n(6, -120)\n(7, -120)\n(8, -120)\n(9, -120)\n};\n\\end{axis}\n\\end{tikzpicture}\n\\end{document}\n\n\n\n\u2022 Ah, interesting. I thought I tried something like this, but the explicit setting of ymin messes things up. I used this because the lines started 'hovering' over the lower axis line, but with a number very different from the lowest value this works. Thanks! \u2013\u00a0pschulz Mar 24 at 20:27\n\u2022 I made a key of this for my settings: comb origin infty\/.style = { yticklabel = {\\pgfmathparse{\\tick-1000}\\pgfmathprintnumber{\\pgfmathresult}}, y filter\/.expression = {y + 1000} } \u2013\u00a0pschulz Mar 24 at 20:28","date":"2019-07-20 16:20:24","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\": 0.7149893641471863, \"perplexity\": 1349.6186639123648}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2019-30\/segments\/1563195526536.46\/warc\/CC-MAIN-20190720153215-20190720175215-00163.warc.gz\"}"}
| null | null |
\section{Introduction}
Internet of Things (IoT) is having its hipe now, as the Internet had its hipe two decades ago. IoT market is expected to grow from more than 15 billion devices three years ago to more than 75 billion in 2025 \cite{dataeconomy}. IoT needs a strong technological foundation for its rapid development and acceptance from the scientific community. Hence, the fog computing is a very strong candidate to provide this foundation for IoT. By providing several advantages, fog computing is expected to be one of the main backbone pillars of the IoT in terms of computational support.
As shown in Fig.~\ref{fig1}, from a conceptual point of view, we are predicting fog computing to serve as an intermediate level of service for seamlessly handshaking the protocols of cloud computing and IoT. This will bring many benefits: 1) Cloud computing servers are super fast in contrast to the IoT devices. Fog computing devices will provide an interface between the two far set of devices. 2) This intermediate layer of fog computing will allow several fixes (such as patch updates, etc.) to be done easier. Instead of making changes on IoT devices, software updates can be pushed on to the fog device(s). 3) Fog computing will bring all the advantages of edge-computing, such as the agility, scalability, decentralization, etc.
As a centralized resource out of users' control, the cloud represents every possible opportunity to violate privacy. Unfortunately, privacy has become a luxury today, a situation that will be exacerbated in the IoT \cite{zhang2015cloud}. Therefore, a remedy is needed to enhance the privacy needs of the users in these services and fog computing is a strong candidate to provide this.
Fog computing actually is a tool for cloud-based services (CBS) that can be thought of as an interface in between the real end-devices and the rest of the CBS. CBS offers three service models, namely Infrastructure as a Service (IaaS), Platform as a Service (PaaS) and Software as a Service (SaaS) \cite{butun2015anomaly}. We are projecting that fog computing paradigm will act as an interface for these CBS service models so that intended services can be used by the front-end users seamlessly and promptly.
The \textit{Security Plane} for CBS proposed by Butun \textit{et al.} \cite{butun2015anomaly} was intended to be used for the front-end IoT devices and to be an interface to the cloud. After the proposal of fog computing, this \textit{Security Plane} kind of solution is highly implementable. Therefore, we think of fog computing to provide extra services such as security to the edge of the cloud for the CBS. For example, the usage of fog computing would bring benefits to the Intrusion Detection Systems (IDS) that are devised for IoT. Hence early detection is important to stop ill effects of intrusions, fog computing would bring early detection opportunities to IDS algorithms working on IoT.
Fog computing brings three immediate advantages over cloud computing: $1)$ Enhanced service quality to mobile users. $2)$ Enhanced efficiency to the network. $3)$ Enhanced location awareness. Among these benefits, the major benefit of fog computing over the cloud is that the support for location awareness which might be very useful for the applications that are employing location based services (LBS) \cite{A9:LuanG0XS15}.
\begin{figure}[b]
\centering
\includegraphics[width=0.5\textwidth]{fog_computing}
\caption{\label{fig1} Fog computing proposed as a gateway in between cloud computing and IoT.}
\end{figure}
\begin{figure*}
\centering
\includegraphics[width=1.0\textwidth]{Fog_diagramV2.png}
\caption{\label{fig2} An illustration of four different possible fog computing applications with IoT: Smart Office, Smart Factory, Smart Home and Smart Traffic.}
\end{figure*}
\section{Background}
Large-scale IoT deployments created situations which cloud computing could not handle efficiently and effectively. For instance, applications which require low latency while processing the data on the edge of the network. In real life, a massive amount of data is being collected by IoT from many different sensors in various environments such as factory production lines, vehicles, machines, elevators etc. or individual purposes such as smart home systems, hobby related sensors, etc.
These sensing devices have different characteristics and features. They are connected to each other via hardwire or WiFi. Large-scale device deployments in heterogeneous environments bring management issues. Hence, intelligent communications approaches are needed in which efficiency and robustness are prioritized.
Using a cloud network to stream data and analyze data has its limitations such as bandwidth consumption and communication costs. If the user data are sensitive, securing the data is another important issue. The data are important for auditing purpose or controlling the assets to improve efficiency or preventing disasters etc.
The data analysis could be done on site by running the software at local stations. The cloud would be used as storing the analysis result for historical and audit purposes. The data aggregation will reduce the bandwidth and also bandwidth related cost.
\begin{table*}
\caption{Comparison of cloud and fog computing concepts.}
\label{table1}
\small
\begin{center}
\begin{tabular}{|l||c|c|}
\hline
\textbf{Feature} & \textbf{Cloud computing} & \textbf{Fog computing}\\
\hline
\hline
Access & Wired or wireless& Wireless\\
\hline
Access to the service & Through server & At the edge device\\
\hline
Availability & Mostly available & Mostly volatile\\
\hline
Content distributed to & Edge device & Anywhere\\
\hline
Content generator & Man made & Sensor made\\
\hline
Content generation at & Central server & Edge device\\
\hline
Control & Centralized & Distributed\\
\hline
Latency & High & Minor\\
\hline
Location of resources (i.e. processing and storage) & Center& Edge\\
\hline
Mobility & Not supported & Supported\\
\hline
Number of users & Millions & Billions \\
\hline
Virtual infrastructure location & Enterprise server & User devices\\
\hline
\end{tabular}
\end{center}
\end{table*}
Fig.~\ref{fig2} presents various possible application fields of fog computing: Smart Office concept can be an example of the generic relation of IoT devices and fog computing. Smart Factory is an example of industrial IoT (IIoT) and fog computing application. There could be many IoT devices, sensors (temperature, pressure etc.), electric actuators or other control devices could be involved. Smart Home concept is emerging with IoT devices and home appliances such as TV, washing machine, dryer, refrigerator etc. as they are getting smarter and intelligent. In Smart Traffic example, data collection on site and immediately analyzing and processing data on the edge may help in fast decision making locally, instead of sending data to a central location. For instance, in case of an emergency, the traffic lights can be controlled to open a way for emergency vehicles such as fire trucks and ambulances based on local IoT devices. In these four different scenarios, the common idea is that the devices generate a massive amount of data and may need to collaborate with each other and to take critical decisions reducing the delay. Hence, an agile response is important and the philosophy of fog computing may help to overcome bandwidth and latency related problems in this manner.
Because of introducing agile response nearby the edge components, we are expecting fast implementation and business growth of fog computing for future IoT applications such as smart-traffic and smart-factories. Thereby, the integration will not remain in just IoT but expand to industrial IoT (IIoT) and further other areas. This will impose its own challenges to IIoT \cite{forsstrom2018challenges}, as well as bringing benefits.
IoT and fog computing can be helpful in designing ``smart'' things such as smart home, smart traffic lights, smart cities, etc. For instance, the sensors in a smart traffic system can detect accidents or sense the road conditions due to weather or some other factors and inform the drivers. A traffic jam can be regulated by a smart traffic system.
In recent years, due to the usage of IoT and other sensors, the data generated by end-devices increased massively. The question is where/when/how should these data be analyzed? In cloud-centric design, IoT devices generate data and send them to the cloud (operates as a central server) for storage and analyses. However, in fog computing, the data is analyzed on the edge stations and just necessary results are being sent to the cloud.
Fog computing concept is recently introduced by CISCO \cite{computing2015internet}, which is a new vision that enables IoT devices to run on the edge of the network. According to Bonomi \textit{et al.} \cite{A1:Bonomi:2012}, ``Fog Computing" is not an alternative for ``Cloud Computing". Fog extends the cloud computing and complements the cloud computing with the concept of smart devices which can work on the edge of the network. According to CISCOs vision, fog computing has following characteristics: 1- Low latency, 2- Location awareness, 3- Geographical distribution, 4- Mobility, 5- Very large number of nodes, 6- A predominant role of wireless access, 7- Streaming and real-time applications, 8- Heterogeneity \cite{computing2015internet}.
OpenFog Consortium \cite{openfogconsortium} is defining the standards of the fog computing with different committees and work-groups. The founding members are Arm, Cisco, Dell, Intel, Microsoft and Princeton University. The focus is to create and promote an open reference architecture for fog computing to solve challenges such as bandwidth, latency, etc. in various areas like AI, IoT, industrial machinery, Robotics, etc. According to OpenFog Consortium, the key pillars of the fog architecture are security, scalability, openness, autonomy, reliability, availability, serviceability, agility, hierarchy, and programmability. Fog computing is helping to the IoT, 5G and AI related systems which need some special unique properties such as security (trusted transactions), cognition (objective awareness), agility (scalable), latency (real-time processing), and efficiency (utilizing unused resources). According to OpenFog, the benefits of using fog computing are; low latency, business agility, security, real-time analytics, reduced cost, less bandwidth usage.
Table \ref{table1} presents fog computing and cloud computing concepts in a comparative way \cite{stallings2018computer}. As can be seen, fog computing presents more agile and rapid response when compared to cloud computing, henceforth represents a strong candidate as a technological solution for future IoT and IIoT based implementations. Fog computing would be a preferable approach with various IoT designs and applications such as Smart Home, Smart Traffic (Transportation and Connected vehicles etc.), Smart Grid, Industrial Automation and integration with IIoT, Smart Health-care Systems, etc.
The benefits of fog computing for IoT (and IIoT) can be summarized as follows:
\begin{itemize}
\item\textbf{Reducing cost:} The data will be processed on edge rather then cloud
\item\textbf{Reducing the delay:} Critical applications require low latency to interpret the data and to take a decision. The cloud computing is not suitable to serve this task.
\item\textbf{Agile response:} Real-time applications may benefit from fog computing concept to gain speed during analysis or decision making phase.
\item\textbf{Increased security:} With fog computing, service providers can filter out sensitive personally identifiable information (PII) and process it locally, sending the non-sensitive information to the cloud for further processing \cite{microsoft}.
\end{itemize}
\section{Practical application scenarios of Fog Computing usage in IoT}
In this section, we introduced a subset of possible attack scenarios from real-life and discussed possible related mitigation methods.
Coin, bill and card-based systems suffer from multiple attack vectors. Asking the following question ``How to hack a vending machine?" on Google returns thousands and thousands of results. Whether the hacks work or not does not really mean anything, the information of possible attack vectors poses a great threat to existing vending machine domain since one attack vector may work with an old (vulnerable) model. In early 80-90s, many coin-operated machines including but not limited to vending machines, arcade machines, public phones etc. suffered from simple Coin-on-a-String trick \cite{tvropes}. Later, this problem is mitigated with the one-way ratchet. Coin or card systems can be bypassed by attacker \cite{youtube:laundry1}, the machine can be tampered \cite{youtube:laundry2}. Paper bills tampered with plastic tape or other materials to get them back from machine after the credit is earned or attacker tries to machine spit the bill back after it is credited.
Even the vending machines in one of the most secured places on the world can be hacked, such as the ones in Central Intelligence Agency (CIA) facilities. According to BuzzFeed News, a group of CIA contractors exploited a vulnerability in the vending machines electronic payment systems to buy snacks at no cost \cite{buzzfeed}. The total loss is around \$3,314, but the most shocking fact is even vending machines in CIA are vulnerable and the unexpected can happen. Apparently, attackers disconnected the FreedomPay network cable connected to vending machines to exploit an ``Availability" issue which resulted in the machine permitting purchases made by unfunded FreedomPay cards. Attackers are later on identified by agency's surveillance cameras.
The extreme example above from real-life has shown us that we may more way to go in securing vending machines kind of payment systems. Luckily, now fog computing IoT technology enables us to implement more secure and agile payment systems as follows:
\subsection{Smart Laundromats}
In the real life, there have been many cases of laundromat system hacking reported \cite{reddit}.
Accordingly, it can be stated that none of the laundromat systems (whether token based, magnetic card based, or even smart card based) that are being used today is secure. One of the main reasons is that the machine usage is not controlled via the server but at the machines. When a machine is hacked, or a token (card) is hacked, there is no mechanism to check the validity (whether it is authentic or not) of the transaction. Therefore, we project real-time (or close to real-time) usage of authentication mechanism that is governed by a central server (Authentication Management) and served to the edge devices (laundromats) with the adoption of fog computing.
\subsection{Smart Vending Machines}
As an application of fog supported-IoT to vending machines, a vending machine can report missing items to the vendor so that they can be shipped on time. Besides, auto status check report can be generated and can be sent to the vendor for maintenance purposes.
Smart vending machines mitigates the security problems but much more can be achieved from a ``smart'' machine. For instance, detailed sales reports can be generated for the vendor. A particular product can be tracked throughout the year, and the vendor can get detailed insight about the sales of a product or user buying habits like which product sales are best, when, where, etc. Product-A sales are best in February, user-1 buys product-B on Monday morning but buys product-C on Tuesday evening, etc.
\begin{table*}
\small
\caption{Possible security considerations when the Fog Computing gateway is compromised}
\label{table2}
\begin{center}
\begin{tabular}{|c||c|c|}
\hline
\hline
\textbf{Feature} & \textbf{Risk on IoT network} & \textbf{Impact on Cloud}\\
\hline
Access control & Moderate & Minor\\
\hline
Authentication & Moderate/Significant & Minor/Major\\
\hline
Availability & Significant& Minor\\
\hline
Confidentiality & Moderate& Minor\\
\hline
Integrity & Minimal & None/Minor\\
\hline
Privacy & Significant & Minor/Major\\
\hline
\end{tabular}
\end{center}
\end{table*}
\subsection{Smart Chip Card Systems}
Smart Chip Card Systems are relatively secure than coin or paper bill based systems. The threat model for these systems is unencrypted or not well-encrypted cards. With a card reader/writer, the value in the card can be easily updated if the necessary preemptive steps are not taken. Another attack vector would be, an attacker may clone the card and get unlimited credit if there is not a mechanism to do validation (checking transaction with a server etc.) If IoT is embedded with the device in the production phase, it is unlikely that it will be physically tampered by an attacker easily (assuming human security details are overseeing the machine). A possible solution is designing a fog computing-based system which runs a micro-service to validate the transactions. Smart Chip Card can be an identifier for a legal user, once it is used in a device, the micro-service on fog computing system validates the user and transaction or escalates the situation and asks mobile application based authentication. For instance, if the card is used in a different location (same apartment complex but different building etc.) this can be reported as an instance.
Another distinct advantage of using fog computing-based IoT is the reduction of possible operational cost. For example, end-devices can connect to the smart grid and negotiate with the grid on the unit electric price. On a queue based approach, they can operate their tasks when the electric price is cheap (during the daytime for instance).
\section{Projections of Fog Computing usage in IoT and its Security Implications}
According to the scientific projections, fog computing is expected to be one of the main backbone structure of IoT in the near future \cite{dataeconomy, openfogconsortium,microsoft}. Inevitably, there will be implications of this integration.
In ideal conditions, extra introduced components are desired to bring no further burden on the overall operation of the existing system. However, this is not the fact in real life scenarios. Sometimes they bring an extra load (e.g. processing and memory storage), and sometimes (preferably) decrease existing load. The same is valid for security features.
Security implications of using fog computing for IoT systems is shown in Table \ref{table2}. There are six features (most important ones being CIA, i.e. confidentiality, integrity and availability) that we considered in the case of a failure (capture) in defending the fog computing gateway (FCG):
\begin{enumerate}
\item \textit{Access Control:} FCG devices provide a gateway between IoT network and cloud. However, the functioning is like providing field data from the IoT to cloud and command messages from the cloud to IoT. There is no way of accessing databases of the cloud from FCG devices, on the other hand, it is possible to command and conquer all IoT devices connected to a rouge FCG device.
\item \textit{Authentication:} Depending on the authentication algorithm design, if several layered authentication methodology (one at FCG, one at the cloud, etc.) is used then this might be a very secure solution. Any compromise would only affect limited number of IoT devices. Otherwise, if all authentication operation is left to the FCG devices, this might create problems when the FCG devices get compromised.
\item \textit{Availability:} Depending on the critical position of the FCG, we expect a significant impact on the availability of the IoT resources if the communications are blocked. However, this will not affect the cloud side marginally.
\item \textit{Confidentiality:} Confidentiality of the data at FCG has a moderate impact on IoT network, whilst has a minor impact on the cloud (all the devices connected to FCG gets affected, however, the rest of the IoT network and the cloud remains safe operation).
\item \textit{Integrity:} Depending on the communications scheme, we expect minor (if end-to-end encryption is not employed) to none (with encryption) effect of FCG on the integrity of the messages.
\item \textit{Privacy:} As in any service, privacy of the users in IoT is critical and any leakage through FCG devices might have serious consequences. This will affect all users that are using the IoT though that hacked FCG device. However, the rest of the data (resulting from other FCGs) at the cloud will still be secure and private.
\end{enumerate}
If these computers (gateways) are installed in a few numbers, they may constitute a single point of failure. Since, if an attacker manages to harm this gateway, all the communication between cloud and IoT would be blocked. Therefore, we suggest several gateways installed network architectural implementation.
As discussed above and also listed in Table \ref{table1}, compromising the fog computing gateway, will have an impact on cloud computing layer from minor to critical levels; and the risk on IoT network will be from minimal to significant levels, depending on the security feature that is observed. Therefore, in some applications, the security of fog computing gateway might be very important. Hardware and/or software security precautions for the fog computing gateway should be considered while deciding security provisioning for the overall network.
The fog computing layer can be leveraged by security services as a proxy server with the firewall and/or IDS capability. Most of the attacks can be prevented by the firewall. Any harmful intrusion attempt can be detected by IDS and mitigated at the fog computers before it can reach to main servers on the cloud.
To improve privacy, a wiser suggested solution would be keeping the private data on the edge while sending the just necessary data to the cloud.
\section{Future Remarks and Conclusions}
The proliferation of IoT devices in our surrounding environment is indispensable and this will be possible due to the dominant usage of fog computing as a backbone supporting architecture. Throughout this article, we have discussed the implications of using fog computing as a backbone architecture for IoT, especially from cyber security point of view.
According to our findings, we have stated that usage of fog computing for cloud-based IoT systems might have several benefits; in terms of cost, QoS and more importantly, security.
\balance
\bibliographystyle{IEEEtran}
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| 4,704
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{"url":"http:\/\/www.ryanhmckenna.com\/2018\/","text":"## Posts\n\nShowing posts from 2018\n\n### Optimal Strategy for Farkle Dice\n\nIn this post, I will discuss my findings in terms of an optimal strategy for Farkle, which is a dice game of chance. \u00a0If you are unfamiliar with the game, I encourage you to skim the linked article so that you can better understand this blog post. \u00a0All of my findings are based on sound mathematical methods and a computer program was used to determine the optimal strategy. \u00a0Before I begin, let me first state the value of different dice rolls I assumed while developing the strategy.\n\nEach 1: 100 points\nEach 5: 50 points\n3 1's: 1000 points\n3 2's: 200 points\n...\n3 6's: 600 points\n4 1's: 2000 points\n...\n4 6's 1200 points\n5 1's 3000 points\n...\n5 6's 1800 points\nStraight 1-6 1500 points\n3 Pairs (e.g., 22 33 44): 750 points\nEverything else is considered to be a Farkle (0 points).\n\nThere are many variants to Farkle, but I like to play by this set of rules.\n\nThe \"game state\" of Farkle roll can roughly be characterized by 2 values: the current score accumulate\u2026\n\n### Writing Efficient Numpy Code\n\nIn this blog post, I am going to talk about writing efficient numpy code in python. I do a good amount of numerical linear algebra for my research and personal projects, and I typically code in python with numpy (and spicy) because it is very easy to use and it is also very efficient when used correctly.\n\nConsider the following problem, which will serve as a concrete task to use as an example throughout this post. Suppose we have a (column) vector $x$ of length $n = 2^k$ and we want to compute $H_k x$ where $H_k$ is a \u201chierarchical\u201d matrix with branching factor $2$, defined by \\begin{align*} H_0 &= \\begin{bmatrix} 1 \\end{bmatrix} \\\\ H_{k+1} &= \\begin{bmatrix} 1 & 1 \\\\ H_k & 0 \\\\ 0 & H_k \\end{bmatrix} \\end{align*} Where the top row is a vector of all ones and 0 denotes a matrix of zeros having the same size as $H_k$. For example, $H_2$ is a $7 \\times 4$ matrix that looks like this: \\begin{bmatrix} 1 & 1 & 1 & 1 \\\\ 1 & 1 & 0 & 0\u2026\n\n### Representing Graphical Model Factors in Numpy\n\nI've been working a bit with graphical models lately for my research.\u00a0 For a while I was using a library called pgmpy for their implementations of factor arithmetic and inference algorithms.\u00a0 However, as my research is progressing I am needing more control than what pgmpy offers, so I decided to re-implement and extend the algorithms that I needed.\u00a0 If you are in a similar situation and are finding yourself implementing your own graphical models algorithms, this post is for you.\n\nIn the setting I am working in, I have a Markov random field model where there are only a small number of variables (no more than a few dozen, often much less).\u00a0 However, each variable can take on a possibly large number of categorical values and the graph may contain relatively large cliques that make exact inference computationally challenging (although still possible).\n\nSuppose the model has $d$ variables and variable $i$ can take on $n_i$ possible values.\u00a0 Usually a factor defined over $k$ variables can\u2026","date":"2020-05-29 19:10:44","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 1, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.9576312899589539, \"perplexity\": 597.9641953842686}, \"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-2020-24\/segments\/1590347406365.40\/warc\/CC-MAIN-20200529183529-20200529213529-00291.warc.gz\"}"}
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Jacob Crouch Suicide Prevention Services consist of suicide awareness and prevention educational programs that can be taught to anyone.
Jacob Crouch Suicide Prevention Services began in November 2006, taking educational programming into schools in the Acadiana area. This program provides students with the tools necessary to help themselves or others who may be showing signs of potential at-risk behavior. In 2010, we expanded our programs to include suicide prevention education for teachers, counselors and school administrators. Now thanks to generous funding from Acadiana Area Human Services District, our programming can be taught to anyone in the Region 4 area!
In addition to education, we also provide counseling services to individuals who are struggling with depression and may be contemplating suicide, to individuals and families experiencing a suicide crisis, to caregivers, and to family and/or friends who are grieving the loss of life as a result of a suicide. We also operate a survivor of suicide support group which is open to any adult who is grieving a loss to suicide.
Please visit our scheduling page or click HERE to schedule a session with one of our clinicians.
|
{
"redpajama_set_name": "RedPajamaC4"
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| 4,949
|
Omanhene ist bei verschiedenen Akanvölkern Ghanas der Titel eines obersten, traditionellen Herrschers ("Königs") einer Region oder größeren Stadt.
Die angelsächsische Übersetzung des Titels Omanhene ist Paramount Chief ("Oberster Regent"). Er ist die zentrale Figur und Institution einer Nation. Zwar hat er heute keine offizielle Funktion im Staat Ghana, besitzt aber erhebliche öffentliche Autorität. Der Omanhene ist auch ein Großgrundbesitzer und Mittelpunkt eines Feudalsystems, der Land an "Caretakers" verteilt.
Ein Omanhene wird von der Queen mother, der Mutter des Königs, bestimmt. Die Nachfolge verläuft also nach matriarchalischen Prinzipien. Eine Ausnahme bildete hiervon nur der Omanhene von Elmina (s. Geschichte Elminas)
Nicht bei allen Akanvölkern ist der Omanhene der oberste Herrscher. So erkennen die Aschanti beispielsweise als obersten traditionelle Herrscher den Asantehene an, dem lokal Omanhene untergeordnet sind.
Der Wortbestandteil "hene" findet sich dabei als Bezeichnung für "Herrscher" auch in anderen Titeln der Völker Ghanas wieder. So ist der Herrscher der Dagomba Nordghanas, die nicht zu den Akanvölkern gehören, der Dagombahene usw.
Siehe auch
Regentschaft bei den Akan
Weblinks/Quellen
Liste traditioneller Herrscher der Akan, darunter Omanhenes
Kultur (Ghana)
Politik (Ghana)
Herrschertitel
|
{
"redpajama_set_name": "RedPajamaWikipedia"
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| 6,485
|
Q: On minimality of semitopological and quasitopological groups The phenomemnon of minimality is well-studied in the realm of topological groups.
Let us recall that a topological group $X$ is minimal if each bijecive continuous homomorphism $h:X\to Y$ to a topological group $Y$ is a topological isomorphism.
I am interested if the notion of minimality was studied in the realm of semi-topological or quasi-topological groups.
A semi-topological group is a group $G$ endowed with a Hausdorff topology making the group operation $G\times G\to G$, $(x,y)\mapsto xy$, separately continuous.
A semitopological group is called a quasi-topological group if the operation of inversion $G\to G$, $x\mapsto x^{-1}$, is continuous.
Let us define a semi-topological group $X$ to be semi-minimal if each continuous bijective homomorphism $h:X\to Y$ to a semitopological group $Y$ is a topological isomorphism.
By analogy we can define a quasi-topological group $X$ to be quasi-minimal if each continuous bijective homomorphism $h:X\to Y$ to a quasi-topological group $Y$ is a topological isomorphism.
It is clear that each semi-minimal quasi-topological group is quasi-minimal and each quasi-minimal topological group is minimal.
It can be shown that each compact Hausdorff semitopological group is semi-minimal (topological group).
Problem 1. Is each semi-minimal semi-topological group compact?
Problem 2. Is each quasi-minimal quasi-topological group compact?
I even cannot prove or disprove the following
Conjecture. No countable Boolean semi-topological group is semi-minimal.
Remark. By the answer to this MO problem, for some submonoid $M$ of the monoid $\omega^\omega$ of self-maps of a countable set $X$ there is no minimal Hausdorff topology on $\omega$ in which all self-maps $f\in M$ are continuous. At the moment this is the unique (known to me) example of an algebraic systems with unary operations, admitting no minimal Hausdorff topology.
