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Q: Symfony2 extract text from javascript to translation I have installed BazingaJsTranslationBundle in my Symfony2 project.
It works fine but is there a way to extract text from javascript file the way done in translation:extract symfony2 command?
The text in javascript file looks like this:
Translator.trans("This is the text that I want to extract")
I do not want to go through adding keys in twig or php files to translate text in javascript files.
Here is the command that I use to extract text from php and twig files:
php app/console translation:extract en
--enable-extractor=jms_i18n_routing --dir=./src/ --output-dir=./app/Resources/translations
A: You can by creating a custom extractor for the JS translations. That's done by implementing the Symfony\Component\Translation\Extractor\ExtractorInterface:
/**
* Extracts translation messages from a template directory to the catalogue.
* New found messages are injected to the catalogue using the prefix.
*
* @author Michel Salib <michelsalib@hotmail.com>
*/
interface ExtractorInterface
{
/**
* Extracts translation messages from a template directory to the catalogue.
*
* @param string $directory The path to look into
* @param MessageCatalogue $catalogue The catalogue
*/
public function extract($directory, MessageCatalogue $catalogue);
/**
* Sets the prefix that should be used for new found messages.
*
* @param string $prefix The prefix
*/
public function setPrefix($prefix);
}
To make it usefull to others, it'll be even more awesome if you submitted your extractor to the BazingaJsTranslationBundles.
A: I opened a PR for this in the main repository: https://github.com/willdurand/BazingaJsTranslationBundle/pull/164
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 4,808
|
\section{Introduction}
\label{sec:1}
Diffusion MRI is a powerful non-invasive imaging technique widely used to explore white matter in the human brain.
Diffusion Tensor Imaging (DTI)~\citep{Basser1994} is used to reconstruct a tensor field from diffusion weighted images (DWIs).
High Angular Resolution Diffusion Imaging~\citep{TuchMRM2002,frank_MRM2002,Descoteaux2007,tournier_NI2007,Cheng_PDF_MICCAI2010,cheng_MICCAI2015,cheng_NI2014,ozarslan_NI13},
which makes no assumption of a 3D Gaussian distribution of the diffusion propagator, is used to reconstruct a general function field from DWIs,
(e.g., an Orientation Distribution Function (ODF) or Ensemble Average Propagator (EAP) field).
Both the ODF and the EAP fields with a given radius are spherical function fields.
Exploring microstructural information from the reconstructed tensor field or spherical function field is of interest in many biological and clinical application areas,
which makes diffusion MRI a powerful means to study white matter.
For example, in an voxel exhibiting anisotropic diffusion, local peaks of the reconstructed spherical function or the principal eigenvector of the reconstructed 2nd-order diffusion tensor normally prescribe the fiber directions in that voxel.
Some scalar indices have been proposed to be estimated voxel-wise from tensors/ODFs/EAPs.
For DTI, well-established tensor scalar indices, including the mean diffusivity and Fractional Anisotropy (FA), are widely used as biologically meaningful descriptors~\citep{Pierpaoli1996}.
\cite{Kindlmann_TMI2007} proposed two sets of scalar indices (three scalar indices per set), which are orthogonal in terms of tensor changes and the Euclidean inner product.
For High Angular Resolution Diffusion Imaging (HARDI), the generalized FA~\citep{Tuch2004}, Orientation Dispersion index (OD)~\citep{zhang_NODDI_NI2012}, return-to-origin probability~\citep{helmer_MRM2003,Wu_NI2007}, and mean-squared displacement~\citep{basser_MRM2002,Wu_NI2007}
were all proposed for ODFs and EAPs.
These indices indicate some information inside a voxel, but cannot describe local geometric or topological information, including fiber crossing, fanning, bending, and twisting, in a local spatial neighborhood.
Some previous works have extracted local geometric information by considering the local spatial change of tensor fields or ODF fields.
\cite{pajevic_JMR2002} demonstrated that the norm of the spatial gradient of the tensor's isotropic and anisotropic parts can detect boundaries between white matter, CSF, and gray matter.
\cite{Kindlmann_TMI2007} proposed tangents of scalar invariants and rotation tangents, which are 2nd-order tensors,
and also proposed projecting the 3rd-order spatial gradient tensor onto these 2nd-order tangents to obtain the spatial direction with the largest change of scalar indices or rotation of tensors.
Based on the rotation tangents of tensors, \cite{Savadjiev_NI2010} proposed fiber curving and fiber dispersion indices.
\cite{tax_NI2016} proposed a sheet probability index to quantify the local sheet structure by using spatial changes of ODF peaks.
\cite{duits:2009,duits:IJCV2011,portegies:PLOSOne2015} proposed spatial and spherical smoothing to enhance an ODF field in a PDE framework, preserving crossing structures.
\cite{reisert_TMI11,JianCheng_NNSD_ISBI13,michailovich:TMI2011} considered spatial coherence in ODF estimation.
The terms ``splay'', ``bend'' and ``twist'' have been used to qualitatively describe complex local white matter structural configuration in literature of diffusion MRI for about 20 years~\citep{basser_ANYAS1997,pajevic_JMR2002,dMRI_book2009}.
However, to our knowledge, there is no existing work that \emph{quantitatively} describes the degree of local orientational change of white matter,
including splay, bend, and twist, from general ODF fields in dMRI,
although the fiber curving and dispersion indices in~\cite{Savadjiev_NI2010} can be seen to quantify ``splay'' and ``bend'' for a tensor field in DTI.
\cite{basser_ANYAS1997} discussed the initial idea to study the torsion and curvature of a fiber tract by using the Frenet frame~\footnote{\href{https://en.wikipedia.org/wiki/Frenet-Serret_formulas}{https://en.wikipedia.org/wiki/Frenet-Serret\_formulas}} along the tract.
Torsion and curvature from the Frenet frame were later used in diffusion data analysis in~\cite{batchelor_MRM2006}.
\cite{savadjiev_ICCV2007} used the Frenet frame as a prior in the relaxation labeling algorithm to regularize the data and estimate ODFs in voxels.
These works on the Frenet frame studied geometric information along a single tract.
However, tractography is known to be sensitive to a large number of parameters,
and any flaws in the reconstructed tracts due to noise or parameter selection will inevitably be reflected in the geometric information that is extracted subsequently.
\cite{piuze_PAMI2015} proposed moving frames determined by the geometry of cardiac data, and calculated Maurer-Cartan connections.
However this method is not applicable to general diffusion MRI data, and does not consider the sign ambiguity in the frame.
There exist some connections between diffusion MRI data analysis and liquid crystals.
Orientational order parameter is well-established to describe the degree of alignment in liquid crystals~\cite{andrienko_2006}.
\cite{lasivc:FrontiersInPhysics2014,szczepankiewicz:NI2015} calculated the order parameter map by estimating variance of microscopic diffusion parameters from the contrast between diffusion signals measured by directional and isotropic diffusion encoding.
However, it cannot be used for general DTI and HARDI data.
\cite{topgaard:PCCP2016} used a diffusion tensor method to estimate the director orientations of a lyotropic liquid crystal as a spatially resolved field of Saupe order tensors.
In this paper, inspired by orientation and distortion analyses applied to liquid crystals,
we propose a unified framework, called Director Field Analysis (DFA), to study the local geometric information of white matter from the reconstructed spherical function field.
DFA works both for tensor fields obtained from DTI and for spherical function fields from HARDI.
At the voxel level,
1) the Orientational Order index (OO) and Orientational Dispersion index (OD) are defined for the spherical function in a voxel with a given axis (e.g., the ODF with its principal direction);
and 2) the principal direction is extracted from the spherical function in such a voxel exhibiting anisotropic diffusion.
At a local neighborhood level,
1) an orthogonal coordinate frame is defined for each voxel with anisotropic diffusion, where the first axis is the extracted principal direction;
2) OO is defined for spherical functions in a local neighborhood with the given principal direction;
and 3) three distortion indices (splay, bend, twist) and a total distortion index are defined based on the spatial directional derivatives of the principal direction.
An overview of the DFA pipeline for a spherical function field is shown in Fig.~\ref{fig:DFA}.
\begin{figure*}[t!]
\centering
\includegraphics[width=1.0\textwidth]{DFA_pipeline2}
\caption{\label{fig:DFA}Director Field Analysis (DFA) pipeline for an ODF field obtained from DTI or HARDI.
DFA provides total six scalar indices calculated from a spherical function field at the voxel level and at the local neighborhood level.}
\end{figure*}
This paper is organized as follows.
Section~\ref{sec:method_tensor} provides a unified overview of existing works on tensor field analysis for exploring local geometric information~\citep{pajevic_JMR2002,Kindlmann_TMI2007,Savadjiev_NI2010},
which is also a motivation for the proposed DFA.
Section~\ref{sec:method} proposes the DFA framework that works for both diffusion tensor fields and ODF fields.
Section~\ref{sec:experiments} demonstrates some results of synthetic and real data experiments by using DFA.
Section~\ref{sec:discussion} discusses some issues on implementing DFA.
\section{Tensor Field Analysis}
\label{sec:method_tensor}
This section provides an overview of existing works exploring the local geometric features of a 2nd-order diffusion tensor field by using the spatial gradient of tensors~\citep{pajevic_JMR2002,Kindlmann_TMI2007,Savadjiev_NI2010} in a unified framework.
It also proposes a new 4th-order structure tensor applied to 2nd-order tensor data that generalizes the conventional 2nd-order structure tensor applied to scalar fields.
\textbf{The 2nd-order diffusion tensor field}.
In a diffusion tensor field denoted as $\BM{D}$, there is a diffusion tensor $\BM{D}(\mathbi{x})$ at the voxel $\mathbi{x}$, where $\BM{D}(\mathbi{x}) \in S_+^3$,
and $S_+^3$ is the set of $3\times 3$ symmetric positive definite matrices.
The diffusion tensor is symmetric with six unique (i.e., independent) elements.
\textbf{The 3rd-order spatial gradient of the diffusion tensor}.
For a tensor field denoted as $\BM{D}$ with elements $[D_{ij}(\mathbi{x})]$ at the voxel $\mathbi{x}$, its spatial gradient at voxel $\mathbi{x}$, denoted as $\nabla_\mathbi{x} \BM{D}(\mathbi{x})$, is a 3rd-order tensor with elements $[D_{ij,k}(\mathbi{x})=\frac{\partial D_{ij}}{\partial x_k}(\mathbi{x})]$, where $i,j,k\in \{1,2,3\}$.
Since the diffusion tensor is symmetric with six unique elements, the 3rd-order spatial gradient has 18 unique elements.
\textbf{Mapping the 3rd-order spatial gradient to a vector}.
Let $\BM{W}=[W_{ij}]$ be a designed 2nd-order weighting tensor in tensor space,
then the tensor inner product
\begin{footnotesize}
\begin{equation}\label{eq:tensor2vector}
\BM{W} : \nabla_\mathbi{x} \BM{D} =\sum_{ij} W_{ij} D_{ij,k} = \sum_{ij} W_{ij} \frac{\partial D_{ij}}{\partial x_k} = \frac{\partial \sum_{ij}W_{ij} D_{ij}}{\partial x_k}
\end{equation}
\end{footnotesize}%
produces a vector in the image space that is the spatial gradient of the scalar field $\sum_{ij}W_{ij} D_{ij}$ at the voxel $\mathbi{x}$.
Note that the inner product in~\EEqref{eq:tensor2vector} is performed at voxel $\mathbi{x}$, and the $\mathbi{x}$ dependency is omitted in the notation if there is no ambiguity.
There are several ways to design a physically meaningful weighting tensor, $\BM{W}$.
$\BM{W}$ could be a constant independent of spatial position, $\mathbi{x}$, or a function of $\mathbi{x}$.
1) If $\BM{W}=\frac{1}{3} \BM{I}$, then $\sum_{ij}W_{ij} D_{ij}$ is the mean diffusivity field, and~\EEqref{eq:tensor2vector} is its spatial gradient.
2) If $f: \BM{D}\in S_+^3 \mapsto f(\BM{D})\in \mathbb{R}^1 $ is a scalar function that maps $\BM{D}$ to a scalar value, then $\frac{\partial f(\BM{D})}{\partial \BM{D}}$ is the gradient of the scalar function, which is also a 2nd-order tensor with elements $[\frac{\partial f(\BM{D})}{\partial D_{ij}}]$.
If we set $\BM{W}=\frac{\partial f(\BM{D})}{\partial \BM{D}}$, then the vector in~\EEqref{eq:tensor2vector} is the spatial gradient of the scalar field $f(\BM{D}(\mathbi{x}))$,
because of $\frac{\partial f}{ \partial \mathbi{x}} = \frac{\partial f}{\partial \BM{D}} : \frac{\partial \BM{D}}{\partial \mathbi{x}}$ by the chain rule.
If $f(\BM{D})$ is the mean diffusivity function, then $\frac{\partial f}{\partial \BM{D}}=\frac{1}{3}\BM{I}$.
We can also use other scalar invariants of tensors, e.g., FA.
3) If we choose $\BM{W}$ as the rotation tangent $\Phi_p(\BM{D})$~\citep{Kindlmann_TMI2007} defined as the change of tensor value due to infinitesimal rotations
around the $p$-th eigenvector, then~\EEqref{eq:tensor2vector} denotes the direction in which the tensor orientation
around the $p$-th eigenvector varies the fastest.
\textbf{Mapping the 3rd-order spatial gradient to a scalar value}.
Let $\BM{W}=[W_{ij}]$ be a 2nd-order weighting tensor in tensor space, and let $\mathbi{v}$ be a vector,
then the tensor inner product
\begin{align}
\BM{W} : \nabla_\mathbi{x} \BM{D} : \mathbi{v} &= \sum_{ijk} W_{ij}v_k \frac{\partial D_{ij}}{\partial x_k} \nonumber \\
&= \frac{\partial \sum_{ij}W_{ij} D_{ij}}{\partial \mathbi{v}} = \sum_{ij} W_{ij} \frac{\partial D_{ij}}{\partial \mathbi{v}} \label{eq:tensor2scalar}
\end{align}
produces a scalar value
that is the directional derivative of the scalar field $\sum_{ij}W_{ij} D_{ij}$ at voxel $\mathbi{x}$ along the vector $\mathbi{v}$,
and is also the weighted mean of the directional derivative of $\BM{D}(\mathbi{x})$ along the vector $\mathbi{v}$.
1) If we set $W_{ij}v_k = D_{ij,k}$, then~\EEqref{eq:tensor2scalar} is the squared norm of the tensor gradient,
which is useful for detecting boundaries of a tensor field~\citep{pajevic_JMR2002}.
2) By choosing $\mathbi{v}$ as the three eigenvectors of $\BM{D}$, and $\BM{W}$ as three rotation tangents around three eigenvectors, respectively,
we have total 9 scalar values to distinguish 9 configurations of tensor fields~\citep{Savadjiev_NI2010}.
3) The above 9 scalar indices can be combined to devise the fiber curving and fiber dispersion indices~\citep{Savadjiev_NI2010}.
\textbf{The 4th-order structure tensor}.
We propose a new 4th-order structure tensor with elements
$D_{ij,kl}= \frac{\partial D_{ij}}{\partial x_k} \frac{\partial D_{ij}}{\partial x_l}$,
which is analogous, but generalizes the structure tensor of a scalar field~\footnote{\href{https://en.wikipedia.org/wiki/Structure_tensor}{https://en.wikipedia.org/wiki/Structure\_tensor}}.
The above 4th-order structure tensor is minor symmetric~\citep{moakher_2009}, i.e., $D_{ij,kl}=D_{ji,kl}$, $D_{ij,kl}=D_{ij,lk}$.
Thus, there are 36 unique elements out of a possible total of $81$ elements, and there is one-to-one mapping between this 4th-order tensor and a 2nd-order tensor (i.e., a $6\times 6$ matrix).
However, since the 4th-order tensor is minor symmetric, the corresponding $6\times 6$ matrix is not symmetric in general.
Thus, eigenvalues and the 2nd-order left and right eigenvectors can be calculated based on eigen-decomposition of the non-symmetric $6\times 6$ matrix.
We may define some scalar invariants from these six eigenvalues of the 4th-order structure tensor, which can be used as features in this high dimensional space.
We can also contract the 4th-order structure tensor to a scalar value by using the tensor inner product
$\sum_{ijkl} W_{ij} D_{ij,kl} v_kv_l$,
which is the weighed mean of the squared spatial directional derivative along vector $\mathbi{v}$.
When setting $\mathbi{v}$ as three eigenvectors and corresponding weighting $\BM{W}$ as rotation tangents divided by the spatial gradient,
then the tensor inner product produces nine scalar indices that are the squares of the corresponding nine indices in~\cite{Savadjiev_NI2010}.
Thus, the curving and dispersion indices in~\cite{Savadjiev_NI2010} can also be obtained by choosing the corresponding $\BM{W}$.
\section{Method: Director Field Analysis}
\label{sec:method}
Section~\ref{sec:method_tensor} provides a unified framework to explore geometric structure information (e.g., boundary, curving, dispersion, etc.) from a tensor field, by considering a different weighting matrix on the spatial gradients.