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 7,670
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Q: Can andoird bluetooth api scan BLE device when the broadcast channel wasn't at 37,38,39 So I had a Nordic nrf52840 using the example periodic_adv in nrf connect sdk v2.0.0, and I wrote an app using android bluetooth api. I can't find my device, but I can find some device then I look up in wireshark sniffer to check what's the diffrence, I found out that the device I can find broadcast at channel 37,38,39 mine broadcast at 0 to 39, can bluetooth api scan device broadcasting at other channel or it can only find device working at 37,38,39?
package com.example.tryble_scanner;
import androidx.activity.result.ActivityResultLauncher;
import androidx.activity.result.contract.ActivityResultContracts;
import androidx.appcompat.app.AppCompatActivity;
import android.Manifest;
import android.bluetooth.BluetoothAdapter;
import android.bluetooth.BluetoothDevice;
import android.bluetooth.BluetoothManager;
import android.bluetooth.le.BluetoothLeScanner;
import android.bluetooth.le.ScanCallback;
import android.bluetooth.le.ScanFilter;
import android.bluetooth.le.ScanResult;
import android.bluetooth.le.ScanSettings;
import android.content.Intent;
import android.os.Build;
import android.os.Bundle;
import android.os.Handler;
import android.util.Log;
import android.view.View;
import android.widget.Button;
import android.widget.Toast;
import java.io.UnsupportedEncodingException;
import java.nio.ByteBuffer;
import java.nio.ByteOrder;
import java.util.ArrayList;
import java.util.List;
import java.util.UUID;
public class MainActivity extends AppCompatActivity {
private BluetoothAdapter mBluetoothAdapter = null;
private BluetoothLeScanner mBluetoothLeScanner = null;
public static final int REQUEST_BT_PERMISSIONS = 0;
public static final int REQUEST_BT_ENABLE = 1;
private boolean mScanning = false;
private Handler mHandler = null;
private ScanCallback mLeScanCallback = new ScanCallback() {
@Override
public void onScanResult(int callbackType, final ScanResult result) {
//super.onScanResult(callbackType, result);
BluetoothDevice btdevice = result.getDevice();
Log.d("BLE", btdevice.getAddress());
}
@Override
public void onScanFailed(int errorCode) {
super.onScanFailed(errorCode);
Log.d("BLE", "error");
}
};
private ScanCallback mLeScanCallback2=new ScanCallback() {
@Override
public void onScanResult(int callbackType, ScanResult result) {
Log.d("BLE","scan stop");
}
@Override
public void onScanFailed(int errorCode) {
Log.d("BLE","stop scan failed");
}
};
@Override
protected void onCreate(Bundle savedInstanceState) {
super.onCreate(savedInstanceState);
setContentView(R.layout.activity_main);
Button btnScan = (Button) findViewById(R.id.btnScan);
BluetoothManager bluetoothManager = getSystemService(BluetoothManager.class);
mBluetoothAdapter = bluetoothManager.getAdapter();
mBluetoothLeScanner = mBluetoothAdapter.getBluetoothLeScanner();
this.mHandler = new Handler();
}
public void stop_scan(View view) {
Log.d("Ble","scan stop pressed");
// mBluetoothLeScanner.stopScan(mLeScanCallback2);
mBluetoothAdapter.getBluetoothLeScanner().stopScan(mLeScanCallback2);
}
public void onBtnScan(View view) {
Log.i("Btn","get click");
checkBTPermission();
String[] names=new String[]{"Auden test"};
List<ScanFilter> filters=null;
if(names != null){
filters=new ArrayList<>();
for(String name:names){
ScanFilter filter=new ScanFilter.Builder().setDeviceName(name).build();
filters.add(filter);
}
}
if(mBluetoothLeScanner==null){
Log.i("BLE","could not get scanner");
}else{
mBluetoothLeScanner.startScan(mLeScanCallback);
}
}
private void checkBTPermission(){
if(Build.VERSION.SDK_INT>Build.VERSION_CODES.LOLLIPOP){
int pc=this.checkSelfPermission("Manifest.permission.ACCESS_FINE_LOCATION");
pc+=this.checkSelfPermission("Manifest.permission.ACCESS_COARSE_LOCATION");
if(pc!=0){
this.requestPermissions(new String[]{Manifest.permission.ACCESS_FINE_LOCATION,Manifest.permission.ACCESS_COARSE_LOCATION},1001);
}else {
Log.d("BLE","checkBT permission");
}
}
}
}
A: Try to use https://developer.android.com/reference/android/bluetooth/le/ScanSettings.Builder#setLegacy(boolean) and set it to false. That should enable scanning for Extended Advertising packets.
I see that you create a filters list that is then never used. If you want to use it, it should be passed into https://developer.android.com/reference/android/bluetooth/le/BluetoothLeScanner#startScan(java.util.List%3Candroid.bluetooth.le.ScanFilter%3E,%20android.bluetooth.le.ScanSettings,%20android.bluetooth.le.ScanCallback) together with your ScanSettings that has legacy=false.
|
{
"redpajama_set_name": "RedPajamaStackExchange"
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| 958
|
\section{Introduction}
$\mathrm{LaCoO_3 }$\ and related compounds have been much studied for
half a century.\cite{Heikes19641600,naiman:1044,raccah}
Their strongly temperature dependent magnetic and
transport properties have been eluding complete theoretical description
so far.\cite{korotin,PhysRevB.67.172401,PhysRevB.71.054420,PhysRevB.77.045130,PhysRevB.77.045123,Hsu,eder}
An apparent band insulator below 50~K, $\mathrm{LaCoO_3 }$\ exhibits a local moment magnetic
response above 100~K while the charge gap continuously disappears
between 450 and 600~K.\cite{PhysRevB.6.1021,saitoh,ISI:000176767100012} This behavior points
to an important role played by the electronic correlations as common
among transition metal oxides. The correlated nature of $\mathrm{LaCoO_3 }$\ reveals
itself also in the formation of atomic-scale inhomogeneities
with large magnetic moments upon moderate hole doping.\cite{PhysRevLett.101.247603,PhysRevB.83.134430}
The picture of thermal evolution of $\mathrm{LaCoO_3 }$\ as
an entropy driven crossover from a non-magnetic to a magnetic state of Co$^{3+}$
ion has been generally accepted.
However, the actual why's and how's are far from settled.
The main open questions include the following. (i) Which atomic multiplet is responsible
for the formation of local moments?
(ii) Does the lattice thermal expansion actively contribute to the spin-state
transition or is it merely a slave to the changes of the electronic structure?
(iii) Why do the crossovers to local moment paramagnet and to bad metal
happen at different temperatures?
In this work we use the combination of the density functional band structures
with the dynamical mean-field theory, known as LDA+DMFT,~\cite{PhysRevLett.62.324,PhysRevB.45.6479,RevModPhys.68.13,0953-8984-9-35-010,PhysRevB.57.6884,0953-8984-20-6-064202}
to address the former two questions. We discuss in detail the attribution
of local moment behavior to a particular atomic multiplet in systems with
covalent bonds, a question of general importance in oxide physics.
The magnetic susceptibility of $\mathrm{LaCoO_3 }$\ is usually analyzed
in terms of the lowest multiplets of an isolated Co$^{3+}$ ion in octahedral
crystal field:\cite{tanabe-sugano}
the low-spin (LS) $^1A_1$ ($t^6e^0$), the intermediate-spin (IS) $^3T_1$
($t^5e^1$), or the high-spin (HS) $^5T_2$ ($t^4e^2$) states.
The energy differences between the multiplets
are controlled by the crystal-field (CF)
splitting and the intra-atomic exchange $J$.
While the LS singlet ground state is undisputed (at least for low temperature
crystal structures)
the nature of the first excited state is still a subject of debate.
Goodenough~\cite{Goodenough1958287} attributed the appearance of
local moment to population of the HS state.
Heikes {\it et al.},~\cite{Heikes19641600} on the other hand, proposed
the IS scenario, which became popular~\cite{radaelli,asai,zobel}
after Korotin {\it et al.}~\cite{korotin} obtained
an IS ground state for expanded lattice with LDA+U
calculation, contrary to a simple ligand-field theory.
More recent experiments make a strong case for the HS scenario.
The electron spin resonance shows a triplet excited state
with a large $g$ factor of 3.35\,--\,3.55 which is consistently explained
in the HS scenario invoking the effect of spin-orbit coupling.\cite{PhysRevB.66.094404,
PhysRevB.67.172401} The x-ray absorption spectra (XAS) and
magnetic circular dichroism at the $L_{2,3}$ edge of Co \cite{haverkort}
also select the HS scenario.
However, several authors \cite{kyomen1,kyomen2,haverkort,eder} pointed out
that in order to interpret the magnetic susceptibility, specific heat,
or XAS data in the HS scenario a rather strong temperature dependence
of the crystal field has to be assumed. In other words, the experimentally
deduced increase of the HS population is significantly slower than anticipated with a fixed
crystal field. The implication is that the apparent crystal field
grows with temperature. This is somewhat unexpected since the expansion
of the Co-O bond which accompanies the spin crossover
should reduce the crystal field. A possible explanation of this
puzzling feature is provided by an interatomic repulsion between the HS states, which
is equivalent to attraction between HS and LS states.~\cite{kyomen1,kyomen2,eder}
Breathing lattice distortion proposed by Raccah and Goodenough,~\cite{raccah}
studied in detail by Bari and Sivardi\`ere,~\cite{bari72} provides
a mechanism. Kn\'{i}\v{z}ek {\it et al.} \cite{knizek1} used LDA+U calculations to argue for
HS-LS attraction. Recently, a purely electronic mechanism of HS-LS attraction
was observed by two of us \cite{krapek} in a two band Hubbard model
and by Zhang {\it et al.}~\cite{zhang} in $\mathrm{LaCoO_3 }$\ specific calculation.
It is well known that the effective crystal field in transition
metal oxides is largely due to hybridization with ligands,
i.e.,~generated by hopping of predominantly $e_g$ electrons between
the metal and oxygen sites. This effect is particularly pronounced in $\mathrm{LaCoO_3 }$\
and leads to charge fluctuations on the Co site.
Therefore, it seems natural to question descriptions based on an isolated ion.
How does one define and distinguish HS and IS states when sizable charge (valence)
fluctuations are present?
Or, more precisely, is it possible to express the $T$-dependent susceptibility as a
sum of contributions from different atomic states?
We will show in Sec.~\ref{method_susceptibility} that in a solid the answer is negative in general.
However, in the insulating phase of $\mathrm{LaCoO_3 }$\ the notion of LS, IS, and HS states can be preserved
when these are generalized to include the hybridization induced charge fluctuations.
The $T$-{\it dependent} paramagnetic moment
can be to a good accuracy approximated by a $T$-{\it
independent} HS moment multiplied by
a $T$-{\it dependent} weight. Importantly, the magnitude of this
effective moment differs from the free ion value.
The role of the lattice thermal expansion poses a chicken-and-egg question
about the relationship between the spin-state transition
and the anomalous lattice expansion.~\cite{PhysRevB.50.3025,zobel,PhysRevB.78.134402} While the transition to
the magnetic state, both IS and HS,
weakens the Co-O bond and thus leads to its expansion, stretching the Co-O
bond reduces the effective
CF splitting and thus favors a magnetic state. Therefore, there is a
positive feedback between these two effects.
To include the lattice response directly into our calculations is not
computationally feasible at the moment.
Therefore, we have performed calculations for several lattices
corresponding to experimental
crystal structures at different temperatures. By comparing the
$T$ dependence of the spin susceptibilities
on these lattices we find that experimentally observed variation of Co-O
bond lengths has a pronounced effect
on the electronic properties.
Inclusion of thermal effects
in ``first principles" density functional approaches is notoriously difficult.
Therefore, the existing studies are either limited to the $T=0$ LS phase~\cite{abbate_lda}
or assume that temperature enters only through the lattice thermal
expansion.~\cite{korotin,knizek1} The only serious attempt to explicitly include temperature
in such a calculation was made by Eder~\cite{eder} using the variational cluster
approximation (VCA). The VCA and DMFT methods share many formal similarities,
but involve different approximations.
VCA is, in principle, an exact method for calculation
of one-particle properties, but, as pointed out in Ref.~\onlinecite{eder},
the relevance of multiplets populations obtained
from a reference CoO$_6$ cluster is a conjecture,
which calls for verification with other methods.
DMFT, on the other hand, treats one-particle
and multiparticle correlations on the same footing,
but becomes exact only in the limit of vanishing
nonlocal correlations (infinite dimension).
Recently, DMFT was applied to $\mathrm{LaCoO_3 }$\ to study
the effect of varying interaction strength and pressure.~\cite{zhang}
\section{Methods}
\label{method}
\subsection{Model}
Multiband Hubbard Hamiltonian with the two-particle interaction within
the Co $3d$ shell is used to describe $\mathrm{LaCoO_3 }$.
The one-particle part of the Hamiltonian, which spans the
Co $3d$ and O $2p$ orbitals, has been constructed from the
local density approximation (LDA) to the density functional theory.
The non-spin-polarized band structure obtained with
\textsc{wien}2k~\cite{wien2k} was transformed into the
Wannier basis representation with \textsc{wien2wannier}~\cite{w2w} and
\textsc{wannier}90~\cite{wannier90} codes. The Hamiltonian in this representation reads
\begin{equation}
\begin{split}
H=\sum_{\bm{k}\sigma}\left(
h_{\bm{k},\alpha\beta}^{dd} d_{\bm{k}\alpha\sigma}^\dagger d_{\bm{k}\beta\sigma}
+
h_{\bm{k},\gamma\delta}^{pp} p_{\bm{k}\gamma\sigma}^\dagger p_{\bm{k}\delta\sigma}
\right .
\\
+\left .
h_{\bm{k},\alpha\gamma}^{dp} d_{\bm{k}\alpha\sigma}^\dagger p_{\bm{k}\gamma\sigma}
+
h_{\bm{k},\gamma\alpha}^{pd} p_{\bm{k}\gamma\sigma}^\dagger d_{\bm{k}\alpha\sigma}
\right)
+
\sum_{i,\sigma,\sigma^\prime} U_{\alpha\beta}^{\sigma\sigma^\prime}
n^d_{i\alpha\sigma} n^d_{i\beta\sigma^\prime}.
\end{split}
\label{Hamiltonian}
\end{equation}
Here $d_{\bm{k}\alpha\sigma}$ and $p_{\bm{k}\gamma\sigma}$ are
Fourier transforms of the annihilation operators
$d_{i\alpha\sigma}$, $p_{i\gamma\sigma}$ which destroy
the $d$ or $p$ electron with the orbital index $\alpha$ or $\gamma$
and the spin index $\sigma$ in the $i$th unit cell,
$n^d_{i\alpha\sigma}$ are the corresponding occupation
number operators, and $h_{\bm{k},\alpha\beta}^{ab}$ are the
corresponding matrix elements of the one-particle LDA Hamiltonian.
The $U_{\alpha\beta}^{\sigma\sigma^\prime}$ are the density-density
matrix elements in the $e_g$-$t_{2g}$ basis of the full Coulomb interaction~\cite{PavariniChapter6}
parametrized with $U$ (Slater parameter $F_0$) and Hund's exchange $J$
[connected with the Slater parameters $F_2$, $F_4$ as $J=(F_2+F_4)/14$,
$F_4/F_2=0.625$].
The screened values of $U=6.0$~eV and $J=0.8$~eV
have been obtained using the constrained density functional theory (cDFT),
described in detail in Ref.~\onlinecite{springerlink:10.1140/epjb/e2008-00326-3}.
To the DFT potential an orbitally dependent term was added, which shifts
by a small amount the energy of
selected Wannier functions (WFs). Due to this shift, occupation of the WFs
in question is changed and the Coulomb
interaction parameters are then determined as a derivative of the site energy with
respect to the occupation.
In the following calculations are performed with cDFT values
of $U$ and $J$.
Calculations, where $U$ and $J$ were varied, are also reported to assess
the stability of the results.
The $h^{dd}$ diagonal elements were modified to account for the static
part of the interaction, double-counting correction,
\begin{equation}
\label{eq_edc}
h_{\bm{k},\alpha\beta}^{dd}=h_{\bm{k},\alpha\beta}^{0,dd}-
(N_\mathrm{orb}-1)\bar{U}\bar{n}\delta_{\alpha\delta},
\end{equation}
where $\bar{n}$ is the average self-consistent occupancy per Co:$d$ orbital,
$\bar{U}$ is the orbital averaged interaction energy,
and $N_\mathrm{orb}$ is the total number of interacting orbitals
on a single site (10 in our case).\cite{fe2o3} This is equivalent to subtracting the
orbitally averaged Hartree potential felt by the $d$ electrons.
\subsection{DMFT calculations and one-particle spectra}
The one-particle Green's function of the Hamiltonian (\ref{Hamiltonian})
is found by iteratively solving the DMFT equations on the Matsubara contour.
The auxiliary impurity problem is solved
by the continuous time quantum Monte Carlo (CT-QMC) method in
the hybridization expansion formulation~\cite{Werner}
using the implementation based on free-access package ALPS.~\cite{alps,ALPS2}
The Wang-Landau reweighting~\cite{PhysRevLett.86.2050,PhysRevE.64.056101}
was employed in order to ensure the ergodicity of the simulations for some parameter values,
in particular at low temperatures and close to the spin state transition.
Once the calculation was converged we have evaluated the one-particle spectra
in real frequency and analyzed the impurity dynamics and spin susceptibility.
For the former analytic continuation is necessary. To this end we have employed
the maximum entropy method in two modes: (i) continuation of the local Green's function
from the imaginary time $\tau$ to real frequency $\omega$ and (ii) continuation
of the local self-energy from the Matsubara frequency $i\omega_n$ to $\omega$.
For the latter we have used the statistical error estimates following Ref.~\onlinecite{PhysRevB.80.045101}.
The former was used to cross-check the results of (ii) and the spectra are not shown in the paper.
Analytic continuation of self-energy has several attractive features, such as
being exact in the noninteracting limit, providing a direct access
to the $k$-resolved and ligand spectra, and smearing out the lifetimes (imaginary
part of the self-energy) but not the quasiparticle dispersions.
\subsection{Susceptibility and local moments}
\label{method_susceptibility}
To analyze the on-site dynamics we have studied two additional quantities.
First, the state weights \cite{PhysRevLett.99.126405} which are the diagonal terms of the
site-reduced density matrix, i.e., expectation values of the projection operators
on the atomic (many-body) states $\hat{P}_\mu=|\mu\rangle\langle \mu|$.
In the present case of the density-density interaction the site-reduced density matrix
is diagonal in the occupation number basis. Therefore, the knowledge of
the state weights allows us to evaluate the expectation value of any local operator, e.g.,
the instantaneous local moment $\langle \hat{m}_z^2
\rangle=\sum_{\mu}m_z^2(\mu)\langle \hat{P}_{\mu} \rangle$ with
$m_z(\mu)=\langle\mu|\hat{m}_z|\mu\rangle$.
Second, we define the imaginary time
state-state correlation matrix $C_{\mu\nu}(\tau)$ and its time average $\Pi_{\mu\nu}$,
\begin{equation}
\begin{split}
C_{\mu\nu}(\tau)=\langle \hat{P}_\mu(\tau) \hat{P}_\nu(0) \rangle, \\
\Pi_{\mu\nu}=T\int_0^{\beta}d\tau C_{\mu\nu}(\tau),
\end{split}
\label{CPi}
\end{equation}
where $\beta=1/T$ is the inverse temperature.
The correlation matrix allows us to analyze the local response functions (via
fluctuation-dissipation theorem) and decompose them into different contributions.
In particular, we will be interested in the local spin susceptibility $\chi$,
which in the paramagnetic state can be expressed as
\begin{equation}
\begin{split}
\label{eq_chi}
\chi&=\int_0^{\beta}d\tau\langle m_z(\tau)m_z(0)\rangle \\
&=\frac{1}{T}\sum_{\mu,\nu}m_z(\mu)m_z(\nu)\Pi_{\mu\nu}.
\end{split}
\end{equation}
The second expression shows that in general the local magnetic response cannot be
decomposed into contributions of atomic states, but pairs of states must be considered.
Decomposition into individual states contributions is possible only if $C_{\mu\nu}(\tau)$ can be made diagonal, e.g., in an isolated atom.
Discussing briefly the physical meaning of these quantities we start by pointing out
that in the course of time a given atom visits various quantum-mechanical states
as a result of statistical (thermal) fluctuations and quantum-mechanical (causal)
evolution. The weight of a given state is a relative measure of the time spent
by the atom in this state, which does not distinguish between thermal fluctuations
and causal evolution. The state-state correlations distinguish to some extent
between these two effects as only states connected by causal evolution can have
a nonzero cross term. The state weights can be obtained as row (or column)
sums over $\Pi_{\mu\nu}$.
\section{Results and discussion}
\label{results}
\subsection{Non-interacting band structure}
We have considered the experimental distorted
perovskite structure with $R\bar{3}c$ space group containing
two formula units per unit cell. The structural parameters were taken
from the x-ray measurements of Ref.~\onlinecite{radaelli}. To assess the effect of
lattice thermal expansion the calculations were repeated for the experimental
structural parameters obtained at three different temperatures (denoted as
$\tau_\mathrm{lattice}$ in the following):
5, 450, and 750~K.
\begin{figure}[ht]
\includegraphics[angle=270,width=\columnwidth,clip]{spec_hyb.ps}
\caption{
(Color online) Calculated orbitally resolved spectral function
[O:$p$ (shaded cyan); Co:$t_\mathit{2g}$ and $e_g$
(red and black line)].
Left panel: the hybridization term $h_{\bm{k},\alpha\gamma}^{dp}$ of the Hamiltonian (\ref{Hamiltonian})
was set to zero. Right panel: the LDA Hamiltonian in the Wannier basis.
}
\label{fig_spec_hyb}
\end{figure}
The octahedral crystal field splits Co:$d$ states into six $t_{2g}$ states
at lower energy and four $e_g$ states at higher energy.
The resulting orbitally resolved spectral density is shown
in Fig.~\ref{fig_spec_hyb}. The splitting is strongly
contributed by the O:$p$--Co:$d$ hybridization, as is clear from the comparison
in Fig.~\ref{fig_spec_hyb}, where the right and left panels show the spectral
function with and without the $p$-$d$ hybridization included.
The on-site contribution to the crystal field splitting $\Delta$ (left panel of Fig.~\ref{fig_spec_hyb})
is close to the value 0.7~eV extracted from the XAS measurements.~\cite{haverkort}
The $p$-$d$ hybridization increases
the distance between the centers of $t_{2g}$ and $e_g$ bands considerably.
The band broadening, more pronounced for the $e_g$ band, is another
consequence.
Matching O:$p$ and Co:$d$ features reflect formation of bonding and
antibonding states.
Stronger hybridization of the $e_g$ orbitals compared to $t_{2g}$ ones
results from their larger spatial overlap with O:$p$ orbitals. The $t_{2g}$
band is further split to the $e_g^\pi$ doublet and the $a_{1g}$ singlet due
to a distortion from the octahedral symmetry. This splitting does not
play an important role in our study, though.
As in previous calculations a metallic ground state
is incorrectly predicted by LDA.~\cite{Hsu}
\subsection{Thermal effects and lattice expansion}
LDA+DMFT calculations were performed for several temperatures between 290 and 2320~K
($\beta$ from 40 to 5~eV$^{-1}$).
If not stated otherwise, the results are shown for $U=6.0$\,eV and $J=0.8$\,eV
(obtained by cDFT calculations).
\begin{figure}[ht]
\includegraphics[width=\columnwidth,clip]{fig_chi_mtm.ps}
\caption{
(Color online)
(a) Local spin susceptibility as a function
of the temperature for different lattices ($\tau_\mathrm{lattice}$).
(b) The corresponding screened spin moment.
The dashed line indicates the value for the IS in the ionic limit.
The letters A, B, C denote the solutions discussed in the text.
(c) The apparent crystal field $\Delta_{\text{eff}}$ obtained
from Eq.~(\ref{eq_cfeff}).
}
\label{fig_chi_mtm}
\end{figure}
{\it Local susceptibility.}
The local spin susceptibility $\chi$, calculated
from Eq.~(\ref{eq_chi}), is shown in
Fig.~\ref{fig_chi_mtm}(a) as a function of $T$.
For $\tau_\mathrm{lattice}=750$~K the huge error bar
at $T=580$~K is due to a long autocorrelation time despite the
Wang-Landau~\cite{PhysRevLett.86.2050,PhysRevE.64.056101} sampling.
For all lattice parameters we observe an emergence of Curie-like susceptibility at high temperatures.
The corresponding average spin moments $m_\mathrm{scr}=\sqrt{T\chi}$
are shown in Fig.~\ref{fig_chi_mtm}(b). As we calculate directly the local
susceptibility (the local response to a field applied to a single site of the infinite
crystal) the intersite exchange does not enter the definition of the local moment.
Figure~\ref{fig_chi_mtm}(b) suggests a gradual thermal population of a magnetic
state. The lattice expansion clearly favors the magnetic state and the experimentally
observed magnitude of the Co-O bond-length expansion has a sizable impact on our results.
Next, we discuss the temperature dependences $\chi(T)$, obtained from Eq.~(\ref{eq_chi}), for a fixed lattice.
To get in touch with experimental observations we adopt the single ion
expression commonly used in analysis of experimental data,
\begin{equation}
\label{eq_cfeff}
\chi(T)=\frac{\mu^2}{T}
\frac{\nu}{\nu+\exp[\Delta_{\text{eff}}(T)/T]},
\end{equation}
where $\mu$ and $\nu$ are the magnitude of the local moment and the multiplicity of
the excited magnetic state and
$\Delta_{\text{eff}}$ is the excitation energy with respect to the LS ground state.
In Fig.~\ref{fig_chi_mtm}(c) we show $\Delta_{\text{eff}}$ obtained
from Eq.~(\ref{eq_cfeff}) using $\nu=6$ and $\mu=3.5$, which correspond to
an Ising HS state, a choice explained later in the text. Like in the experiments \cite{kyomen1,kyomen2,haverkort}
and the VCA theory,~\cite{eder} our $\Delta_{\text{eff}}$ for a fixed lattice increases with temperature
for the reason discussed below.
In particular, the increase of $\Delta_\mathrm{eff}$
by a factor of 3--5 over the spin-state crossover was deduced from
XAS~\cite{haverkort} and measurements of the magnetic susceptibility and heat
capacity.~\cite{kyomen2}
It is quite clear that our results do not provide an accurate quantitative description
of $\mathrm{LaCoO_3 }$\ as the spin crossover takes place at a too high temperature. This is not surprising
since the present theory is unlikely to achieve the necessary $\sim$10~meV accuracy without
fine-tuning the material
parameters by hand.
The approximations of the model (such as the restriction to
the density-density Coulomb interaction and neglect of the long-range
or $p$-$d$ interactions) limits the accuracy further.
However, two important trends are revealed. First, the lattice response (expansion
of O$_6$ octahedra around moment-carrying sites) acts as a positive feedback for generation
of local moments. Second, this is countered by a purely electronic effect,
making addition of a local moment the harder the more local moments are
already in the system; this is reflected in the increase of
$\Delta_{\text{eff}}$ with the temperature [Fig.~\ref{fig_chi_mtm}(c)].
This is another way of saying that there is a repulsive interaction between the magnetic
sites in the LS background.
We point out that in our calculation all O$_6$ octahedra are the same, which excludes
a possible contribution of the breathing distortion.~\cite{raccah} Indeed, Ky\^omen {\it
et al.}~\cite{kyomen1} substituting Co with Al, Ga, and Rh came to the conclusion that electronegativity
rather than ionic radius of the neighbors is the parameter which controls $\Delta_{\text{eff}}$.
We suggest the following picture based on the observation that
a strong Co:$e_g$-O bond favors LS state, and that in the Co-O-Co trimer the Co-O bonds share the
central $p_{\sigma}$ orbital. Due to this sharing the energy gain per Co-O bond in the trimer
is less than the energy gain for an isolated Co-O bond. Therefore, breaking (weakening) one bond
in the trimer makes the other bond stronger. Introducing a local moment on one Co site provides
this bond breaking and strengthening the other bond favors the LS state.
\begin{figure}[ht]
\includegraphics[width=\columnwidth,clip]{fig_spec_band.ps}
\caption{
(Color online)
(Upper panels) Orbital-resolved spectral functions
(states/eV/formula unit). $t_\mathit{2g}$ ($e^{\pi}_g$): solid red line;
$t_\mathit{2g}$ ($a_\mathit{1g}$): dashed red line; $e_g$: black line;
O:$p$: cyan shaded area. The O:$p$ spectral function is downscaled
by the factor of 2 to fit in the graph.
(Lower panels) $k$-resolved spectral function $A_\mathbf{k}(\omega)$ along
the high symmetry directions.
Left panels show the interacting
nonmagnetic (low-spin) solution [denoted as A in Fig.~\ref{fig_chi_mtm}(b);
$T=580$~K, $\tau_\mathrm{lattice}=5$~K] and the right panels
display the paramagnetic solution (with a large content of high-spin atomic states)
[denoted as C in Fig.~\ref{fig_chi_mtm}(b); $T=1160$~K; $\tau_\mathrm{lattice}=750$~K]
.
}
\label{fig_spectral_k}
\end{figure}
{\it Spectral functions.}
In Fig.~\ref{fig_spectral_k} we compare the one-particle spectra of the low-$T$ nonmagnetic state (left panels)
and the high-$T$ paramagnetic state (right). The orbital resolved spectra are displayed in the upper panels
and the $k$-resolved spectra along the high symmetry directions are
shown in the lower panels.
The nonmagnetic spectrum resembles the LDA solution (Fig.~\ref{fig_spec_hyb}), the main difference being a uniform (Hartree) shift
of the $e_g$ band. There is very little dynamical renormalization since the LS state is
an approximate eigenstate of both the kinetic term (dominated by the crystal field) and the interaction term
taken separately. The correlated nature of $\mathrm{LaCoO_3 }$\ is revealed at elevated temperature.
The thermal population of the excited atomic multiplets leads to a formation of local moments, which
are incompatible with dispersive bands. As a result incoherent features appear in the spectra. The nature
of the charge gap changes from a semiconductor like gap between coherent valence $t_{2g}$ and conduction
$e_g$ bands (left panel of Fig.~\ref{fig_spectral_k}) to a $t_{2g}$-$t_{2g}$ gap (right panel of Fig.~\ref{fig_spectral_k}).
The bottom of the conduction band is now defined by an incoherent
$t_{2g}$ excitation, the tail of which gradually fills the gap with the increasing temperature.