However, it is challenging to generalize this framework to ODFs in HARDI, where ODFs are normally general spherical functions with antipodal symmetry.
In this section, we propose a novel mathematical framework, called Director Field Analysis (DFA).
Section~\ref{sec:dir} defines director related concepts to deal with vectors with sign ambiguity.
Section~\ref{sec:math_dir} provides a set of mathematical tools for DFA.
Section~\ref{sec:order} proposes OO and OD for tensors and ODFs in voxels and in a spatial neighborhood, and gives closed-form results in some specific cases.
Section~\ref{sec:frame} extracts the principal direction and its related local orthogonal frame in voxels exhibiting anisotropic diffusion.
Section~\ref{sec:distortion} defines four orientational distortion indices and demonstrates the implementation of the calculation by using the local orthogonal frame in Section~\ref{sec:frame}.
Section~\ref{sec:dir} and~\ref{sec:math_dir} are the theory part of DFA.
Section~\ref{sec:order}, \ref{sec:frame}, and~\ref{sec:distortion} are the application part of DFA in diffusion MRI.
Fig.~\ref{fig:DFA} demonstrates DFA to a spherical function field obtained from DTI or HARDI.
\subsection{Director and Director Field}
\label{sec:dir}
We define a \emph{director} as a unit norm vector $\mathbi{v}$ that is equivalent to $-\mathbi{v}$.
The \emph{director} term is borrowed from studies of liquid crystals~\footnote{\label{nt:lq}\href{https://en.wikipedia.org/wiki/Liquid_crystal}{https://en.wikipedia.org/wiki/Liquid\_crystal}}.
We also define a \emph{director with weight}, or \emph{weighted directors}, as a vector associated with a weight $(\mathbi{v}, w)$, which is equivalent to $(-\mathbi{v}, w)$, where $\|\mathbi{v}\|=1$, $w\in \mathbb{R}^1$.
If $w\geq 0$, then a weighted director $(\mathbi{v},w)$ can be represented as $w\mathbi{v}$. See Fig.~\ref{fig:directors}.
A director $\mathbi{v}$ can be uniquely represented as a dyadic tensor, $\mathbi{v}\Vv^T$, which avoids the sign ambiguity,
and a director with weight $(\mathbi{v},w)$ can be uniquely represented as a dyadic tensor, $w\mathbi{v} \mathbi{v}^T$.
We define a \emph{director field} as $\{(\ensuremath{\mathbf{u}}_i(\mathbi{x}), w_i(\mathbi{x}))\}$, where there are some weighted directors in each voxel $\mathbi{x}$.
Directors occur very often in diffusion MRI studies.
Eigenvectors of diffusion tensors, local maxima of ODFs, and local fiber directions are all directors.
Based on eigen-decomposition, a tensor $\BM{D}=\sum_{i=1}^3 \lambda_i \mathbi{v}_i\mathbi{v}_i^T$ is the sum of three dyadic tensors that represent three weighted directors.
A spherical function $f(\ensuremath{\mathbf{u}})$ which satisfies antipodal symmetry, i.e., $f(\ensuremath{\mathbf{u}})=f(-\ensuremath{\mathbf{u}})$, can be seen as infinite weighted directors $\{(\ensuremath{\mathbf{u}}_i, f(\ensuremath{\mathbf{u}}_i))\}$.
Thus, a spherical function field, $f(\ensuremath{\mathbf{u}},\mathbi{x})$, is a director field by definition.
An ODF in a voxel exhibiting anisotropic diffusion is anisotropic, and the orientations where the ODF takes its local peak (i.e., local maximal values) are normally considered to be local fiber directions in that voxel.
A normal peak detection algorithm for ODFs performs a grid search in a spherical mesh, and then refines the solution by using a gradient ascent on the continuous sphere~\citep{TournierNI2004}.
Note that peak detection is only performed for voxels exhibiting anisotropic diffusion (e.g., where ODFs have Generalized FA (GFA)~\citep{Tuch2004} values larger than $0.3$).
Moreover, in order to avoid including small peaks produced by noise, only peaks whose values are larger than a threshold percentage (e.g., $0.5$) of the largest ODF value are counted.
After peak detection, for each voxel $\mathbi{x}$, we obtain a discrete spherical function $g(\ensuremath{\mathbf{u}},\mathbi{x})=\sum_{i}f(\ensuremath{\mathbf{u}}_i,\mathbi{x})\delta(\ensuremath{\mathbf{u}}-\ensuremath{\mathbf{u}}_i)$ from the continuous spherical function $f(\ensuremath{\mathbf{u}},\mathbi{x})$, where $\{\ensuremath{\mathbf{u}}_i\}$ are local peaks.
This discrete spherical function field is also a \emph{director field}, or called a \emph{peak field}, which emphasizes local peaks and suppresses weights for other directors.
A peak field can also be extracted from a tensor field.
In each voxel for a tensor field, there is 0 or 1 peak,
and the principal eigenvector of the tensor is considered as a peak, if the tensor has a large FA value (e.g., larger than $0.3$).
\subsection{Mathematical Tools for Directors}
\label{sec:math_dir}
We provide a set of mathematical tools for analyzing directors and director fields.
These tools are useful not only for this paper, but also for other applications which deal with continuous or discrete director data.
\begin{figure*}[t!]
\begin{tabular}{c@{\hskip 0.1in} c@{\hskip 0.1in} c }
\includegraphics[scale=2,draft=false]{my_director_two} &
\includegraphics[scale=.26,draft=false]{director_3} &
\includegraphics[scale=2,draft=false]{my_director_two_difference} \\
(a) & (b) & (c)
\end{tabular}
\caption{\label{fig:directors}The mean, main, and difference of directors.
Directors $(\mathbi{v}_i, w_i)$, $i=1,2,3$, are visualized as vectors $w_i\mathbi{v}_i$ and $-w_i\mathbi{v}_i$, and the length of $w_i\mathbi{v}_i$ is the positive weight $w_i$.
(a) the mean (in blue) and main (in red) directors of two directors (in black), where $w_1=w_2$, and $\mathbi{v}_1^T\mathbi{v}_2>0$. See Proposition~\ref{prop:twodir}.
(b) the mean (in blue) and main directors of three directors (in black) which are orthogonal to each other, where $w_1>w_2>w_3$, $\mathbi{v}_i^T\mathbi{v}_j=\delta_i^j$.
The mean director is not unique, because $\frac{1}{3}\sum_i s_iw_i\mathbi{v}_i$ with an arbitrary sign assignment $\{s_i\}$ can be the mean director.
The main director is $w_1\mathbi{v}_1$.
(c) the difference of two directors.
The two blue vectors denote the director representation of the difference, i.e., $\texttt{Diff}_\text{d}(w_1\mathbi{v}_1, w_2\mathbi{v}_2)=w_1\mathbi{v}_1-w_2\mathbi{v}_2$.
The two red arcs denote the rotation matrix representation of the difference, i.e., $\texttt{Diff}_\text{r}(w_1\mathbi{v}_1, w_2\mathbi{v}_2)= \BM{R}$, where $\BM{R}$ is a scaled rotation matrix such that $w_1\mathbi{v}_1=\BM{R}w_2\mathbi{v}_2$.
The rotation matrix representation has no sign ambiguity, while the director representation has the sign ambiguity.}
\end{figure*}
\subsubsection{Mean Director of a Set of Directors}
For a given $N$ weighted directors $\{(\mathbi{v}_i, w_i)\}_{i=1}^N$, if we convert a director to a vector by assigning a sign, we have total $2^N$ possible sign assignments.
Thus, we have $2^N$ possible Euclidean mean vectors for the $N$ vectors.
We define a \emph{mean weighted director} of a set of weighted directors as the Euclidean mean vector with the sign assignment that takes the maximal norm among the $2^N$ mean vectors.
\begin{definition}{Mean director of weighted directors.}\label{def:mean_dir}
A mean weighted director of a set of weighted directors $\{(\mathbi{v}_i, w_i) \}_{i=1}^N$ is defined as $\texttt{Mean}(\{(\mathbi{v}_i, w_i) \}_{i=1}^N)=\frac{1}{N}\sum_{i=1}^N w_i s_i \mathbi{v}_i$,
where the signs $\{s_i\}= \argmax_{s_i=\{1,-1\}} \|\sum_{i=1}^N w_i s_i \mathbi{v}_i\|^2$, and $\texttt{Mean}(\cdot)$ is the mean director operator.
\end{definition}
It is obvious that $\{(\mathbi{v}_i, w_i)\}_{i=1}^N$ and $\{(\mathbi{v}_i, |w_i|)\}_{i=1}^N$ have the same mean director.
Thus, without loss of generality, we assume non-negative weights for calculating the mean director.
If the angle between any two vectors $\mathbi{v}_i$ and $\mathbi{v}_j$ is no more than $90^\circ$, then the sign assignment for the mean director can be proved to be $s_i=1$, $\forall i$.
See Proposition~\ref{prop:mean_cone_dir} whose proof is based on the proof of the mean director of two directors, which is trivial.
The mean director may be not unique when some directors are orthogonal.
\begin{proposition}{Mean director of weighted directors in a $90^\circ$ cone.}\label{prop:mean_cone_dir}
For a set of weighted directors $\{(\mathbi{v}_i, w_i)\}_{i=1}^N$ with non-negative weights,
if all directors are in a $90^\circ$ cone, i.e., $\mathbi{v}_i^T\mathbi{v}_j \geq 0 $, $\forall i, j$,
then the mean weighted director is $\frac{1}{N}\sum_{i=1}^N w_i \mathbi{v}_i$.
\end{proposition}
\subsubsection{Main Director of a Set of Directors}
We define the \emph{main director} of a set of weighted directors as the main axis in Principal Component Analysis (PCA) by using eigen-decomposition.
This concept is from the average director of molecule orientations in liquid crystals.
\begin{definition}{Main director of weighted directors.}\label{def:main_dir}
A \emph{main weighted director} of a set of weighted directors $\{(\mathbi{v}_i, w_i) \}_{i=1}^N$ is defined as $ \texttt{Main}(\{(\mathbi{v}_i, w_i) \}_{i=1}^N) = (\mathbi{v}_0, \lambda_0)$,
where $\lambda_0$ is the eigenvalue of the tensor $\sum_{i=1}^N w_i \mathbi{v}_i\mathbi{v}_i^T$ which has the largest absolute value among all eigenvalues, and $\mathbi{v}_0$ is its corresponding eigenvector, and $\texttt{Main}(\cdot)$ is the main director operator.
\end{definition}
Note
1) The dyadic tensor of the largest eigenvalue and eigenvector $\lambda_0 \mathbi{v}_0\mathbi{v}_0^T $ is the best rank-1 approximation of $\sum_{i=1}^N w_i \mathbi{v}_i\mathbi{v}_i^T$ in terms of the L2 norm.
2) The main director may not be unique, considering there may be more than one eigenvalues which are equal, and are all the largest eigenvalue.
3) Unlike the mean director which is independent of the signs of the weights, the main director is dependent on the weight signs.
4) Although, in general, the mean and the main directors are not the same, in some cases, they may give the same direction.
See Proposition~\ref{prop:twodir} and Fig.~\ref{fig:directors} (a) for the two director case.
See Fig.~\ref{fig:directors} (a) and (b) for an illustration of the mean and main directors.
\begin{proposition}{Two weighted directors with the same weight.}\label{prop:twodir}
For two weighted directors with the same weight, denoted as $(\mathbi{v}_1, w)$ and $(\mathbi{v}_2, w)$,
the main director is $(\frac{\mathbi{v}_1+\mathbi{v}_2}{\|\mathbi{v}_1+\mathbi{v}_2\|}, w(1+ \mathbi{v}_1^T\mathbi{v}_2))$ if $w \mathbi{v}_1^T\mathbi{v}_2 \geq 0$,
and is $(\frac{\mathbi{v}_1-\mathbi{v}_2}{\|\mathbi{v}_1-\mathbi{v}_2\|}, w(1 - \mathbi{v}_1^T\mathbi{v}_2))$ if $w \mathbi{v}_1^T\mathbi{v}_2 \leq 0 $.
The mean director is $\frac{|w|}{2}(\mathbi{v}_1+\mathbi{v}_2)$, if $\mathbi{v}_1^T\mathbi{v}_2 \geq 0$,
and is $\frac{|w|}{2}(\mathbi{v}_1-\mathbi{v}_2)$, if $\mathbi{v}_1^T\mathbi{v}_2 \leq 0$.
\end{proposition}
The mean director and main director describe different meaningful information about directors.
Take a diffusion tensor $\BM{D}=\sum_{i=1}^3 \lambda_i \mathbi{v}_i \mathbi{v}_i^T$, $\lambda_1>\lambda_2>\lambda_3>0$, as an example.
There are three weighted directors $\{(\mathbi{v}_i, \lambda_i)\}_{i=1}^3$, or represented as $\{\lambda_i\mathbi{v}_i\}$ because of non-negative weights.
The mean director is $\frac{1}{3}\sum_{i=1}^3 s_i \lambda_i \mathbi{v}_i$ with any sign assignment of $\{s_i\}$,
while the main director is $\lambda_1\mathbi{v}_1$.
See Fig.~\ref{fig:directors} (b).
Additionally, small changes in $\lambda_2$, $\lambda_3$, $\mathbi{v}_2$, and $\mathbi{v}_3$ may change the mean director, but not the main director, if $\lambda_1$ and $\mathbi{v}_1$ are still the largest eigenvalue and eigenvector.
This example clearly shows that the mean director concept is a generalization of the mean vector concept in vector space, while the main director emphasizes the main axis in PCA.
Please note that in a general case, the change of any director $(\mathbi{v}_i, w_i)$ may cause the change of the main director (i.e., the largest eigenvector and eigenvalue of the tensor $\sum_{i}w_i \mathbi{v}_i \mathbi{v}_i^T$) and also the mean director.
\subsubsection{Two Representations of the Difference between Two Directors}
\label{sec:diff_dir}
We aim to generalize the tensor field analysis in Section~\ref{sec:method_tensor} to director fields (i.e., ODF fields),
and explore geometric structure information by using spatial derivatives which rely on the concept of difference between two directors.
We propose two ways, i.e., the director representation denoted as $\texttt{Diff}_\text{d}$ and the rotation matrix representation denoted as $\texttt{Diff}_\text{r}$,
to represent the difference between two weighted directors with non-negative weights, $w_1\mathbi{v}_1$ and $w_2\mathbi{v}_2$.
These two directors can be converted to the vectors $w_1s_1\mathbi{v}_1$ and $w_2s_2\mathbi{v}_2$ by assigning the sign $s_1=1$ (or $s_1=-1$), and $s_2$ such that $w_1w_2s_1s_2\mathbi{v}_1^T\mathbi{v}_2 \geq 0$.
Thus, there are two different cases because of the sign ambiguity.
We can represent the difference as a director, i.e., $\texttt{Diff}_\text{d}(w_1\mathbi{v}_1, w_2\mathbi{v}_2) = w_1s_1\mathbi{v}_1 - w_2s_2\mathbi{v}_2$.
We can also represent the difference as a scaled rotation matrix $\BM{R}$ which rotates $w_2s_2\mathbi{v}_2$ to $w_1s_1\mathbi{v}_1$, i.e., $w_1s_1\mathbi{v}_1 = \BM{R}w_2s_2\mathbi{v}_2 $, $\texttt{Diff}_\text{r}(w_1\mathbi{v}_1, w_2\mathbi{v}_2) = \BM{R}$.
The rotation matrix can be calculated from the rotation axis $s_2\mathbi{v}_2\times s_1\mathbi{v}_1$ (i.e., the cross product of $s_2\mathbi{v}_2$ and $s_1\mathbi{v}_1$) and the rotation angle (i.e., the angle between $s_2\mathbi{v}_2$ and $s_1\mathbi{v}_1$)~\footnote{\url{https://en.wikipedia.org/wiki/Rotation_matrix\#Rotation_matrix_from_axis_and_angle}},
and the scale can be calculated from the weights $w_1$ and $w_2$.
Note that this rotation matrix is the same for the above two cases, without sign ambiguity.
See Fig.~\ref{fig:directors} (c) as an illustration.
The director representation of the difference has the sign ambiguity, but it gives a vector without a sign which can be projected onto axes.
The rotation matrix representation is unique without sign ambiguity, but cannot be projected onto axes.
\subsubsection{Spatial Gradient and Directional Derivative of a Director Field}
\label{sec:derivative_dir}
Considering a director field where there is only one director with non-negative weight $w\mathbi{v}(\mathbi{x})$ (simplified notation for $w(\mathbi{x})\mathbi{v}(\mathbi{x})$) at each position $\mathbi{x} \in \mathbb{R}^3$,
a directional derivative along $\ensuremath{\mathbf{u}}$ at $\mathbi{x}$ is defined as
\begin{equation}\label{eq:derivative_dir}
\frac{\partial w \mathbi{v}}{ \partial \ensuremath{\mathbf{u}}} = \lim_{k\to 0} \frac{\texttt{Diff}( w\mathbi{v}(\mathbi{x}+ k\ensuremath{\mathbf{u}}), w\mathbi{v}(\mathbi{x}- k\ensuremath{\mathbf{u}}))}{ 2 k}.