The top of the valence manifold is formed by a renormalized dispersive $t_{2g}$ band. This
spectrum is consistent with the positive Seebeck coefficient \cite{senaris,jirak} indicating
holes to dominate the electronic transport.
Like the VCA results,~\cite{eder} the photoemission part of the spectra exhibits a transfer
of spectral weight from the low-energy peak ($\sim$1~eV) to higher energies ($\sim$3~eV)
observed experimentally.~\cite{abbate}
\begin{figure}[ht]
\includegraphics[angle=270,width=\columnwidth,clip]{AW_XAS_PES.ps}
\caption{
(Color online)
Comparison of the calculated density of states (lines) with the PES measurements (symbols). The calculated density denoted as LS is taken for $T=580$~K, $\tau_\mathrm{lattice}=5$~K [A in Fig.~\ref{fig_chi_mtm}(b)] and that denoted as HS
for $T=1160$~K, $\tau_\mathrm{lattice}=750$~K [C in Fig.~\ref{fig_chi_mtm}(b)].
The measurements were taken at 65 K (denoted as LS) and 300 K (denoted as HS+LS,
as the temperature is not high enough for the full spin-state crossover).
}
\label{fig_XASPES}
\end{figure}
In Fig.~\ref{fig_XASPES} we compare the calculated spectral functions to the photoemission spectra (corrected for surface
effect) of Ref.~\onlinecite{Koethe} (Fig.~2.8).
We have tuned the relative weights of the O:$p$ and Co:$d$ (1:5) spectra to
mimic the
effect of different absorption cross sections and added a Gaussian
broadening of 0.2 eV
to account for the experimental resolution. We find a good match
of the major spectral features. We also observe consistent trends
in both the O:$d$ and Co:$d$ parts of the spectra. The more pronounced
difference
of the two theoretical spectra reflects most likely a higher degree of
LS to HS crossover.
\subsection{Spin state analysis}
{\it Local state statistics.}
Next, we address the local dynamics at Co sites and
whether it is meaningful to describe it in terms of the
atomic states (such as LS, IS, and HS). The hybridization
expansion CT-QMC solver is well suited to this task as it
provides the site-reduced statistical operator (density matrix),
referred to as state statistics.~\cite{PhysRevLett.99.126405} This quantity
describes the probability of finding an atom in a particular
many-body state and the expectation value of any local operator
can be easily obtained from it.
We display schematically the atomic states important for the
forthcoming discussion in Fig.~\ref{fig_splitting}.
\begin{figure}[ht]
\includegraphics[width=\columnwidth,clip]{hybridization.eps}
\caption{
(Color online)
Atomic states at LS and HS configurations.
The blue lines depict the oxygen $p$ orbitals and the black
circles denote the hole in O:$p$ shell. The effects
of the crystal field and Co:$d$-O:$p$ hybridization
are schematically depicted.
In total four atomic states belong to LS block
(see the left part of figure): (LS) all $t_\mathit{2g}$ orbitals occupied, all $e_g$ orbitals
empty, (LS+$e_g$) in addition to LS single $e_g$ orbital occupied,
[LS+2$e_g$($\uparrow\downarrow$)] in addition to LS two $e_g$ orbitals with
the opposite spin orientation occupied, and
[LS+2$e_g$($\uparrow\uparrow$)] in addition to LS two $e_g$ orbitals with
the parallel spin orientation occupied.
All these states are accessible from LS via the $e_g$ hybridization.
For brevity we do not distinguish
LS+2$e_g$($\uparrow\downarrow$) from LS+2$e_g$($\uparrow\uparrow$)
from now on.
In total three atomic states belong to HS block
(see the right part of figure): (HS) all majority spin orbitals and one minority spin
$t_\mathit{2g}$ occupied, (HS+$e_g$) in addition to HS another minority spin
$e_g$ orbital occupied, and (HS+$t_\mathit{2g}$) in addition to HS another
minority spin $t_\mathit{2g}$ orbital occupied.
All these states are accessible from HS via the $e_g$ or $t_\mathit{2g}$
hybridization.
}
\label{fig_splitting}
\end{figure}
There are many atomic states with non-negligible weights
contributing to the partition function
(Fig.~\ref{fig_stat2}).
The total contributions of different charge states (insets
of Fig.~\ref{fig_stat2}) point to sizable valence fluctuations,
which is related to finding substantial admixtures
of $\mathrm{d^7\underline{L}}$ and $\mathrm{d^8\underline{L}^2}$ state
to the $\mathrm{d^6}$ ground state in the cluster calculations.\cite{saitoh}
Unlike the cluster calculations, in DMFT the ligand hole does not
remain coherent with the central Co atom due to the influence of the
rest of the crystal. Therefore, one cannot use the CoO$_6$ eigenstates
to analyze the local dynamics. Instead, we use the statistical
description and also analyze the temporal evolution of the atomic states.
Clearly, the $\mathrm{d^6}$ atomic multiplets denoted as LS, IS, and HS in Fig.~\ref{fig_stat2}
are not sufficient to describe the local physics in $\mathrm{LaCoO_3 }$.
We distribute the atomic states into LS, IS, and HS
blocks (Fig.~\ref{fig_splitting}). Although an {\it a priori} assignment
of the blocks is not unique we later present an {\it a posteriori}
justification of our choice. For example, the state denoted as HS+$t_{2g}$ can,
in principle, be reached by adding a $t_{2g}$ electron to the HS state as well as by adding
an $e_g$ electron to the IS state.
\begin{figure}[ht]
\includegraphics[width=\columnwidth]{fig_stat2.ps}
\caption{
(Color online)
Weight of dominant atomic states for
$U=6.0$~eV, $J=0.8$~eV.
(a) Nonmagnetic solution [A in Fig.~\ref{fig_chi_mtm}(b), $T=580$~K, $\tau_\mathrm{lattice}=5$~K],
(b) low-$T$ solution [B in Fig.~\ref{fig_chi_mtm}(b), $T=580$~K, $\tau_\mathrm{lattice}=450$~K],
(c) high-$T$ solution [C in Fig.~\ref{fig_chi_mtm}(b), $T=1160$~K, $\tau_\mathrm{lattice}=750$~K].
}
\label{fig_stat2}
\end{figure}
In Fig.~\ref{fig_stat2} we present the state statistics for various lattice parameters and
temperatures corresponding to the solutions denoted as A, B, and C in Fig.~\ref{fig_chi_mtm}(b).
Besides substantial weights of the $\mathrm{d^7}$ states the figure reveals that
the increasing local moment response is connected to the growing weight of the HS block,
while the IS block has only minor weight.
\begin{figure}[t!]
\includegraphics[width=\columnwidth]{chi_triangle_B20_ham450.ps}
\caption{
(Color online)
Correlation matrix $\Pi_{\mu\nu}$ between the dominant atomic states
for $T=580$~K and $\tau_\mathrm{lattice}=450$~K [solution B of Fig.~\ref{fig_chi_mtm}(b)].
Color-coded values show the state-by-state relative
contributions in \% to the sum over all pairs. The
numbers within blocks of atomic states indicate the contribution of the blocks
to the local susceptibility in the units of $\mathrm{10^{-3}emu\cdot Oe^{-1}\cdot mol^{-1}}$.
}
\label{fig_chi_matrix1}
\end{figure}
{\it Correlation matrix of local states.}
Although we have identified the atomic states with large weights,
a question arises whether their appearance is due to a unitary evolution
or rather due to statistical averaging. This question
is closely connected to the decomposition of susceptibility into atomic
states contributions [Eqs.~(\ref{CPi}) and (\ref{eq_chi})]. States connected by a unitary evolution
lead to a large off-diagonal element of the time averaged state-state
correlation matrix $\Pi$ and thus their individual
contributions to the susceptibility cannot be well defined. On the other hand,
if the weights of different states originate in statistical averaging the
corresponding off-diagonal element of $\Pi$ is vanishing as is its contribution
to the susceptibility.
Analysis of the correlation matrix $\Pi_{\mu\nu}$,
displayed in Figs.~\ref{fig_chi_matrix1} and~\ref{fig_chi_matrix2}, reveals
nonzero off-diagonal elements, indicating a unitary evolution
between the corresponding states.
Nevertheless, the matrices can be arranged in a block diagonal form, which
justifies our choice of the LS, IS, and HS blocks.
As the unitary evolution between the most populated LS and HS blocks
has a vanishing probability,
their simultaneous population is a result of the statistical averaging
and blocks generalize the notion of atomic multiplets in isolated atoms.
The block-summed contributions to the local susceptibility are indicated
by numbers inside the respective blocks in Figs.~\ref{fig_chi_matrix1} and
\ref{fig_chi_matrix2}.
In both cases the contribution of the HS diagonal block amounts around 97\%
of the total susceptibility. Inspecting the
block contributions to the spin-spin correlation (Fig.~\ref{fig_sts})
we find a finite $\tau$-constant part of the HS contribution,
leading to Curie-type susceptibility, in contrast to the rapidly
decaying LS contribution. This allows us to define an effective
HS moment as
\begin{equation}
\mu_{\text{HS}}=\frac{
\sqrt{
\sum_{\mu\nu\in \mathrm{HS}}
m_z(\mu)m_z(\nu)\Pi_{\mu\nu}
}
}
{
\sum_{\mu\nu\in \mathrm{HS}}\Pi_{\mu\nu}
}.
\end{equation}
We obtain $\mu_{\text{HS}}$ of 3.52 and 3.56~$\mu_B$ in the low-$T$ and
the hight-$T$ solutions, respectively.
The weak $T$ dependence of the effective moment and its dominant contribution
to the susceptibility $\chi$ justifies expressing $\chi$ as a product
of Curie term $\mu^2/T$ and a $T$-dependent weight. Covalent
Co-O bonding results in about 10\% reduction of the effective
moment from its atomic value of 4~$\mu_B$.
\begin{figure}[t!]
\includegraphics[width=\columnwidth,clip]{chi_triangle_B10_ham750.ps}
\caption{
(Color online)
Same as Fig.~\ref{fig_chi_matrix1} for
$T=1160$~K and $\tau_\mathrm{lattice}=750$~K [solution C of Fig.~\ref{fig_chi_mtm}(b)].
}
\label{fig_chi_matrix2}
\end{figure}
\begin{figure}
\includegraphics[height=0.5\columnwidth,angle=270,clip]{block.ps}
\caption{(Color online)
HS (upper panel) and LS (lower panel) normalized block
contributions to the spin-spin correlation function,
$\sum_{\mu,\nu\in \mathrm{block}}m_z(\mu )m_z(\nu )C_{\mu\nu}(\tau)/\sum_{\mu,\nu \in \mathrm{block}}\Pi_{\mu\nu}$,
for $T=1160$~K and $\tau_\mathrm{lattice}=750$~K [solution C of Fig.~\ref{fig_chi_mtm}(b)].
}
\label{fig_sts}
\end{figure}
\subsection{Role of $U$, $J$, double counting correction}
Since the form and construction of the Hamiltonian (1) is to some
extent an {\it ad hoc} procedure it is important to
understand the sensitivity of our conclusions to the particular values
of $U$, $J$, and the double counting energy [$E_\mathrm{dc}$ is the second
term on right-hand side~of Eq.~(\ref{eq_edc})]. Although these are not adjustable
parameters in the present theory, their determination is not unique, which
holds in particular for the double counting correction $E_\mathrm{dc}$.
\begin{figure}[ht]
\includegraphics[width=\columnwidth,clip]{fig_UJ_vertical.ps}
\caption{
(Color online)
Weight of the LS block of atomic states (LS, $\mathrm{LS}+e_g$, and $\mathrm{LS}+2e_g$)
for various
$U$ and $J$ at $\tau_\mathrm{lattice}=5$~K, (a) $T=580$~K, (b) $T=1160$~K.
}
\label{fig_UJ}
\end{figure}
In Fig.~\ref{fig_UJ} we show the dependence of the weight of the LS block
for various values of $U$ and $J$. As expected $J$ is the more important
parameter. Its cDFT value falls into the spin state crossover range of
0.8\,--\,0.9~eV. Outside this range the results are insensitive to temperature
or the variation of $U$. Variation of $U$ inside the crossover regime
has some impact as higher $U$ suppresses the fluctuations
to the $\mathrm{d^7\underline{L}}$ and $\mathrm{d^8\underline{L}^2}$ states.
As a result the LS state is destabilized.
\begin{figure}[ht]
\includegraphics[angle=270,width=\columnwidth,clip]{fig_eDC.ps}
\caption{
(Color online)
Weight of LS, IS, and HS blocks of atomic multiplets for various
double counting energies for high-$T$ solution
($U=6$~eV, $J=0.8$~eV, $T=1160$~K, and $\tau_\mathrm{lattice}=750$~K).
The values indicate the offset from the self-consistent value of 33.56~eV.
}
\label{fig_eDC}
\end{figure}
We have examined the role of double counting correction ($E_\mathrm{dc}$) for
$U=6.0$~eV, $J=0.8$~eV, $T=1160$~K, and $\tau_\mathrm{lattice}=750$~K.
We varied $E_\mathrm{dc}$ by $\delta E_\mathrm{dc}$ in the range of $\pm 5$~eV
around the self-consistent value of 33.56~eV.
The positive values of $\delta E_\mathrm{dc}$ mean that the Co:$d$ states
are shifted downward in energy closer to the O:$p$
state, which in turn enhances the hybridization and leads to
preference of a LS metallic phase.
The weights of the LS, IS and HS blocks
are shown in Fig.~\ref{fig_eDC}. For $\delta E_\mathrm{dc}>1$
the system becomes metallic
and the definition of the LS, IS, and HS blocks loses
its justification.
Based on the one-particle spectra and the overall behavior
of our results we conclude that the self-consistent value of $E_\mathrm{dc}$
provides a rather good description of the actual material.
We also point out the $T$-dependent variation of $E_\mathrm{dc}$
is rather small and had minor effect on the $T$ dependence of
both $\chi$ and $\Delta_{\text{eff}}$.
\section{Conclusion}
We have studied the temperature dependence of the magnetic
and spectral properties of $\mathrm{LaCoO_3 }$\ using the LDA+DMFT approach. Our results show that
the local moment response at elevated temperatures is associated
with the HS state of Co ion and that there is an effective interatomic
repulsion between HS atoms in the LS matrix. Our findings at this point agree
with the VCA calculations \cite{eder} and LDA+U cluster calculations \cite{knizek1}
as well as with the conclusions of the experimental studies.~\cite{kyomen1,kyomen2,haverkort}
On the other hand, our results are inconsistent with interpretation
of the intermediate temperature phase in terms of the IS state.~\cite{korotin,radaelli,zobel}
Furthermore, since a purely electronic mechanism
of the HS-LS attraction exists \cite{krapek} the absence or smallness
of the breathing mode distortion \cite{radaelli}
does not exclude HS-LS short range ordering.
The experimentally observed (anomalous) lattice expansion has
a pronounced effect on the LS-HS crossover, which leads to the conclusion
that the lattice provides a strong positive feedback for the
crossover.
To account for the strong covalent bonding
with O the notion of Co HS state has to be generalized
to include not only $d^6$, but also $d^7$ and $d^8$ electron configurations.
This leads
to reduction of the magnitude of the effective HS moment. Therefore,
using the apparent magnitude of the magnetic moment as an indicator
of the underlying atomic state may lead to incorrect conclusions.
Although we have varied the computational parameters in a wide range
we have not found a phase that would be dominated by the IS state.
Therefore, scenarios invoking HS-IS crossover \cite{knizek1,kyomen2}
are not consistent with our results. Guided by a similar LDA+DMFT study
on metallic SrCoO$_3$, it is plausible that in the metallic
phase of $\mathrm{LaCoO_3 }$\ observed experimentally above 600~K distinction between IS and HS state
is not possible as those are connected by unitary evolution of the system.
Based on the above observations we propose the following scenario of $\mathrm{LaCoO_3 }$\
physics, which is in many aspects similar to Refs.~\onlinecite{knizek1,eder}.
At the lowest temperatures most Co ions are in the LS state with isolated Co ions in the
excited HS state. Increasing temperature assisted by the lattice feedback
leads to growing density of the HS sites. Effective repulsion
keeps the HS sites apart leading to a short range HS-LS order, which is
responsible for the insulating behavior in the 100--500~K range, similar
to the observation made in Ref.~\onlinecite{krapek}. We speculate that the second
crossover experimentally observed around 500~K is associated with ``melting" of the LS-HS order.
This leads to an anomalous lattice expansion due to the breaking of attractive
LS-HS bonds. The experimentally observed onset of metallicity
changes the local moment character by coherently admixing some
IS-like states to the dominant HS configuration. The distinction
between HS and IS in the high $T$ metallic phase is thus not possible.
\subsection*{Acknowledgments}
We thank Z. Jir\'ak for numerous discussion and suggestions
concerning the manuscript.
This work was supported by Grant No.~P204/10/0284 of the
Grant Agency of the Czech Republic, by the Deutsche
Forschungsgemeinschaft through FOR1346,
by the Russian Foundation for Basic Research (Project Nos.~10-02-00046 and 10-02-96011) and by the Fund of the President of the Russian Federation for the Support of Scientific Schools NSH-6172.2012.2.
|
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"redpajama_set_name": "RedPajamaArXiv"
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NELH usually focus on North East events and history, but Grunwick was a key moment in British Labour History. In 1976, six workers walked out of Grunwick Film Processing Laboratory in Willesden and ignited an historic two-year dispute which united thousands to demand better rights for poorly treated workers.
Two murals to commemorate the event have been unveiled in Willisden, the site of the dispute.
More information on the Grunwick Strike at the Grunwick 40 site.
This entry was posted in Uncategorized on 10/10/2017 by Peter Nicklin.
Keith Armstrong has made a five minute video, 'Thomas Spence in London' (1750 -1814) which includes renditions of two of his polemics, 'The Hive of Liberty' and 'Pigs Meat', the latter including the delightful lines, 'Pigs meat, pigs meat / We piss on the elite'.
Those of a nervous disposition or members of the ruling class should not watch it.
Leanne Smith has been awarded the annual Sid Chaplin Memorial Prize for the best undergraduate dissertation on North East history. Leanne, who graduated from the University of Sunderland this summer with a First Class degree, chose as her subject a little known, almost forgotten aspect of social history.
Leanne's dissertation, 'The Struggle over Female Labour in the Durham Coalfield, 1914-1918', has unearthed original research into how the Durham Mining Association (DMA) resisted pressure from colliery owners and the government to accept the introduction of female labour during the First World War.
This entry was posted in Uncategorized on 06/09/2017 by Peter Nicklin.
This entry was posted in Uncategorized on 07/02/2017 by Peter Nicklin.
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{
"redpajama_set_name": "RedPajamaC4"
}
| 2,515
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<?php
namespace test\g\g;
class n { }
|
{
"redpajama_set_name": "RedPajamaGithub"
}
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Aquaman (2018)
By Kaptain Carbon
It has been a while since I wrote a movie review to the point of being nervous if I still knew how to review a film. I do not think I ever knew how to write a review rather I just enjoy throwing my opinion out regardless if people want to to hear it. The last film I reviewed was the Indian epic Balhubali and before that the Spielberg retrotrip Ready Player One. There have been a handful of movies I could have done before the newest DC superhero movie but this is the one which is getting me back into the act. What was it about Aquaman that brought me back to the art of reviewing fantasy cinema? I do not want to spoil anything for people so suffice to say the spectacle which was on display that New Years night was one of the most insane moments of 2018 and was the perfect way to being in the new year. In the absence of any countdown and large gathering, I chose a showtime that would allow the new year to pass in the arms of one of the most bonkers films I can remember. I understand we are on a fantasy website so let me weave you a tale of Arthur Curry and the kingdom of Atlantis.
When we look back on the first 20 years of film, comic book movies will be a topic of discussion. Though I do not know what the next 20 years will hold, I can say for certain the films adapted from comic books will be a cultural watershed for both the comic and film industry. We would be remiss of me not to mention the Marvel Cinematic Universe (MCU) as one of the most ambitious comic adaptations in history. This set of films, wherever it may end, will be the metric in which furture movies will be judged. We would also be equally at a loss in not mentioning the almost comedic performance of the DC Extended Universe (DCEU) and their awkward role in the establishing franchises. This set of films will be the cautionary tale for franchsies who do not want to bungle handfuls of IP content. I could discuss the merits and pitfalls of DC Comics and its adaptions for many articles but suffice to say has not been able to match the magnitude of the MCU. Perhaps this why Aquaman, the sixth movie from the DCEU, works so well since in the absence of matching the pace of Marvel, the films of DC have become the antithesis of a stable franchise with each of them unconnected in tone and quality.In their wild experimentation, DC may have achieved something ironically unexpected in that these films are becoming the best adaptations of comic books. I do not mean they are the best films, far from it, rather movies like Justice League (2017) Wonder Woman (2017) and now Aquaman (2018) have the feeling of reading DC superhero comics and all of the ridiculous fantasy which is tied to its franchises.
Aquaman is difficult to take seriously much less adapt into a film. Arthur Curry is the reluctant heir to Atlantis imbued with the powers of the underwater kingdom but with the care and diplomacy of the surface dwellers. His father is an ordinary lighthouse keeper, his mother is the rogue queen of Atlantis, while his half brother his arch nemesis dubbed Ocean Master. Aside from having a long history of publications back to the golden age, Aquaman, as a character, struggled to be as cool as his contemporary superheroes who possessed cohesive identitys. This is perhaps why the lead was given to Jason Momoa whose roles as Khal Drago, in Game of Thrones, has made him a household heartthrob. Momoa's gruff charm and chiseled frame is perhaps one of closest adaptations of the mythical comic physique we have seen as Aquaman, even without the iconic costume, looms on screen. Visually he is a magnet of attention with stunning eyes and various body tattoos This larger than life attitude for the film continues with a spectacle that not only feels like reading a comic but is akin to reading a silver age comic.
One of the biggest reasons for my recommendation for this film lies on the fact that the film enters into a groove where it continues to double down on the ridiculous nature without looking back. The premise of Aquaman is already slightly ridiculous but getting to the point where lasers are being shot from the eyes of a robot villain takes some extra steps in lunacy. At no point does the tone of Aquaman become self referential nor becomes too self important for ridicule. Aquaman nears the essence of pulp with an unbelievable character doing inconceivable things in scenarios which are too mythic to even comprehend. The viewer is never given a chance to process all of this as ever downtime in the film is rear-ended by more action at which point the only stable thing to grasp is the impossible shoulders of our lead actor. I would use this opportunity to express the virtues of film-making in the fantasy realm but at this point I did not care. Lasers underwater? Yes. Giant Godzilla like monster guarding a treasure? Yes. Plot that feels like a three month D&D campaign? Absolutely. Battle scene at the end where you think nothing can be as silly as what you have seen before? Lets fucking do this.
This tone which is taken throughout the film gives me hope for the future of DC. In the absence of a lockstep franchise which is interconnected with actors directors and narrative, DC is now just doing anything. The bizarre success of Aquaman could potentially work for other larger than life heroes such as the Green Lantern Corps, Shazam, Birds of Prey, or lets hope it can be better, Suicide Squad. Aquaman, along with Justice League, is a story of a comic book movie in a world where Marvel films do not exist. Marvel has changed the playing field for how other comic book movies are judged and the over the top nature emerging from DC movies gives hope of a possible alternative to the continual seriousness of the MCU. Either that of the DCEU will continue to be en enigma with varying success and failure stories throughout the 2020s. What is known now is Aquamna is intended for an audience and who that audience is will be determined in the theater. It was for me but I am just one person who loves watching ridiculous fantasy movies. If that is your deal as well, go see this and buy the biggest popcorn bucket.
Tags: Aquaman, Jason Mamoa, Kaptain Carbon, Movie Review
Categorised in: Film
MOUSE GUARD – The Black Axe
WAR OF THE VIKINGS
DAWN OF WAR III
ASSASSIN'S CREED – Rogue
Pathfinder Log (2nd Entry): Into The Shingles
BLIND STARE – The Dividing Line
Ōkami
|
{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
| 6,094
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Q: Sort LastUpdateTime not working quickbooks API PHP I am getting the below error when i tried to retrieve Objects with the Simple Query in Quickbooks.
Invalid Property Name in Sort Criteria: LastUpdatedTime
BAD_QUERY_REQUESTQUERY_INVALID_SORT_CRITERIA
I tried to sort using name and some other values, its working but for CreateTime and LastUpdatedTime, its not working.
Used the below function to get the Quickbooks Vendors
QuickBooks_IPP_Service_Vendor
Query used to retrieve in order:
PageNum=1&ResultsPerPage=50&Sort=LastUpdatedTime OldestToNewest
Response:
Invalid Property Name in Sort Criteria: LastUpdatedTime
BAD_QUERY_REQUESTQUERY_INVALID_SORT_CRITERIA
A: Looking at the documentation:
https://developer.intuit.com/docs/0025_quickbooksapi/0050_data_services/v2/0400_quickbooks_online/0100_calling_data_services/0030_retrieving_objects
It appears as if you're trying to sort by a field that doesn't exist.
Are you sure you didn't mean:
LastUpdatedTime
Instead of:
LastUpdateTime
A: My Code and Response
Code:
$ServiceName = "QuickBooks_IPP_Service_".$module;
$Service = new $ServiceName();
if ($creds['qb_flavor'] == QuickBooks_IPP_IDS::FLAVOR_ONLINE)
{
$qbmodule = "QB".$module;
$updatedtime = getLastSyncDetails($qbmodule);
$query = "";
if(!empty($updatedtime) && trim($updatedtime) != '')
{
$time = str_replace(" ", "T", $updatedtime);
$time = $time."-07:00";
$query = array('Sort' => 'LastUpdatedTime OldestToNewest');
}
}
$list = array();
$responseQuery = array();
$responseQuery = $Service->findAll($Context, $realm, $query, $page, $limit);
Response
Content-Type: application/xml
Invalid Property Name in Sort Criteria: LastUpdatedTime
BAD_QUERY_REQUEST
QUERY_INVALID_SORT_CRITERIA
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 291
|
Tres Lomas ist ein Partido im Westen der Provinz Buenos Aires in Argentinien. Laut einer Schätzung von 2019 hat der Partido 8801 Einwohner auf 1270 km². Der Verwaltungssitz ist die Ortschaft Tres Lomas.
Orte
Tres Lomas ist in 2 Ortschaften und Städte, sogenannte Localidades, unterteilt.
Tres Lomas
Ingeniero Thompson
Wirtschaft
Die Wirtschaft von Tres Lomas wird von der Landwirtschaft dominiert, die Hauptanbaupflanzen sind Sonnenblumen, Weizen und Mais. Daneben wird Rinderzucht betrieben.
Einzelnachweise
Tres Lomas
|
{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 9,822
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New findings of various local and international researchers focusing on commercial ties between the Nabataean Kingdom and Rome, India and Egypt, as well as the economy and agriculture of Petra's hinterland were discussed at the third international conference on Petra and the Nabataean heritage, which concluded on Thursday.
The four-day event, titled "The Nabataeans-Economy and Culture", was organised by the University of Jordan and Petra Development and Tourism Region Authority (PDTRA) in Petra between June 18 and June 21.
"Petra is the main tourist site in Jordan, but it is the symbol of the Jordanian national identity," said Chief Commissioner of PDTRA Falah Al Omoush, adding that despite the conflicts and turmoil gripping the region since the Arab Spring, the last two years have witnessed an increase in the number of tourists, and PDTRA works with both local and international stakeholders to boost Petra's tourism.
"Being an important city under Roman occupation and Byzantine influence up to the Crusaders' period in the 12th century, Petra seems to have been away from the world until it was rediscovered to the Western world in 1812, marking a new era of interest and rediscovery," Farajat said, underlining that most travellers at that time entered it from the south as they came from Egypt accompanied by bedouins from Aqaba and Sinai who acted as "the first Jordanian tour operators".
The encounters of those travellers with the diverse local groups at that time showed the fragmentation the society lived in, but also the lack of appreciation and understanding of the site, he explained, adding that there is "no wonder that most of early travellers were certain that Petra was a Roman city".
"It was only after the archaeologists started to study Petra in the early 20th century that the Arab identity of Petra was clear," Farajat noted.
The importance of Petra in the incense trade was very well documented by ancient sources.
"The Nabataean role in the trade of frankincense and myrrh for use as incense and in perfumes is well documented," said David Johnson from Brigham Young University in Utah, in the US.
According to Johnson, who worked as a researcher in Jordan from 1977, the role the Nabataeans played in the trade of these products and other plant products, such as rock rose in medicine and magic, was not often discussed.