\end{equation}
Thus, there are also director and rotation matrix representations of the directional derivative because of the two representations of $\texttt{Diff}$, i.e., $\texttt{Diff}_\text{d}$ and $\texttt{Diff}_\text{r}$.
For the director field where directors $w\mathbi{v}(\mathbi{x})$ are only obtained in a integer lattice,
the central difference can be used to approximate the spatial gradient $[\frac{\partial w\mathbi{v} }{ \partial x_i}]$, where
\begin{small}
\begin{equation}\label{eq:central_diff}
\frac{\partial w\mathbi{v} }{ \partial x_i} \approx \texttt{Diff}(w\mathbi{v}(\mathbi{x}+\VV{o}_i) , \texttt{Mean}(\{w\mathbi{v}(\mathbi{x}-\VV{o}_i), w\mathbi{v}(\mathbi{x}+\VV{o}_i)\}) ),
\end{equation}
\end{small}%
$\VV{o}_1=[1,0,0]^T$, $\VV{o}_2=[0,1,0]^T$, $\VV{o}_3=[0,0,1]^T$ are the unit norm vectors along spatial axes,
and $\texttt{Mean}$ is the mean director operator in Definition~\ref{def:mean_dir}.
We normally use the rotation matrix representation for the spatial gradient $\frac{\partial w\mathbi{v}}{ \partial x_i}$, considering this representation is unique.
Let $\{\BM{R}_i(\mathbi{x})\}$ be the rotation matrices for the spatial gradient at $\mathbi{x}$ along axes $\{\mathbi{x}_i\}$, i.e., $\texttt{Diff}_\text{r}$ is used in~\EEqref{eq:central_diff}.
Then, analogously to the spatial gradient of a vector field, the director $w\mathbi{v}(\mathbi{x}+ k\ensuremath{\mathbf{u}})$ at position $\mathbi{x}+k\ensuremath{\mathbf{u}}$ with small $k$ can be approximated as the sum of rotated directors, i.e.,
\begin{footnotesize}
\begin{equation}\label{eq:weighted_ratation_dir}
\sum_{i=1}^3 k\VV{p}_i(\mathbi{x}), \quad \text{where} \ \ \VV{p}_i(\mathbi{x}) = \left\{ \begin{aligned}
& u_i\BM{R}_i(\mathbi{x}) w(\mathbi{x})\mathbi{v}(\mathbi{x}), \ \text{if}\ \ u_i \geq 0\\
& -u_i\BM{R}_i^T(\mathbi{x}) w(\mathbi{x})\mathbi{v}(\mathbi{x}), \ \text{if}\ \ u_i<0.
\end{aligned}
\right.
\end{equation}
\end{footnotesize}%
Note that every director in the above sum is a rotated $w(\mathbi{x})\mathbi{v}(\mathbi{x})$ in a small local rotation,
thus we assume all directors are in a $90^\circ$ cone to obtain a simple sum of vector representation.
See Proposition~\ref{prop:mean_cone_dir}.
If~\EEqref{eq:central_diff} is used to approximate $\BM{R}_i$, then the angle between $w\mathbi{v}(\mathbi{x})$ and $\BM{R}_i w\mathbi{v}(\mathbi{x})$ is no more than $45^\circ$,
thus all three rotated directors are indeed in a $90^\circ$ cone.
\subsection{Orientational Order and Dispersion}
\label{sec:order}
Before working on a field of ODFs, an ODF in a voxel can provide some geometric information at the voxel level,
including GFA~\citep{Tuch2004}, and orientation dispersion~\citep{zhang_NODDI_NI2012}.
\subsubsection{Orientational Order Transform and Orientational Tensor}
The NODDI model is increasingly used to study neurite orientation dispersion~\citep{zhang_NODDI_NI2012}.
NODDI uses the Watson distribution in~\EEqref{eq:watson} to model the ODF with a single orientation,
where $M$ is the confluent hypergeometric function,
$\mathbi{n}_0 \in \mathbb{S}^2$ is a given axis
\begin{footnotesize}
\begin{equation}\label{eq:watson}
f(\ensuremath{\mathbf{u}} \mid (\mathbi{n}_0, \kappa) ) = \frac{1}{4 \pi \ M(1/2, 3/2, \kappa) } \exp(\kappa (\ensuremath{\mathbf{u}}^T \mathbi{n}_0 )^2), \quad \ensuremath{\mathbf{u}} \in \mathbb{S}^2.
\end{equation}%
\end{footnotesize}%
Note that the original formula of the Watson distribution in~\cite{zhang_NODDI_NI2012} has no unit integral in $\mathbb{S}^2$, because it missed $4\pi$.
An orientation dispersion index (OD) was defined as~\EEqref{eq:OD_w}, where we denote it as $\text{OD}_{\text{w}}$ because it only applies to the Watson distribution.
\begin{equation}\label{eq:OD_w}
\text{OD}_{\text{w}}=\frac{2}{\pi}\arctan(\frac{1}{\kappa})
\end{equation
Note that in order to obtain good contrast in the dispersion index map,
in the NODDI toolbox provided by the authors, a scaled $\kappa$ ($10\kappa$ in the codes) is used in~\EEqref{eq:OD_w} to calculate $\text{OD}_{\text{w}}$, instead of the estimated $\kappa$ from the NODDI model.
$\text{OD}_{\text{w}}$ can not be used for ODFs that have general shapes, have more than one peak, or are not antipodally symmetric.
Some other works also proposed dispersion indices based on different models of ODFs, e.g., Bingham distributions~\cite{tariq:NI2016} and mv-$\Gamma$ distributions in DIAMOND~\cite{scherrer:MRM2015}.
These dispersion indices cannot work for general ODFs.
Inspired by liquid crystals, we would like to define the degree of dispersion for general ODFs, independent of microscopic diffusion signal models.
For a general spherical function $f(\ensuremath{\mathbf{u}})$, $\ensuremath{\mathbf{u}}\in\mathbb{S}^2$,
we define the \emph{orientational tensor} as
\begin{equation}\label{eq:OO_qtensor}
\BM{Q}(f) = \int_{\mathbb{S}^2} \ensuremath{\mathbf{u}}\uu^T f(\ensuremath{\mathbf{u}}) \rmd \ensuremath{\mathbf{u}},
\end{equation}
which is related to the Q-tensor in liquid crystal modeling~\citep{andrienko_2006}~\textsuperscript{\ref{nt:lq}}.
$\BM{Q}(f)$ is a $3\times 3$ symmetric matrix dependent on $f(\ensuremath{\mathbf{u}})$.
If $f(\ensuremath{\mathbf{u}})$ is non-negative, then $\BM{Q}(f)$ is positive semidefinite.
We propose the \emph{orientational order index (OO)} from the theory of liquid crystals~\citep{andrienko_2006}
to describe the orientation or dispersion of a general spherical function along a given axis $\VV{n}$:
\begin{footnotesize}
\begin{align}
\text{OO}(\VV{n}) &= \int_{\ensuremath{\mathbf{u}}\in\mathbb{S}^2} P_2(\ensuremath{\mathbf{u}}^T\VV{n}) f(\ensuremath{\mathbf{u}}) \rmd \ensuremath{\mathbf{u}} \nonumber \\
&=\int_{\ensuremath{\mathbf{u}}\in\mathbb{S}^2} \frac{3(\ensuremath{\mathbf{u}}^T \VV{n})^2 -1}{2} f(\ensuremath{\mathbf{u}}) \rmd \ensuremath{\mathbf{u}} \label{eq:OO}
\end{align}
\end{footnotesize}%
where $P_2$ is the 2nd-order Legendre polynomial.
By definition, \EEqref{eq:OO} is an \emph{integral transform} in $\mathbb{S}^2$ which converts the spherical function $f(\ensuremath{\mathbf{u}})$ to another spherical function $\text{OO}(\VV{n})$, and the kernel is $P_2(\ensuremath{\mathbf{u}}^T\VV{n})$,
similar to the Funk-Radon transform used in Q-Ball imaging~\citep{Tuch2004}, where the kernel is $\delta(\ensuremath{\mathbf{u}}^T\VV{n})$.
We call~\EEqref{eq:OO} the \emph{Orientational Order Transform (OOT)}, i.e., $\text{OOT}(f)=\text{OO}(\mathbi{n})$.
Note that we have
\begin{equation}
\text{OO}(\mathbi{n})= \frac{3}{2}\mathbi{n}^T \BM{Q}(f) \mathbi{n} - \frac{1}{2} \int_{\mathbb{S}^2} f(\ensuremath{\mathbf{u}}) \rmd \ensuremath{\mathbf{u}}.
\end{equation}
By definition, $\text{OO}(\mathbi{n})$ is antipodally symmetric and has a global maximum and a global minimum on the unit sphere, which correspond to the largest and smallest eigenvectors of $\BM{Q}(f)$, respectively.
Based on Definition~\ref{def:main_dir}, the main director of infinite weighted directors $\{(\ensuremath{\mathbf{u}}_i, f(\ensuremath{\mathbf{u}}_i))\}$ is the maximum point of $\text{OO}(\mathbi{n})$.
\begin{figure}[t!]
\centering
\includegraphics[scale=1.4,draft=false]{my_orientationalOrder_2d}
\caption{\label{fig:OO}A cross-section view of a spherical function $f(\ensuremath{\mathbf{u}})$ along axis $\mathbi{n}$. The projection of $f(\ensuremath{\mathbf{u}})\ensuremath{\mathbf{u}}$ onto $\mathbi{n}$ is $(f(\ensuremath{\mathbf{u}})\cos\theta)\mathbi{n}$.}
\end{figure}
Although $\text{OO}(\VV{n})$ is a spherical function, it is a scalar index when $\VV{n}$ is chosen as a physically meaningful axis, e.g., $f(\ensuremath{\mathbf{u}})$ takes its maximal value at $\VV{n}$.
Let $\theta$ be the angle between $\ensuremath{\mathbf{u}}$ and axis $\VV{n}$, then $P_2(\ensuremath{\mathbf{u}}^T\VV{n})=\frac{3\cos^2\theta-1}{2}$.
Thus, if $f(\ensuremath{\mathbf{u}})$ is a Probability Density Function (PDF) on the unit sphere,
then $\text{OO}(\VV{n})$ is $\langle \frac{3\cos^2\theta -1}{2} \rangle $, where $\langle \cdot \rangle$ signifies the expectation operation.
As shown in Fig.~\ref{fig:OO}, $\langle \cos^2\theta \rangle $ is the expectation of the squared projected length of $\ensuremath{\mathbf{u}}$ onto the axis $\VV{n}$.
If $f(\ensuremath{\mathbf{u}})$ is more concentrated along $\mathbi{n}$, then $\langle \cos^2\theta \rangle $ is larger, so is $\text{OO}(\mathbi{n})$.
Based on the definition, when $f(\ensuremath{\mathbf{u}})$ is a PDF, then we have $\text{OO}(\VV{n})\in[-0.5, 1]$.
If $f(\ensuremath{\mathbf{u}})=\delta(\ensuremath{\mathbf{u}}^T\VV{n}_0-1)$, i.e., the delta function along a given $\VV{n}_0$ axis, then $\text{OO}(\VV{n}_0)=1$.
If $f(\ensuremath{\mathbf{u}})=0$, $\forall \ensuremath{\mathbf{u}} \in \mathbb{S}^2$ such that $\ensuremath{\mathbf{u}}^T\VV{n}_0\neq 0 $, i.e., $f(\ensuremath{\mathbf{u}})$ is non-zero only in the plane orthogonal to $\VV{n}_0$, then $\text{OO}(\VV{n}_0)=-0.5$.
If $f(\ensuremath{\mathbf{u}})$ is the isotropic PDF, i.e., $f(\ensuremath{\mathbf{u}})=\frac{1}{4\pi}$, then $\text{OO}(\VV{n})=0$.
In practice, if we choose the axis $\VV{n}$ such that $f(\ensuremath{\mathbf{u}})$ takes its local or global maximal value, then $\text{OO}(\VV{n})$ is normally non-negative.
We define the \emph{orientational dispersion} along axis $\VV{n}$ as
\begin{equation}
\text{OD}(\VV{n})=1-\text{OO}(\VV{n}).
\end{equation}
Then $\text{OD}(\VV{n})\in [0,1.5]$.
Note that the proposed OO is different from the order parameter in~\cite{lasivc:FrontiersInPhysics2014,szczepankiewicz:NI2015} which was also inspired by liquid crystals~\citep{andrienko_2006}.
In \cite{lasivc:FrontiersInPhysics2014,szczepankiewicz:NI2015}, the order parameter is calculated by estimating the variance of microscopic diffusion parameters from the contrast between signals measured by directional and isotropic diffusion encoding.
However, it cannot be used for general DTI and HARDI data.
The proposed OO in this paper is defined for general spherical functions (i.e., ODFs) along a given axis, independent of microscopic diffusion models and reconstruction of the ODFs.
\subsubsection{Axisymmetric Spherical Functions}
When $f(\ensuremath{\mathbf{u}})$ is axisymmetric, and its axis is given by $\VV{n}_0$, i.e., $f(\ensuremath{\mathbf{u}})=f'(\ensuremath{\mathbf{u}}^T\VV{n}_0)$, where $f'(x)$ is the corresponding scalar function defined in $[-1,1]$,
then OOT has a closed form:
\begin{footnotesize}
\begin{align}
\text{OO}(\VV{n}) & = \int_0^\pi \left( \int_0^{2\pi} P_2( \cos\theta \cos t + \sin\theta\cos t \sin \phi)\rmd t \right)f'(\cos\theta) \rmd \theta \nonumber \\
&= \frac{(1+3\cos(2\phi))\pi}{2} a_2 = \frac{1+3\cos(2\phi)}{4} \text{OO}(\mathbi{n}_0) \label{eq:OOT_sym}
\end{align}
\end{footnotesize}
where $\phi=\arccos(|\mathbi{n}^T\mathbi{n}_0|)$ is the angle between $\mathbi{n}$ and the axis $\mathbi{n}_0$,
and $a_2 = \int_{-1}^1 P_2(x) f'(x) \rmd x$ is the 2nd-order Legendre coefficient of $f'(x)$.
Note that if $a_2>0$, when $\VV{n}=\VV{n}_0$, $\phi=0$, then $\text{OO}=2\pi a_2$ is the global maximum of $\text{OO}(\VV{n})$.
When $\VV{n}^T\VV{n}_0=0$, $\phi=\pi/2$, then $\text{OO}=-\pi a_2$ is the global minimum of $\text{OO}(\VV{n})$.
In the following development, without any ambiguity, we will use OO to denote $\text{OO}(\mathbi{n}_0)$, and OD to denote $\text{OD}(\mathbi{n}_0)$, for axisymmetric spherical functions.
\subsubsection{Watson Distributions}
The Watson distribution defined in~\EEqref{eq:watson} is axisymmetric with the axis $\VV{n}_0$.
Thus, based on the above analysis of axisymmetric spherical functions,
we have $\text{OO}(\mathbi{n})= \frac{1+3\cos(2\phi)}{4} \text{OO}$, and
\begin{equation}\label{eq:OO_watson}
\text{OO} = \frac{3 e^\kappa}{2\sqrt{\kappa\pi}\ \texttt{Erfi}(\sqrt{\kappa}) } - \frac{3+2\kappa}{4\kappa}
\end{equation}
where $\texttt{Erfi}(x)=\frac{2}{\sqrt{\pi}}\int_0^x \exp(t^2)\rmd t$ is the imaginary error function.
Then $\text{OD}=1-\text{OO}$.
The left part of Fig.~\ref{fig:OO_tensor_watson} shows the above $\text{OD}$ and $\text{OD}_{\text{w}}$ as functions of $\kappa$,
where the axis $\VV{n}$ is set as the Watson distribution's axis $\mathbi{n}_0$.
Both dispersion indices decrease as $\kappa$ increases.
Based on the derivation of $\kappa$, $\text{OD}_{\text{w}}$ is more sensitive to changes of $\kappa$ when $\kappa$ is small ($<2$),
while it is less sensitive when $\kappa$ is large ($>2$).
Compared with $\text{OD}_{\text{w}}$, the change of $\text{OD}$ is smoother for the change of $\kappa$ over the entire range of $\kappa$.
\subsubsection{Tensors}
For the tensor model in DTI, denoted as $\BM{D}$, OOT is defined for its ODF, i.e.,
\begin{equation}\label{eq:odf_tensor}
\Phi (\ensuremath{\mathbf{u}} \mid \BM{D}) = \frac{1}{4\pi |\BM{D}|^{\frac{1}{2}} } \frac{1}{(\ensuremath{\mathbf{u}}^T \BM{D}^{-1}\ensuremath{\mathbf{u}})^{\frac{3}{2}}},
\end{equation}%
which is a PDF on the unit sphere.
When the three eigenvalues of $\BM{D}$ satisfy $\lambda_1>\lambda_2=\lambda_3>0$,
$\Phi (\ensuremath{\mathbf{u}} \mid \BM{D})$ is an axisymmetric function with the axis $\mathbi{v}_1$ that is the principal eigenvector of $\BM{D}$.
OOT has a closed-form expression in~\EEqref{eq:OOT_sym}, and
\begin{footnotesize}
\begin{equation}\label{eq:OO_tensor}
\text{OO} = \frac{ \sqrt{\lambda_1-\lambda_2}(2\lambda_1+\lambda_2) - 3\lambda_1\sqrt{\lambda_2}\arctan\left(\sqrt{\frac{\lambda_1-\lambda_2}{\lambda_2}}\right) }{2(\lambda_1 - \lambda_2)^{\frac{3}{2}}}.
\end{equation}%
\end{footnotesize}%
The right panel of Fig.~\ref{fig:OO_tensor_watson} shows $\text{OO}$ and FA as functions of $\lambda_1/\lambda_2$, where we set $\VV{n}=\mathbi{v}_1$.
Both $\text{OO}$ and FA increases as $\lambda_1/\lambda_2$ increases.
Thus, $\text{OO}$ can be seen as a type of anisotropy index for tensors.
For general tensors with $\lambda_1>\lambda_2>\lambda_3>0$, no such closed form solution like~\EEqref{eq:OO_tensor} and~\EEqref{eq:OOT_sym} exists,
but we can calculate OO using the spherical harmonic representation of the ODF.
\begin{figure*}[t!]
\centering
\includegraphics[width=0.48\linewidth]{OO_watson}
\includegraphics[width=0.48\linewidth]{OO_symmetricTensor}
\caption{\label{fig:OO_tensor_watson}
Left: dispersion indices ($\text{OD}$ and $\text{OD}_{\text{w}}$) of a Watson distribution as functions of $\kappa$.
Right: OO and FA of prolate tensors ($\lambda_2 = \lambda_3$) as functions of $\frac{\lambda_1}{\lambda_2}$.}
\end{figure*}
\subsubsection{Spherical Harmonic Representation}
For a general spherical function $f(\ensuremath{\mathbf{u}})$, OO and OD can be analytically calculated from the spherical harmonic coefficients of the rotated function.
Considering $f(\ensuremath{\mathbf{u}})$ is a real function on the unit sphere, it can always be linearly represented by the real Spherical Harmonic (SH) basis $\{Y_l^m(\ensuremath{\mathbf{u}})\}$, i.e.,
\begin{equation}\label{eq:SH_f}
f(\ensuremath{\mathbf{u}}) = \sum_{l,m} c_{l,m} Y_l^m(\ensuremath{\mathbf{u}})
\end{equation}
\begin{equation}\label{eq:SH}
Y_l^m(\theta,\phi) =
\left\{ \begin{array}{lcl}
\sqrt{2}\mbox{Re}(y_l^{|m|}(\theta,\phi)) & \mbox{if} & -l\leq m<0 \\
y_l^m(\theta,\phi) & \mbox{if} & m=0 \\
\sqrt{2}\mbox{Im}(y_l^m(\theta,\phi)) & \mbox{if} & l\geq m>0
\end{array}\right.
\end{equation}%
where $y_l^m(\theta,\phi) = \sqrt{\frac{2l+1}{4\pi}\frac{(l-m)!}{(l+m)!}} e^{im\phi} P_l^m(\cos\theta)$ is the complex SH basis, $P_l^m(\cdot)$ is the associated Legendre polynomial.
For any rotation matrix, the SH coefficients of the rotated function can be calculated with very high accuracy
based on the Wigner D-matrix~\footnote{\href{https://en.wikipedia.org/wiki/Spherical_harmonics}{https://en.wikipedia.org/wiki/Spherical\_harmonics}},
or based on fitting rotated function samples~\citep{lessig_2012}.
Let $\BM{R}$ be the rotation matrix which rotates the axis $\VV{n}$ to $\VV{z}$-axis,
and $\{a_{l,m}\}$ be the real SH coefficients of the rotated function $(Rf)(\ensuremath{\mathbf{u}})=f(\BM{R}^{-1}\ensuremath{\mathbf{u}})$,
considering the orthogonality of the real SH basis and $Y_2^0(\theta,\phi)=\sqrt{\frac{5}{4\pi}}P_2(\cos\theta)$, we have
\begin{footnotesize}
\begin{align}
\text{OO}(\VV{n}) &= \int_{\ensuremath{\mathbf{u}}\in\mathbb{S}^2} P_2(\ensuremath{\mathbf{u}}^T\VV{n}) f(\ensuremath{\mathbf{u}}) \rmd \ensuremath{\mathbf{u}} \nonumber \\
&=\int_{\ensuremath{\mathbf{u}}\in\mathbb{S}^2} P_2(\cos \theta) \sum_{l,m} a_{l,m} Y_l^m(\theta, \phi) \rmd \ensuremath{\mathbf{u}} = \sqrt{\frac{4\pi}{5}} a_{2,0}. \label{eq:OO_sh}
\end{align}
\end{footnotesize}%
Note that $\text{OO}(\VV{n})$ is only determined by the rotated SH coefficient $a_{2,0}$ that is only related to $\{c_{2,m}\}_{-2\leq m \leq 2}$ and the axis $\VV{n}$, based on the rotation property of the SH basis.
Thus, $\text{OO}(\VV{n})$ is only related to the SH coefficients of $f(\ensuremath{\mathbf{u}})$ with $l=2$, and also the axis $\mathbi{n}$.
\subsubsection{Relationship Between OO, OD, and GFA}
\label{sec:gfa_OO}
For an ODF in an SH representation in~\EEqref{eq:SH_f}, its GFA~\citep{Tuch2004} is
\begin{equation}\label{eq:GFA}
\text{GFA} = \sqrt{ 1- \frac{ c_{0,0}^2}{\sum_{lm} c_{l,m}^2}}.
\end{equation}
Note that the rotation of a spherical function does not change the shape of the function and the norm of SH coefficients,
thus we have $\sum_{m}a_{2,m}^2=\sum_{m}c_{2,m}^2$.
Based on~\EEqref{eq:OO_sh}, we have
\begin{align}
\text{OO}(\mathbi{n}) &= \sqrt{\frac{4\pi}{5}} a_{2,0} \leq\sqrt{\frac{4\pi}{5}} \sqrt{\sum_{m=-2}^2 c_{2,m}^2} \nonumber \\
& \leq \sqrt{\frac{4\pi}{5}} \sqrt{\sum_{l\geq 2}\sum_{m=-l}^l c_{l,m}^2}.\label{eq:OO_ineq_1}
\end{align}
Combining~\EEqref{eq:GFA} and~\EEqref{eq:OO_ineq_1}, we have
\begin{equation}
\text{OO}(\mathbi{n}) \leq \sqrt{\frac{4\pi c_{0,0}^2}{5}} \sqrt{ \frac{1}{1-\text{GFA}^2} - 1}.
\end{equation}
The above inequality gives an upper bound of $\text{OO}(\mathbi{n})$ as a function of $\text{GFA}$ which is independent of $\mathbi{n}$.
Note that the above upper bound is tight, and the equality holds when $c_{l,m}=0$, for $l>2$, and $a_{2,m}=0$ for $m\neq 0$ after rotation.
If the ODF has unit integral, i.e., $\int_{\mathbb{S}^2} f(\ensuremath{\mathbf{u}}) \rmd \ensuremath{\mathbf{u}} =1$~\footnote{Note that ODFs estimated by some methods (e.g., constrained spherical deconvolution~\citep{tournier_NI2007}, Q-Ball Imaging~\citep{Tuch2004,Descoteaux2007}),
do not have the unit integral, if there is no normalization after estimation.}, then $c_{0,0}= \frac{1}{\sqrt{4\pi}}$, and we have
\begin{equation}\label{eq:OO_ineq}
\text{OO}(\mathbi{n}) \leq \sqrt{\frac{1}{5}} \sqrt{ \frac{1}{1-\text{GFA}^2} - 1 }
\end{equation}
Thus, for an ODF with low GFA, OO is also low, and OD is high, no matter how we choose the axis $\mathbi{n}$.
Note that~\EEqref{eq:OO_ineq} does not imply that an ODF with high GFA tends to have high OO, because it is an upper bound of $\text{OO}(\mathbi{n})$, not a lower bound.
\subsubsection{Mixture Model}
OOT in~\EEqref{eq:OO} is a linear transform.
Thus, if $f(\ensuremath{\mathbf{u}})=\sum_i w_i f_i(\ensuremath{\mathbf{u}})$ is the PDF of a mixture of models, where $f_i(\ensuremath{\mathbf{u}})$ is the PDF for the $i$-th model, and $w_i$ is the weight,
then $\text{OO}(\VV{n})= \sum_i w_i \text{OO}_i(\VV{n})$ is also a mixture of OO functions.
Fig.~\ref{fig:OO_tensor2} illustrates OO for a two-tensor model with a crossing angle $\phi$,
where two tensors share the same eigenvalues $[1.7,0.2,0.2]\times 10^{-3} mm^2/s$, the weights are $0.5$ and $0.5$,
and one tensor component is along the $y$-axis and the other one rotates from the $y$-axis to the $x$-axis.
Based on~\EEqref{eq:OOT_sym} and~\EEqref{eq:OO_tensor}, OO for the mixture model can be analytically calculated.
\subsubsection{OO and OD for a General ODF Along the Principal Peak}
\label{sec:OO_OD_odf}
In the above context, we focus on $\text{OO}(\mathbi{n})$ and $\text{OO}(\mathbi{n})$ as spherical functions.
A physically meaningful axis $\mathbi{n}_0$ is needed to obtain scalar indices of OO and OD from $\text{OO}(\mathbi{n})$ and $\text{OO}(\mathbi{n})$.
For an axisymmetric function $f(\ensuremath{\mathbf{u}})$, its axis can be used as described above.
For a general function (e.g., an ODF), we can set the axis as the local maxima of $f(\ensuremath{\mathbf{u}})$ (e.g., detected peaks of the ODF),
because the peaks of ODFs are considered as local fiber directions in dMRI.
A general ODF may have more than one peak.
The \emph{principal peak} of the anisotropic ODF $f(\ensuremath{\mathbf{u}})$, where the ODF takes its global maximum $\ensuremath{\mathbf{u}}_1$, i.e., $f(\ensuremath{\mathbf{u}}_1,\mathbi{x})>f(\ensuremath{\mathbf{u}},\mathbi{x})$, $\forall \ensuremath{\mathbf{u}}\in \mathbb{S}^2$,
is used to calculate OO and OD for the ODF.
Note that peaks are detected from ODFs with all orders of SH coefficients, not only SH coefficients with $l=2$.
Thus, the scalar indices of OO and OD are actually dependent on SH coefficients of ODFs with all orders.
See Algorithm~\ref{alg:OO_OD} for the pipeline to calculate OO and OD maps from a given ODF field with SH representation,
where peaks are detected for voxels whose GFA values are larger than a given threshold (e.g., $0.3$).
It is also possible to calculate OO and OD for all voxels by setting the GFA threshold as $0$.
As shown in Section~\ref{sec:gfa_OO}, for the voxels with $\text{GFA}<0.3$, we have $\text{OO}<0.14$ and $\text{OD}>0.86$.
\begin{algorithm}[!h]
\caption{\label{alg:OO_OD}\textbf{Calculation of OO and OD for ODFs with SH representation along principal peaks:}}
\SetAlgoLined
\KwIn{ODF field $f(\ensuremath{\mathbf{u}},\mathbi{x})=\sum_{lm}c_{l,m}(\mathbi{x})Y_l^m(\ensuremath{\mathbf{u}})$ in SH representation.}
\KwOut{OO map, OD map.}
Peak detection using gradient ascent for ODFs in voxels with the anisotropy higher than a given threshold (e.g., $\text{GFA}>0.3$). See Section~\ref{sec:dir}\;
\For{each voxel $\mathbi{x}$ with detected peaks $\{\ensuremath{\mathbf{u}}_i(\mathbi{x})\}$}{
1) Find the principal peak $\ensuremath{\mathbf{u}}_1$ with the largest ODF value, i.e., $f(\ensuremath{\mathbf{u}}_1,\mathbi{x})>f(\ensuremath{\mathbf{u}}_i,\mathbi{x})$, $\forall i$ \;
2) Calculate rotation matrix $\BM{R}$, which rotates $\ensuremath{\mathbf{u}}_1$ to the $\VV{z}$-axis \;
3) Calculate the rotated SH coefficient $a_{2,0}$ from $\{ c_{2,m}\}_{-2\leq m \leq 2}$ under the rotation $\BM{R}$ \;
4) $\text{OO}=\sqrt{\frac{4\pi}{5}} a_{2,0}$ as shown in~\EEqref{eq:OO_sh}, and $\text{OD}=1-\text{OO}$ \;
}
\end{algorithm}
\begin{figure*}[t!]
\centering
\includegraphics[width=0.45\linewidth]{OO_tensor2_phi}
\includegraphics[width=0.45\linewidth]{OO_tensor2_angles}
\caption{\label{fig:OO_tensor2}
OO for the mixture tensor model.
Left: OO as a function of the angle between two tensor components.
Right: ODF glyphs of the two-tensor model for different crossing angles,
where the yellow tube shows the $y$-axis which is used to calculate OO. }
\end{figure*}
\subsubsection{OO, OD and the Orientational Tensor in a Spatial Region}
The above OO, OD, and the orientational tensor are defined for a single voxel.
They can also be defined for voxels in a spatial region of voxels.
A linear weighting generalization of OO can be defined as
\begin{equation}\label{eq:OOT_region}
\text{OO}(\VV{n}) = \int_{\mathbi{x}\in \Omega} \int_{\ensuremath{\mathbf{u}}\in\mathbb{S}^2} P_2(\ensuremath{\mathbf{u}}^T\VV{n}) w(\mathbi{x}) f(\ensuremath{\mathbf{u}}, \mathbi{x}) \rmd \ensuremath{\mathbf{u}} \rmd \mathbi{x}.
\end{equation}
The orientational tensor in a spatial region is
\begin{equation}\label{eq:OO_qtensor_region}
\BM{Q}(f) = \int_{\mathbi{x}\in \Omega} \int_{\ensuremath{\mathbf{u}}\in\mathbb{S}^2} w(\mathbi{x}) f(\ensuremath{\mathbf{u}}, \mathbi{x}) \ensuremath{\mathbf{u}}\uu^T \rmd \ensuremath{\mathbf{u}} \rmd \mathbi{x}.
\end{equation}
Because of the linearity of the integration,
\EEqref{eq:OOT_region} is actually OOT in~\EEqref{eq:OO} performed on the region smoothed spherical function $\int_{\mathbi{x}\in \Omega} w(\mathbi{x}) f(\ensuremath{\mathbf{u}}, \mathbi{x}) \rmd \mathbi{x}$,
and \EEqref{eq:OO_qtensor_region} is the orientational tensor for the region smoothed function.
The largest eigenvector of $\BM{Q}(f)$ in~\EEqref{eq:OO_qtensor_region} indicates the main orientation of all ODFs $f(\ensuremath{\mathbf{u}},\mathbi{x})$ in the region $\Omega$.
\subsection{Local Orthogonal Frame}
\label{sec:frame}
\begin{figure*}[t!]
\centering
\begin{tabular}{c@{\hskip 0.1in} c@{\hskip 0.1in} c }
\includegraphics[width=0.31\textwidth]{fodf_splay_demo_odf_peaks} &
\includegraphics[width=0.31\textwidth]{fodf_splay_demo_odf_peaks_plane} &
\includegraphics[width=0.31\textwidth]{fodf_splay_demo_odf_peaks_plane_frame} \\
(a) & (b) & (c)
\end{tabular}
\caption{\label{fig:demo_frame}Sketch to determine local orthogonal frames from an ODF field, where an ODF may have 0, 1, or more than 1 peaks.
(a) an ODF field with peaks, where yellow tubes denote peaks.
(b) the orthogonal plane for the principal peak, where red tubes denote principal peaks.
(c) local orthogonal frames, where three tubes in red, green, and blue colors denote three directors in local orthogonal frames.}
\end{figure*}
As described in Section~\ref{sec:dir}, after peak detection on a spherical function field or a tensor field, the obtained peak field is also a director field.
We propose extracting a local orthogonal frame in each voxel exhibiting anisotropic diffusion from the detected peak field.
The orthogonal frame has three orthogonal orientations.
Denote the peaks at voxel $\mathbi{x}$ as $\{\ensuremath{\mathbf{u}}_i(\mathbi{x})\}$.
The first orientation is the \emph{principal peak} where the ODF takes its global maximum $\ensuremath{\mathbf{u}}_1(\mathbi{x})$, i.e., $f(\ensuremath{\mathbf{u}}_1,\mathbi{x})>f(\ensuremath{\mathbf{u}},\mathbi{x})$, $\forall \ensuremath{\mathbf{u}} \in \mathbb{S}^2$.
We call it the \emph{principal director} of the voxel $\mathbi{x}$.
The other two orientations are in the orthogonal plane of the principal direction.
Considering $f(\ensuremath{\mathbf{u}})$ is normally antipodally symmetric in diffusion MRI, all these orientations are equivalent to their antipodal ones.