In his paper titled "Magic, Medicine and Fraud", the role of the Nabataeans in this trade is examined by looking at the literary evidence from ancient writers such as Dioscorides(40-90 AD), Pliny( 23-79AD) and Galen(130-210 AD); the Greek magical papyri; and a botanical survey of present day plants found at Petra.
Another source is the archaeological evidence from Nabataean sculpture, painting and pottery, the American scholar continued, adding that among the major plant products exported by the Nabataeans, three — frankincense, myrrh and bdellium — originated outside of Nabataean state while others, iunctus odoratus, gum labdanum, balanites oil and terebinth resin were produced in Nabataea itself.
Laurant Tholbecq, from University of Brussels, talked about a complex system of taxes which Nabataeans imposed on caravans, land property and individuals. According to Pliny, "the Nabataean Kingdom taxed caravans, which was a significant source of income".
French architect Thibaud Fournet studied Petra's baths, which are located on 300-metre cliff edges above the valley, and were excavated in 2012 and 2015.
"The breathtaking location of both buildings, on the edge of the cliff, overhanging the valley by circa 300 metres, was deliberately decided," Fournet said.
Swiss researcher and resident of Petra Ulrich Bellwald presented the number of winery presses in the Beidha area and speculated that wine production was the state monopoly in ancient Nabataean Kingdom.
"In Nabataean period it was absolutely impossible to influence the fermentation process by cooling. Due to the still very high temperatures during harvest and due to the fact that the bedrock, into which the wine presses were inserted, was at least warm, fermentation concluded within a timeframe of not more than one day," Balwald highlighted.
Hence, the production process in the presses took place in short intervals to avoid a secondary fermentation which would have turned the wine into vinegar, the harvest only started at the last possible moment, with the grapes having a high concentration of sugar, the veteran scholar explain, adding that "as there is only a very limited number of shards belonging to locally manufactured wine amphorae known from the Petra area, the wine was most probably stored and transported in wineskins from goats or/and sheep".
|
{
"redpajama_set_name": "RedPajamaC4"
}
| 7,937
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Kebonsari kan syfta på följande platser:
Indonesien
Desa Kebonsari (administrativ by i Indonesien, Jawa Tengah, lat -7,73, long 109,60),
Desa Kebonsari (administrativ by i Indonesien, Jawa Timur, lat -8,05, long 111,08),
Desa Kebonsari (administrativ by i Indonesien, Jawa Tengah, lat -7,21, long 110,03),
Desa Kebonsari (administrativ by i Indonesien, Jawa Tengah, lat -6,93, long 110,07),
Desa Kebonsari (administrativ by i Indonesien, Banten),
Desa Kebonsari (administrativ by i Indonesien, Jawa Tengah, lat -6,95, long 109,68),
Desa Kebonsari (administrativ by i Indonesien, Jawa Timur, lat -7,49, long 112,72),
Desa Kebonsari (administrativ by i Indonesien, Jawa Tengah, lat -6,95, long 110,73),
Desa Kebonsari (administrativ by i Indonesien, Jawa Timur, lat -7,09, long 112,31),
Kelurahan Kebonsari (administrativ by i Indonesien, lat -7,64, long 112,91), Jawa Timur,
Desa Kebonsari (administrativ by i Indonesien, Jawa Timur, lat -7,73, long 111,49),
Kecamatan Kebonsari, distrikt, Jawa Timur,
Kelurahan Kebonsari (administrativ by i Indonesien, lat -7,33, long 112,72), Jawa Timur,
Desa Kebonsari (administrativ by i Indonesien, Jawa Timur, lat -8,21, long 112,08),
Kelurahan Kebonsari (administrativ by i Indonesien, lat -8,02, long 112,62), Jawa Timur,
Kelurahan Kebonsari (administrativ by i Indonesien, lat -8,19, long 113,71), Jawa Timur,
Robotskapade Indonesienförgreningar
|
{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 5,259
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\section{Introduction}
One of the most visible applications of \emph{Intelligent Transport Systems} (ITS), within the field of public transportation, is the display of real-time traffic information. This has happened, traditionally, in the form of arrival and departure times on digital departure boards at stops and stations, and more recently in smart-phone apps and in-vehicle infotainment screens. It is widely deployed in most major cities, and is now considered a standard method to deliver an attractive and competitive public transport service. To an increasing extent, real-time information channels constitute the only source of passenger information.
Public transport authorities have long found that GPS trajectory data from already deployed \emph{Automatic Vehicle Location} systems (AVL) can be used in the production of arrival and departures times \citep{Tcrp48}.
Our motivation is to improve the accuracy yielded by current prediction methods by exploiting spatio-temporal correlations present in public transport networks. Our focus is especially on urban bus networks that often share considerable parts of the infrastructure with other modes of transport, and therefore are prone to ripple effects. The proposed system, and the information it produces, can be integrated into ITSs in various ways with different applications. In its most basic form, the system can simply substitute current methods as a data source in passenger information systems, presenting real-time arrival and departure times. Passengers presented with reliable travel times can make use of this information in their decision-making \citep{Cats2011}, e.g.\ choose alternative routes or modes of transport to avoid prolonged travel time on their current route. The availability of the information produced by the system to awaiting passengers can also simply function as a comforting assurance, as studies have shown that reliable real-time information at bus stops has a statistically significant dampening effect on the perceived waiting time \citep{Fan2016}.
A more \textit{intelligent} use of the information can be in the context of automated trip planners. These already accept this kind of real-time information, e.g.\ using the General Transit Feed Specification (GTFS). This allows for alerting passengers or proposing alternative routes earlier on the passenger's trip when a connecting service might be unreachable due to prolonged travel times.
Operating more sophisticated ITS applications successfully requires, to an even larger extent, accurate and reliable travel time predictions, since the cost of making erroneous decisions based on the predictions increases. \citet{Lo2012} present a decision support system for bus holding that requires accurate estimated arrival times to function optimally. Other advanced ITS examples include \emph{connection assurance} between two low frequency public transport services, where an expert system advises the driver or the traffic management system that one of the services should wait for the other, based on the arrival time predictions for both services. If the travel time predictions are too optimistic, the expert system ends up advising to hold the connecting service for longer than anticipated, introducing a prolonged delay for the other service and the passengers already present on that service. The use of our proposed system in this context can be achieved with a simple rule-based decision engine on top of the travel time model presented in detail in this paper.
\subsection{Bus travel time prediction}
Arrival/departure time prediction is commonly approached as a specialization of travel time prediction as illustrated in \Cref{fig:bus-arrival-links}. The predicted travel time for each link is simply accumulated downstream the route to yield the arrival/departure time predictions at each stop point of the rest of the current journey. Thus, in the example, to predict when the bus at stop A will arrive at stop B, we would just sum up our predicted link travel times for Links 1 to 3. Besides the link travel time, estimations of dwell time (i.e.\ when a bus is holding at a stop point) should also be accumulated downstream.
\begin{figure}[!ht]
\center
\includegraphics[width=\columnwidth]{bus-arrival-links-narrow.pdf}
\caption{Arrival- and departure using link travel time.}
\label{fig:bus-arrival-links}
\end{figure}
Producing precise bus travel time predictions in areas with little external influence, e.g.\ rural areas, can to a large extent be solved with historical averaging or simple regression methods \citep{Williams2003,Altinkaya2013}. The problem becomes much more complex in urban areas where congestion, special events, roadworks, weather, etc.\ highly influence the traffic flow and passenger demand. As on-board GPS and AVL systems have become more affordable and common, data has both grown in coverage, i.e.\ number of vehicles with AVL installed, and frequency, i.e.\ number of GPS positions collected for each vehicle per time-unit.
Using geofencing techniques the raw GPS trajectory data can be converted into arrivals and departures at stop points, and subsequently, travel times on the links between the stop points. The objective is an intelligent expert system that utilizes this data in order to produce precise short-term predictions (e.g.\ up to\ $0-1.5$ hours in the future) for link travel time, specifically for bus traffic in urban areas.
Our contribution is an intelligent model for bus travel time prediction that takes advantage of the non-static spatio-temporal correlations present in urban bus traffic. We leverage on recent state-of-the-art techniques from \textit{machine learning} by combining \emph{convolutional} and \emph{long short-term memory} (LSTM) \citep{Lstm1,Lstm2} neural networks, thus allowing the discovery of patterns across both time and space. Our proposed model is also multidimensional in its output with respect to both spatial and temporal aspects, i.e.\ we predict travel times for all links, for multiple time-steps ahead.
The porposed method is empirically evaluated and compared to other popular approaches for link travel time prediction, including the model currently deployed in production by the public transport authority for the Greater Copenhagen Area, \emph{Movia}. Furthermore, the method is compared to Google Traffic (part of Google Maps), a popular online service for travel time prediction.
This paper is structured in the following manner: In the next section, related work and relevant literature are reviewed. \Cref{sec:convlstm} introduces Convolutional LSTM neural networks in general, and in \Cref{sec:model} we present the proposed multi-output model in more detail, including e.g.\ network topology, and data preparation. \Cref{sec:experiments} introduces the Copenhagen dataset, which the model has been evaluated on, and our results are presented and discussed in~\Cref{sec:results}. Finally, we conclude on the work in~\Cref{sec:conclusion}.
\section{Related work}
Bus link travel time prediction has been explored in research as GPS and AVL data has become increasingly available. The problem overlaps with other research areas such as general traffic flow and speed estimation. But the problem has also unique constraints and opportunities that follow from servicing a fixed route with fixed stop points. The improvement in computational power in recent decades has gradually allowed more complex link travel time models with increased precision.
Early approaches for bus travel time prediction rely on historical average models \citep{Dailey1999,Sun2007}, and linear regression \citep{Patnaik2004}. Recent research presents this type of models only for comparison purposes, and in all cases, these are outperformed by the proposed alternatives \citep{Shalaby2004,Jeong2005}. The major disadvantage of historical average models is that they will only slowly converge to changes in the travel time, which of course is undesired with short, but highly impacting, external influences (e.g.\ a traffic incident or a large event). However, their simplicity, both with respect to computational cost and need of input data, has made them widely used in the industry. In rural areas, where traffic patterns are quite static, they can actually perform reasonably.
By their capabilities of maintaining state between predictions, Kalman filters (KF) have been the topic of several studies either as an independent model \citep{Chen2001,Shalaby2004}, or in combination with other models \citep{Yo2010,Bai2015}. In all cases, the applied filters are traditional linear KFs, and applied independently to each link. Because of the linearity, these models are computationally still quite cheap, but likewise, their disadvantage is that they are very limited in capturing and forecasting the complex non-linear dynamics of travel and dwell time in a metropolitan bus system. For example, the KF's state is only directly accessible for the leading time-step and thus is not capable of finding long-distance patterns spanning over several links and/or over several time-steps. In order to overcome this, KFs can be generalized to \textit{extended Kalman filters} (EKFs), allowing nonlinearities, but they still do not consider multiple links simultaneously. Making EKFs output travel times for all links simultaneously, with possible nonlinear interactions between them, would dramatically increase the computational cost.
The above analysis is substantiated by \citet{Lin2013} and \citet{Kumar2014} who find \textit{artificial neural networks} (ANN) to outperform independent Kalman filter models. However, the computational challenges of fully connected ANNs are also limiting the number of neurons of the network, and thus the complexity of the patterns it can learn to recognize. This has sparked the interest in studying \textit{composite} or \textit{hybrid} models. \citet{Bai2015} use a two-stage approach by combining an offline ANN model with an adaptable/online Kalman filter to yield a dynamic model. The advantage is the balance between computational complexity and the ability to adapt to smaller deviations quickly. The model is able to adapt to temporal variations in the current travel time on a journey, but it is still not able to recognize long distance patterns. The model proposed in this work uses \emph{long short-term memory} cells (LSTM), a apecial form of recurrent neural network (RNN) cells. \citet{Ma2015} use LSTM cells for highway speed prediction, and find it to significantly outperform KFs. Our proposal differs from existing research in bus link travel time prediction by combining the capability for maintaining state-space over multiple time-steps, while allowing the deep neural network to be efficiently trained.
Some recent research recognizes that several routes can benefit from each other's predictions if they share some partial route segment, e.g.\ \citep{Yu2011,Gal2014,Bai2015}. However, none of these approaches consider cross-temporal correlations between different route segments, and they only use a small window for correlation with upstream links (e.g.\ max.\ 3 links). Likewise, \citet{YanjieDuan2016} propose the use of an LSTM model for general highway travel time prediction, and to predict multiple time-steps ahead, but only for a single link at a time, i.e.\ cross link (spatial) correlations are lost. Another non-public transport study estimates travel times on road segments \citep{Tang2018}, and actually incorporates the spatial correlation, but the temporal aspect is very coarse and does not predict multiple time-steps ahead. In contrast, the combination of both \emph{LSTM} cells and \emph{convolutional} filters for bus travel time prediction, proposed in this paper, allows the learned patterns to generalize beyond a single link and time, i.e.\ multi-output and multi-time-step. Furthermore, this reduces the computational complexity by orders of magnitude compared to fully connected ANNs capable of capturing similar complex patterns.
We can identify the following strengths of the proposed system compared to existing approaches:
\begin{enumerate}[i)]
\item Unlike previous contributions in bus arrival prediction, it has the ability to learn spatio-temporal correlations as a coherent structure. The learned patterns can generalize over time and network links since the \emph{convolutional} filters are shared.
\item We predict multiple time-steps ahead using a recurrent structure and an encoder-decoder architecture that allows the time-steps ahead to follow more complex patterns compared to existing approaches that just use a fully connected ANN layer as the final layer to split the prediction into multiple output time-steps.
\item The input data needed for the method is easily obtained from the raw GPS traces that the AVL systems output, given the relatively fixed road network and location of bus stop points and stations.
\end{enumerate}
In contrast, the following possible weakness should also be considered:
\begin{enumerate}[i)]
\item The computational complexity of the training is still a concern. Even though the computational complexity is reduced greatly with \emph{convolutional} filters compared to pure ANN models, it is still time-consuming to train the proposed model. That said, we have successfully trained models for complete routes using commodity-grade hardware within reasonable time. With our test setup we could do retraining on a daily basis without computational complications. The training can easily be distributed across multiple computational instances, so we argue that the scalability issue can be overcome.
\end{enumerate}
\section{Convolutional LSTM neural networks}
\label{sec:convlstm}
A long short-term memory (LSTM) neural network is a special type of Recurrent Neural Network~(RNN) which has been proven robust for capturing long-term dependencies \citep{Lstm1,Lstm2}. The important feature of an LSTM network is its capability to maintain a cell state, $\matr{c}_t$, from previous observations across sequences of input (e.g.\ time), but also to eliminate information considered irrelevant. To allow this mechanism, the maintenance of information is controlled by three gates: \emph{input gate}, \emph{forget gate}, and \emph{output gate}. Each gate yields a state variable at time $t$, respectively $\matr{i}_t$, $\matr{f}_t$, and $\matr{o}_t$, along with the cell output, $\matr{h}_t$, cf.~\cref{eq:lstm}, where $\circ$ denotes the element-wise product.
\begin{equation}
\begin{aligned}
\matr{i}_t &= \sigma \left( \matr{W}^i \matr{x}_t + \matr{R}^i \matr{h}_{t-1} + \matr{U}^i \circ \matr{c}_{t-1} + \matr{b}^i \right) \\
\matr{f}_t &= \sigma \left( \matr{W}^f \matr{x}_t + \matr{R}^f \matr{h}_{t-1} + \matr{U}^f \circ \matr{c}_{t-1} + \matr{b}^f \right) \\
\matr{c}_t &= \matr{f}_t \circ \matr{c}_{t-1} + \matr{i}_t \circ \mathrm{tanh} \left( \matr{W}^c \matr{x}_t + \matr{R}^c \matr{h}_{t-1} + \matr{b}^c \right) \\
\matr{o}_t &= \sigma \left( \matr{W}^\mathit{o} \matr{x}_t + \matr{R}^\mathit{o} \matr{h}_{t-1} + \matr{U}^\mathit{o} \circ \matr{c}_{t} + \matr{b}^o \right) \\
\matr{h}_t &= \matr{o}_t \circ \mathrm{tanh} \left( \matr{c}_t \right)
\end{aligned}
\label{eq:lstm}
\end{equation}
\vspace{.5em}
\Cref{fig:lstm-peepholes} illustrates the inner structure of an LSTM cell with peephole as proposed by \cite{LstmPeephole}.
It has especially grown popular for predicting time series using methods evolved from \cite{LstmTs}, where fixed-length windows of time-series are generated and feed into an LSTM network. Multiple LSTMs can be stacked such that more complex patterns of sequential information (e.g.\ temporal patterns) can be learned.
\begin{figure}[!ht]
\centering
\includegraphics[scale=.45]{lstm-peepholes.pdf}
\caption{Structure of LSTM cell with peephole.}
\label{fig:lstm-peepholes}
\end{figure}
Convolutional Neural Networks~(CNNs), on the other hand, have been widely used for capturing spatial relationships, e.g.\ the importance of neighboring pixels in an image. As opposed to fully connected layers, where each unit $i$ in the layer has a dedicated scalar weight $w_{ij}$ for all input values $x_j$, convolutional units are only locally connected and reuse the same weights to produce several outputs. Instead of considering the entire input-vector, only a fixed-size window, or \emph{convolution}, around each input is considered. The weights are therefore referred to as the \emph{filters} or \emph{kernels} of the layer. \Cref{fig:conv} illustrates a single convolutional filter of size $3$ being applied to one-dimensional data.
\begin{figure}[!ht]
\centering
\includegraphics[scale=.5]{conv.pdf}
\caption{Application of convolutional filter onto 1D data.}
\label{fig:conv}
\end{figure}
Special care needs to be taken at the boundaries, i.e.\ where the convolutional filter will exceed the input. To avoid that the size of the output decreases, an approach is to pad the input, e.g.\ with zeros. This ensures that the output shape of each convolutional unit will always be identical to the input shape, which is often desirable. One of the key benefits of convolutional networks is that the number of weights that needs to be learned is considerably reduced compared to fully connected networks, and also that learned patterns can be transferred across space. I.e.,\ the convolutional filters become feature detectors that, in our case, can detect spatial patterns across links, e.g.\ congestion forming, etc.
\citet{ConvLSTM} introduced the novel combination of convolutional and LSTM layers into a single structure, the \emph{Convolutional LSTM}, or simply \emph{ConvLSTM}. Specifically, the method applies convolutional filters in the \emph{input-to-state} and \emph{state-to-state} transitions of the LSTM cf.~\cref{eq:convlstm}, where $*$ denotes the convolution operator.
\begin{equation}
\begin{aligned}
\matr{i}_t &= \sigma \left( \matr{W}^i * \matr{x}_t + \matr{R}^i * \matr{h}_{t-1} + \matr{U}^i \circ \matr{c}_{t-1} + \matr{b}^i \right) \\
\matr{f}_t &= \sigma \left( \matr{W}^f * \matr{x}_t + \matr{R}^f * \matr{h}_{t-1} + \matr{U}^f \circ \matr{c}_{t-1} + \matr{b}^f \right) \\
\matr{c}_t &= \matr{f}_t \circ \matr{c}_{t-1} + \matr{i}_t \circ \mathrm{tanh} \left( \matr{W}^c * \matr{x}_t + \matr{R}^c * \matr{h}_{t-1} + \matr{b}^c \right) \\
\matr{o}_t &= \sigma \left( \matr{W}^\mathit{o} * \matr{x}_t + \matr{R}^\mathit{o} * \matr{h}_{t-1} + \matr{U}^\mathit{o} \circ \matr{c}_{t} + \matr{b}^o \right) \\
\matr{h}_t &= \matr{o}_t \circ \mathrm{tanh} \left( \matr{c}_t \right)
\end{aligned}
\label{eq:convlstm}
\end{equation}
\vspace{.5em}
\begin{figure*}[!t]
\centering
\includegraphics[scale=.75]{conv_lstm.pdf}
\caption{Convolutional LSTM network topology.}
\label{fig:ConvLSTM}
\end{figure*}
As with traditional CNN layers, the output dimensionality of a \emph{ConvLSTM} layer is determined by the number of filters applied. However, \emph{ConvLSTMs} require a total of eight filters for each desired output, i.e.\ four \emph{input-to-state} filters ($\matr{W}^i$, $\matr{W}^f$, $\matr{W}^c$, and $\matr{W}^o$) and four \emph{state-to-state} filters ($\matr{R}^i$, $\matr{R}^f$, $\matr{R}^c$, and $\matr{R}^o$).
Still, it is important to emphasize that the application of convolutional filters to the LSTM model greatly reduces the number of parameters/weights that need to be learned, compared to a \emph{pure LSTM} approach. This allows for even deeper networks.
\vspace{-.3em}
\section{Multi-output model}
\label{sec:model}
In this section, we present the multi-output, multi-time-step model for bus travel time prediction that uses the \emph{ConvLSTM} layer introduced in the previous section.
\vspace{-.2em}
\subsection{Network topology}
\Cref{fig:ConvLSTM} shows the overall network topology, where blue boxes illustrate \emph{input-to-state} convolutions and yellow boxes \emph{state-to-state} convolutions. The network uses a sequence encoder/decoder technique, which is an extension of the encoder/decoder presented by \citet{ConvLSTM}. The encoder block consists of two \emph{ConvLSTM} layers, where the resultant sequence (last $k$ values of the sequence) is fed into a decoder, or prediction block. The decoder block also consists of two \emph{ConvLSTM} layers, and a fully connected (FC) layer. The proposed architecture allows unequal $w$ and $k$, e.g.\ it predicts the next $3$ time-steps based on a window size of $20$ previous time-steps.
Therefore, convolutional filters are applied to each input, at each time-step, to the respective LSTM cell and also between LSTM cells in the state-transition. Since the time-steps are one-dimensional~(i.e.\ link travel times across links), the filters are also one-dimensional. In each of the two blocks, the \emph{ConvLSTMs} are arranged with filter sizes of respectively $10\times1$ and $5\times1$ for each of the layers in the block. This size is used both for the \emph{input-to-state} and \emph{state-to-state} convolutional filters. Lastly, each \emph{ConvLSTM} layer has 64~outputs, yielding a total need of 512 convolutional filters.
In order to avoid over-fitting during training \emph{Dropout} \citep{Dropout} is used between the \emph{ConvLSTM} layers, and \emph{Batch Normalization} \citep{BatchNorm} is also performed before each \emph{ConvLSTM} layer to ensure reasonable inputs for the activations and speed-up learning. The dropout probability is adjusted to 20\%, 10\%, and 10\%, respectively.
Each of the \emph{ConvLSTM} layers uses linear activation functions, and the output from the last layer in the decoder block is fed into a fully connected (FC) layer using the \emph{ReLU} activation function, which also ensures that only positive travel times are predicted.
\subsection{Data preparation}
We expect link travel times from AVL systems to be available in a tabular form, where each link travel time measurement has a timestamp, and a reference to the link as illustrated in~\Cref{tab:data}. This output is standard for most AVL systems used in public transport systems, thus allowing the proposed system to generalize to other regions.
\begin{table}[!ht]
\centering
\footnotesize
\begin{tabular}{llr}
Timestamp & Linkref. & Link travel time (s) \\ \hline \hline
2017-10-10 00:20:02 & 29848:1254 & 63 \\ \hline
2017-10-10 00:21:07 & 1254:1255 & 65 \\ \hline
2017-10-10 00:21:51 & 1255:10115 & 44 \\ \hline
\vdots & \vdots & \vdots
\end{tabular}
\caption{Example of raw travel time measurements.}
\label{tab:data}
\end{table}
For the \emph{ConvLSTM} model to be able to capture the desired spatio--temporal patterns, the input data must be arranged in a suitable manner, i.e.\ in $N$ samples, each with a window of the $w$ lagging time-steps $t-w+1, \ldots, t$, and each time-step with $u$ link travel times $1, \ldots, u$, as illustrated in~\Cref{fig:data_shape}.
As for the output, it consists of $N$ predictions for each of the $k$ time-steps ahead, $t+1, \ldots, t+k$. Thus the input is a 4D-tensor, $\matr{X}$ with dimensionality $N \times w \times u \times 1$, and the output, $\matr{Y}$, a 4D-tensor with dimensionality $N \times k \times u \times 1$. In both cases, the last one refers to the single link travel time for each time-step/link combination. It is emphasized that each prediction consists of travel time predictions for all links for the next $k$ time-steps, i.e.\ multi-output, multi-time-step-ahead prediction.
The $N$ samples are sampled at a fixed time resolution since we need a shared time reference across all links. \Cref{sec:experiments} elaborates on some of the considerations for choosing an adequate resolution.
\begin{figure}[!ht]
\centering
\includegraphics[scale=1.0]{data_shape.pdf}
\caption{Shapes of the input and output data.}
\label{fig:data_shape}
\end{figure}
\subsection{Detrending}
Urban bus travel times vary throughout the day, and the day of the week due to \emph{recurring congestion}. In order to reduce the need for the deep neural network to learn this recurring variation, the travel times for link $\mathit{ln} \in \{ 1,\ldots,u\}$, at time-step $t$, $x_{\mathit{ln},t}$, are normalized to focus on deviations from the normal and expected pattern. Travel times are centered with the mean for each link, at the time of day, and day of week, $\mathit{\bar{x}_{\mathit{ln},\mathit{dow},\mathit{tod}}}$, and scaled with the standard deviation for each link, $\sigma_\mathit{ln}$:
\begin{equation}
x'_{\mathit{ln},t} = \frac{x_{\mathit{ln},t} - \bar{x}_{\mathit{ln},\mathit{dow},\mathit{tod}}}{\sigma_\mathit{ln}}
\label{eq:normalization}
\end{equation}
A similar normalization is applied to the predicted travel times, $y_{\mathit{ln},t}$, but only using the historical mean and standard deviation, since the true mean and standard deviation are obviously unavailable in real-time prediction scenarios.
When calculating the mean and standard deviation, it can be beneficial to exclude extreme outliers, since both mean and standard deviations are highly sensitive to such measurements. A suggested method is to apply \emph{absolute deviation around the median} (MAD; see \cite{Olewuezi2011}) when calculating $\mathit{\bar{x}_{\mathit{ln},\mathit{dow},\mathit{tod}}}$ and $\sigma_\mathit{ln}$.
\subsection{Implementation and training}
The proposed network model was implemented in Python using the Keras Framework \citep{Keras}, and trained using the \emph{RMSprop} algorithm \citep{RMSprop}. The source code for the proposed method is publicly available at GitHub: \cite{gh-bus-arrival-convlstm}.
During training, the variables $\mathit{\bar{x}_{\mathit{ln},\mathit{dow},\mathit{tod}}}$ and $\sigma_\mathit{ln}$ should be calculated solely based on the training set, to emulate the real-world application.
\section{Experiments}
\label{sec:experiments}
For the purpose of evaluation, the proposed method is applied to a dataset from Copenhagen's public transport authority, \emph{Movia}. The dataset consists of 1,2M~travel time observations for the ``4A'' bus line in the period May to October 2017. The data points were collected using the real-time AVL system installed in every vehicle servicing the line.
\begin{figure}[!ht]
\centering
\includegraphics[width=0.4\textwidth]{map.png}
\caption{Geography of the 4A bus line in Copenhagen.}
\label{fig:4a_map}
\end{figure}
The geography of the route is shown in \Cref{fig:4a_map}. As the line circles Central Copenhagen, it is highly sensitive to congestion to/from the city since it intersects with several large corridors along its route. Southeast of the city center, the line splits into different destination patterns (gray), therefore only the first 32 links are considered for the purposes of this experiment (red).
\begin{figure*}[!t]
\centering
\begin{subfigure}[t]{0.31\textwidth}
\centering
\includegraphics[width=\textwidth]{decycle_5min_n.pdf}
\caption{5 min}
\end{subfigure}%
~
\begin{subfigure}[t]{0.31\textwidth}
\centering
\includegraphics[width=\textwidth]{decycle_15min_n.pdf}
\caption{15 min}
\end{subfigure}
~
\begin{subfigure}[t]{0.31\textwidth}
\centering
\includegraphics[width=\textwidth]{decycle_30min_n.pdf}
\caption{30 min}
\end{subfigure}
\caption{Examples of travel time for a single link over a single day, at various time resolutions.}
\label{fig:resolutions}
\end{figure*}
\subsection{Time resolution}
In order to allow predictions for fixed time-steps ahead, the data is aggregated at a fixed time resolution. The choice of time resolution is a hyper-parameter for the proposed system, and should be tuned for the specific dataset. \Cref{fig:resolutions} shows examples of travel time for a single link over a single day at various time resolutions. The black dots are actual measurements, and the lines the aggregated mean link travel time at the given resolution. Several considerations should be made when choosing the time resolution:
\begin{itemize}
\item The \emph{expected} frequency of the line, since a choice far from this will lead to either 1) sparse measurements, and low probability of actually using a prediction, because no service runs in the predicted time step; or 2) an overly smooth time-series, with too much detail about variability being lost. Thus it is a balance between capturing the details and still having a reasonable number of measurements of each time-step to avoid overfitting.