Thus, we project all peaks in a spatial local neighborhood onto the orthogonal plane,
and define a weighted sum of dyadic tensors in voxel $\mathbi{x}$:
\begin{equation}\label{eq:dyadic_tensor}
\BM{Q}_\mathbi{x} = \sum_{\mathbi{y}\in\Omega_\mathbi{x}}\sum_{i} w(\mathbi{y},\mathbi{x}) f(\ensuremath{\mathbf{u}}_i(\mathbi{y}),\mathbi{y})\ensuremath{\mathbf{u}}_{i,\perp}(\mathbi{y})\ensuremath{\mathbf{u}}_{i,\perp}^T(\mathbi{y})
\end{equation}
\begin{equation}\label{eq:dyadic_tensor_2}
\ensuremath{\mathbf{u}}_{i,\perp}(\mathbi{y})=\ensuremath{\mathbf{u}}_i(\mathbi{y})-(\ensuremath{\mathbf{u}}_i^T(\mathbi{y})\ensuremath{\mathbf{u}}_1(\mathbi{x})) \ensuremath{\mathbf{u}}_1(\mathbi{x})
\end{equation}
where $\Omega_\mathbi{x}$ is a local neighborhood of voxel $\mathbi{x}$, $w(\mathbi{y},\mathbi{x})$ is the spatial weight which is normally set to be proportional to $\exp(-\frac{\|\mathbi{y}-\mathbi{x}\|^2}{2\sigma^2})$,
$\delta$ which is normally set as 1 voxel controls spatial weight concentration,
$\ensuremath{\mathbf{u}}_{i,\perp}(\mathbi{y})$ is the projected orientation $\ensuremath{\mathbf{u}}_i(\mathbi{y})$ onto the orthogonal plane of $\ensuremath{\mathbf{u}}_1(\mathbi{x})$.
The above $3\times 3$ matrix $\BM{Q}_\mathbi{x}$ is actually the orientational tensor of all projected peaks in region $\Omega_\mathbi{x}$ based on~\EEqref{eq:OO_qtensor_region},
where the continuous integral is replaced by a discrete summation over all projected peaks in region $\Omega_\mathbi{x}$.
Note that although we can define $\BM{Q}_\mathbi{x}$ using continuous ODF $f(\ensuremath{\mathbf{u}},\mathbi{x})$ with projected directors in a continuous integration like~\EEqref{eq:OO_qtensor_region},
we choose a discrete summation over peaks, which actually focuses only on peaks and sets zero weights for orientations that are not peaks in the continuous integration.
$\BM{Q}_\mathbi{x}$ in~\EEqref{eq:dyadic_tensor} has at most two non-zero eigenvalues, because it is defined by using $\{\ensuremath{\mathbf{u}}_{i,\perp}(\mathbi{y})\}$ in the orthogonal plane.
The eigenvector for the largest absolute eigenvalue of $\BM{Q}_\mathbi{x}$ is set as the second orientation of the orthogonal frame,
which is the main director of directors $\{(\ensuremath{\mathbf{u}}_{i,\perp},\ w(\mathbi{y},\mathbi{x})f(\ensuremath{\mathbf{u}}_i,\mathbi{y}) )\}$, and indicates the main orientation of the local spatial change of $\ensuremath{\mathbf{u}}_1(\mathbi{x})$ in the orientational plane.
Note that we define $\BM{Q}_\mathbi{x}$ using the isotropic spatial weight $w(\mathbi{y},\mathbi{x})$ to capture the general spatial change of the principal director $\ensuremath{\mathbf{u}}_1(\mathbi{x})$ in the orthogonal plane.
If one has a good motivation and specific spatial prior knowledge (e.g., to capture local change only in a specific region like hippocampus),
an anisotropic spatial weight $w(\mathbi{y},\mathbi{x})$ with consideration of spatial prior knowledge may be useful.
The third orientation in the orthogonal frame is set as the cross product of the first two orientations.
These three orientations are three orthogonal directors due to sign ambiguity.
Please see the sketch map in Fig.~\ref{fig:demo_frame} to determine local orthogonal frames from a given ODF field.
If these two eigenvalues of $\BM{Q}_\mathbi{x}$ are equal or their difference is very small, then we set the second and third orientations in the orthogonal frame to zero,
which means any two orthogonal vectors in the orthogonal plane can be the second and third axes in the orthogonal frame.
\subsection{Local Distortion Indices: Splay, Bend, and Twist}
\label{sec:distortion}
\begin{figure*}[t!]
\centering
\begin{tabular}{c@{\hskip 0.5in} c@{\hskip 0.5in} c }
\includegraphics[scale=.15]{dti_splay_demo} &
\includegraphics[scale=.15]{dti_bend_demo} &
\includegraphics[scale=.15]{dti_twist_demo} \\
splay & bend & twist
\end{tabular}
\caption{\label{fig:distortion}Demonstration of three types of distortions, i.e., splay, bend, and twist.}
\end{figure*}
\textbf{Three types of orientational distributions in liquid crystals}.
Based on the liquid crystal analogy, there are three fundamental types of distortions~\textsuperscript{\ref{nt:lq}} for the director field as demonstrated in Fig.~\ref{fig:distortion}.
1) \emph{splay}: bending occurs perpendicular to the director;
2) \emph{bend}: bending is parallel to the director and molecular axis;
3) \emph{twist}: neighboring directors are rotated with respect to one another, rather than aligned.
These three fundamental distortions can be used to describe a myriad of complex geometric patterns that liquid crystals can assume.
We would like to quantify these fundamental distortion patterns in dMRI by exploring the local spatial changes of principal directors.
\textbf{Spatial derivatives of the local orthogonal frame}.
With the local orthogonal frame $\{\ensuremath{\mathbf{u}}_1(\mathbi{x}),\ensuremath{\mathbf{u}}_2(\mathbi{x}),\ensuremath{\mathbf{u}}_3(\mathbi{x})\}$ at each voxel $\mathbi{x}$ obtained above,
we can define the spatial directional derivatives of $\ensuremath{\mathbf{u}}_i(\mathbi{x})$ along a direction $\mathbi{v}$ as
\begin{equation}\label{eq:derivative_dir_v}
\frac{\partial \ensuremath{\mathbf{u}}_i}{\partial \mathbi{v}}= \lim_{k\to 0} \frac{ \texttt{Diff}_\text{d}(\ensuremath{\mathbf{u}}_i(\mathbi{x}+ k\mathbi{v}), \ensuremath{\mathbf{u}}_i(\mathbi{x}- k\mathbi{v}))}{ 2 k}.
\end{equation}%
$\texttt{Diff}_\text{d}$ is the director representation of the difference of two directors as described in Section~\ref{sec:diff_dir}.
Note that we use the director representation for the spatial derivative, instead of a rotation matrix representation, because we would like to project the director onto different axes.
See Section~\ref{sec:diff_dir} and~\ref{sec:derivative_dir}.
\textbf{Spatial derivatives of vectors, and Maurer-Cartan connection forms in the moving frame method}.
If we assume $\{\ensuremath{\mathbf{u}}_i\}$ are all well-aligned unit vectors (i.e., no sign ambiguity),
then we have
\begin{equation}\label{eq:frame_derivative}
\frac{\partial \ensuremath{\mathbf{u}}_i}{\partial \mathbi{v}} = \nabla_\mathbi{x} \BM{\ensuremath{\mathbf{u}}}_i : \mathbi{v}
\end{equation}
with elements $\left[ \sum_k \frac{\partial u_{il}}{\partial x_k} v_k \right]$,
where $u_{il}$ is the $l$-th element of $\ensuremath{\mathbf{u}}_i$, and $\nabla_\mathbi{x} \BM{\ensuremath{\mathbf{u}}}_i=[ \frac{\partial u_{il}}{\partial x_k}]$ is the spatial gradient matrix of $\ensuremath{\mathbf{u}}_i(\mathbi{x})$.
Similarly with~\EEqref{eq:tensor2scalar}, we can extract some features by devising $\mathbi{v}$ and a weighting vector $\mathbi{w}$:
\begin{equation}\label{eq:frame_weight}
\mathbi{w}^T\frac{\partial \ensuremath{\mathbf{u}}_i}{\partial \mathbi{v}}= \mathbi{w} : \nabla_\mathbi{x} \BM{\ensuremath{\mathbf{u}}}_i : \mathbi{v} = \sum_{lm} w_l v_m \frac{\partial u_{il}}{\partial x_m}.
\end{equation}
\EEqref{eq:frame_weight} can be seen as a generalization of~\EEqref{eq:tensor2scalar} in tensor field analysis.
When we set $\mathbi{w}=\ensuremath{\mathbf{u}}_j$ and $\mathbi{v}=\ensuremath{\mathbf{u}}_k$, \EEqref{eq:frame_weight} is the projection of the directional derivatives onto $\ensuremath{\mathbf{u}}_j(\mathbi{x})$, denoted as $c_{ijk}$:
\begin{equation}
c_{ijk} = \ensuremath{\mathbf{u}}_j^T\frac{\partial \ensuremath{\mathbf{u}}_i}{\partial \ensuremath{\mathbf{u}}_k}= \ensuremath{\mathbf{u}}_j : \nabla_\mathbi{x} \BM{\ensuremath{\mathbf{u}}}_i : \ensuremath{\mathbf{u}}_k = \sum_{lm}u_{jl} u_{km} \frac{\partial u_{il}}{\partial x_m}.
\end{equation}%
$\{c_{ijk}(\mathbi{x})\}$ is the Maurer-Cartan connection form in the moving frame method~\footnote{\url{https://en.wikipedia.org/wiki/Maurer-Cartan_form}}.
$c_{ijk}(\mathbi{x})$ denotes the spatial change rate of frame vector $\ensuremath{\mathbf{u}}_i$ towards $\ensuremath{\mathbf{u}}_j$ when moving the frame along $\ensuremath{\mathbf{u}}_k$ at voxel $\mathbi{x}$~\citep{piuze_PAMI2015}.
\textbf{Orientational distortion indices}.
In this paper, instead of directly using the connections $\{c_{ijk}(\mathbi{x})\}$,
we propose three scalar indices to describe the relative prevalence of each of the three types of local distortions of white matter, inspired by liquid crystals~\citep{andrienko_2006}.
We define three indices and a total distortion index as
\begin{footnotesize}
\begin{align}
\text{Splay index:} \ \ & s = \sqrt{c_{122}^2 + c_{133}^2} = \sqrt{ (\ensuremath{\mathbf{u}}_2^T\frac{\partial \ensuremath{\mathbf{u}}_1}{\partial \ensuremath{\mathbf{u}}_2} )^2 + (\ensuremath{\mathbf{u}}_3^T\frac{\partial \ensuremath{\mathbf{u}}_1}{\partial \ensuremath{\mathbf{u}}_3} )^2} \label{eq:splay}\\
\text{Bend index:} \ \ & b = \sqrt{c_{121}^2 + c_{131}^2} = \sqrt{ (\ensuremath{\mathbf{u}}_2^T\frac{\partial \ensuremath{\mathbf{u}}_1}{\partial \ensuremath{\mathbf{u}}_1} )^2 + (\ensuremath{\mathbf{u}}_3^T\frac{\partial \ensuremath{\mathbf{u}}_1}{\partial \ensuremath{\mathbf{u}}_1} )^2} \label{eq:bend}\\
\text{Twist index:} \ \ & t = \sqrt{c_{123}^2 + c_{132}^2} = \sqrt{ (\ensuremath{\mathbf{u}}_2^T\frac{\partial \ensuremath{\mathbf{u}}_1}{\partial \ensuremath{\mathbf{u}}_3} )^2 + (\ensuremath{\mathbf{u}}_3^T\frac{\partial \ensuremath{\mathbf{u}}_1}{\partial \ensuremath{\mathbf{u}}_2} )^2} \label{eq:twist}
\end{align}
\begin{equation}\label{eq:distortion}
\text{Total distortion index:} \ \ d = \sqrt{s^2+b^2+t^2}.
\end{equation}
\end{footnotesize}%
\begin{algorithm}[t!]
\caption{\label{alg:derivative}\textbf{Calculation of spatial directional derivatives of the principal director:}}
\SetAlgoLined
\KwIn{A local orthogonal frame field $\{\ensuremath{\mathbf{u}}_1(\mathbi{x}), \ensuremath{\mathbf{u}}_2(\mathbi{x}), \ensuremath{\mathbf{u}}_3(\mathbi{x})\}$.}
\KwOut{Three spatial directional derivatives $\frac{\partial \ensuremath{\mathbf{u}}_1}{\partial \ensuremath{\mathbf{u}}_i}$, $i=1,2,3$.}
// Calculate three rotational matrices $\{\BM{R}_i\}$, $i=1,2,3$ \;
$\VV{o}_1=[1,0,0]^T$, $\VV{o}_2=[0,1,0]^T$, $\VV{o}_3=[0,0,1]^T$ \;
\For{$i=1,2,3$}{
$\mathbi{v}_1 = \ensuremath{\mathbf{u}}_1(\mathbi{x}+\VV{o}_i)$, $\mathbi{v}_0 = \ensuremath{\mathbf{u}}_1(\mathbi{x}-\VV{o}_i)$ \;
\eIf{$\mathbi{v}_1^T\mathbi{v}_0\geq 0$}
{$\mathbi{v}_2 = (\mathbi{v}_1+\mathbi{v}_0)/2$}
{$\mathbi{v}_2 = (\mathbi{v}_1-\mathbi{v}_0)/2$}
$\mathbi{v}_2= \frac{\mathbi{v}_2}{\|\mathbi{v}_2\|}$ \ \ \ \ // $\mathbi{v}_2$ is the normalized mean director of $\mathbi{v}_1$ and $\mathbi{v}_0$ \;
Calculate rotation matrix $\BM{R}_i$ which rotates $\mathbi{v}_2$ to $\mathbi{v}_1$ \;
}
// Calculate spatial directional derivatives from rotation matrices \;
\For{$i=1,2,3$}{
\For{$j=1,2,3$}{
$u_{i,j}=\ensuremath{\mathbf{u}}_i^T\VV{o}_j$ \;
\eIf{$u_{i,j}\geq 0$}
{ $\VV{p}_j = u_{i,j} \BM{R}_j \ensuremath{\mathbf{u}}_1 $, $\VV{n}_j = u_{i,j} \BM{R}_j^T \ensuremath{\mathbf{u}}_1 $ \;}
{ $\VV{p}_j = -u_{i,j} \BM{R}_j^T \ensuremath{\mathbf{u}}_1 $, $\VV{n}_j = -u_{i,j} \BM{R}_j \ensuremath{\mathbf{u}}_1 $ \; }
}
$\VV{p}_0 = \VV{p}_1+\VV{p}_2+\VV{p}_3$, $\VV{p}_0=\frac{\VV{p}_0}{\|\VV{p}_0\|}$ \;
$\VV{n}_0 = \VV{n}_1+\VV{n}_2+\VV{n}_3$, $\VV{n}_0=\frac{\VV{n}_0}{\|\VV{n}_0\|}$ \;
\eIf{$\|\VV{p}_0-\VV{n}_0\| \leq \|\VV{p}_0+\VV{n}_0\|$ }
{ $\frac{\partial \ensuremath{\mathbf{u}}_1}{\partial \ensuremath{\mathbf{u}}_i} = \VV{p}_0-\VV{n}_0$}
{ $\frac{\partial \ensuremath{\mathbf{u}}_1}{\partial \ensuremath{\mathbf{u}}_i} = \VV{p}_0+\VV{n}_0$}
}
\end{algorithm}
\textbf{Numerical calculation of spatial derivatives of directors and orientational distortion indices}.
Note that the above definitions of four indices and the formulae from~\EEqref{eq:frame_derivative} to~\EEqref{eq:distortion} are for a general vector frame in a vector field without sign ambiguity.
We would like to calculate the above four indices for the local orthogonal frames in Section~\ref{sec:frame} with sign ambiguity.
Squared values of $\ensuremath{\mathbf{u}}_j^T\frac{\partial \ensuremath{\mathbf{u}}_1}{\partial \ensuremath{\mathbf{u}}_i}$ in definitions are used to avoid the sign ambiguity of $\{\ensuremath{\mathbf{u}}_i\}$ and $\{\frac{\partial \ensuremath{\mathbf{u}}_1}{\partial \ensuremath{\mathbf{u}}_i}\}$.
The difficulty of numerically calculating the above four indices is that it is challenging to calculate the three spatial directional derivatives $\{\frac{\partial \ensuremath{\mathbf{u}}_1}{\partial \ensuremath{\mathbf{u}}_i}\}$, $i=1,2,3$,
because the local orthogonal frame $\{\ensuremath{\mathbf{u}}_i(\mathbi{x})\}$ with three directors is ambiguous with respect to its sign.
In other words, $\ensuremath{\mathbf{u}}_i$ is equivalent to $-\ensuremath{\mathbf{u}}_i$, considering the ODF and its peaks are antipodally symmetric.
We propose calculating the above spatial directional derivatives using a rotation matrix representation and a central difference approximation as described in Section~\ref{sec:derivative_dir}.