\item The computational cost of training the system, since smaller time-steps will require further iterations over the training data and larger values of $w$ and $k$ to include the same lagging time window, and time horizon for predictions.
\end{itemize}
\Cref{fig:runtime_stats} shows how the choice of resolution influences the training time of our proposed deep neural network architecture on commodity hardware (blue). It also shows how the portion of time-steps with missing values (yellow) also increases as more fine-grained resolutions are considered. For instance, using a 2-minute resolution will cause 89\% of all time steps to not include any measurements.
\begin{figure}[!ht]
\centering
\includegraphics[width=0.4\textwidth]{runtime_stats.pdf}
\caption{Choice of resolution influence on training time and missing values.}
\label{fig:runtime_stats}
\end{figure}
For this experiment, the AVL data was aggregated into 15-minute intervals and normalized as described in~\Cref{sec:model}. This resolution was chosen based on the above-mentioned considerations. The ``4A'' bus line had a measured mean \emph{headway} (the time between two vehicles during daytime) of $7.5$ minutes between 06:00 and 22:00, and thus there is a reasonably high probability that 15-minute time-steps will include 1-2 measurements. Indeed, the average number of measurements in each time step was $1.7$ for the training set.
Given the time resolution, we set the fixed window size, $w = 32$, equivalent to 8 hours, to allow patterns in the morning peak to affect patterns in the afternoon peak. We set $k = 3$ to allow predictions of up to 45 minutes into the future.
\subsection{Evaluation}
The evaluation of the proposed model and all the considered baselines is based on the following statistics: \emph{mean absolute error} (MAE), \emph{root mean square error} (RMSE), and \emph{mean absolute percentage error} (MAPE), as formalized in \cref{eq:mae,eq:rmse,eq:mape}, where $\matr{Y}_i$ is the true link travel times for sample $i$ and $\matr{\widehat{Y}}_i$ is the predicted travel times. Since the multi-output, multi-time-step model predicts link travel times for all $u$ links for the next $k$ time-steps, $\matr{Y}_i$ and $\matr{\widehat{Y}}_i$ have the same dimensionality: $w \times u \times 1$.
\begin{equation}
\textrm{MAE}(\matr{Y}, \matr{\widehat{Y}}) = \frac{\sum_{i = 1}^{N} \left| \matr{Y}_i - \matr{\widehat{Y}}_i \right| }{N}
\label{eq:mae}
\end{equation}
\begin{equation}
\textrm{RMSE}(\matr{Y}, \matr{\widehat{Y}}) = \sqrt{\frac{\sum_{i = 1}^{N} \left(\matr{Y}_i - \matr{\widehat{Y}}_i \right)^2}{N}}
\label{eq:rmse}
\end{equation}
\begin{equation}
\textrm{MAPE}(\matr{Y}, \matr{\widehat{Y}}) = \frac{1}{N} \sum_{i = 1}^{N} \left| \frac{\matr{Y}_i - \matr{\widehat{Y}}_i}{\matr{Y}_i} \right|
\label{eq:mape}
\end{equation}
\vspace{.5em}
\begin{table*}[!t]
\center
\begin{tabular}{ll|rrr}
Model & Time ahead & RMSE (min) & MAE (min) & MAPE (\%) \\
\hline
\hline
Historical average & & 4.35 & 3.23 & 6.51 \% \\
\hline
Current model & t + 1 (15 min) & 4.92 & 3.90 & 8.05 \% \\
& t + 2 (30 min) & 4.91 & 3.46 & 6.82 \% \\
& t + 3 (45 min) & 5.47 & 4.15 & 8.68 \% \\
\hline
Pure LSTM & t + 1 (15 min) & 3.48 & 2.48 & 5.02 \% \\
& t + 2 (30 min) & 3.56 & 2.51 & 5.08 \% \\
& t + 3 (45 min) & 3.68 & 2.62 & 5.34 \% \\
\hline
Google Traffic & t + 1 (15 min) & 3.67 & 2.96 & 6.32 \% \\
\hline
ConvLSTM & t + 1 (15 min) & 2.66 & 1.99 & 4.19 \% \\
& t + 2 (30 min) & 2.89 & 2.11 & 4.44 \% \\
& t + 3 (45 min) & 3.11 & 2.27 & 4.75 \% \\
\hline
\end{tabular}
\caption{Results of the proposed and the baseline models}
\label{tab:results}
\end{table*}
To allow a clear comparison, we reduce $\matr{Y}_i$ and $\matr{\widehat{Y}}_i$ by summing over all links:
\begin{align}
\matr{Y}_i' = \sum_{\mathit{ln} = 1}^{u} \matr{Y}_{i,\mathit{ln}} & & \matr{\widehat{Y}}_i' = \sum_{\mathit{ln} = 1}^{u} \matr{\widehat{Y}}_{i,\mathit{ln}}
\label{eq:total}
\end{align}
This is equivalent to predicting the total travel time of all 32 links, and follows the initial approach for arrival/departure time prediction by accumulating link travel times.
The output of each of the evaluation functions is thus simply a vector of size $k$, i.e.\ the evaluation of the different time-steps for all links accumulated.
The model is trained on the prepared data using a sliding window approach in order to simulate real-world conditions, in which real-time travel time measurements arrive as a continuous data stream. We use 23 weeks of data for training, and one week of data for testing. The window is advanced for 1 week at a time for a total of 4 test weeks. The trained models are available, alongside the source code, at \citep{gh-bus-arrival-convlstm} and include a test dataset (4 weeks). For replicating our results, the full dataset used in this experiment is available from Movia upon request.
\section{Results and discussion}
\label{sec:results}
The performance of our proposed model for link travel time prediction, based on \emph{ConvLSTM}, is compared against several baseline models and services:
\begin{enumerate}
\item a naïve historical average model, i.e.\ equivalent to just predicting the normalized value, $\bar{x}_{\mathit{ln},\mathit{dow},\mathit{tod}}$;
\item the traffic prediction model currently deployed by Movia;
\item a pure LSTM model for link travel time prediction, i.e.\ without applying convolutional filters in state transitions;
\item travel time predictions from Google Traffic (part of Google Maps).
\end{enumerate}
\Cref{tab:results} shows the overall performance of the proposed model and the baseline methods. Predictions are limited to daytime, i.e.\ between 06:00 and 22:00, and are accumulated downstream on a journey level to simulate the use for real-time bus arrival/departure time prediction, cf.~\cref{eq:total}.
Before going into a direct comparison, it is important to understand some aspects of the baseline models, and how measurements were collected.
\subsection{Historical average}
The performance of the historical average is independent with respect to the number of time-steps ahead in time it predicts, as it just represents a weekly cycle of mean link travel times.
\subsection{Current model}
Measurements from the currently deployed bus prediction model were collected at a 5-minute frequency using a non-publicly accessible endpoint at the transport authority. The model is based on a historical average model, but also has a rule-based mechanism on top that can override or adjust the historical link travel times. For instance, it will, to some extent, assume that a delayed vehicle will recover from its delay by traversing links a bit faster. Of course, such an assumption can be problematic in an urban area with many external traffic effects.
\subsection{Pure LSTM}
The pure LSTM model for link travel time prediction is similar to the model proposed by \citet{YanjieDuan2016}. The model was trained on the exact same dataset as the \emph{ConvLSTM} model and has a similar architecture, but without the convolutional filters.
\subsection{Google Traffic}
Measurements from the Google Traffic model were collected using the Google Maps Distance Matrix API~\citep{GMaps_DistanceMatrixAPI}. Google uses crowd-sourced road congestion data collected from smart-phones with the \emph{Google Maps App} installed \citep{GMaps_Crowdsourcing}. While the exact model powering the service is not publicly described in detail, the documentation states that ``the returned \emph{duration in traffic} should be the best estimate of travel time given what is known about both historical traffic conditions and live traffic''. Furthermore, it states that ``live traffic becomes more important the closer the departure time is to now'' \citep{GMaps_DistanceMatrixAPI}.
Because there is a limit to the number of requests that one can freely make to the API over a 24-hour period, it has only been possible to collect link travel times for the $t + 1$ time-step (i.e.,\ next 15 minutes). Travel times for each link were collected at a 15-minute interval between 06:00 and 22:00.
Another important aspect is that the Google Traffic model is primarily designed for estimating car travel times, and therefore it can be biased and not ideal for estimating the bus travel times used in our experiments. Since we only consider link travel times and collect data for each link individually, the bus dwell time will not be an issue, as it is not included in either measurement.
\begin{figure*}
\centering
\includegraphics[width=\textwidth]{comparison_day.pdf}
\caption{Accumulated link travel time over a single day (a Thursday) in the test set.}
\label{fig:comparison_day}
\end{figure*}
\begin{figure*}
\centering
\includegraphics[width=\textwidth]{comparison_day_2.pdf}
\caption{Accumulated link travel time over a single day (a Friday) in the test set.}
\label{fig:comparison_day_2}
\end{figure*}
\subsection{Comparison}
We compare the performance of the proposed \emph{ConvLSTM} model for bus link travel time prediction against the baseline models mentioned above. The overall results from \Cref{tab:results} show that the \emph{ConvLSTM} model outperforms all the other methods. The \emph{current model} performs the worst, even compared to the \emph{historical average} model, on which it is based on. This is most likely due to the rule-based enforcement put on top of the historical average.
Although the difference in performance might seem small, it should be emphasized that the evaluation measurements are averaging their errors, and thus the increased accuracy can be much higher on individual journeys, especially if they experience very irregular travel times. To investigate this, we focus our analysis on periods when the transport system is most vulnerable, and even small changes in regularity can propagate, since recovery is not an option, i.e.\ during morning and afternoon peaks.
\begin{table}[!ht]
\center
\begin{tabular}{l|rrr}
Model & RMSE & MAE & MAPE \\
\hline
\hline
Historical Average & 6.40 & 5.57 & 10.62 \% \\ \hline
Current Model & 6.69 & 5.88 & 11.22 \% \\ \hline
Pure LSTM & 3.80 & 3.16 & 6.01 \% \\ \hline
Google Traffic & 5.25 & 4.62 & 9.17 \% \\ \hline
ConvLSTM & 2.64 & 2.09 & 4.04 \% \\ \hline
\end{tabular}
\caption{Results: Morning peak (7h--9h)}
\label{tab:morning_peak}
\end{table}
\begin{table}[!ht]
\center
\begin{tabular}{l|rrr}
Model & RMSE & MAE & MAPE \\
\hline
\hline
Historical Average & 5.90 & 4.65 & 8.28 \% \\ \hline
Current Model & 6.28 & 5.20 & 9.37 \% \\ \hline
Pure LSTM & 5.26 & 3.97 & 7.08 \% \\ \hline
Google Traffic & 4.16 & 3.34 & 6.21 \% \\ \hline
ConvLSTM & 3.79 & 3.02 & 5.61 \% \\ \hline
\end{tabular}
\caption{Results: Afternoon peak (14h--18h)}
\label{tab:afternoon_peak}
\end{table}
Tables~\ref{tab:morning_peak} and \ref{tab:afternoon_peak} show the evaluation results for \emph{morning peaks (weekdays, 7h--9h)} and \emph{afternoon peaks (weekdays, 14h--18h)}, respectively, for the time-step $t + 1$.
The peak hour evaluation shows that the \emph{ConvLSTM} model increases its performance over the baseline models when the transport network is put under stress. In the morning peak, the \emph{ConvLSTM} model does not degrade in performance compared to the overall daytime results, whereas all the baseline models experience a decrease in performance of up to several minutes according to both RMSE and MAE, and an increase in MAPE of roughly one third.
Similarly, the afternoon peak evaluation shows improvements with respect to the baseline models, even though the \emph{ConvLSTM} model also decreases its performance when compared to the overall results. However, in this case, the difference in performance with the baseline methods is not as significant as in the morning peak. We can also observe that the \emph{Google Traffic} model performs rather well in the afternoon peak, which reduces the gap in error to less than a minute to the proposed ConvLSTM-based approach.
To obtain a more detailed view of how the different models perform at the micro-level (i.e.\ the specific journey), we can inspect a single day of predictions. A random weekday from the test dataset is plotted in~\Cref{fig:comparison_day} which shows the accumulated travel time of all 32 links and the predicted travel time at time-step $t + 1$, both for the proposed model and the baseline model.
On this particular day (a Thursday), the peak hour traffic was worse than normal, which leads both the \emph{historical average} model and the \emph{current model} to underestimate travel time in the peak periods. Please recall that the \emph{current model} is based on the \emph{historical average} model. Therefore, it is not surprising that they perform similarly. There is also a small peak in travel time in the afternoon, which none of the historical average models is able to predict.
On the other hand, both the \emph{Google Traffic} model and the proposed \emph{ConvLSTM} model get much closer to the ground truth in the peak hours. The \emph{Google Traffic} model seems to predict more accurately than the \emph{ConvLSTM} model in the afternoon peak, whereas the opposite occurs in the morning peak. However, both models are able to detect the irregular peak in the afternoon and adjust to it, at least to some degree.
\Cref{fig:comparison_day_2}~shows another example day - a Friday. Here the difference between the proposed model and the historical average and current model baselines is slightly less significant, simply because the day to a larger degree follows the average pattern for a ``normal'' Friday (especially around the afternoon peak). Nonetheless, the proposed model still performs the best, and this also supports our claim that the proposed model is strongest when the traffic pattern deviates from the normal pattern, i.e.\ when the transport network is under stress.
Finally, we compare the computational complexity of training the different models. Obviously, we cannot include metrics for the \textit{Google Traffic}, as the model is not public. Likewise, it is not sensible to compare with the \textit{Current Model}, since it is ``trained'' on a dataset of different size and on hardware using in production at the transport authority. But, since we know it is essentially an \textit{historical average} approach, we can expect a similar computational complexity. The historical average can be calculated within seconds for the full 23-week training dataset. The training of the \textit{Pure LSTM} and \textit{ConvLSTM} model can be achieved in both cases, for the full 23-week training dataset and the full 32-links, in less than 20 minutes on commodity hardware (8 cores, 64 GB RAM, GTX 1070 GPU). This might indicate why the \textit{historical average} models are still popular in the industrial systems, but we, however, argue that the more complex models are indeed scalable and the improved accuracy desirable, even though it is more computationally expensive.
\section{Conclusion}
\label{sec:conclusion}
This paper proposed a multi-output, multi-time-step system for bus travel time prediction. The proposed system uses a deep neural network model consisting of convolutional and long short-term memory (LSTM) layers, that is able to capture the non-static spatio-temporal correlations of variability in urban bus travel times. This allows the model to generalize patterns learned in predictions across space and time. Also, our approach for multi-time-step prediction using an encoder/decoder architecture is, to the best of our knowledge, new in the context of bus travel time prediction. The proposed approach allows accurate predictions further into the future compared to traditional approaches where subsequent time-steps are predicted independently. Our empirical results demonstrate that the proposed model outperforms other popular and recent methods from the state-of-the-art. This includes Google's Traffic model based on crowd-sourced live traffic data, and the current model deployed by Movia, the public transport authority in the Greater Copenhagen Area. The increased accuracy when compared to the baseline approaches is even more significant in the peak hours, where the urban bus transport network is under stress. The data required for the proposed system is simply the standard output that most AVL systems used in the public transport industry produce. We are aware that public transport agencies in Singapore, London, New York, Stockholm, Oslo, and Helsinki all have deployed AVL systems that fulfill this requirement, and thus the proposed system indeed generalizes trivially across different cities in different countries.
Although the proposed method is more computationally expensive than simple \textit{historical average} models, given the state of modern computational hardware, it is indeed scalable to be applied to an urban bus network for independent routes. Even with commodity hardware, we are able to retrain the route used in this experiment in less than 20 minutes, and we can thus easily retrain the model on a daily basis. Given the results of our proposed model, we are currently actively pursuing deployment of the model in the Greater Copenhagen region, in close collaboration with the transport authority - Movia. We do however consider this route-independent approach a limitation of the current system, and below we provide some research opportunities to extend the proposed system by handling correlations between different routes.
\subsection{Future work}
As future work, we would like to extend the presented systems in the following directions:
\begin{enumerate}[i)]
\item The integration of our proposed system to different control strategies for enhancing the regularity and reliability of the bus service, e.g.\ as suggested by \citet{Lo2012}. This would create a possible feedback loop from the predicted travel times that could possibly affct the travel times on a short-term basis. This is a non-trivial task, since it requires either simulation, which is complex for urban public transport networks in the detail needed here, or integration directly into currently running services, which is organizationally and technically challenging.
\item In order for the prediction accuracy to be increased further, it would be interesting to include more contextual features in the input data and not only the observed travel times. This could include features from the road network that the link consists of, e.g. whether intersections on the link are regulated by a traffic signal or not. In order to achieve this, map matching of at least the link geometry to the road network is necessary. However, this should be easily overcome, and many interesting crowd-sourced data are freely available (e.g.\ Open Street Map). Additional data sources such as weather conditions have shown to impact bus travel time \citep{Chen2004} and could also be included. More rare, but highly impacting deviations, such as traffic incidents, service-outage, holidays and large events, could also be considered in this research direction.
\item Currently, the model only uses convolutions over a single bus route, i.e.\ 1D-convolutions. We believe that it would be interesting to see the effect on the accuracy of the system if this was generalized to a network of bus routes. This would require extending the convolutions into a multi-dimensional space. Popular approaches used traditionally in conjunction with convolutions, such as overlaying the geographical map with a 2D-grid, have not shown good results. The challenge seems to be that bus networks are relatively sparse, and that travel times do not aggregate well in cells, e.g. compared to travel demand. Recent state-of-the-art proposes \textit{graph convolutional neural networks} \citep{Li2017}, i.e.\ where the convolutions are done over graph structures. We plan to pursue this approach - with the complications and development needed - to adapt the method to bus networks and the bus travel time prediction problem.
\item A final direction we have identified is to include the proposed system in an ensemble/multi-model approach. In this case, the proposed model can be included and used as a sub-model for the ensemble. The challenge here is the coordination between the different sub-models that can be seen as autonomous agents in an expert and intelligent system context. Especially, how to solve disagreements. Different approaches have been proposed by \citet{Weng2018}, and we expect to explore these approaches in our research.
\end{enumerate}
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 4,794
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For media enquiries, please contact Lucienne Bamford, Media and Communications Manager.
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To mark the one-year anniversary of the passing of Professor David Cooper AC, much-beloved inaugural...
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{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
| 2,973
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{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
| 7,976
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Q: Spring Quartz Dynamically adding Jobs I am trying to add a job to the scheduler dynamically by
schedulerFactoryBean.getScheduler().scheduleJob(cronTrigger);
But this is giving an error
Caused by: org.quartz.JobPersistenceException: The job (JOB_GROUP.JOB_NAME) referenced by the trigger does not exist.
at org.quartz.simpl.RAMJobStore.storeTrigger(RAMJobStore.java:422)
at org.quartz.core.QuartzScheduler.scheduleJob(QuartzScheduler.java:932)
at org.quartz.impl.StdScheduler.scheduleJob(StdScheduler.java:258)
at com.abc.RescheduleJob.rescheduleJobs(RescheduleJob.java:32)
at sun.reflect.NativeMethodAccessorImpl.invoke0(Native Method)
at sun.reflect.NativeMethodAccessorImpl.invoke(NativeMethodAccessorImpl.java:62)
at sun.reflect.DelegatingMethodAccessorImpl.invoke(DelegatingMethodAccessorImpl.java:43)
at java.lang.reflect.Method.invoke(Method.java:498)
at org.springframework.util.MethodInvoker.invoke(MethodInvoker.java:269)
at org.springframework.scheduling.quartz.MethodInvokingJobDetailFactoryBean$MethodInvokingJob.executeInternal(MethodInvokingJobDetailFactoryBean.java:257)
Please help.
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
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{
"redpajama_set_name": "RedPajamaC4"
}
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{
"redpajama_set_name": "RedPajamaC4"
}
| 2,470
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//-------------------------------------
//-- Form - Base
//-------------------------------------
//= **require vendor/node_modules/inputmask/dist/inputmask/inputmask
//= **require vendor/node_modules/inputmask/dist/inputmask/inputmask.extensions
//= **require vendor/node_modules/inputmask/dist/inputmask/inputmask.date.extensions
//= **require vendor/node_modules/inputmask/dist/inputmask/inputmask.numeric.extensions
//= **require vendor/node_modules/inputmask.phone/dist/inputmask.phone/inputmask.phone.extensions
//= **require vendor/node_modules/inputmask/dist/inputmask/jquery.inputmask
(() => {
'use strict';
const local = {};
//-- Input mask
const bindInputMask = (/* $context = __.$body */) => {
/**
// Numeric
$context.find('input[data-mask="numeric-integer"]').inputmask('integer', {
allowPlus: false,
allowMinus: false,
min: 1,
integerDigits: 3
});
$context.find('input[data-mask="numeric-integer-nomin"]').inputmask('integer', {
allowPlus: false,
allowMinus: false,
integerDigits: 3
});
$context.find('input[data-mask="numeric-decimal"]').inputmask('decimal', {
allowPlus: false,
allowMinus: false,
min: 1,
integerDigits: 5,
digits: 2
});
// Phone
$context.find('input[type="tel"]').inputmask('(999) 999-9999');
$context.find('input[type="tel"][data-mask="ext"]').inputmask('(999) 999-9999 [ext: 99999]');
// Postal code
$context.find('input[data-mask="postalcode"]').inputmask('A9A 9A9');
// Date
if (!Modernizr.inputtypes.date) {
$context.find('input[type="date"]').inputmask('yyyy-mm-dd', { placeholder: app.env.lang === 'fr' ? 'aaaa-mm-jj' : 'yyyy-mm-dd' });
}
// Time
$('input[data-mask="time"]').inputmask('hh:mm:ss');
// Credit card
$('input[data-mask="credit-card"]').inputmask('9{10}');
$('input[data-mask="credit-card-cvv"]').inputmask('9{4}');
/**/
};
//-- Numeric keyboard
const bindNumericKeyboard = (/* $context = __.$body */) => {
/**
$context.find(`
input[data-mask="numeric-integer"],
input[data-mask="numeric-integer-nomin"],
input[data-mask="credit-card"],
input[data-mask="credit-card-cvv"]
`)
.attr('pattern', '\\d*')
;
/**/
};
//-- Form events
const rebindFormEvent = ($context) => {
bindInputMask($context);
bindNumericKeyboard($context);
pinki.message.publish(`${PROJECT}.form.rebindFormEvent`);
};
//-- Cache data instantly
local.cache = () => {
//
};
//-- Cache data once DOM is loaded
local.cacheDOM = () => {
//
};
//-- Bind events once DOM is loaded
local.bind = () => {
rebindFormEvent();
/**
// Bind on text field change
$('input:text').on('input paste cut keyup', () => {});
/**/
};
//-- Subscribe to topics
local.subscribe = () => {
/**
pinki.message.subscribe('sample', (message, data) => {
rebindFormEvent(data.$context);
});
/**/
};
//-- Execute once DOM is loaded
local.start = () => {
//
};
//-- Execute once page is loaded
local.delayedStart = () => {
//
};
// Outline
local.cache();
local.subscribe();
// DOM Ready
pinki.vow.when(DOM_PARSED).then(() => {
local.cacheDOM();
local.bind();
local.start();
});
// Document loaded
pinki.vow.when(DOCUMENT_LOADED).then(() => {
local.delayedStart();
});
})();
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 764
|
\section{Introduction}
We are concerned with the stochastic $p$-Laplace equation on the torus $\T^d= \R^d/ \Z^d\, (d\geq 2)$, perturbed by multiplicative noise of transport type:
\begin{equation}\label{stoch-p-laplace}
d u = \Delta_p u \,d t + \nabla u \circ d W, \quad u(0,\cdot) = u_0,
\end{equation}
where $\Delta_p u= \div(|\nabla u|^{p-2} \nabla u)$ with $p>2$, $\circ\, dW$ means the stochastic differential is understood in the Stratonovich sense and $W= W(t,x)$ is a space-time noise, white in time and coloured in space, modelling some background random perturbation. We shall also assume that $W(t,x)$ is divergence free in the space variable; see Section \ref{sec: noise setting} for its precise form. At least formally, one has the energy balance: $\P$-a.s. for all $t\geq 0$,
\begin{equation}\label{energy identity}
\|u(t)\|_{L^2}^2 + 2 \int_0^t \|\nabla u(s)\|_{L^p}^p\,d s = \|u_0 \|_{L^2}^2;
\end{equation}
similarly as in the non-perturbed case, this immediately implies decay of solutions:
\begin{equation}\label{deterministic-dissip}
\|u(t) \|_{L^2} \leq \frac{\|u_0 \|_{L^2} }{\big(1+ (p-2)\lambda_1^{p/2} t \|u_0 \|_{L^2}^{p-2} \big)^{1/(p-2)}},
\end{equation}
where $\lambda_1$ is the spectral gap of $\T^d$.
Inspired by the recent work \cite{FHX21} in the deterministic setting, we will show that suitably chosen random noise can greatly enhance the rate of decay, both in averaged sense and in the pathwise sense. Before stating the precise results, let us briefly recall some literature related to the phenomena of dissipation enhancement.
It has been known for a long time, in the physics and engineering communities, that certain perturbations speed up the mixing of fluids. In the mathematical literature, early influential studies of such phenomena date back to the works \cite{BHN05, CKRZ08}. In particular, for a bounded divergence free vector field $v$ on a smooth bounded domain $D\subset \R^N$, the authors of \cite{BHN05} studied the eigenvalue problem for the operators $-\Delta + A v \cdot \nabla\ (A>0)$ with Dirichlet boundary condition; it was shown that the principal eigenvalue $\lambda_A$ remains bounded as $A\to +\infty$ if and only if $v$ admits a first integral $w\in H^1_0(D)$, namely, $v\cdot \nabla w=0$. Later on, Constantin et al. \cite{CKRZ08} extended such ideas to the setting of compact manifolds $M$, and proved that an incompressible flow $v$ on $M$ fulfils the relaxation-enhancing property if and only if $v\cdot \nabla$ has no nontrivial eigenfunction in $H^1(M)$. Recall that an incompressible flow $v$ on $M$ is called relaxation enhancing if the solution $\phi^A$ to the advection-diffusion equation
$$\partial_t \phi - \Delta \phi + A v\cdot \nabla \phi =0 ,\quad \phi(0,\cdot) =\phi_0 \in L^2(M) $$
satisfies, for any $t>0$,
$$\lim_{A\to \infty} \|\phi^A(t,\cdot)- \bar\phi_0 \|_{L^2(M)}= 0, $$
where $\bar\phi_0 = \frac1{|M|}\int_M \phi_0 \,\d x$ is the average of $\phi_0$, a quantity preserved by the above equation.
Motivated by the pioneering works \cite{BHN05, CKRZ08}, the phenomena of dissipation enhancement have been studied intensively in the past years, see for instance \cite{Zla10, BCZ17, FI19, CZDE20, BBPS21, GY21, IXZ21} among many others. In particular, Feng and Iyer \cite{FI19} gave some explicit estimates on the dissipation time of $v$ in terms of the so-called mixing rate; the former means the minimal time needed for the solutions of advection-diffusion equations to dissipate a fraction (e.g. $1/2$) of their $L^2$-norms. Later on, Iyer et al. \cite{IXZ21} have used the dissipation time to formulate conditions on incompressible flows so that the blow-up in certain nonlinear systems is suppressed by the perturbation of such flows. In the remarkable work \cite{BBPS21}, Bedrossian et al. have shown that the incompressible flows $v$ can be chosen as the sample paths of certain stationary solutions of stochastic 2D Navier-Stokes equations, showing the generality of flows with dissipation-enhancing properties. Their arguments are quite technical, relying on the quantitative Harris theorem from ergodic theory and delicate studies of the underlying projective process. More recently, Gess and Yaroslavtsev \cite{GY21} obtained some related results for the famous Kraichnan model from turbulence theory.