See Algorithm~\ref{alg:derivative} for a detailed implementation.
The algorithm first calculates three rotation matrices respectively along the $x$, $y$, $z$ axes, which is analogous to the spatial gradient of a vector field.
Then $\frac{\partial \ensuremath{\mathbf{u}}_1}{\partial \ensuremath{\mathbf{u}}_i}$ is numerically approximated by the director representation of the difference, i.e., $\texttt{Diff}_\text{d}(\ensuremath{\mathbf{u}}_1(\mathbi{x}+ \ensuremath{\mathbf{u}}_i), \ensuremath{\mathbf{u}}_1(\mathbi{x}- \ensuremath{\mathbf{u}}_i))$,
where $\ensuremath{\mathbf{u}}_1(\mathbi{x}+ \ensuremath{\mathbf{u}}_i)$ and $\ensuremath{\mathbf{u}}_1(\mathbi{x}- \ensuremath{\mathbf{u}}_i)$ are approximated by the weighted mean of three rotated vectors along three axes, as shown in~\EEqref{eq:weighted_ratation_dir}.
After $\{\frac{\partial \ensuremath{\mathbf{u}}_1}{\partial \ensuremath{\mathbf{u}}_i}\}$ are obtained, we can calculate the above four indices in~\EEqref{eq:splay},~\EEqref{eq:bend},~\EEqref{eq:twist}, and~\EEqref{eq:distortion}, from the directional derivatives.
Note that Algorithm~\ref{alg:derivative} avoids alignment of local frames in~\cite{piuze_PAMI2015} that does not work for general dMRI data.
\section{Experiments}
\label{sec:experiments}
\subsection{Synthetic Data Experiments}
\begin{figure*}[t!]
\resizebox{\textwidth}{!}{%
\begin{tabular}{c | c | c | c | c}
& splay & bend & twist & total distortion \\
\hline
\includegraphics[width=0.19\textwidth]{dti_splay} & \includegraphics[width=0.19\textwidth]{dti_splay_frame_splay} & \includegraphics[width=0.19\textwidth]{dti_splay_frame_bend} & \includegraphics[width=0.19\textwidth]{dti_splay_frame_twist} & \includegraphics[width=0.19\textwidth]{dti_splay_frame_distortion}\\
\hline
\includegraphics[width=0.19\textwidth]{dti_bend} & \includegraphics[width=0.19\textwidth]{dti_bend_frame_splay} & \includegraphics[width=0.19\textwidth]{dti_bend_frame_bend} & \includegraphics[width=0.19\textwidth]{dti_bend_frame_twist} & \includegraphics[width=0.19\textwidth]{dti_bend_frame_distortion}\\
\hline
\includegraphics[width=0.18\textwidth]{dti_twist} & \includegraphics[width=0.18\textwidth]{dti_twist_frame_splay} & \includegraphics[width=0.18\textwidth]{dti_twist_frame_bend} & \includegraphics[width=0.18\textwidth]{dti_twist_frame_twist} & \includegraphics[width=0.18\textwidth]{dti_twist_frame_distortion}\\
\end{tabular}
}
\caption{\label{fig:distortion_exp}Distortion indices calculated from different tensor fields.
In each row, the four distortion indices are calculated from the ODF field obtained from the tensor field.
The tensors were visualized by using superquadric tensor glyphs~\citep{kindlmann_2004}.}
\end{figure*}
Fig.~\ref{fig:distortion_exp} demonstrates these four orientational distortion indices (i.e, splay, bend, twist, and total distortion) calculated from idealized tensor fields.
The tensors are visualized by using superquadric tensor glyphs~\citep{kindlmann_2004}.
The first column of Fig.~\ref{fig:distortion_exp} shows different tensor fields.
The middle area of the first tensor field is the splaying area, while the middle area of the second tensor field is the bending region.
These two tensor fields are generated by rotating a tensor from left to right around the $z$-axis perpendicular to the page and decreasing the tensor mode~\citep{Kindlmann_TMI2007} from the bottom row to the top row.
The third tensor field shows the twist of tensor orientations,
which is generated by rotating a tensor around the $x$-axis (i.e., the left-to-right axis), and decreasing the tensor mode from bottom to top.
Fig.~\ref{fig:distortion_exp} shows that
1) the four indices only depend on the orientations (i.e., local orthogonal frame), not on the tensor or ODF shape;
2) splay, bend, twist indices provide complementary information about the orientational change, and demonstrate different types of orientational distortions.
Note that the twist index for the third tensor field is actually a constant, and the index value around the boundary is different due to the Neumann boundary condition used in the calculation.
Although the results in Fig.~\ref{fig:distortion_exp} are for tensor fields,
the distortion indices are actually determined by the local orthogonal frame field that can be calculated from a general spherical function field as described in Section~\ref{sec:frame}.
\begin{figure*}[t!]
\centering
\includegraphics[scale=.3]{dti_rotation}
\vspace{2mm} \\
\begin{tabular}{c c c}
\includegraphics[scale=.4]{dti_rotation_splay} & \includegraphics[scale=.4]{dti_rotation_bend} & \includegraphics[scale=.4]{dti_rotation_twist}\\
splay & bend & twist
\vspace{2mm}
\\
\includegraphics[scale=.4]{dti_rotation_dispersion} & \includegraphics[scale=.4]{dti_rotation_curving} & \includegraphics[scale=.4]{dti_rotation_distortion}\\
dispersion & curving & total distortion
\end{tabular}
\caption{\label{fig:distortion_exp_compare}Dispersion, curving~\citep{Savadjiev_NI2010}, and the proposed four orientational distortion indices calculated from a tensor field.}
\end{figure*}
We would like to compare the four orientational distortion indices with the curving and dispersion indices proposed for tensor fields in~\cite{Savadjiev_NI2010}.
The tensor field in Fig.~\ref{fig:distortion_exp_compare} was used in~\cite{Savadjiev_NI2010}.
It has three areas where the tensors rotate about its three eigenvectors, respectively.
From bottom to top, the mode of the tensors changes linearly.
Fig.~\ref{fig:distortion_exp_circle} shows two other synthetic tensor fields used in~\cite{Savadjiev_NI2010}.
Fig.~\ref{fig:distortion_exp_compare} and Fig.~\ref{fig:distortion_exp_circle} demonstrate all six scalar indices.
It can be seen that
1) the splay index is similar to the dispersion index;
2) the bend index is similar to the curving index;
3) the four orientational distortion indices are independent of tensor shapes, while curving and dispersion indices are dependent on the tensor mode;
4) when the principal directions are well aligned (e.g., in the left part of the tensor field in Fig.~\ref{fig:distortion_exp_compare}), all distortion indices are close to zero, because they are calculated based on the spatial difference of principal directions;
5) the definition and calculation of distortion indices are rotationally invariant.
Note that the singular values of these scalar indices around the central point in the tensor fields in Fig.~\ref{fig:distortion_exp_circle} are attributable to the singularity of the tensor orientation in the central point.
Although the proposed splay and bend indices have similar contrast compared with the dispersion and curving indices that are only for tensor fields,
the proposed distortion indices can be defined for both tensor fields and ODF fields.
Moreover, the proposed splay and bend indices are independent of tensor shapes, while the dispersion and curving indices are related with tensor shapes.
\begin{figure*}[t!]
\centering
\begin{tabular}{c c c c}
\includegraphics[scale=.18]{dti_circleBend} & \includegraphics[scale=.3]{dti_circleBend_dispersion} & \includegraphics[scale=.3]{dti_circleBend_curving} & \includegraphics[scale=.3]{dti_circleBend_distortion} \\
tensor field & dispersion & curving & total distortion
\vspace{2mm} \\
& \includegraphics[scale=.3]{dti_circleBend_splay} & \includegraphics[scale=.3]{dti_circleBend_bend} & \includegraphics[scale=.3]{dti_circleBend_twist} \\
& splay & bend & twist \\
\end{tabular}
\begin{tabular}{c c c c}
\includegraphics[scale=.18]{dti_circleSplay} & \includegraphics[scale=.3]{dti_circleSplay_dispersion} & \includegraphics[scale=.3]{dti_circleSplay_curving} & \includegraphics[scale=.3]{dti_circleSplay_distortion}\\
tensor field & dispersion & curving & total distortion
\vspace{2mm} \\
& \includegraphics[scale=.3]{dti_circleSplay_splay} & \includegraphics[scale=.3]{dti_circleSplay_bend} & \includegraphics[scale=.3]{dti_circleSplay_twist} \\
& splay & bend & twist
\end{tabular}
\caption{\label{fig:distortion_exp_circle}Dispersion, curving~\citep{Savadjiev_NI2010}, and the proposed four orientational distortion indices calculated from two tensor fields.}
\end{figure*}
\subsection{Real Data Experiments}
The experimental data are from Human Connectome Project (HCP), Q3 release~\citep{sotiropoulos_HCP_NI13,van:NI2013:HCP}.
This data set is acquired using three shells, with 90 staggered directions per shell, and at $b=1000,2000$, and $3000\,\text{s}/\text{mm}^2$.
We perform NODDI on the HCP multi-shell data using the released matlab toolbox by the authors~\citep{zhang_NODDI_NI2012}.
The first row in Fig.~\ref{fig:HCP_oo} shows the parameter maps by the NODDI toolbox, i.e., the $\kappa$ map and $\text{OD}_\text{w}$ map.
It should be noted that $\text{OD}_\text{w}$ is calculated based on $10\kappa$ and~\EEqref{eq:OD_w} in the author-released toolbox.
Fig.~\ref{fig:HCP_oo} also shows OO and OD from NODDI, based on the closed form in~\EEqref{eq:OO_watson} with a scaled $25\kappa$.
The scale on $\kappa$ is used for a better contrast in the obtained dispersion index map.
The obtained $\kappa$ map has intensities that are less than $0.4$ in most voxels.
Thus, as shown in Fig.~\ref{fig:OO_tensor_watson}, $10\kappa$ obtains the range $[0,4]$ which is good for $\text{OD}_\text{w}$,
and $25\kappa$ obtains the range $[0,10]$ which is good for $\text{OD}$.
The two dispersion index maps calculated in two ways from $\kappa$ visually have similar contrast.
Non-negative spherical deconvolution (NNSD)~\citep{cheng_NI2014} is performed to estimate non-negative fiber ODFs from three-shell DWI data.
NNSD works for multi-shell data.
It is more robust to noise, and the obtained fiber ODFs (fODFs) in isotropic regions are closer to the isotropic spherical PDF, compared with conventional constrained spherical deconvolution~\citep{tournier_NI2007}.
After obtaining the fODFs by NNSD, the peaks are detected from the estimated fODFs with GFA larger than $0.3$, as described in Section~\ref{sec:frame}.
OO and OD are calculated from the spherical harmonic representation of fODFs along their principal peaks as shown in Algorithm~\ref{alg:OO_OD}.
The second row in Fig.~\ref{fig:HCP_oo} demonstrates FA from tensors estimated by DTI, OO and OD from fODFs estimated by NNSD,
and the total distortion map estimated from the local orthogonal frames of fODFs.
Fig.~\ref{fig:HCP_region1} and~\ref{fig:HCP_region2} show the close-up views of
fODFs, local orthogonal frames, and the six proposed indices for the red and blue regions in Fig.~\ref{fig:HCP_oo},
where the region shown in~\ref{fig:HCP_region1} is also visualized in the DFA pipeline in Fig.~\ref{fig:DFA}.
The fODF glyphs are colored by using its sampled directions.
The three orientations in the local orthogonal frame in each voxel are visualized by using three tubes in red, green, and blue colors respectively.
There is no local orthogonal frame in some voxels because those voxels have GFA values lower than $0.3$.
These figures show the following:
1) OO is high in anisotropic areas with well-aligned directions, while OD is high in isotropic or crossing areas.
2) The four orientational distortion indices are low in areas with well aligned principal directions, and zero in isotropic voxels without peaks.
Distortion indices are high in voxels where the principal directions in its local neighborhood change largely.
3) The central voxels in red region is the crossing area of the Corpus Callosum from left to right and Fornix that goes through the coronal slice.
The twist index showed high value in this crossing area as expected.
OO and OD by NODDI are different from OO and OD by NNSD in Fig.~\ref{fig:HCP_oo}.
We propose OO and OD as general properties (i.e., the degree of aligment and dispersion along peaks) for general ODFs, like GFA for ODFs, independent of diffusion signal models.
OO and OD can be calculated from ODFs estimated by the NODDI model~\citep{zhang_NODDI_NI2012}, the tensor model in DTI~\citep{Basser1994}, and various spherical deconvolution methods~\citep{tournier_NI2007,cheng_NI2014}, etc.
In this sense, we claim that the proposed OD (and OO) inspired from liquid crystals is more general than the dispersion index in NODDI that only works for Watson distributions.
\begin{figure*}[t!]
\small
\centering
\begin{subfigure}[b]{0.24\textwidth}
\includegraphics[width=\textwidth]{hcp_c88_noddi_kappa}
\caption*{$\kappa$ from NODDI}
\end{subfigure}
\begin{subfigure}[b]{0.24\textwidth}
\includegraphics[width=\textwidth]{hcp_c88_noddi_odi}
\caption*{$\text{OD}_{\text{w}}$ from NODDI ($10\kappa$)}
\end{subfigure}
\begin{subfigure}[b]{0.24\textwidth}
\includegraphics[width=\textwidth]{hcp_c88_noddi_kappa25_OO}
\caption*{$\text{OO}$ from NODDI ($25\kappa$)}
\end{subfigure}
\begin{subfigure}[b]{0.24\textwidth}
\includegraphics[width=\textwidth]{hcp_c88_noddi_kappa25_OD}
\caption*{$\text{OD}$ from NODDI ($25\kappa$)}
\end{subfigure}
\begin{subfigure}[b]{0.24\textwidth}
\includegraphics[width=\textwidth]{hcp_c88_fa_with2regions}
\caption*{FA from tensors}
\end{subfigure}
\begin{subfigure}[b]{0.24\textwidth}
\includegraphics[width=\textwidth]{hcp_c88_NNSD_distortion}
\caption*{total distortion from fODFs}
\end{subfigure}
\begin{subfigure}[b]{0.24\textwidth}
\includegraphics[width=\textwidth]{hcp_c88_NNSD_OO}
\caption*{OO from fODFs}
\end{subfigure}
\begin{subfigure}[b]{0.24\textwidth}
\includegraphics[width=\textwidth]{hcp_c88_NNSD_OD}
\caption*{OD from fODFs}
\end{subfigure}
\caption{\label{fig:HCP_oo}
First row: NODDI results for multi-shell HCP data,
where $\kappa$ is estimated from NODDI model,
$\text{OD}_\text{w}$ is calculated from~\EEqref{eq:OD_w} by using $10\kappa$,
and $\text{OO}$ and $\text{OD}$ are calculated from~\EEqref{eq:OO_watson} by using $25\kappa$.
Second row: DTI and NNSD results for HCP data, where OO, OD and total orientational distortion are calculated from fODFs by NNSD.
The close-up views of red and blue regions are in Fig.~\ref{fig:HCP_region1} and~\ref{fig:HCP_region2}.
}
\end{figure*}
\begin{figure*}[t!]
\begin{tabular}{c c c }
\includegraphics[width=0.32\textwidth]{hcp_c88_region1_fODF_OO} & \includegraphics[width=0.32\textwidth]{hcp_c88_region1_fODF_OD} & \includegraphics[width=0.32\textwidth]{hcp_c88_region1_fODF_distortion} \\
OO & OD & total distortion\\
\includegraphics[width=0.32\textwidth]{hcp_c88_region1_fODF_splay} & \includegraphics[width=0.32\textwidth]{hcp_c88_region1_fODF_bend} & \includegraphics[width=0.32\textwidth]{hcp_c88_region1_fODF_twist}\\
splay & bend & twist
\end{tabular}
\caption{\label{fig:HCP_region1}fODFs, local orthogonal frames, and six scalar indices for the red region in Fig.~\ref{fig:HCP_oo}.
Local orthogonal frames are visualized using tubes in red, green, and blue colors.
The scalar indices are shown in the background. This region is also used in the DFA pipeline in Fig.~\ref{fig:DFA}
}
\end{figure*}
\begin{figure*}[t!]
\begin{tabular}{c c c }
\includegraphics[width=0.32\textwidth]{hcp_c88_region2_fODF_OO} & \includegraphics[width=0.32\textwidth]{hcp_c88_region2_fODF_OD} & \includegraphics[width=0.32\textwidth]{hcp_c88_region2_fODF_distortion}\\
OO & OD & total distortion\\
\includegraphics[width=0.32\textwidth]{hcp_c88_region2_fODF_splay} & \includegraphics[width=0.32\textwidth]{hcp_c88_region2_fODF_bend} & \includegraphics[width=0.32\textwidth]{hcp_c88_region2_fODF_twist}\\
splay & bend & twist
\end{tabular}
\caption{\label{fig:HCP_region2}fODFs, local orthogonal frames, and six scalar indices for the blue region in Fig.~\ref{fig:HCP_oo}.
Local orthogonal frames are visualized using tubes in red, green, and blue colors.
The scalar indices are shown in the background.