On the other hand, there is a series of studies on stochastic fluid dynamical equations with transport noise; the latter might be regarded as random analogues of the vector field $v$ appearing above. The vorticity form of stochastic 2D Euler equation driven by transport noise has been studied for various types of initial data, see e.g. \cite{BFM16, BM19, FL19, FL20, LC22}. In particular, it was shown in \cite{FL20} that white noise solutions of a sequence of stochastic 2D Euler equations converge weakly to the unique stationary solution of the 2D Navier-Stokes equation driven by space-time white noise. A remarkable fact is that the approximating equations are formally conservative, while the limit equation is dissipative. Partly inspired by this work, Galeati \cite{Galeati20} established a scaling limit result for $L^2$-solutions of stochastic linear transport equations on $\T^d$ with transport noise:
$$d u + b\cdot \nabla u\,d t + \sqrt{\nu}\, \nabla u\circ d W=0, $$
where $b$ is a time-dependent vector field with appropriate regularity and $\nu$ is the intensity of noise. Rescaling the noise in a suitable way, he showed that the solutions converge to the unique solution of the deterministic parabolic equation
$$\partial_t u + b\cdot \nabla u = \nu \Delta u.$$
Loosely speaking, small scale transport noise produces in the limit an extra dissipative term, which can be called eddy dissipation. Such scaling limit result was later on extended in \cite{FGL21a, Luo21} to some stochastic fluid equations with transport noise, see \cite{FGL21c} for quantitative convergence rates. Note that the bigger the noise intensity, the stronger the dissipation term in the limit equation; we have made use of this fact to show that transport noise suppresses possible explosion of solutions to some deterministic equations, cf. \cite{FL21, FGL21b}.
Moreover, using mild formulations of both approximating stochastic equations and the limit equation, we have shown in \cite{FGL21c} dissipation enhancement for stochastic heat equations with transport noise, i.e. \eqref{stoch-p-laplace} with $p=2$. In this paper, we aim to extend such result to the case $p>2$ corresponding to nonlinear diffusions. The same problem has already been studied in the deterministic setting by Feng, Hu and Xu in \cite{FHX21}, to which we also refer for discussions on the motivation of studying $p$-Laplace evolution systems. Following some ideas in \cite{CZDE20, FI19}, they introduced a nonlinear version of dissipation time $\kappa_d$ (see \cite[Definition 2.3]{FHX21}), and estimated the latter via the mixing rate of the time-dependent incompressible flow $v$, see \cite[Theorems 2.5 and 2.7]{FHX21} for general estimates and Corollaries 2.6 and 2.8 therein for more explicit results when $v$ is assumed to be strongly/weakly mixing.
Our first result gives enhanced dissipation in the average sense.
\begin{theorem}\label{thm-average}
Let $p\in (2, 2d/(d-2))$ and $R>0$ be given. For any $\mu>0$, there exists a space-time noise $W(t,x)$ such that for any $\| u_0\|_{L^2}\leq R $, the solution $\{u(t) \}_{t>0}$ to \eqref{stoch-p-laplace} with initial condition $u_0$ satisfies
\begin{equation}\label{thm-average.eq}
\|u(t) \|_{L^2(\Omega,L^2)} \leq \frac{\|u_0 \|_{L^2} }{\big(1+ (p-2)\mu t \|u_0 \|_{L^2}^{p-2} \big)^{1/(p-2)}}, \quad \mbox{for any } t\geq 1.
\end{equation}
\end{theorem}
Taking $\mu$ big enough, we see that the above estimate improves \eqref{deterministic-dissip}, though in a weaker averaged sense. We can also prove a pathwise assertion on dissipation enhancement.
\begin{theorem}\label{thm-pathwise}
Let $p\in (2, 2d/(d-2))$ and $R>0$ be given. For any $\mu>0$ and $0<q< \frac{\ln 2}{\mu(p-2)^2}$, there exists a noise $W(t,x)$ such that for any $\| u_0\|_{L^2}\leq R,\, u_0\neq 0 $, there is a random constant $C(\omega)>0$ with finite $q$-th moment such that the solution $\{u(t) \}_{t>0}$ to \eqref{stoch-p-laplace} satisfies, $\P$-a.s. for all $t>0$,
\begin{equation}\label{thm-pathwise.eq}
\exp\bigg(- \frac1{\|u(t) \|_{L^2}^{p-2}} \bigg) \leq C(\omega) e^{-(p-2)\mu t} \exp\bigg(- \frac1{\|u_0 \|_{L^2}^{p-2}} \bigg);
\end{equation}
if $t>\frac{\ln C(\omega)}{(p-2)\mu}$, the above inequality can be rewritten as
\[\|u(t) \|_{L^2} \leq \frac{\|u_0 \|_{L^2}}{\big(1+\left((p-2)\mu t-\ln C(\omega) \right) \|u_0 \|_{L^2}^{p-2}\, \big)^{1/(p-2)}}. \]
\end{theorem}
To prove Theorem \ref{thm-pathwise}, we will use the Borel-Cantelli lemma as in the proof of \cite[Theorem 1.9]{FGL21c} (see also \cite[Section 7]{BBPS22}), and apply the dissipation estimates obtained in the proof of Theorem \ref{thm-average} on the intervals $[n-1,n],\, n\geq 1$. Here, the main new idea is that we choose noises with some fixed intensity $\kappa$ but different coefficients $\{\theta_{k,n} \}_{k\in \Z^d_0}$ on these intervals; as $n$ increases, the noises produce stronger and stronger dissipation enhancement in the averaged sense. By Chebyshev's inequality, this implies the probabilities of a sequence of events are summable, and then Borel-Cantelli's lemma gives us dissipation enhancement in the pathwise sense, see Section \ref{sec:dissipation.2} for details.
We also mention that, for fixed $p>2$, the range of $q$ provided by Theorem \ref{thm-pathwise} becomes very small for big $\mu>0$. In fact, we can enlarge the range by replacing the exponential $2^n$ in the proof with $a^n$ for some big $a>2$; then we can get an upper bound like $\frac{\ln a}{\mu(p-2)^2}$. We omit the details in this work.
\iffalse
In contrast to the noise selection in Theorem \ref{thm-average}, the coefficient of noise we choose in Theorem \ref{thm-pathwise} is a step function of time rather than a constant function. Under the noise selection of Theorem \ref{thm-pathwise}, we can obtain a faster dissipation estimate in the average sense than Theorem \ref{thm-average}, as seen in Theorem \ref{dissipation.2}. Further, we can get dissipation enhancement in the pathwise sense, as seen in Theorem \ref{thm-pathwise} and Remark \ref{remark}.
\fi
We finish the short introduction with the organization of the paper. In Section \ref{sec: Preliminaries}, we introduce some notations for functional setting and recall some useful results. We present in Section \ref{sec: noise setting} the exact choice of the space-time noise $W(t,x)$, for which the stochastic $p$-Laplace equation \eqref{stoch-p-laplace} admits variational solutions; we also show that the solution can be rewritten in mild formulation. Using the semigroup approach as in \cite{FGL21c}, Theorems \ref{thm-average} and \ref{thm-pathwise} will be proved in Sections \ref{sec:dissipation.1} and \ref{sec:dissipation.2}, respectively. Finally, we demonstrate in the appendix the well posedness of \eqref{stoch-p-laplace} using the variational framework.
\section{Preliminaries}\label{sec: Preliminaries}
In this section, we introduce some notations and provide several important tools that will be used in the paper.
Let $\T^d=\R^d/\Z^d$ be the $d$-dimensional torus, $d \geq 2$; $\Z_0^d = \Z^d \setminus \{0\}$ is the set of nonzero integer points. The brackets $\langle \cdot, \cdot \rangle$ stand for the inner product in $L^2(\T^d)$ and the duality between elements in $W^{1, p}(\T^d)$ and $W^{-1, \frac{p}{p-1}}(\T^d)$.
The norms in spaces $L^2(\T^d)$, $L^p(\T^d)$, and $W^{1,p}(\T^d)$ are denoted as $\|\cdot\|_{L^2}$, $\|\cdot\|_{L^p}$, and $\|\cdot \|_{W^{1, p}}$, respectively. Let $\{e_k \}_k$ be the usual complex basis of $L^2(\T^d, \mathbb C)$.
As the $p$-Laplace equations considered below preserve the means of solutions, we assume in this paper that the function spaces consist of functions on $\T^d$ with zero average.
Let $\Delta$ be the Laplace operator on $\T^d$ and $\lambda_k =4 \pi^2 |k|^2, \, k \in \Z^d,$ are the eigenvalues of $\Delta$. We denote by $\{ e^{t \Delta} \}_{t \geq 0}$ the usual heat semigroup on $\T^d$. For $p >2,$ we define the $p$-Laplace operator as below:
\begin{equation}\label{def,p-laplace}
\Delta_p u \assign \div (|\nabla u|^{p - 2} \nabla u).
\end{equation}
We recall some properties of $\Delta_p$.
\begin{proposition}\label{pro.p-laplace}
The p-Laplace operator $\Delta_p $ satisfies the estimates below:
\begin{equation*}
\begin{split}
\langle \Delta_p u - \Delta_p v, u - v \rangle \leq 0 , &\\
\langle \Delta_p v, v \rangle \leq - c \| v \|_{W^{1,p}}^p, &
\end{split}
\end{equation*}
for all $u,v \in W^{1,p}(\T^d),$ where $c$ is a positive number that only depends on $p$. In particular, the second assertion implies
$$\|\Delta_p v \|_{W^{-1, \frac{p}{p-1}}} \leq c \| v \|_{W^{1,p}}^{p-1}. $$
\end{proposition}
\begin{proof}
The proof can be found in \cite[Example 4.1.9]{liu15}.
\end{proof}
The next lemma (see \cite[Lemma 4.2]{FHX21}) is an easy fact, but is necessary to deal with Case 1 of Section \ref{sec:dissipation.1}.
\begin{lemma}\label{P1}
For any $p>2$, it holds
\[ 1 - \frac{2 x}{p - 2} \leq \frac{1}{(1 + x)^{\frac2{p-2}}} , \quad x > 0.\]
\end{lemma}
\begin{proof}
The linear function $1 - \frac{2 x}{p - 2}$ and the convex function $\frac{1}{(1 + x)^{\frac2{p-2}}}$
are tangent at $x=0$. The inequality holds by the property of convex functions.
\end{proof}
The following lemma is motivated by \cite[Lemma 4.5]{FHX21}; it is the key ingredient for running the iteration argument when proving Theorem \ref{thm-average} and Proposition \ref{dissipation.2}.
\begin{lemma}\label{P2}
For $p>2,$ suppose $a,b \geq 0$, and
\begin{equation*}
y \leq \frac{x}{\big(1+a x^{\frac{p-2}{2}} \big)^{\frac{2}{p-2}}}, \quad z \leq \frac{y}{\big(1+b y^{\frac{p-2}{2}}\big)^{\frac{2}{p-2}}} ,\quad x,y,z \geq 0,
\end{equation*}
then we have
\[z \leq \frac{x}{\big(1+(a+b) x^{\frac{p-2}{2}} \big)^{\frac{2}{p-2}}} .\]
\end{lemma}
\begin{proof}
Note that $\frac{y}{\big(1+b y^{\frac{p-2}{2}} \big)^{\frac{2}{p-2}}}$ is monotonically increasing in the variable $y$, it holds
\begin{equation*}
\begin{split}
z &\leq \frac{y}{\big(1+b y^{\frac{p-2}{2}} \big)^{\frac{2}{p-2}}} \leq \frac{x}{\big(1+a x^{\frac{p-2}{2}} \big)^{\frac{2}{p-2}}} \cdot \frac{1}{\left(1+b \frac{x^{\frac{p-2}{2}}}{1+a x^{\frac{p-2}{2}}}\right)^{\frac{2}{p-2}}} \\
& \leq \frac{x}{\big(1+(a+b) x^{\frac{p-2}{2}} \big)^{\frac{2}{p-2}}} .
\end{split}
\end{equation*}
\end{proof}
We state the classical heat semigroup estimates (see e.g. \cite{Lun95}) for later use.
\begin{lemma}
\label{P3} For any $ 2 \leq p \leq + \infty, $ and any $T>0$, there exists a
constant $C>0,$ such that
\[ \|e^{ t \Delta} \phi \|_{W^{1, p}} \leq C t^{-\big( \frac{1}{2} + \frac{d}{2} ( \frac{1}{2} - \frac{1}{p}) \big)} \| \phi \|_{L^2} \]
for any $\phi \in L^{2}(\mathbb{T}^d), $ and any $t \in (0, T].$
\end{lemma}
\section{Choice of noise $W$ and mild formulation of \eqref{stoch-p-laplace}}\label{sec: noise setting}
In Section \ref{subsec:noise setting}, we describe the choice of the noise $W(t, x)$ and show the existence and uniqueness of variational solutions of SPDE \eqref{stoch-p-laplace}, by using a general result proved in the appendix.
Section \ref{subsec:mild form} explains the link between variational solution and mild form of SPDE \eqref{stoch-p-laplace}.
\subsection{Choice of noise}\label{subsec:noise setting}
As in \cite[Section 1.3]{FGL21c}, we perturb the equations \eqref{stoch-p-laplace} with the space-time noise below:
\begin{equation}\label{noise}
W(t, x)= \sqrt{C_{d} \kappa}\, \sum_{k\in \Z^d_0}\sum_{i=1}^{d-1} \theta_{k} \sigma_{k, i}(x) W^{k, i}_{t} ,
\end{equation}
where $C_{d}=d/(d-1)$ is a normalizing constant, $\kappa>0$ is the noise intensity and $\theta\in\ell^{2} =\ell^{2}(\Z^d_0)$,
the space of square summable sequences indexed by $\Z_0^d$. $\{W^{k, i}:k\in\mathbb{Z}^{d}_{0}, i=1, \ldots, d-1\}$
are standard complex Brownian motions defined on a filtered probability space $(\Omega, \mathcal F, (\mathcal F_t), \P)$, satisfying
\begin{equation}\label{noise.1}
\overline{W^{k, i}} = W^{-k, i}, \quad\big[W^{k, i}, W^{l, j} \big]_{t}= 2t \delta_{k, -l} \delta_{i, j} .
\end{equation}
$\{\sigma_{k, i}: k\in\mathbb{Z}^{d}_{0}, i=1, \ldots, d-1\}$ are divergence free vector fields on $\T^d$ defined as
\begin{equation}\label{sigma}
\sigma_{k, i}(x) = a_{k, i} e_{k}(x) ,
\end{equation}
where $\{a_{k, i}\}_{k, i}$ is a subset of the unit sphere $\mathbb{S}^{d-1}$ such that: (i) $a_{k, i}=a_{-k, i}$ for all
$k\in \mathbb{Z}^{d}_{0}, \, i=1, \ldots, d-1$; (ii) for fixed $k$, $\{a_{k, i}\}_{i=1}^{d-1}$ is an ONB of
$k^{\perp}=\{y\in\mathbb{R}^{d}:y\cdot k=0 \}$. It holds that $\sigma_{k, i}\cdot \nabla e_k = \sigma_{k, i}\cdot \nabla e_{-k} \equiv 0$
for all $k\in \Z^d_0$ and $1\leq i\leq d-1$.
We shall always assume that
\begin{itemize}
\item $\theta \in \ell^2$ is symmetric, i.e. $\theta_k = \theta_l$ for all $k, l\in \Z^d_0$ satisfying $|k|=|l|$;
\item $\|\theta \|_{\ell^2} = \big(\sum_{k \in \Z_0^d} \theta_k^2 \big)^{1/2} =1$.
\end{itemize}
In fact, under the assumptions above, the noise is $W$ real. We rewrite \eqref{noise} as
\begin{equation}\label{noise.2}
\begin{split}
W(t, x) &= \sqrt{C_{d} \kappa}\, \sum_{k\in \Z^d_0}\sum_{i=1}^{d-1}
\theta_{k} \left\{ {\rm Re}(\sigma_{k, i}(x)) {\rm Re}(W^{k, i}_{t})-{\rm Im}(\sigma_{k, i}(x)) {\rm Im}(W^{k, i}_{t}) \right\} \\
& = \sum_{k\in \Z^d_0}\sum_{i=1}^{d-1} \xi_{k, i}(x) B_t^{k, i} .
\end{split}
\end{equation}
$\{\xi_{k, i}: k\in\mathbb{Z}^{d}_{0}, i=1, \ldots, d-1\}$ are also divergence free vector fields on $\T^d$ defined as
\begin{equation}\label{xi}
\xi_{k, i}(x) =\left\{\begin{array}{ll}
2 \sqrt{C_{d} \kappa}\, \theta_{k} {\rm Re}(\sigma_{k, i}(x)), & \quad k \in \Z_{+}^d, \\
2 \sqrt{C_{d} \kappa}\, \theta_{k} {\rm Im}(\sigma_{k, i}(x)), & \quad k \in \Z_{-}^d,
\end{array} \right.
\end{equation}
where $\Z^d_0 = \Z^d_+ \cup \Z^d_-$ is a partition of $\Z^d_0$ satisfying $\Z^d_+ = -\Z^d_-$. $\{B^{k, i}: k\in\mathbb{Z}^{d}_{0}, i=1, \ldots, d-1\}$ are independent standard Brownian motions defined as
\begin{equation}\label{B_t}
B_t^{k, i}=\left\{\begin{array}{ll}
{\rm Re}(W^{k, i}_{t}) ,& \quad k \in \Z_{+}^d, \\
-{\rm Im}(W^{k, i}_{t}) ,& \quad k \in \Z_{-}^d.
\end{array} \right.
\end{equation}
The SPDE \eqref{stoch-p-laplace} can be rewritten as the form \eqref{stoch-p-laplace.2} in the appendix.
By the above assumptions of noise, we have $\| \sigma_{k,i} \|_{L^\infty} \leq 1 , $ thus
\begin{equation*}
\sum_{k \in \Z^d} \sum_{i=1}^{d-1} \| \xi_{k, i} \|_{L^{\frac{2p}{p-2}}(\T^d)}^{2} \leq 4C_d \kappa (d-1)\sum_{k \in \Z^d} \theta_k^2=4\kappa d <+\infty .
\end{equation*}
By Theorem \ref{well-posed}, we conclude that variational solutions of SPDE \eqref{stoch-p-laplace} exist and are unique; moreover, the energy identity \eqref{energy identity} holds.
\subsection{Mild formulation}\label{subsec:mild form}
Under the noise setting in Section \ref{subsec:noise setting}, we will show that the solution of SPDE \eqref{stoch-p-laplace} has a mild formulation, see \eqref{mild form.eq} below.
Transforming \eqref{stoch-p-laplace} into the It\^o form and similarly to the discussion in \cite[Section 2]{Galeati20}, we know that the Stratonovich-It\^o corrector is $S(u)=\kappa \Delta u$, hence we obtain
\begin{equation}\label{p-Laplace.3}
d u = \Delta_p u \, d t + \kappa \Delta u \, d t +
\sum_{k \in \Z^d} \sum_{i=1}^{d-1} \xi_{k, i} (x) \cdot \nabla u \, d B_t^{k, i} ,
\end{equation}
where $\{\xi_{k, i}\}$ and $\{B_t^{k, i}\}$ are defined in \eqref{xi} and \eqref{B_t} respectively.
\begin{theorem}\label{mild form}
Let $u$ be the variational solution of the SPDE \eqref{p-Laplace.3} in the sense of Definition \ref{variational solution}. Then $\P $-a.s. for any $t \in [0, T], $ it holds
\begin{equation}\label{mild form.eq}
u (t) = e^{\kappa t \Delta} u_0 + \int_0^t e^{\kappa (t - s) \Delta} \Delta_p u(s) \, \, d s
+\sum_{k \in \Z^d} \sum_{i = 1}^{d - 1} \int_0^t e^{\kappa (t - s) \Delta} \xi_{k, i} \cdot \nabla u(s) \, d B_s^{k, i} .
\end{equation}
\end{theorem}
\begin{proof}
For fixed $t \in [0, T], $ the variational solution $u$ of \eqref{stoch-p-laplace} satisfies
\[ u (t) = u_0 + \int_0^t \Delta_p u \, \, d s +
\int_0^t \kappa \Delta u \, \, d s + \sum_{k \in \Z^d} \sum_{i=1}^{d-1}
\int_0^t \xi_{k, i} (x) \cdot \nabla u (s) \, \, d B_s^{k, i} \]
in $W^{- 1, \frac{p}{p - 1}} (\T^d)$. By the property of Bochner integral, for any $\varphi \in C^{\infty} (\T^d), $ we have
\begin{equation*}
\begin{split}
\langle u (t), \varphi \rangle =&\, \langle u_0, \varphi \rangle + \int_0^t \langle \Delta_p u(s), \varphi \rangle \, \, d s
+ \kappa \int_0^t \langle \Delta u(s), \varphi \rangle \, \, d s \\
& + \sum_{k \in \Z^d} \sum_{i = 1}^{d - 1} \int_0^t \langle \xi_{k, i} \cdot \nabla u(s), \varphi \rangle \, d B_s^{k, i} .
\end{split}
\end{equation*}
For fixed $l \in \Z^d$, we take $\varphi=e_l$, then
\begin{equation*}
d \langle u (s), e_l \rangle = \langle \Delta_p u(s), e_l \rangle \, d s- \kappa \lambda_l \langle u(s), e_l \rangle \, \, d s
+ \sum_{k \in \mathbb{Z}^d} \sum_{i = 1}^{d - 1} \langle \xi_{k, i} \cdot \nabla u(s), e_l \rangle \, d B_s^{k, i} .
\end{equation*}
Applying the It\^o formula to the process $e^{\kappa t \lambda_l} \langle u(t) , \varphi \rangle ,$ we have
\begin{equation*}
\begin{split}
d \big( e^{\kappa \lambda_l s} \langle u (s), e_l \rangle \big) & = \kappa \lambda_l e^{\kappa \lambda_l s}
\langle u (s), e_l \rangle \, \, d s + e^{\kappa \lambda_l s} d \langle u (s), e_l \rangle \\
& = e^{\kappa \lambda_l s} \langle \Delta_p u(s), e_l \rangle \, \, d s + e^{\kappa \lambda_l s}
\sum_{k \in \Z^d} \sum_{i = 1}^{d - 1} \langle \xi_{k, i} \cdot \nabla u(s), e_l \rangle \, d B_s^{k, i} ,
\end{split}
\end{equation*}
integrating in time yields, $\P$-a.s. for all $t \in [0, T]$,
\begin{equation*}
\begin{split}
\langle u (t), e_l \rangle &= e^{- \kappa \lambda_l t} \langle u_0, e_l \rangle +
\int_0^t e^{- \kappa \lambda_l (t - s)} \langle \Delta_p u(s), e_l \rangle \, \, d s \\
&+\sum_{k \in Z^d} \sum_{i = 1}^{d - 1} \int_0^t e^{-\kappa \lambda_l (t - s)}
\langle \xi_{k, i} \cdot \nabla u(s), e_l \rangle \, d B_s^{k, i} .
\end{split}
\end{equation*}
We can then find $\Gamma\subset \Omega$ of full probability such that the above equality holds
for all $t\in [0, T]$ and all $l\in \Z^2$. But this is exactly \eqref{mild form.eq} written in Fourier modes.
\end{proof}
\section{Averaged dissipation enhancement}\label{sec:dissipation.1}
This section is devoted to showing dissipation enhancement in the sense of expectation by choosing appropriate intensity
$\kappa$ and coefficients $\{ \theta_{k} \}_{k \in \Z_0^d}$ of the noise. We fix $\beta>\frac{d}{2}+1,\, t_0>0$, let $t_n \assign n t_0, \, n \in \Z_+$,
and assume $p \in \big(2,\frac{2d}{d-2} \big),$ initial data $\|u_0\|_{L^2} \leq R.$
Then we define two constants $C_1(\kappa,t_0,R)$ and $C_2(\theta,\mu)$ as below:
\begin{equation}\label{C_1,C_2}
\begin{split}
C_1(\kappa,t_0,R) & \assign \Bigg\{ \frac{6}{(4 \pi^{2} \kappa t_{0})^2} +
6 \bigg(\frac{C_{0}}{\kappa^{\eta} t_{0}^{\eta-1/p}} \bigg)^{2} R^{2-4/p} \Bigg\}^{\frac{p-2}{2}} ,\\
C_2(\theta,\mu) & \assign \Big( 6 d \mu^{\frac{2}{p}\big(1-\frac{1}{\beta} \big)} \Lambda_{\beta} \|\theta\|_{\infty}^{\frac{2}{\beta}} \Big)^{\frac{p-2}{2}} ,
\end{split}
\end{equation}
where $\eta \assign \frac{1}{2}+\frac{d}{2}(\frac{1}{2}-\frac{1}{p})<1,$ $\| \theta \|_{\infty} \assign \sup_{k \in \Z_0^d} |\theta_k| , $ and $\Lambda_{\beta} \assign \big(\sum_{l \in \Z^d} \lambda_l^{1-\beta} \big)^{\frac{1}{\beta}} < + \infty.$
To prove Theorem \ref{thm-average}, we first prove
\begin{proposition}\label{dissipation.1}
For any $ p \in \big(2,\frac{2d}{d-2} \big),$ $\mu >0$, and initial data satisfying $\| u_0 \|_{L^2} \leq R $, we can find appropriate noise intensity $\kappa$ and coefficients $\{\theta_k\}_{k \in \Z_0^d}$, depending only on parameters $d, p, t_0, R, \mu$ and verifying
\begin{equation}\label{condition.1}
\max \{ C_1(\kappa,t_0,R),C_2(\theta,\mu) \} \leq \frac{1}{1+\mu (p-2) t_0 R^{p-2} } ,
\end{equation}
such that the solution $u(t)$ of SPDE \eqref{p-Laplace.3} satisfies
\begin{equation}\label{dissipation.1.eq}
\E \|u(t_{n+1}) \|_{L^2}^2 \leq \frac{\E \|u(t_n) \|_{L^2}^2}{\Big(1+\mu (p-2) t_0 \left(\E \|u(t_n) \|_{L^2}^2 \right)^{\frac{p-2}{2}} \!\Big)^\frac{2}{p-2} }, \quad \forall\, n \in \N.
\end{equation}
\end{proposition}
For any fixed $n \in \N$, we will prove inequality \eqref{dissipation.1.eq} in the following two different cases:
\begin{itemize}
\item Case 1:
\begin{equation} \label{case1.assumption}
\E \int_{t_{n}}^{t_{n+1}}\|\nabla u(s)\|_{L^p}^{p} \, d s \geq \mu t_{0} \big(\E\left\|u\left(t_{n}\right)\right\|_{L^2}^{2} \!\big)^{\frac{p}{2}} ;
\end{equation}
\item Case 2:
\begin{equation} \label{case2.assumption}
\E \int_{t_{n}}^{t_{n+1}}\|\nabla u(s)\|_{L^p}^{p} \, d s < \mu t_{0} \big(\E\left\|u\left(t_{n}\right)\right\|_{L^2}^{2}\! \big)^{\frac{p}{2}} .
\end{equation}
\end{itemize}
In Case 1, inequality \eqref{dissipation.1.eq} follows immediately from the lower bound \eqref{case1.assumption} and the energy identity, cf. \eqref{energy identity.n} below. Case 2 is much more complicated; we will use the assumption \eqref{case2.assumption} to estimate the stochastic convolutional term, which is essential for estimating $\E \| u (t_{n+1}) \|_{L^2}^2$.
We first deal with Case 1. The solution $u(t)$ satisfies energy identity \eqref{energy identity}; as a result,
\begin{equation}\label{energy identity.n}
\| u (t_{n+1}) \|_{L^2}^2 = \| u (t_n) \|_{L^2}^2 - 2\int_{t_n}^{t_{n+1}} \| \nabla u(s) \|_{L^p}^p \, d s .
\end{equation}
It implies that the estimate of $\| u (t_{n+1}) \|_{L^2}^2$ can be obtained by estimating $\int_{t_n}^{t_{n+1}} \| \nabla u(s) \|_{L^p}^p \, d s$,
and the assumption \eqref{case1.assumption} gives a lower bound of its expectation. In this case, we can prove the inequality \eqref{dissipation.1.eq} without the assumptions in Proposition \ref{dissipation.1}.
\begin{proposition}\label{case.1}
The inequality \eqref{dissipation.1.eq} holds in Case 1.
\end{proposition}
\begin{proof}
Taking expectation on both sides of \eqref{energy identity.n} and using assumption \eqref{case1.assumption}, we get
\begin{equation*}
\begin{split}
\E \| u (t_{n+1}) \|_{L^2}^2 &= \E \| u (t_n) \|_{L^2}^2 - 2 \E \int_{t_n}^{t_{n+1}} \| \nabla u(s) \|_{L^p}^p \, d s \\
& \leq \E \| u (t_n) \|_{L^2}^2 - 2 \mu t_{0} \big(\E\left\|u\left(t_{n}\right)\right\|_{L^2}^{2}\! \big)^{\frac{p}{2}} .
\end{split}
\end{equation*}
Applying Lemma \ref{P1} with $x=\mu (p-2) t_0 \big(\E\left\|u\left(t_{n}\right)\right\|_{L^2}^{2} \! \big)^{\frac{p-2}{2}},$ we obtain inequality \eqref{dissipation.1.eq}.
\end{proof}
In Case 2, we will follow the idea of proof of \cite[Theorem 1.9]{FGL21c} and estimate $\|u(t_{n+1})\|_{L^2}$ by using the mild formulation of $u(t)$ and the semigroup method,
where the inequality \eqref{case2.assumption} is necessary to estimate the stochastic convolution term.
\begin{proposition}\label{case.2}
Under the assumptions in Proposition \ref{dissipation.1}, inequality \eqref{dissipation.1.eq} holds in Case 2.
\end{proposition}
Firstly, we give some notations that are frequently used in proving Proposition \ref{case.2}; for any $r \in [t_n, t_{n+1}]$, we define
\begin{equation}\label{mild solution's component}
\begin{split}
v_{1}(r) &:=e^{\kappa\left(r-t_{n}\right) \Delta} u\left(t_{n}\right) , \\
v_{2}(r) &:=\int_{t_{n}}^{r} e^{\kappa(r-\tau) \Delta} \Delta_p u(\tau) , \\
v_{3}(r) &:=\sum_{k \in \Z^{d}} \sum_{i=1}^{d-1} \int_{t_{n}}^{r} e^{\kappa(r-\tau) \Delta}
\xi_{k, i} \cdot \nabla u(\tau) \, d B_{\tau}^{k, i} .
\end{split}
\end{equation}
By Theorem \ref{mild form}, we know $u(r)=v_1(r)+v_2(r)+v_3(r)$ for $t\geq r_n$. Then by the decreasing property of $t \rightarrow \|u(t)\|_{L^2}$, it holds
\begin{equation}\label{case2.1}
\| u(t_{n+1}) \|_{L^2} \leq \frac{1}{t_0} \int_{t_n}^{t_{n+1}} \|u(r)\|_{L^2} \, d r
\leq \frac{1}{t_0} \int_{t_n}^{t_{n+1}} \big( \|v_1(r)\|_{L^2}+\|v_2(r)\|_{L^2}+\|v_3(r)\|_{L^2} \big) \, d r .
\end{equation}
The estimate of $\|v_1(r)\|_{L^2}$ is straightforward, thus we focus on the last two terms.
\begin{lemma}\label{v_2}
If $p \in \big(2,\frac{2d}{d-2} \big)$, then it holds $\P$-a.s.
\begin{equation}\label{v_2.eq}
\frac{1}{t_0} \int_{t_n}^{t_{n+1}} \|v_2(r)\|_{L^2} \, d r \leq \frac{C_{0}}{\kappa^{\eta} t_{0}^{\eta-1/p}}
R^{1-2/p} \left\|u\left(t_{n}\right)\right\|_{L^2} ,
\end{equation}
where $\eta =\frac{1}{2}+ \frac{d}{2}(\frac{1}{2}-\frac{1}{p})<1, $ and the constant $C_0$ depends only on parameters $p$ and $d$.
\end{lemma}
\begin{proof}
For fixed $r \in [t_n, t_{n+1}], $ it holds $\P$-a.s.
\begin{equation*}
\begin{split}
\left\|v_{2}(r)\right\|_{L^2} &=\sup _{\|\phi\|_{L^2}=1}\left\langle\int_{t_{n}}^{r} e^{\kappa(r-\tau) \Delta} \Delta_p u(\tau) \, d \tau, \phi\right\rangle \\
& \leq \sup _{\|\phi\|_{L^2}=1} \int_{t_{n}}^{r} \big\langle e^{\kappa(r-\tau) \Delta }\Delta_p u(\tau), \phi \big\rangle \, d \tau \\
& \leq \sup _{\|\phi\|_{L^2}=1} \int_{t_{n}}^{r} \| e^{\kappa(r-\tau) \Delta} \phi \|_{W^{1, p}}\left\|\Delta_{p} u(\tau)\right\|_{W^{-1, \frac{p}{p-1}}} \, d \tau .
\end{split}
\end{equation*}
Applying Lemma \ref{P3} and Proposition \ref{pro.p-laplace}, we have
\begin{equation*}
\begin{split}
\left\|v_{2}(r)\right\|_{L^2} & \leq C \sup _{\|\phi\|_{L^2}=1} \int_{t_{n}}^{r}(\kappa(r-\tau))^{-\eta}
\|\phi\|_{L^2} \left\|\Delta_{p} u(\tau)\right\|_{W^{-1, \frac{p}{p-1}}} \, d \tau \\
& \leq C \int_{t_{n}}^{r}(\kappa(r-\tau))^{-\eta}\|\nabla u(\tau)\|_{L^p}^{p-1} \, d \tau;
\end{split}
\end{equation*}
integrating in $r\in [t_n, t_{n+1}]$ yields
\begin{equation*}
\begin{split}
\frac{1}{t_{0}} \int_{t_{n}}^{t_{n+1}}\left\|v_{2}(r)\right\|_{L^2} \, d r & \leq \frac{C}{t_{0}} \int_{t_{n}}^{t_{n+1}} \!\! \int_{t_{n}}^{r}(\kappa(r-\tau))^{-\eta}\|\nabla u(\tau)\|_{L^p}^{p-1} \, d \tau \, d r \\
& =\frac{C}{t_{0}} \int_{t_{n}}^{t_{n+1}}\|\nabla u(\tau)\|_{L^p}^{p-1} \int_{\tau}^{t_{n+1}}(\kappa(r-\tau))^{-\eta} \, d r \, d \tau \\
& \leq \frac{C}{\kappa^{\eta} t_{0}^{\eta} (1-\eta)} \int_{t_n}^{t_{n+1}} \| \nabla u(\tau) \|_{L^p}^{p-1} \, d \tau .
\end{split}
\end{equation*}
Let $C_0=\frac{C}{1-\eta}$, by H\"older's inequality and energy identity \eqref{energy identity.n}, we have
\[ \frac{1}{t_{0}} \int_{t_{n}}^{t_{n+1}}\left\|v_{2}(r)\right\|_{L^2} \, d r \leq
\frac{C_{0}}{\kappa^{\eta} t_{0}^{\eta-1/p}} \left\|u\left(t_{n}\right)\right\|_{L^2}^{ 2(p-1)/p} .\]
Recalling that $\| u(t_n)\|_{L^2} \leq \|u_0\|_{L^2} \leq R, $ we get the inequality \eqref{v_2.eq}.
\end{proof}
Next we deal with the most difficult term $v_3$ for which we will need \eqref{case2.assumption}.
\begin{lemma}\label{v_3}
Let $\frac{1}{\alpha}+ \frac{1}{\beta}=1 , \beta > \frac{d}{2} +1$; under the assumption \eqref{case2.assumption}, we have
\begin{equation}\label{v_3.eq}
\mathbb{E}\left(\frac{1}{t_0} \int_{t_{n}}^{t_{n+1}}\left\|v_{3}(r)\right\|_{L^2} d r\right)^{2} \leq d \mu^{\frac{2\alpha}{p}}
\Lambda_{\beta} \|\theta\|_{\infty}^{\frac{2}{\beta}} \, \E \left\|u\left(t_{n}\right)\right\|_{L^2}^{2} ,
\end{equation}
where $\| \theta \|_{\infty} \assign \sup_{k \in \Z_0^d} |\theta_k| , $
and $\Lambda_{\beta} \assign \big(\sum_{l \in \Z^d} \lambda_l^{1-\beta} \big)^{\frac{1}{\beta}} < + \infty.$
\end{lemma}
\begin{proof}
By Jensen's inequality and It\^o's isometry, it holds
\begin{equation*}
\begin{split}
\E\left(\frac{1}{t_{0}} \int_{t_{n}}^{t_{n+1}}\left\|v_{3}(r)\right\|_{L^2} d r\right)^{2} & \leq \frac{1}{t_{0}} \int_{t_{n}}^{t_{n+1}} \E\left\|v_{3}(r)\right\|_{L^2}^{2} \, d r \\
&=\frac{1}{t_{0}} \E \int_{t_{n}}^{t_{n+1}} \sum_{k \in \Z^{d}} \sum_{i=1}^{d-1} \int_{t_{n}}^{r} \left\|e^{\kappa(r-\tau) \Delta} \xi_{k, i} \cdot \nabla u(\tau)\right\|_{L^2}^{2} \, d \tau \, d r \\
&=\frac{1}{t_{0}} \E \sum_{i=1}^{d-1} \sum_{k, l \in \Z^{d}} \int_{t_{n}}^{t_{n+1}} \!\! \int_{t_{n}}^{r} e^{-2 \kappa \lambda_{l}(r-\tau)} \left| \left\langle \xi_{k, i} \cdot \nabla u(\tau), e_{l}\right\rangle \right|^{2} \, d \tau \, d r .
\end{split}
\end{equation*}
By the H\"older inequality with exponents $\frac{1}{\alpha}+\frac{1}{\beta}=1$, we get
\begin{equation}\label{v_3.1}
\begin{split}
\E & \left(\frac{1}{t_{0}} \int_{t_{n}}^{t_{n+1}}\left\|v_{3}(r)\right\|_{L^2} d r\right)^{2} \\
& \leq \frac{1}{t_0} \E \sum_{i=1}^{d-1} \sum_{k, l \in \Z^{d}} \int_{t_{n}}^{t_{n+1}} \!\! \int_{t_{n}}^{r} \left(e^{-2 \kappa \lambda_{l}(r-\tau)} \lambda_l \left| \left\langle a_{k, i} e_k \cdot \nabla u(\tau), e_{l}\right\rangle \right|^{2}\right)^{\frac{1}{\alpha}} \\
& \hskip70pt \times \left(e^{-2 \kappa \lambda_{l}(r-\tau)} \lambda_l^{1-\beta} \left| \left\langle a_{k, i} e_k \cdot \nabla u(\tau), e_{l}\right\rangle \right|^{2}\right)^{\frac{1}{\beta}} \, d \tau \, d r \\
& \leq\frac{1}{t_0} I_1^{\frac{1}{\alpha}} I_2^{\frac{1}{\beta}} ,
\end{split}
\end{equation}
where $I_1$ and $I_2$ are defined as below:
\begin{equation*}
\begin{split}
I_1 &\assign \E \sum_{k \in \Z^{d}} \sum_{i=1}^{d-1} \sum_{l \in \Z^{d}} \int_{t_{n}}^{t_{n+1}} \!\! \int_{t_{n}}^{r} e^{-2 \kappa \lambda_{l}(r-\tau)}
\lambda_l \left| \left\langle\xi_{k, i} \cdot \nabla u(\tau), e_{l}\right\rangle \right|^{2} \, d \tau \, d r , \\
I_2 &\assign \E \sum_{k \in \Z^{d}} \sum_{i=1}^{d-1} \sum_{l \in \Z^{d}} \int_{t_{n}}^{t_{n+1}} \!\! \int_{t_{n}}^{r} e^{-2 \kappa \lambda_{l}(r-\tau)}
\lambda_l^{1-\beta} \left| \left\langle\xi_{k, i} \cdot \nabla u(\tau), e_{l}\right\rangle \right|^{2} \, d \tau \, d r .
\end{split}
\end{equation*}
Recalling the definition of the vector fields $\{\xi_{k,i}\}$ and using the Fubini theorem, it holds
\begin{equation*}
\begin{split}
I_1 & =\frac{2 \kappa d}{d-1} \E \sum_{k \in \Z^{d}} \theta_k^2 \sum_{i=1}^{d-1} \sum_{l \in \Z^{d}} \int_{t_{n}}^{t_{n+1}} \!\! \int_{\tau}^{t_{n+1}} e^{-2 \kappa \lambda_{l}(r-\tau)}
\lambda_l \left| \left\langle a_{k, i} e_k \cdot \nabla u(\tau), e_{l}\right\rangle \right|^{2} \, d r \, d \tau , \\
I_2 & =\frac{2 \kappa d}{d-1} \E \sum_{k \in \Z^{d}} \theta_k^2 \sum_{i=1}^{d-1} \sum_{l \in \Z^{d}} \int_{t_{n}}^{t_{n+1}} \!\! \int_{\tau}^{t_{n+1}} e^{-2 \kappa \lambda_{l}(r-\tau)}
\lambda_l^{1-\beta} \left| \left\langle a_{k, i} e_k \cdot \nabla u(\tau), e_{l}\right\rangle \right|^{2} \, d r \, d \tau .
\end{split}
\end{equation*}
We estimate $I_1$ and $I_2$ separately. Firstly,
\begin{equation*}
\begin{split}
I_1 & = \frac{2 \kappa d}{d-1} \E \sum_{k \in \Z^{d}} \theta_k^2 \sum_{i=1}^{d-1} \sum_{l \in \Z^{d}} \int_{t_{n}}^{t_{n+1}} \left| \left\langle a_{k, i} e_k \cdot \nabla u(\tau), e_{l}\right\rangle \right|^{2}
\int_{\tau}^{t_{n+1}} e^{-2 \kappa \lambda_{l}(r-\tau)} \lambda_l \, d r \, d \tau \\
& \leq \frac{d}{d-1} \E \sum_{k \in \Z^{d}} \theta_k^2 \sum_{i=1}^{d-1} \int_{t_{n}}^{t_{n+1}} \sum_{l \in \Z^{d}} \left| \left\langle a_{k, i} e_k \cdot \nabla u(\tau), e_{l}\right\rangle \right|^{2} \, d \tau.
\end{split}
\end{equation*}
By Parseval's identity, we get
\[\sum_{l \in \Z^{d}} \left| \left\langle a_{k, i} e_k \cdot \nabla u(\tau), e_{l} \right\rangle \right|^{2} = \| a_{k, i} e_k \cdot \nabla u(\tau) \|_{L^2}^2 \leq \| \nabla u(\tau) \|_{L^2}^2 \]
for all $k \in \Z^d, i=1, 2, \dots, d-1$ and all $\tau \in [t_n, t_{n+1}].$ Recalling that $\sum_{k \in \Z_0^d} \theta_k^2 =1,$ so
\begin{equation*}
I_1 \leq d \, \E \int_{t_{n}}^{t_{n+1}} \| \nabla u(\tau) \|_{L^2}^2 \, d \tau
\leq \, d t_0^{\frac{p-2}{p}} \left( \E \int_{t_{n}}^{t_{n+1}} \| \nabla u(\tau) \|_{L^p}^p \, d \tau \right)^{\frac{2}{p}} .
\end{equation*}
By the assumption \eqref{case2.assumption}, it holds
\begin{equation}\label{v_3.2}
I_1 \leq d \mu^{\frac{2}{p}} t_0 \E \| u(t_n) \|_{L^2}^2 .
\end{equation}
Now we turn to estimate $I_2.$ Recalling that $\{a_{k,i} e_k\}$ are divergence free, by integration by parts, we have
\begin{equation*}
\left| \left\langle a_{k, i} e_k \cdot \nabla u(\tau), e_{l}\right\rangle \right|^{2}
=\left| \left\langle e_k u(\tau), a_{k, i} \cdot \nabla e_{l}\right\rangle \right|^{2}
= (2\pi)^2 (a_{k, i} \cdot l)^2 \left| \left\langle e_k u(\tau), e_{l}\right\rangle \right|^{2}
\end{equation*}
for all $k \in \Z^d, i=1, 2, \dots, d-1$ and all $\tau \in [t_n, t_{n+1}].$ So
\begin{equation*}
\begin{split}
I_2 & \leq \frac{2 \kappa d}{d-1} \E \sum_{k \in \Z^{d}} \theta_k^2 \sum_{i=1}^{d-1} \sum_{l \in \Z^{d}} \int_{t_{n}}^{t_{n+1}} \!\! \int_{\tau}^{t_{n+1}} e^{-2 \kappa \lambda_{l}(r-\tau)}
\lambda_l^{1-\beta} (2\pi)^2 (a_{k, i} \cdot l)^2 \left| \left\langle e_k u(\tau), e_{l}\right\rangle \right|^{2} \, d r \, d \tau \\
& \leq \frac{2 \kappa d}{d-1} \E \sum_{k \in \Z^{d}} \theta_k^2 \sum_{i=1}^{d-1} \sum_{l \in \Z^{d}} \int_{t_{n}}^{t_{n+1}} \left| \left\langle e_k u(\tau), e_{l}\right\rangle \right|^{2}
\int_{\tau}^{t_{n+1}} e^{-2 \kappa \lambda_{l}(r-\tau)} \lambda_l^{2-\beta} \, d r \, d \tau \\
& \leq d \, \E \sum_{k \in \Z^{d}} \theta_k^2 \sum_{l \in \Z^{d}} \lambda_l^{1-\beta} \int_{t_{n}}^{t_{n+1}} \left| \left\langle e_k u(\tau), e_{l}\right\rangle \right|^{2} \, d \tau .
\end{split}
\end{equation*}
By the Bessel inequality, we have
\begin{equation}\label{v_3.3}
\begin{split}
I_2 & \leq d \| \theta \|_{\infty}^2 \E \sum_{l \in \Z^{d}} \lambda_l^{1-\beta} \int_{t_{n}}^{t_{n+1}} \sum_{k \in \Z^{d}} \left| \left\langle e_k u(\tau), e_{l}\right\rangle \right|^{2} \, d \tau \\
& \leq d \| \theta \|_{\infty}^2 \E \sum_{l \in \Z^{d}} \lambda_l^{1-\beta} \int_{t_{n}}^{t_{n+1}} \| u(\tau) e_l \|_{L^2}^2 \, d \tau \\
& \leq d \| \theta \|_{\infty}^2 \Lambda_{\beta}^{\beta} t_{0}\, \E \| u(t_{n}) \|_{L^2}^2 .
\end{split}
\end{equation}
Combining the estimate \eqref{v_3.1}, \eqref{v_3.2}, and \eqref{v_3.3}, we arrive at the inequality \eqref{v_3.eq}.
\end{proof}
We have estimated the last two terms of the inequality \eqref{case2.1} respectively in Lemmas \ref{v_2} and \ref{v_3}; now we are ready to prove Proposition \ref{case.2}.
\begin{proof}[Proof of Proposition \ref{case.2}]
By the property of heat semigroup, we have
\begin{equation}\label{v_1}
\begin{split}
\frac{1}{t_{0}} \int_{t_{n}}^{t_{n+1}}\left\|v_{1}(r)\right\|_{L^2} \, d r & \leq \frac{1}{t_{0}} \int_{t_{n}}^{t_{n+1}}
\exp \left\{-4 \pi^{2} \kappa\left(r-t_{n}\right)\right\}\left\|{u}\left(t_{n}\right)\right\|_{L^2} \, d r \\
&=\frac{1-\exp \left\{-4 \pi^{2} \kappa t_{0}\right\}}{4 \pi^{2} \kappa t_{0}} \left\|u\left(t_{n}\right)\right\|_{L^2}
\leq \frac{1}{4 \pi^2 \kappa t_{0}} \left\|u\left(t_{n}\right)\right\|_{L^2} .
\end{split}
\end{equation}
By \eqref{case2.1} and the Cauchy inequality, $\| u(t_{n+1})\|_{L^2}^2$ is dominated by
\[ 3 \left\{ \left( \frac{1}{t_0} \int_{t_n}^{t_{n+1}} \|v_1(r)\|_{L^2} \, d r \right)^2
+ \left( \frac{1}{t_0} \int_{t_n}^{t_{n+1}} \|v_2(r)\|_{L^2} \, d r \right)^2
+ \left( \frac{1}{t_0} \int_{t_n}^{t_{n+1}} \|v_3(r)\|_{L^2} \, d r \right)^2 \right\} . \]
Taking expectation and applying Lemmas \ref{v_2} and \ref{v_3}, we get
\begin{equation}\label{case2.2}
\begin{split}
\E \| u(t_{n+1}) \|_{L^2}^2 & \leq \frac{1}{2} \left( C_1(\kappa,t_0,R)^{\frac{2}{p-2}} + C_2(\theta,\mu)^{\frac{2}{p-2}} \right) \E \| u({t_n}) \|_{L^2}^2 \\
& \leq \max \{ C_1(\kappa,t_0,R),C_2(\theta,\mu) \}^{\frac{2}{p-2}} \E \| u({t_n}) \|_{L^2}^2 .
\end{split}
\end{equation}
By the condition \eqref{condition.1} and recalling $\| u(t_n) \|_{L^2} \leq \| u_0 \|_{L^2} \leq R ,$ we get
\begin{equation*}\label{case2.3}
\E \|u(t_{n+1}) \|_{L^2}^2 \leq \frac{\E \|u(t_n) \|_{L^2}^2}{\left(1+\mu (p-2) t_0 R^{p-2}\right)^\frac{2}{p-2} } \leq \frac{\E \|u(t_n) \|_{L^2}^2}{\Big(1+\mu (p-2) t_0 \left(\E \|u(t_n) \|_{L^2}^2 \right)^{\frac{p-2}{2}}\Big)^\frac{2}{p-2} } .
\end{equation*}
Therefore, inequality \eqref{dissipation.1.eq} holds also in Case 2.
\end{proof}
To prove Theorem \ref{thm-average}, we will choose appropriate $t_0$, $\mu',$ and use Lemma \ref{P2} to iterate over the estimates in Proposition \ref{dissipation.1}, finally we get the estimate on $\|u(t)\|_{L^2}$ for any $t \geq 1$.
\begin{proof}[Proof of Theorem \ref{thm-average}]
Given any $\mu>0,$ we choose $0<t_0<1$ and $\mu'$ satisfying $\mu'(1-t_0)>\mu.$ For any $t \geq 1$, there exists $n \in \N$ such that $nt_0= t_n \leq t < t_{n+1} =(n+1)t_0 ,$ then
\[\mu' \frac{t_n}{t} \geq \mu' \left(1-\frac{t_0}{t}\right) \geq \mu'(1-t_0) \geq \mu .\]
Applying Proposition \ref{dissipation.1} with $t_0$ and $\mu'$, we obtain
\[ \E \|u(t_m) \|_{L^2}^2 \leq \frac{\E \|u(t_{m-1}) \|_{L^2}^2}{\Big(1+ 2 \mu'(p-2) t_0 \left( \E \|u(t_{m-1}) \|_{L^2}^2 \right)^{\frac{p-2}{2}} \!\Big)^\frac{2}{p-2}} , \quad m=1,2,\dots,n .\]
Applying Lemma \ref{P2} with $x=\E \|u(t_{n-2}) \|_{L^2}^2$, $y=\E \|u(t_{n-1}) \|_{L^2}^2$, $z=\E \|u(t_n) \|_{L^2}^2$, and $a=b=\mu'(p-2)t_0$, we have
\[ \E \|u(t_n) \|_{L^2}^2 \leq \frac{\E \|u(t_{n-2}) \|_{L^2}^2}{\Big(1+ 2 \mu'(p-2) t_0 \left( \E \|u(t_{n-2}) \|_{L^2}^2 \right)^{\frac{p-2}{2}} \! \Big)^\frac{2}{p-2}} .\]
Continuing in this way, we finally get
\begin{equation*}
\E \|u(t_n) \|_{L^2}^2 \leq \dots \leq \frac{\|u_0 \|_{L^2}^2}{\big(1+\mu' (p-2) n t_0 \|u_0 \|_{L^2}^{p-2} \big)^\frac{2}{p-2} } .
\end{equation*}
By decreasing property of $t\to \|u(t)\|_{L^2}$, we have
\begin{equation*}
\begin{split}
\E \|u(t) \|_{L^2}^2 &\leq \E \|u(t_n) \|_{L^2}^2 \leq \frac{\|u_0 \|_{L^2}^2}{\big(1+\mu' \times \frac{t_n}{t} (p-2) t \|u_0 \|_{L^2}^{p-2} \big)^\frac{2}{p-2} } \\
& \leq \frac{\|u_0 \|_{L^2}^2}{\big(1+\mu (p-2) t \|u_0 \|_{L^2}^{p-2} \big)^\frac{2}{p-2} } .
\end{split}
\end{equation*}
So for any $t \geq 1$, the inequality \eqref{thm-average.eq} holds.
\end{proof}
From the previous proof and choice of the noise intensity $\kappa$ and coefficient $\{\theta_k\}_{k \in \Z_0^d}$ in Proposition \ref{dissipation.1},
we can easily conclude that Theorem \ref{thm-average} holds when the intensity $\kappa$ and coefficient $\{\theta_k\}_{k \in \Z_0^d}$ satisfy
\begin{equation}\label{thm-average.condition}
\max \left\{ C_1(\kappa,t_0,R),C_2\left(\theta,\frac{\mu}{1-t_0}\right) \right\} \times \left(1+ \frac{\mu t_0}{1-t_0} (p-2) R^{p-2}\right) < 1
\end{equation}
for some $t_0 \in (0,1),$ where $C_1(\kappa,t_0,R)$ and $C_2(\theta,\mu)$ are defined in \eqref{C_1,C_2}.
\section{Almost sure dissipation enhancement}\label{sec:dissipation.2}
This section deals with the dissipation enhancement in $\P$-a.s. sense. For this purpose,
we set $t_0=1$ and choose time-dependent coefficients $\{\theta_{k,i}\}$ which are step functions of time $t$ as below:
\begin{equation}\label{noise.3}
\theta_{k} (t) \assign \theta_{k,n} , \quad t\in [n,n+1) .
\end{equation}
For any fixed $n \in \N,$ the parameters $\theta^{(n)} \assign \{ \theta_{k,n} \}_{k \in \Z_0^d}$ satisfy the requirements in Section \ref{subsec:noise setting}. We consider the following SPDE:
\begin{equation}\label{p-Laplace.4}
d u = \Delta_p u \, d t + \kappa \Delta u \, d t +
\sum_{k \in \Z^d} \sum_{i=1}^{d-1} \xi_{k, i} (x,t) \cdot \nabla u \, d B_t^{k, i} ,
\end{equation}
where the vector fields $\{\xi_{k,i}(x,t)\}$ are defined as below:
\begin{equation}\label{xi.1}
\xi_{k, i}(x,t) =\left\{\begin{array}{ll}
2 \sqrt{C_{d} \kappa}\, \theta_{k}(t) {\rm Re}(\sigma_{k, i}(x)), & \quad k \in \Z_{+}^d, \\
2 \sqrt{C_{d} \kappa}\, \theta_{k}(t) {\rm Im}(\sigma_{k, i}(x)), & \quad k \in \Z_{-}^d,
\end{array} \right.
\end{equation}
with $\{\sigma_{k,i}\}$ being defined in \eqref{sigma}. Because $\theta_k(t)$ is a step function, for any $n \in \N$,
the variational solution $u(t)$ of SPDE \eqref{p-Laplace.4} exists and is unique in each interval $[n,n+1]$,
then we get the existence and uniqueness of solution $u(t)$ in $[0,+\infty).$
Similarly, the energy identity \eqref{energy identity} and mild form \eqref{mild form.eq} also hold.
\begin{proposition}\label{dissipation.2}
For any $2 < p < \frac{2d}{d-2}, $ $\mu_0 >0$ and initial data satisfying $\| u_0 \|_{L^2} \leq R , $
we can find appropriate intensity $\kappa$ and coefficients $\{\theta^{(n)}\}_{n \in \N}$ (only depending on parameters $d, p, R, \mu_{0}$) so that the solution $u(t)$ of SPDE \eqref{p-Laplace.4} satisfies
\begin{equation}\label{dissipation.2.eq1}
\E \|u(n) \|_{L^2}^2 \leq \frac{\E \|u(n-1) \|_{L^2}^2}{\Big(1+\mu_0 (p-2) 2^{n-1}\! \left(\E \|u(n-1) \|_{L^2}^2\right)^{\frac{p-2}{2}} \Big)^\frac{2}{p-2} } , \quad \forall\, n \in \Z_+ .
\end{equation}
Moreover, we can get the following dissipation estimate:
\begin{equation}\label{dissipation.2.eq2}
\E \|u(n) \|_{L^2}^2 \leq \frac{\|u_0 \|_{L^2}^2}{\big(1+\mu_0 (p-2) (2^n-1) \|u_0 \|_{L^2}^{p-2} \big)^\frac{2}{p-2} } , \quad \forall\, n \in \Z_+ .