}
\end{figure*}
We perform whole brain streamline tractography on the estimated fODF field using mrtrix~\citep{tournier_mrtrix_12}~\footnote{\href{http://www.mrtrix.org}{http://www.mrtrix.org/}}.
The voxels with GFA larger than $0.3$ are used as seed voxels to generate $10000$ tracts by using \texttt{tckgen} in mrtrix.
All other parameters are default parameters in mrtrix.
The obtained fiber tracts are then visualized by using trackvis~\footnote{\href{http://trackvis.org}{http://trackvis.org}}.
Fig.~\ref{fig:HCP_roi2_track} and~\ref{fig:HCP_roi_track} demonstrate the tracts respectively cross two given ball ROIs.
The tracts are colored by using the proposed six scalar indices.
Note that the proposed scalar indices are calculated based on estimated fODFs, not based on fiber tracts.
It can be seen that
1) OO is high in areas with well aligned fibers, while OD is high in crossing areas and distortion areas;
2) distortion indices are low when fibers are well aligned;
3) the total distortion index is high in areas with highly curved fibers or crossing fibers.
4) although splay, bend, twist indices may be separable (e.g., one is large while another one is close to zero) in synthetic data,
in real data, these three types of distortions normally occur together, especially for bending and splaying.
5) the ROI in Fig.~\ref{fig:HCP_roi2_track} is the crossing area of the Corpus Callosum and the Fornix, where all distortion indices have high values, especially for twist and total distortion indices.
This finding agrees with Fig.~\ref{fig:HCP_region1}.
\begin{figure*}[t!]
\centering
\includegraphics[width=0.265\textwidth]{hcp_track_roi2} \\
ROI on OO map \\
\begin{tabular}{c c c }
\includegraphics[width=0.32\textwidth]{hcp_track_roi2_OO} & \includegraphics[width=0.32\textwidth]{hcp_track_roi2_OD} & \includegraphics[width=0.32\textwidth]{hcp_track_roi2_distortion}\\
OO & OD & total distortion \\
\includegraphics[width=0.32\textwidth]{hcp_track_roi2_splay} & \includegraphics[width=0.32\textwidth]{hcp_track_roi2_bend} & \includegraphics[width=0.32\textwidth]{hcp_track_roi2_twist}\\
splay & bend & twist
\end{tabular}
\caption{\label{fig:HCP_roi2_track}Fiber tracts cross a given ROI are colored by the six indices, respectively.}
\end{figure*}
\begin{figure*}[t!]
\centering
\includegraphics[width=0.265\textwidth]{hcp_track_roi} \\
ROI on OO map \\
\begin{tabular}{c c c }
\includegraphics[width=0.32\textwidth]{hcp_track_roi_OO} & \includegraphics[width=0.32\textwidth]{hcp_track_roi_OD} & \includegraphics[width=0.32\textwidth]{hcp_track_roi_distortion}\\
OO & OD & total distortion \\
\includegraphics[width=0.32\textwidth]{hcp_track_roi_splay} & \includegraphics[width=0.32\textwidth]{hcp_track_roi_bend} & \includegraphics[width=0.32\textwidth]{hcp_track_roi_twist}\\
splay & bend & twist
\end{tabular}
\caption{\label{fig:HCP_roi_track}Fiber tracts cross a given ROI colored by the six indices, respectively.}
\end{figure*}
\section{Discussion}
\label{sec:discussion}
\subsection{Effect of the Spatial Resolution on Directional Derivatives and Distortion Indices}
The definition of the spatial directional derivative in~\EEqref{eq:derivative_dir} is for continuous spatial domain.
The unit of the spatial directional derivative $\frac{\partial \ensuremath{\mathbf{u}}_i}{ \partial \mathbi{v}}$ in~\EEqref{eq:derivative_dir_v} is $[\si{mm^{-1}},\si{mm^{-1}},\si{mm^{-1}}]$,
considering the numerator is a director with no unit and the unit of the denominator is $\si{mm}$.
Thus, the four distortion indices have the unit of $\si{mm^{-1}}$.
The spherical function field and peak field in diffusion MRI are obtained in a discrete integer lattice.
In Algorithm~\ref{alg:derivative}, rotation matrices $\{\BM{R}_i\}$ are calculated based on directors in neighborhood voxels $\ensuremath{\mathbf{u}}_1(\mathbi{x}+\VV{o}_i)$ and $\ensuremath{\mathbf{u}}_1(\mathbi{x}-\VV{o}_i)$.
Thus, these three rotation matrices are dependent on spatial resolution of the diffusion image.
So are the central difference approximation of the spatial gradient in~\EEqref{eq:central_diff}, and the distortion indices calculated based on directional derivatives.
Consider the twisting synthetic tensor image in Fig.~\ref{fig:distortion_exp} as an example, where the tensor from left to right rotates about the $x$-axis by the angle $\pi$.
If there are $N+1$ tensors from left to right, then the spatial gradient along the $x$-axis is the rotation matrix with a rotation angle of $\pi/N$.
With the local linear change assumption of rotation angles, finer spatial resolution will produce smaller rotation angles of central differences in the three rotation matrices,
which results in smaller spatial gradients, directional derivatives and smaller distortion indices.
An improved version of calculation of the spatial directional derivatives in Algorithm~\ref{alg:derivative} is to consider the spatial resolution of the image as the step size of the central difference.
The image resolution should be used to normalize the rotation angles in the rotation matrices.
We can approximate the spatial gradient using all directors within a given physical resolution, e.g., $3\times 3\times 3~\si{mm}$.
If the image resolution is also $3~\si{mm}$ isotropic, then we just use the central difference described in Section~\ref{sec:derivative_dir}.
If the image resolution is $1.5~\si{mm}$ isotropic, then we can use a mean of rotation matrices from two central differences.
One rotation matrix is generated by $w\mathbi{v}(\mathbi{x}+\VV{o}_i)$, $w\mathbi{v}(\mathbi{x}-\VV{o}_i)$, then we keep the rotation axis, but scale the rotation angle by $2$, based on the local linear rotation angle assumption.
The other one is generated by $w\mathbi{v}(\mathbi{x}+2\VV{o}_i)$, $w\mathbi{v}(\mathbi{x}-2\VV{o}_i)$.
The Riemannian mean is used to calculate the mean of rotation matrices~\citep{moakher_SIAM2002}.
In this way, for the twisting synthetic tensor image in Fig.~\ref{fig:distortion_exp}, the rotation matrix representation of the central difference along the $x$-axis remains the same for different spatial resolutions of the synthetic image.
Note that the local linear rotation change assumption only holds in a small local neighborhood, not for a large spatial scale.
With the correct consideration of image resolution in calculation,
the image resolution effect can be reduced in numerical calculation of the proposed distortion indices.
\subsection{DFA For General Spherical Functions Without Antipodal Symmetry}
\label{sec:DFA_vector}
Considering spherical functions obtained in diffusion MRI are normally antipodally symmetric, the detected principal directors and local orthogonal frames all have sign ambiguity.
Thus, the proposed DFA is mainly for director data analysis.
However, if the reconstructed spherical function in a voxel (e.g., an ODF) is not antipodally symmetric, the detected principal peak field is a traditional vector field.
Then DFA can be modified for vector field analysis.
The difference of two vectors, the spatial gradient, and the spatial derivative of a vector field are all well defined.
Note that the definitions of OO and OD in Section~\ref{sec:order} work for a general spherical function $f(\ensuremath{\mathbf{u}})$ without requiring antipodal symmetry, although $\text{OO}(\mathbi{n})$ is always antipodally symmetric by definition.
If the detected peaks have no sign ambiguity, then the three orientations in the local orthogonal frame can all be traditional vectors.
The first orientation is the principal peak at voxel $\mathbi{x}$.
As described in Section~\ref{sec:frame}, after projecting all peaks onto the orthogonal plane, we obtain vectors $\{w(\mathbi{y},\mathbi{x}) f(\ensuremath{\mathbf{u}}_i,\mathbi{y}) (\ensuremath{\mathbf{u}}_i-(\ensuremath{\mathbf{u}}_i^T\ensuremath{\mathbf{u}}_1) \ensuremath{\mathbf{u}}_1)\}$ without sign ambiguity.
Then, the second orientation can be set as the orientation in the orthogonal plan with maximal value among $|w(\mathbi{y},\mathbi{x}) f(\ensuremath{\mathbf{u}}_i,\mathbi{y})|\|\ensuremath{\mathbf{u}}_i-(\ensuremath{\mathbf{u}}_i^T\ensuremath{\mathbf{u}}_1) \ensuremath{\mathbf{u}}_1)\|$, $\forall i$, or the mean orientation of the projected vectors in the orthogonal plane,
and the third orientation is the cross product of the first and second orientations.
Finally, the four orientational distortion indices in~\EEqref{eq:splay}, \EEqref{eq:bend}, \EEqref{eq:twist}, and~\EEqref{eq:distortion} can still be used for the spatial derivatives of the vector field, which are actually functions of Maurer-Cartan connections in the moving frame method.
\section{Conclusion}
\label{sec:conclusion}
In this paper, we propose a unified mathematical framework, called Director Field Analysis (DFA), to analyze a spherical function field and its extracted peak field.
See Fig.~\ref{fig:DFA} for an overview of the DFA pipeline.
First, in DFA, we detect peaks from the spherical function field, and define the Orientational Order (OO) and the Orientational Dispersion (OD) indices in voxels or within spatial regions.
Closed-form solutions of OO and OD are obtained for some specific spherical functions.
We propose OO and OD as properties for general ODFs along peaks, independent of diffusion signal models.
Second, we define a local orthogonal frame in each voxel exhibiting anisotropic diffusion, where the principal peak is its first axis, and the other two axes describe the local spatial change directions of principal peaks.
Third, from the extracted local orthogonal frames in voxels, DFA estimates three distortion indices (splay, bend, twist) that are able to distinguish three types of distortions and a total orientational distortion index.
To our knowledge, this paper is the first work to \emph{quantitatively} describe orientational distortion (splay, bend, and twist) in general spherical function fields from DTI or HARDI data.
The experiments demonstrate the following:
1) The proposed OO is another type of anisotropy index for spherical functions, and OD is more general and natural than the previous dispersion index proposed in NODDI~\citep{zhang_NODDI_NI2012}, which only works for Watson distributions.
2) The proposed splay and bend indices can be seen as a generalization of the dispersion and curving indices in~\cite{Savadjiev_NI2010}, considering they have similar contrast in the same tensor field.
The proposed four orientational distortion indices work not only for tensors but also for general spherical functions.
3) The proposed distortion indices demonstrate good sensitivity for the three different types of orientational distortion in Fig.~\ref{fig:distortion}.
4) Orientational distortion indices normally have large values in areas with fiber curving and crossing.
Note that the proposed DFA and its related mathematical tools can be used not only for diffusion MRI data, but also for general director data.
Moreover, there are many applications in which vector fields, like velocity fields, can be easily processed by using the modified DFA described in Section~\ref{sec:DFA_vector} to analyze their spatial features.
Considering the proposed scalar indices are sensitive to different distortions of principal directions,
these indices have potential in voxel-based analysis and tract-based analysis for group studies and longitudinal studies~\citep{smith:TBSS:NI2006,liu:NI2013},
which is a goal of future work.
We will release the related codes and demos for DFA in DMRITool~\footnote{\label{fn:dmritool}\url{https://diffusionmritool.github.io}}, which is an open source toolbox for diffusion MRI data processing.
\section*{Acknowledgments}
The authors thank Dr.~Carlo Pierpaoli and Dr.~Elizabeth Hutchinson for useful discussions on the Fornix, and Ms. Liz Salak for editing the manuscript.
This work was supported by funds provided by the Intramural Research Program of the \emph{Eunice Kennedy Shriver} National Institute of Child Health and Human Development (NICHD) (ZIA-HD000266).
The data were provided in part by the Human Connectome Project, WU-Minn Consortium
(Principal Investigators: David Van Essen and Kamil Ugurbil; 1U54MH091657) funded by the 16 NIH institutes and centers that support the NIH Blueprint for Neuroscience Research;
and by the McDonnell Center for Systems Neuroscience at Washington University.
|
{
"redpajama_set_name": "RedPajamaArXiv"
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{{Infobox rugby biography
| name = Alan Tait
| image =
| birth_name = Alan Victor Tait
| birth_date =
| birth_place = Kelso, Scottish Borders, Scotland
| height =
| weight =
| school =
| university =
| relatives =
| rl_position = Full back /
| rl_clubyears1 = 1988–92
| rl_clubyears2 = 1992–96
| rl_proclubs1 = Widnes
| rl_proclubs2 = Leeds
| rl_clubapps1 = 136
| rl_clubapps2 = 126
| rl_clubpoints1 = 225
| rl_clubpoints2 = 176
| rl_nationalyears1 = 1989-93
| rl_nationalyears2 = 1995-96
| rl_nationalteam1 = Great Britain
| rl_nationalteam2 = Scotland
| rl_nationalapps1 = 10+4
| rl_nationalapps2 = 4
| rl_nationalpoints1 = 24
| rl_nationalpoints2 = 20
| ru_position = Centre
| amatyears1 =1987–88
| amatteam1 = Kelso
| ru_amupdate =
| ru_province = South of ScotlandReds Trial
| ru_provinceyears = -
| ru_provincecaps =
| ru_provincepoints =
| ru_provinceupdate=
| years1 = 1996–98
| years2 = 1998–2000
| clubs1 = Newcastle Falcons
| clubs2 = Edinburgh
| apps1 = 19
| points1 = 10
| repyears1 = 1987–99
| repyears2 = 1997
| repteam1 =
| repteam2 = British and Irish Lions
| repcaps1 = 27
| repcaps2 = 2
| reppoints1 = 81
| reppoints2 =5
| repupdate =
| coachyears1 = 2009–122022-
| coachteams1 = Newcastle FalconsSouthern Knights (
}}
Alan Victor Tait (born 2 November 1964) is a former Scottish dual-code rugby footballer, and now coach. He is a defence coach at the Super 6 side Southern Knights. He was previously head coach at Newcastle Falcons and a former rugby union and professional rugby league footballer. He played outside centre for Scotland (RU), and the British and Irish Lions. He played club rugby union for Kelso, Edinburgh and the Newcastle Falcons; and club rugby league for Widnes and Leeds.
Tait changed codes twice in his life, once going from rugby union to rugby league, and then going the other way after union became professional during the mid-1990s.
Rugby Union
Unlike many other cross-code converts of the period, Tait had the benefit of growing up in Cumbria, where his father, Alan Senior, was playing for Workington Town. However, Tait played Union first and made his Test début for Scotland in the inaugural 1987 World Cup held in New Zealand where, he came on after seven minutes as a replacement in a 20–20 draw with France in Christchurch.
Tait played for the Reds Trial side in their match against Blues Trial on 3 January 1987.
Rugby League
The following year Tait switched codes to rugby league where he was to spend the next eight years playing club rugby for Widnes and Leeds, as well as representing Great Britain and Scotland.
During the 1989–90 season, Tait played for defending champions Widnes at fullback in their 1989 World Club Challenge victory against the visiting Canberra Raiders.
Tait won the Harry Sunderland Trophy in both 1989 and 1990.
Alan Tait played in Widnes' 24–18 victory over Salford in the 1990 Lancashire Cup Final during the 1990–91 season at Central Park, Wigan on Saturday 29 September 1990.
Alan Tait played in Widnes' 6-12 defeat by Wigan in the 1988–89 John Player Special Trophy Final during the 1988–89 season at Burnden Park, Bolton on Saturday 7 January 1989, and played , and scored a try in the 24-0 victory over Leeds in the 1991–92 Regal Trophy Final during the 1991–92 season at Central Park, Wigan on Saturday 11 January 1992.
In the 1992 World Cup Final at Wembley Stadium Tait was selected to play for Great Britain from the reserve bench in their defeat by Australia. He also made appearances in the Challenge Cup Finals in 1994 and 1995
Return to Union
With the advent of Rugby Union turning profession Tait along with many other converts switched codes back to union in 1996 signing for the Newcastle Falcons with whom he won the Premiership in 1998, making 19 appearances that season.
At first he was ignored by the Scotland selectors but eventually made his return for Scotland after a nine-year absence in 1997 and went on to represent the British & Irish Lions in South Africa also in the summer of that year. Surprising many Tait was selected to start the first two Tests on the wing, even though his favoured and more recognised position was at centre. This was due to coach Ian McGeechan believing that Tait would add extra defensive capabilities to the backline over the other wingers, in what would be a tight test series. He famously scored a try in the first match as the Lions won 25–16 in Cape Town. He also played in the second match which saw the Lions clinch the series after a dramatic 18–15 victory in Durban. Injured before the 3rd Test he did not play as the Lions lost the last match 35–16.