\end{equation}
To be precise, the intensity $\kappa$ and coefficients $\{\theta^{(n)}\}_{n \in \N}$ satisfy
\begin{equation}\label{condition.2}
\begin{split}
C_1(\kappa,1,R) & \leq \min \left\{ \frac{1}{1+\mu_0 (p-2) R^{p-2} },\frac{1}{3} \right\}, \\
\sup_{n \in \N}C_2(\theta^{(n)},\mu_n) & \leq \min \left\{ \frac{1}{1+\mu_0 (p-2) R^{p-2} },\frac{1}{3} \right\} ,
\end{split}
\end{equation}
where $\mu_n =2^{n} \mu_0 $ and $\theta^{(n)} \assign \{ \theta_{k,n} \}_{k \in \Z_0^d}.$
\end{proposition}
\begin{proof}
We are going to prove Proposition \ref{dissipation.2} by induction. Because $\theta_{k}(t)$ is a step function, the estimates in Section \ref{sec:dissipation.1} also hold with $t_0=1,$ $ \mu=\mu_n$ and $\theta =\theta^{(n)}.$
For $n=1$, by the estimate \eqref{dissipation.1.eq}, we have
\begin{equation*}
\E \| u(1) \|_{L^2}^2\leq \frac{\|u_0 \|_{L^2}^2}{ \big(1+\mu_0 (p-2) \|u_0 \|_{L^2}^{p-2} \big)^\frac{2}{p-2}} .
\end{equation*}
Thus, inequalities \eqref{dissipation.2.eq1} and \eqref{dissipation.2.eq2} hold for $n=1$.
For $n=N \geq 2,$ we assume the inequality \eqref{dissipation.2.eq2} holds for $n=N-1,$ i.e.
\begin{equation}\label{inductive assumption}
\E \|u(N-1) \|_{L^2}^2 \leq \frac{\|u_0 \|_{L^2}^2}{\big(1+\mu_0 (p-2) (2^{N-1}-1) \|u_0 \|_{L^2}^{p-2} \big)^\frac{2}{p-2} }.
\end{equation}
If $\E \|u(N) \|_{L^2}^2 = 0$, then obviously the inequalities \eqref{dissipation.2.eq1} and \eqref{dissipation.2.eq2} hold with $n=N$, hence we assume $\E \|u(N) \|_{L^2}^2 > 0$. In this case, $\E \|u(N-1) \|_{L^2}^2 \geq \E \|u(N) \|_{L^2}^2 > 0$, thus we can rewrite \eqref{inductive assumption} as
\begin{equation}\label{inductive assumption.1}
\frac{1}{ \left( \E \| u(N-1) \|_{L^2}^2 \right)^{\frac{p-2}{2}} } \geq \frac{1}{ \| u_0 \|_{L^2}^{p-2} } +\mu_0 (p-2) (2^{N-1}-1).
\end{equation}
We only need to prove \eqref{dissipation.2.eq1} in Case 2 of the last section, with $\mu=\mu_{N-1}$ and $\theta=\theta^{(N-1)}$. By the estimate \eqref{case2.2}, we have
\begin{equation*}
\begin{split}
\frac{1}{ \left( \E \| u(N) \|_{L^2}^2 \right)^{\frac{p-2}{2}} } & -\frac{1}{ \left( \E \| u(N-1) \|_{L^2}^2 \right)^{\frac{p-2}{2}} } \\
\geq & \left(\frac{1}{ \max \{ C_1(\kappa,1,R),C_2(\theta^{(N-1)},\mu_{N-1}) \}}-1 \right) \frac{1}{ \left( \E \| u(N-1) \|_{L^2}^2 \right)^{\frac{p-2}{2}} } .
\end{split}
\end{equation*}
Condition \eqref{condition.2} implies $\max \big\{ C_1(\kappa,1,R),C_2(\theta^{(N-1)},\mu_{N-1}) \big\} \leq \frac{1}{3}$, which, combined with inequality \eqref{inductive assumption.1}, gives us
\begin{equation*}
\begin{split}
\frac{1}{ \left( \E \| u(N) \|_{L^2}^2 \right)^{\frac{p-2}{2}} } -\frac{1}{ \left( \E \| u(N-1) \|_{L^2}^2 \right)^{\frac{p-2}{2}} }
& \geq 2 \bigg( \frac{1}{ \| u_0 \|_{L^2}^{p-2} } +\mu_0 (p-2) (2^{N-1}-1) \bigg) \\
& \geq \mu_0 (p-2) 2^{N-1} .
\end{split}
\end{equation*}
It is equivalent to
\[ \E \|u(N) \|_{L^2}^2 \leq \frac{\E \|u(N-1) \|_{L^2}^2}{\left(1+\mu_0 (p-2) 2^{N-1} (\E \|u(N-1) \|_{L^2}^2)^{\frac{p-2}{2}}\right)^\frac{2}{p-2}}, \]
so inequality \eqref{dissipation.2.eq1} holds for $n=N.$ By inequality \eqref{inductive assumption} and Lemma \ref{P2}, we have
\[\E \|u(N) \|_{L^2}^2 \leq \frac{\|u_0 \|_{L^2}^2}{\left(1+\mu_0 (p-2) (2^{N}-1) \|u_0 \|_{L^2}^{p-2}\right)^\frac{2}{p-2} } ,\]
then inequality \eqref{dissipation.2.eq2} holds for $n=N.$ The induction holds.
\end{proof}
Now we will use Proposition \ref{dissipation.2} and Borel-Cantelli lemma to prove Theorem \ref{thm-pathwise}.
\begin{proof}[Proof of Theorem \ref{thm-pathwise}]
Assume $u_0 \ne 0$; for any $n \geq 1,$ we define
\begin{equation}\label{def.A_n}
A_n \assign \left\{ \omega \in \Omega : \sup_{t \in [n, n + 1]} \| u (t) \|_{L^2}^2 \geq \frac{\| u_0 \|^2_2}{\big( 1+ \mu_0 (p - 2) n \left\| u_0 \right\|_{L^2}^{p - 2} \! \big)^{\frac{2}{p - 2}}} >0 \right\} .
\end{equation}
Then by Chebyshev's inequality and Proposition \ref{dissipation.2}, it holds
\begin{equation}\label{P A_n}
\begin{split}
\P(A_n) & \leq \frac{\big( 1 + \mu_0 (p - 2) n \left\| u_0 \right\|_{L^2}^{p - 2} \!\big)^{\frac{2}{p - 2}}}{\| u_0 \|^2_2} \E \bigg(\sup_{t \in [n,n+1]} \|u(t)\|_{L^2}^2 \bigg) \\
& \leq \left( \frac{1 + \mu_0 (p - 2) n \left\| u_0 \right\|_{L^2}^{p - 2} }{1+\mu_0 (p-2) (2^n-1) \|u_0 \|_{L^2}^{p-2}} \right)^{\frac{2}{p - 2}} .
\end{split}
\end{equation}
So $\sum_{n \in \Z_{+}} \P(A_n) < + \infty ,$ and by Borel-Cantelli lemma, for $\P$-a.s. $\omega \in \Omega,$ there exists a big $N(\omega) >1$ such that for any $n>N(\omega),$ it holds
\begin{equation*}
\sup_{t \in [n, n + 1]} \| u (t) \|_{L^2}^2 \leq \frac{\| u_0 \|^2_2}{\big( 1 + \mu_0 (p - 2) n \left\| u_0 \right\|_{L^2}^{p - 2} \!\big)^{\frac{2}{p - 2}}} ;
\end{equation*}
If we set $\exp \{-\frac{1}{0_+}\}=0,$ then equivalently,
\begin{equation}\label{dissipation3.eq.2}
\sup_{t \in [n,n + 1]} \exp \left( -\frac{1}{\| u (t) \|_{L^2}^{p - 2}} \right) \leq e^{-\mu_0 (p-2)n} \exp \left( -\frac{1}{\| u_0 \|_{L^2}^{p - 2}} \right) .
\end{equation}
For $ 0 \leq n \leq N(\omega) ,$ by the decreasing property of $t \rightarrow \|u(t)\|_{L^2}$, we have
\begin{equation*}
\begin{split}
\sup_{t \in [n,n + 1]} & \exp \left( -\frac{1}{\| u (t) \|_{L^2}^{p - 2}} \right) \leq \exp \left( -\frac{1}{\| u_0 \|_{L^2}^{p - 2}} \right) \\
& \leq e^{\mu_0 (p-2) N(\omega)} e^{-\mu_0 (p-2) n} \exp \left( -\frac{1}{\| u_0 \|_{L^2}^{p - 2}} \right) .
\end{split}
\end{equation*}
Thus, if we take $C(\omega) = e^{\mu_0 (p-2) (1+N(\omega))} ,$ it is easy to show
\[\exp\left(- \frac1{\|u(t) \|_{L^2}^{p-2}} \right) \leq C(\omega) e^{-(p-2)\mu_0 t} \exp\left(- \frac1{\|u_0 \|_{L^2}^{p-2}} \right).\]
And for any $t>\frac{\ln C(\omega)}{(p-2)\mu_0}$, we can get the decay of $\|u(t)\|_{L^2}$ as below:
\[\|u(t) \|_{L^2} \leq \frac{\|u_0 \|_{L^2}}{\big(1+\left((p-2)\mu_0 t-\ln C(\omega) \right) \|u_0 \|_{L^2}^{p-2} \big)^{1/(p-2)}}.\]
It remains to estimate the $q$-th moment of the random variable $C(\omega)$; to this end, we need to estimate the tail probability $\P(\{N(\omega) \geq k\})$.
Note that $N(\omega)$ may be defined as the largest integer $n$ such that
$$\sup_{t \in [n, n + 1]} \| u (t) \|_{L^2}^2 \geq \frac{\| u_0 \|^2_2}{\big( 1 + \mu_0 (p - 2) n \left\| u_0 \right\|_{L^2}^{p - 2} \! \big)^{\frac{2}{p - 2}}};$$
hence
\[\{\omega\in \Omega: N(\omega) \geq k\} = \bigcup_{n=k}^\infty A_n. \]
Since $\|u_0\|_2>0,$ for any $\mu_0>0$, there exists $M=M(\mu_0, \|u_0\|_{L^2})$ such that
\[ \frac{1 + \mu_0 (p - 2) n \left\| u_0 \right\|_{L^2}^{p - 2} }{1+\mu_0 (p-2) (2^n-1) \|u_0 \|_{L^2}^{p-2}} \leq \frac{2n}{2^n-1} \leq \bigg(\frac{1}{2} \bigg)^{\frac{n}{2}} ,\quad \forall\, n \geq M.\]
By the estimate \eqref{P A_n}, for any $n \geq M,$ $\P(A_n) \leq \frac{1}{2^{\frac{n}{p-2}}},$ then for any $k \geq M$, it holds
\[ \P(\{N(\omega) \geq k\}) \leq \sum_{n=k}^\infty \P(A_n) \leq \sum_{n=k}^\infty \left(\frac{1}{2}\right)^{\frac{n}{p-2}} = \frac{2^{\frac{1}{p-2}}}{2^{\frac{1}{p-2}}-1} \times \frac{1}{2^{\frac{k}{p-2}}} .\]
As a result,
\begin{equation}
\begin{split}
\E & e^{\mu_0 (p-2) q N(\omega)} = \sum_{k=0}^\infty e^{\mu_0 (p-2) q k} \P(\{N(\omega) = k\}) \\
& \leq \sum_{k=0}^{M} e^{\mu_0 (p-2) q k} + \frac{2^{\frac{1}{p-2}}}{2^{\frac{1}{p-2}}-1} \sum_{k=M+1}^\infty e^{ \big(\mu_0(p-2) q-\frac{\ln 2}{p-2} \big) k } .
\end{split}
\end{equation}
When $q < \frac{\ln 2}{\mu_0 (p-2)^2} ,$ we get $ \E e^{\mu_0 (p-2) q N(\omega)} < +\infty $ and $C(\omega)$ has finite $q$-th moment.
\end{proof}
\begin{remark}\label{remark}
In fact, in the same way, we can prove that for any positive coefficient polynomial $P(t)$, there exists a random constant $C(\omega)$ such that
\begin{equation*}
\exp \left(-\frac{1}{\| u(t) \|_{L^2}^{p-2}} \right) \leq C(\omega) e^{- \mu_0 (p-2) P(t)} \exp \left( -\frac{1}{\| u_0 \|_{L^2}^{p-2}} \right)
\end{equation*}
holds $\P$-a.s. and for all $t \geq 0$.
\end{remark}
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 2,133
|
package com.gs.collections.impl.tuple;
import java.io.Serializable;
import java.util.Map;
import com.gs.collections.api.block.function.Function;
import com.gs.collections.api.tuple.Pair;
import com.gs.collections.impl.block.factory.Functions;
public class AbstractImmutableEntry<K, V> implements Map.Entry<K, V>, Serializable
{
private static final long serialVersionUID = 1L;
private static final PairFunction<?, ?> TO_PAIR = new PairFunction<Object, Object>();
protected final K key;
protected final V value;
public AbstractImmutableEntry(K key, V value)
{
this.key = key;
this.value = value;
}
/**
* @deprecated Since 6.2 - Use {@link Functions#getKeyFunction()} instead.
*/
@Deprecated
public static <K> Function<Map.Entry<K, ?>, K> getKeyFunction()
{
return Functions.getKeyFunction();
}
/**
* @deprecated Since 6.2 - Use {@link Functions#getValueFunction()} instead.
*/
@Deprecated
public static <V> Function<Map.Entry<?, V>, V> getValueFunction()
{
return Functions.getValueFunction();
}
public static <K, V> Function<Map.Entry<K, V>, Pair<K, V>> getPairFunction()
{
return (Function<Map.Entry<K, V>, Pair<K, V>>) (Function<?, ?>) TO_PAIR;
}
public K getKey()
{
return this.key;
}
public V getValue()
{
return this.value;
}
/**
* {@inheritDoc}
* <p>
* This implementation throws an {@link UnsupportedOperationException}. Override this method to support mutable
* map entries.
*/
public V setValue(V value)
{
throw new UnsupportedOperationException("Cannot call setValue() on " + this.getClass().getSimpleName());
}
/**
* Returns a string representation of the form {@code {key}={value}}.
*/
@Override
public String toString()
{
return this.key + "=" + this.value;
}
/**
* @deprecated Since 6.2 - Kept for serialization compatibility only.
*/
@Deprecated
private static class KeyFunction<K> implements Function<Map.Entry<K, ?>, K>
{
private static final long serialVersionUID = 1L;
public K valueOf(Map.Entry<K, ?> entry)
{
return entry.getKey();
}
}
/**
* @deprecated Since 6.2 - Kept for serialization compatibility only.
*/
@Deprecated
private static class ValueFunction<V> implements Function<Map.Entry<?, V>, V>
{
private static final long serialVersionUID = 1L;
public V valueOf(Map.Entry<?, V> entry)
{
return entry.getValue();
}
}
private static class PairFunction<K, V> implements Function<Map.Entry<K, V>, Pair<K, V>>
{
private static final long serialVersionUID = 1L;
public Pair<K, V> valueOf(Map.Entry<K, V> entry)
{
return Tuples.pairFrom(entry);
}
}
}
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 7,372
|
Aylesworth is an unincorporated community in Marshall County, Oklahoma, United States. It is located east of Kingston. A post office operated in Aylesworth from June 6, 1903 to October 15, 1943. The community was named after a Dawes Commission official named Allison Aylesworth. Most of the original Aylesworth community is now underneath Lake Texoma.
References
Unincorporated communities in Marshall County, Oklahoma
Unincorporated communities in Oklahoma
Populated places established in 1903
Populated places disestablished in 1943
1903 establishments in Indian Territory
|
{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 1,655
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{"url":"http:\/\/www.mathcancer.org\/blog\/tag\/tutorials\/","text":"## Working with PhysiCell MultiCellDS digital snapshots in Matlab\n\nPhysiCell 1.2.1 and later saves data as a specialized MultiCellDS digital snapshot, which\u00a0includes chemical substrate fields, mesh information, and a readout of the cells and their phenotypes at single simulation time point. This tutorial will help you learn to use the matlab processing files included with PhysiCell.\n\nThis tutorial assumes you know (1) how to work at the shell \/ command line of your operating system, and (2) basic plotting and other functions in Matlab.\n\n### Key elements of a PhysiCell digital snapshot\n\nA PhysiCell digital snapshot (a customized form of the MultiCellDS digital simulation snapshot) includes the following elements saved as XML and MAT files:\n\n1. output12345678.xml : This is the \u201cbase\u201d output file, in MultiCellDS format. It includes key metadata such as when the file was created, the software, microenvironment information, and custom data saved at the simulation time. The Matlab files read this base file to find other related files (listed next). Example: output00003696.xml\n2. initial_mesh0.mat : This is the computational mesh information for BioFVM at time 0.0. Because BioFVM and PhysiCell do not use moving meshes, we do not save this data at any subsequent time.\n3. output12345678_microenvironment0.mat : This saves each biochemical substrate in the microenvironment at the computational voxels defined in the mesh (see above).\u00a0Example:\u00a0output00003696_microenvironment0.mat\n4. output12345678_cells.mat : This saves very basic cellular information related to BioFVM, including cell positions, volumes, secretion rates, uptake rates, and secretion saturation densities.\u00a0Example:\u00a0output00003696_cells.mat\n5. output12345678_cells_physicell.mat : This saves extra PhysiCell data for each cell agent, including volume information, cell cycle status, motility information, cell death information, basic mechanics, and any user-defined custom data.\u00a0Example:\u00a0output00003696_cells_physicell.mat\n\nThese snapshots make extensive use of Matlab Level 4 .mat files, for fast, compact, and well-supported saving of array data. Note that even if you cannot ready MultiCellDS XML files, you can work to parse the .mat files themselves.\n\n### The PhysiCell Matlab .m files\n\nEvery PhysiCell distribution includes some matlab functions to work with PhysiCell digital simulation snapshots, stored in the matlab subdirectory. The main ones are:\n\n1. composite_cutaway_plot.m : provides a quick, coarse 3-D cutaway plot of the discrete cells, with different colors for live (red), apoptotic (b), and necrotic (black) cells.\n2. read_MultiCellDS_xml.m\u00a0: reads the \u201cbase\u201d PhysiCell snapshot and its associated matlab files.\n3. set_MCDS_constants.m : creates a data structure MCDS_constants that has the same constants as PhysiCell_constants.h. This is useful for identifying cell cycle phases, etc.\n4. simple_cutaway_plot.m : provides a quick, coarse 3-D cutaway plot of user-specified cells.\n5. simple_plot.m\u00a0: provides, a quick, coarse 3-D plot of the user-specified cells, without a cutaway or cross-sectional clipping plane.\n\n#### A note on GNU Octave\n\nUnfortunately, GNU octave does not include XML file parsing without some significant user tinkering. And one you\u2019re done, it is approximately one order of magnitude slower than Matlab. Octave users can directly import the .mat files described above, but without the helpful metadata in the XML file. We\u2019ll provide more information on the structure of these MAT files in a future blog post. Moreover, we plan to provide python and other tools for users without access to Matlab.\n\n### A sample digital snapshot\n\nWe provide a 3-D simulation snapshot from the final simulation time of the cancer-immune example\u00a0in Ghaffarizadeh et al. (2017, in review)\u00a0at:\n\nThe corresponding SVG cross-section for that time (through\u00a0= 0\u00a0\u03bcm) looks like this:\n\nUnzip the sample dataset in any directory, and make sure the matlab files above are in the same directory (or in your Matlab path). If you\u2019re inside matlab:\n\n!unzip 3D_PhysiCell_matlab_sample.zip\n\n\nMCDS = read_MultiCellDS_xml( 'output00003696.xml');\n\n\nThis will load the mesh, substrates, and discrete cells into the MCDS data structure, and give a basic summary:\n\nTyping \u2018MCDS\u2019 and then hitting \u2018tab\u2019 (for auto-completion) shows the overall structure of MCDS, stored as metadata, mesh, continuum variables, and discrete cells:\n\nTo get simulation metadata, such as the current simulation time, look at MCDS.metadata.current_time\n\nHere, we see that the current simulation time is 30240 minutes, or 21 days. MCDS.metadata.current_runtime gives the elapsed walltime to up to this point: about 53 hours (1.9e5 seconds), including file I\/O time to write full simulation data once per 3 simulated minutes after the start of the adaptive immune response.\n\n### Plotting chemical substrates\n\nLet\u2019s make an oxygen contour plot through z = 0 \u03bcm. First, we find the index corresponding to this\u00a0z-value:\n\nk = find( MCDS.mesh.Z_coordinates == 0 );\n\n\nNext, let\u2019s figure out which variable is oxygen. Type \u201cMCDS.continuum_variables.name\u201d, which will show the array of variable names:\n\nHere, oxygen is the first variable, (index 1). So, to make a filled contour plot:\n\ncontourf( MCDS.mesh.X(:,:,k), MCDS.mesh.Y(:,:,k), ...\nMCDS.continuum_variables(1).data(:,:,k) , 20 ) ;\n\n\nNow, let\u2019s set this to a correct aspect ratio (no stretching in x or y), add a colorbar, and set the axis labels, using\n\naxis image\ncolorbar\nxlabel( sprintf( 'x (%s)' , MCDS.metadata.spatial_units) );\nylabel( sprintf( 'y (%s)' , MCDS.metadata.spatial_units) );\n\n\nLastly, let\u2019s add an appropriate (time-based) title:\n\ntitle( sprintf('%s (%s) at t = %3.2f %s, z = %3.2f %s', MCDS.continuum_variables(1).name , ...\nMCDS.continuum_variables(1).units , ...\nMCDS.mesh.Z_coordinates(k), ...\n\n\nHere\u2019s the end result:\n\nWe can easily export graphics, such as to PNG format:\n\nprint( '-dpng' , 'output_o2.png' );\n\n\nFor more on plotting BioFVM data, see the tutorial\nat\u00a0http:\/\/www.mathcancer.org\/blog\/saving-multicellds-data-from-biofvm\/\n\n### Plotting cells in space\n\n#### 3-D point cloud\n\nFirst, let\u2019s plot all the cells in 3D:\n\nplot3( MCDS.discrete_cells.state.position(:,1) , MCDS.discrete_cells.state.position(:,2), ...\nMCDS.discrete_cells.state.position(:,3) , 'bo' );\n\n\nAt first glance, this does not look good: some cells are far out of the simulation domain, distorting the automatic range of the plot:\n\nThis does not ordinarily happen in PhysiCell (the default cell mechanics functions have checks to prevent such behavior), but this example includes a simple Hookean elastic adhesion model for immune cell attachment to tumor cells. In rare circumstances, an attached tumor cell or immune cell can apoptose on its own (due to its background apoptosis rate),\nwithout \u201cknowing\u201d to detach itself from the surviving cell in the pair. The remaining cell attempts to calculate its elastic velocity based upon an invalid cell position (no longer in memory), creating an artificially large velocity that \u201cflings\u201d it out of the simulation domain. Such cells \u00a0are not simulated any further, so this is effectively equivalent to an extra apoptosis event (only 3 cells are out of the simulation domain after tens of millions of cell-cell elastic adhesion calculations). Future versions of this example will include extra checks to prevent this rare behavior.\n\nThe plot can simply be fixed by changing the axis:\n\naxis( 1000*[-1 1 -1 1 -1 1] )\naxis square\n\n\nNotice that this is a very difficult plot to read, and very non-interactive (laggy) to rotation and scaling operations. We can make a slightly nicer plot by searching for different cell types and plotting them with different colors:\n\n% make it easier to work with the cell positions;\nP = MCDS.discrete_cells.state.position;\n\n% find type 1 cells\nind1 = find( MCDS.discrete_cells.metadata.type == 1 );\n% better still, eliminate those out of the simulation domain\nind1 = find( MCDS.discrete_cells.metadata.type == 1 & ...\nabs(P(:,1))' < 1000 & abs(P(:,2))' < 1000 & abs(P(:,3))' < 1000 );\n\n% find type 0 cells\nind0 = find( MCDS.discrete_cells.metadata.type == 0 & ...\nabs(P(:,1))' < 1000 & abs(P(:,2))' < 1000 & abs(P(:,3))' < 1000 );\n\n%now plot them\nP = MCDS.discrete_cells.state.position;\nplot3( P(ind0,1), P(ind0,2), P(ind0,3), 'bo' )\nhold on\nplot3( P(ind1,1), P(ind1,2), P(ind1,3), 'ro' )\nhold off\naxis( 1000*[-1 1 -1 1 -1 1] )\naxis square\n\n\nHowever, this isn\u2019t much better. You can use the scatter3 function to gain more control on the size and color of the plotted cells, or even make macros to plot spheres in the cell locations (with shading and lighting), but Matlab is very slow when plotting beyond 103 cells. Instead, we recommend the faster preview functions below for data exploration, and higher-quality plotting (e.g., by POV-ray) for final publication-\n\n#### Fast 3-D cell data previewers\n\nNotice that plot3 and scatter3 are painfully slow for any nontrivial number of cells. We can use a few fast previewers to quickly get a sense of the data. First, let\u2019s plot all the dead cells, and make them red:\n\nclf\nsimple_plot( MCDS, MCDS, MCDS.discrete_cells.dead_cells , 'r' )\n\n\nThis function creates a coarse-grained 3-D indicator function (0 if no cells are present; 1 if they are), and plots a 3-D level surface. It is very responsive to rotations and other operations to explore the data. You may notice the second argument is a list of indices: only these cells are plotted. This gives you a method to select cells with specific characteristics when plotting. (More on that below.) If you want to get a sense of the interior structure, use a cutaway plot:\n\nclf\nsimple_cutaway_plot( MCDS, MCDS, MCDS.discrete_cells.dead_cells , 'r' )\n\n\nWe also provide a fast \u201ccomposite\u201d cutaway which plots all live cells as red, apoptotic cells as blue (without the cutaway), and all necrotic cells as black:\n\nclf\ncomposite_cutaway_plot( MCDS )\n\n\nLastly, we show an improved plot that uses different colors for the immune cells, and Matlab\u2019s \u201cfind\u201d function to help set up the indexing:\n\nconstants = set_MCDS_constants\n\n% find the type 0 necrotic cells\nind0_necrotic = find( MCDS.discrete_cells.metadata.type == 0 & ...\n(MCDS.discrete_cells.phenotype.cycle.current_phase == constants.necrotic_swelling | ...\nMCDS.discrete_cells.phenotype.cycle.current_phase == constants.necrotic_lysed | ...\nMCDS.discrete_cells.phenotype.cycle.current_phase == constants.necrotic) );\n\n% find the live type 0 cells\nind0_live = find( MCDS.discrete_cells.metadata.type == 0 & ...\n(MCDS.discrete_cells.phenotype.cycle.current_phase ~= constants.necrotic_swelling & ...\nMCDS.discrete_cells.phenotype.cycle.current_phase ~= constants.necrotic_lysed & ...\nMCDS.discrete_cells.phenotype.cycle.current_phase ~= constants.necrotic & ...\nMCDS.discrete_cells.phenotype.cycle.current_phase ~= constants.apoptotic) );\n\nclf\n% plot live tumor cells red, in cutaway view\nsimple_cutaway_plot( MCDS, ind0_live , 'r' );\nhold on\n% plot dead tumor cells black, in cutaway view\nsimple_cutaway_plot( MCDS, ind0_necrotic , 'k' )\n% plot all immune cells, but without cutaway (to show how they infiltrate)\nsimple_plot( MCDS, ind1, 'g' )\nhold off\n\n\n### A small cautionary note on future compatibility\n\nPhysiCell 1.2.1 uses the <custom> data tag (allowed as part of the MultiCellDS specification) to encode its cell data, to allow a more compact data representation, because the current PhysiCell daft does not support such a formulation, and Matlab is\u00a0painfully\u00a0slow at parsing XML files larger than ~50 MB. Thus, PhysiCell snapshots are\u00a0not yet fully compatible with general MultiCellDS tools, which would by default ignore custom data. In the future, we will make available converter utilities to transform \u201cnative\u201d custom PhysiCell snapshots to MultiCellDS snapshots that encode all the cellular information in a more verbose but compatible XML format.\n\n### Closing words and future work\n\nBecause Octave is not a great option for parsing XML files (with critical MultiCellDS metadata), we plan to write\u00a0similar functions to read and plot PhysiCell snapshots in Python, as an open source alternative. Moreover, our lab in the next year will focus on creating further MultiCellDS configuration, analysis, and visualization routines. We also plan to provide additional 3-D functions for plotting the discrete cells and varying color with their properties.\n\nIn the longer term, we will develop open source, stand-alone analysis and visualization tools for MultiCellDS snapshots (including PhysiCell snapshots). Please stay tuned!","date":"2019-09-15 20:40:06","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\": 0.3779003620147705, \"perplexity\": 6547.0502978940085}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2019-39\/segments\/1568514572289.5\/warc\/CC-MAIN-20190915195146-20190915221146-00451.warc.gz\"}"}
| null | null |
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