For Scotland Tait developed a devastating partnership with John Leslie. Many saw their pairing as instrumental in Scotland's good performances of the time and allowing stand-off Gregor Townsend to exploit gaps in the opposition defence. This was no more apparent that in the 1999 Five Nations Championship where Townsend would become the fifth and last player in history to score a try against each of the other countries in the five nations tournament. The championship culminated in Scotland narrowly finished ahead of England on points difference thanks to Wales' last minute victory over England at Wembley. Tait scored two tries in Scotland's last match of the tournament as they put in an historic performance to beat France 36–22. Later that year he represented Scotland for the last time at the 1999 World Cup finishing with a defeat by the All Blacks. He scored a try against South Africa in the pool stages at Murrayfield in a 46–29 reverse to the holders. In 2000 Tait retired from professional rugby finishing his last game playing for his last club Edinburgh Reivers.
In all Tait played 27 times, scoring 17 tries, for Scotland, this is an exceptional record especially when considering that the bulk of these games were played after his return from League at the age of 32. Partly because of this it is why Tait is still regarded in high esteem by Scotland supporters to this day.
Coaching
Following his retirement from playing he then moved into coaching, initially working with Scotland as a defence coach. Although he was dismissed from the post by Matt Williams he was later restored to that role by Williams successor Frank Hadden. In 2004 he joined the Borders as a defensive coach.
Tait then moved back to the Falcons as an assistant before assuming the top job at the Premiership club as part of a restructuring process in the wake of Steve Bates' departure from the post of director of rugby in 2009. He held the position of Head Coach of Newcastle from 2009 to 2012 until club owner, Semore Kurdi announced Tait was 'taking a break from rugby' following a series of poor results.
On 7 July 2022 he was appointed the Defence Coach to the Southern Knights.
Family
Tait has a son, Michael, who was also a professional rugby union player. Michael appeared for the Scotland national under-20 rugby union team in 2010 and signed for Edinburgh in 2014 before retiring later the same year due to injury.
References
Tait, Alan & Lothian, Bill Rugby Rebel: The Alan Tait Story'' (1998 Mainstream, Edinburgh, )
External links
Profile & Statistics on ESPN Scrum
(archived by web.archive.org) Profile at leedsrugby
SCOTLAND RUGBY LEAGUE INTERNATIONAL HONOURS BOARD
When Widnes muscled in on Wigan's Rugby League trophy romp
1964 births
Living people
British & Irish Lions rugby union players from Scotland
Dual-code rugby internationals
Edinburgh Rugby players
Great Britain national rugby league team players
Kelso RFC players
Leeds Rhinos players
Newcastle Falcons players
Reds Trial players
Rugby league centres
Rugby league fullbacks
Rugby league players from Kelso
Rugby union centres
Rugby union players from Kelso
Scotland international rugby union players
Scotland national rugby league team captains
Scotland national rugby league team players
Scottish rugby league players
Scottish rugby union coaches
Scottish rugby union players
South of Scotland District (rugby union) players
Widnes Vikings players
|
{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 2,910
|
Noah Mittman
UX Strategy & Execution • Est. 1996 • Made in NYC
RoadRunner Portal
The Kick-Off
Building off our relationship with Time Warner Cable, we were asked by RoadRunner to redesign their subscriber portal and bring it up to modern publishing standards after having been an all-Flash application for many years.
I began working with the Art Director on the project on some concept explorations of what a new site could be, given the opportunity to restart from the ground up, the research we had on the usage of the previous site, and the trends in the market on the decline of portal usage in general. We presented these options to the client but in the end fell back to a more traditional portal due to the lack of interest in changing their existing web business structure of content partnerships and syndication.
We then began working on developing a design language that would be modular and fit their new multicolumnal needs for layouts and free configuration of page contents. Once a core langauge was established, we then rode down to their location where we spent a week working side by side with the key stakeholders in joint BRD-writing sprints. These sessions, based on the core experience, walked through each navigable section of the site and established both content and functionality needs. We then would retire and immediately begin producing FSDs for that specific section, and deliver them for review as the next BRD was begun.
After completing the primary sections of the website, we returned back to New York where smaller sections on the periphery of the experience were developed and reviewed. All documents were taken in by the in-house technology teams and the site was coded in house by RoadRunner.
Let's talk. work@teradome.com
|
{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
| 6,580
|
Transfers from Rome Fiumicino Airport to Marino
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How far is it from Rome Fiumicino Airport to Marino?
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How long will it take to get to Marino with a private transfer?
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Vehicles with capacity for more than 4 people or for passengers with special luggage needs are rare at Rome Fiumicino Airport. If you book in advance with us, the transfer from Rome Fiumicino Airport to Marino will be carried out with a vehicle that will be perfectly adapted to your needs.
You can reserve any type of vehicle for this route. Use the availability search on this page to see all the vehicles available for transfer between Rome Fiumicino Airport and Marino.
How much does a transfer from Rome Fiumicino Airport to Marino cost?
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|
{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
| 4,480
|
/**
* @fileoverview Event statistics table.
*
* @author benvanik@google.com (Ben Vanik)
*/
goog.provide('wtf.db.EventDataEntry');
goog.provide('wtf.db.EventStatistics');
goog.provide('wtf.db.InstanceEventDataEntry');
goog.provide('wtf.db.ScopeEventDataEntry');
goog.provide('wtf.db.SortMode');
goog.require('goog.Disposable');
goog.require('goog.object');
goog.require('wtf.data.EventClass');
goog.require('wtf.data.EventFlag');
goog.require('wtf.events.EventType');
/**
* Sorting mode to use when retrieving entries.
* @enum {number}
*/
wtf.db.SortMode = {
ANY: 0,
COUNT: 1,
TOTAL_TIME: 2,
MEAN_TIME: 3,
OWN_TIME: 4
};
goog.exportSymbol(
'wtf.db.SortMode',
wtf.db.SortMode);
goog.exportProperty(
wtf.db.SortMode, 'ANY',
wtf.db.SortMode.ANY);
goog.exportProperty(
wtf.db.SortMode, 'COUNT',
wtf.db.SortMode.COUNT);
goog.exportProperty(
wtf.db.SortMode, 'TOTAL_TIME',
wtf.db.SortMode.TOTAL_TIME);
goog.exportProperty(
wtf.db.SortMode, 'MEAN_TIME',
wtf.db.SortMode.MEAN_TIME);
goog.exportProperty(
wtf.db.SortMode, 'OWN_TIME',
wtf.db.SortMode.OWN_TIME);
/**
* Event data table.
* Caches detailed aggregate information about events.
*
* @param {!wtf.db.Database} db Event database.
* @constructor
* @extends {goog.Disposable}
*/
wtf.db.EventStatistics = function(db) {
goog.base(this);
/**
* Event database.
* @type {!wtf.db.Database}
* @private
*/
this.db_ = db;
/**
* The cached full table for the entire database.
* Since it's a common operation to select everything this is kept around.
* @type {wtf.db.EventStatistics.Table}
* @private
*/
this.fullTable_ = null;
/**
* A cache dtable for a time range.
* Currently we only stash one of these, but we could do many if there is
* regular thrashing.
* This is populated on-demand by {@see #getTable}.
* @type {wtf.db.EventStatistics.Table}
* @private
*/
this.selectedTable_ = null;
db.addListener(wtf.events.EventType.INVALIDATED, function() {
this.fullTable_ = null;
this.selectedTable_ = null;
}, this);
};
goog.inherits(wtf.db.EventStatistics, goog.Disposable);
/**
* Gets a table covering the requested time range or, if the times are omitted,
* the entire database.
* @param {number=} opt_startTime Starting time.
* @param {number=} opt_endTime Ending time.
* @return {!wtf.db.EventStatistics.Table} A table covering the requested time
* range.
*/
wtf.db.EventStatistics.prototype.getTable = function(
opt_startTime, opt_endTime) {
var startTime = goog.isDef(opt_startTime) ? opt_startTime : -Number.MAX_VALUE;
var endTime = goog.isDef(opt_endTime) ? opt_endTime : Number.MAX_VALUE;
if (startTime == -Number.MAX_VALUE && endTime == Number.MAX_VALUE) {
if (!this.fullTable_) {
this.fullTable_ = new wtf.db.EventStatistics.Table(
this.db_, -Number.MAX_VALUE, Number.MAX_VALUE);
this.fullTable_.rebuild();
}
return this.fullTable_;
} else {
if (this.selectedTable_) {
if (this.selectedTable_.getStartTime() == startTime &&
this.selectedTable_.getEndTime() == endTime) {
return this.selectedTable_;
}
}
this.selectedTable_ = new wtf.db.EventStatistics.Table(
this.db_, startTime, endTime);
this.selectedTable_.rebuild();
return this.selectedTable_;
}
};
/**
* Gets all of the event type names found in all of the given tables.
* @param {!Array.<!wtf.db.EventStatistics.Table>} tables Tables.
* @param {wtf.data.EventClass=} opt_eventClass Class to limit to.
* @return {!Array.<string>} All event type names.
*/
wtf.db.EventStatistics.getAllEventTypeNames = function(tables, opt_eventClass) {
var names = {};
for (var n = 0; n < tables.length; n++) {
var table = tables[n];
for (var m = 0; m < table.list_.length; m++) {
var eventType = table.list_[m].eventType;
if (opt_eventClass === undefined ||
eventType.eventClass == opt_eventClass) {
names[eventType.name] = true;
}
}
}
return goog.object.getKeys(names);
};
goog.exportSymbol(
'wtf.db.EventStatistics',
wtf.db.EventStatistics);
goog.exportProperty(
wtf.db.EventStatistics.prototype, 'getTable',
wtf.db.EventStatistics.prototype.getTable);
goog.exportSymbol(
'wtf.db.EventStatistics.getAllEventTypeNames',
wtf.db.EventStatistics.getAllEventTypeNames);
/**
* A result table generated from an event statistics build.
* The results of the table can be retreived in any order or with a filter
* very quickly, but a new table must be generated to change the time range.
* @param {!wtf.db.Database} db Event database.
* @param {number} startTime Starting time.
* @param {number} endTime Ending time.
* @constructor
*/
wtf.db.EventStatistics.Table = function(db, startTime, endTime) {
/**
* Event database.
* @type {!wtf.db.Database}
* @private
*/
this.db_ = db;
/**
* Starting time.
* @type {number}
* @private
*/
this.startTime_ = startTime;
/**
* Ending time.
* @type {number}
* @private
*/
this.endTime_ = endTime;
/**
* Total number of events.
* @type {number}
* @private
*/
this.eventCount_ = 0;
/**
* Event data keyed on event name.
* @type {!Object.<!wtf.db.EventDataEntry>}
* @private
*/
this.table_ = {};
/**
* A list of all event entries.
* @type {!Array.<!wtf.db.EventDataEntry>}
* @private
*/
this.list_ = [];
/**
* The current sort mode of the list.
* This is used to prevent successive sorts of the list.
* @type {wtf.db.SortMode}
* @private
*/
this.listSortMode_ = wtf.db.SortMode.ANY;
};
/**
* Rebuilds from the database.
* @param {wtf.db.Filter=} opt_filter Filter.
* @protected
*/
wtf.db.EventStatistics.Table.prototype.rebuild = function(opt_filter) {
this.eventCount_ = 0;
var tableById = {};
var list = [];
var eventTypeFilter = opt_filter ? opt_filter.getEventTypeFilter() : null;
var argumentFilter = opt_filter ? opt_filter.getArgumentFilter() : null;
// Setup entries for everything.
// We do this here so that the expensive work happens per-type and
// not per-event.
var eventTypeTable = this.db_.getEventTypeTable();
var eventTypeList = eventTypeTable.getAll();
for (var n = 0; n < eventTypeList.length; n++) {
var type = eventTypeList[n];
// Skip system events/etc.
if (type.flags & wtf.data.EventFlag.INTERNAL ||
type.flags & wtf.data.EventFlag.BUILTIN) {
continue;
}
if (eventTypeFilter && !eventTypeFilter(type)) {
continue;
}
var entry;
if (type.eventClass == wtf.data.EventClass.SCOPE) {
entry = new wtf.db.ScopeEventDataEntry(type);
} else {
entry = new wtf.db.InstanceEventDataEntry(type);
}
tableById[type.id] = entry;
list.push(entry);
}
// Find all events that match.
var zones = this.db_.getZones();
for (var n = 0; n < zones.length; n++) {
var eventList = zones[n].getEventList();
var it = eventList.beginTimeRange(this.startTime_, this.endTime_);
for (; !it.done(); it.next()) {
var typeId = it.getTypeId();
var entry = tableById[typeId];
if (entry) {
if (!argumentFilter || argumentFilter(it)) {
entry.appendEvent(it);
this.eventCount_++;
}
}
}
}
// Build a table by type name.
// Also remove any types that had no matching events.
var tableByName = {};
var validList = [];
for (var n = 0; n < list.length; n++) {
var entry = list[n];
if (entry.count) {
tableByName[entry.eventType.name] = entry;
validList.push(entry);
}
}
this.table_ = tableByName;
this.list_ = validList;
this.listSortMode_ = wtf.db.SortMode.ANY;
};
/**
* Gets the time the table starts at.
* @return {number} Starting time. May be MIN_VALUE to indicate a min.
*/
wtf.db.EventStatistics.Table.prototype.getStartTime = function() {
return this.startTime_;
};
/**
* Gets the time the table ends at.
* @return {number} Ending time. May be MAX_VALUE to indicate a max.
*/
wtf.db.EventStatistics.Table.prototype.getEndTime = function() {
return this.endTime_;
};
/**
* Gets the total number of events included in the table.
* @return {number} Event count.
*/
wtf.db.EventStatistics.Table.prototype.getEventCount = function() {
return this.eventCount_;
};
/**
* Gets all entries.
* The result should not be modified.
* @return {!Array.<!wtf.db.EventDataEntry>}
*/
wtf.db.EventStatistics.Table.prototype.getEntries = function() {
return this.list_;
};
/**
* Gets the entry for an event type, if it exists.
* @param {string} eventName Event name.
* @return {wtf.db.EventDataEntry} Event entry, if it exists.
*/
wtf.db.EventStatistics.Table.prototype.getEventTypeEntry = function(eventName) {
return this.table_[eventName] || null;
};
/**
* Gets all entries from the table of the given type.
* @param {wtf.data.EventClass} eventClass Event class.
* @return {!Object.<!wtf.db.EventDataEntry>} All entries of the given
* class, keyed by event type name.
*/
wtf.db.EventStatistics.Table.prototype.getEntriesByClass = function(
eventClass) {
var result = {};
for (var n = 0; n < this.list_.length; n++) {
var entry = this.list_[n];
if (entry.eventType.eventClass == eventClass) {
result[entry.eventType.name] = entry;
}
}
return result;
};
/**
* Enumerates all event type entries in the data table.
* @param {function(this: T, !wtf.db.EventDataEntry)} callback
* A function called for each entry.
* @param {T=} opt_scope Callback scope.
* @param {wtf.db.SortMode=} opt_sortMode Sort mode.
* @template T
*/
wtf.db.EventStatistics.Table.prototype.forEach = function(
callback, opt_scope, opt_sortMode) {
// Sort before enumerating if the sort order does not match the cached
// value.
if (opt_sortMode && this.listSortMode_ != opt_sortMode) {
this.listSortMode_ = opt_sortMode;
switch (this.listSortMode_) {
case wtf.db.SortMode.COUNT:
this.list_.sort(function(a, b) {
return b.count - a.count;
});
break;
case wtf.db.SortMode.TOTAL_TIME:
this.list_.sort(function(a, b) {
if (a instanceof wtf.db.ScopeEventDataEntry &&
b instanceof wtf.db.ScopeEventDataEntry) {
return b.totalTime_ - a.totalTime_;
} else if (a instanceof wtf.db.ScopeEventDataEntry) {
return -1;
} else if (b instanceof wtf.db.ScopeEventDataEntry) {
return 1;
} else {
return b.count - a.count;
}
});
break;
case wtf.db.SortMode.MEAN_TIME:
this.list_.sort(function(a, b) {
if (a instanceof wtf.db.ScopeEventDataEntry &&
b instanceof wtf.db.ScopeEventDataEntry) {
return b.getMeanTime() - a.getMeanTime();
} else if (a instanceof wtf.db.ScopeEventDataEntry) {
return -1;
} else if (b instanceof wtf.db.ScopeEventDataEntry) {
return 1;
} else {
return b.count - a.count;
}
});
break;
case wtf.db.SortMode.OWN_TIME:
this.list_.sort(function(a, b) {
if (a instanceof wtf.db.ScopeEventDataEntry &&
b instanceof wtf.db.ScopeEventDataEntry) {
return b.ownTime_ - a.ownTime_;
} else if (a instanceof wtf.db.ScopeEventDataEntry) {
return -1;
} else if (b instanceof wtf.db.ScopeEventDataEntry) {
return 1;
} else {
return b.count - a.count;
}
});
break;
}
}
for (var n = 0; n < this.list_.length; n++) {
callback.call(opt_scope, this.list_[n]);
}
};
/**
* Filters the event entries in the table based on the given filter.
* @param {!wtf.db.Filter} filter Filter.
* @return {!wtf.db.EventStatistics.Table} New table.
*/
wtf.db.EventStatistics.Table.prototype.filter = function(filter) {
if (!filter.isActive()) {
return this;
}
var newTable = new wtf.db.EventStatistics.Table(
this.db_, this.startTime_, this.endTime_);
if (filter.getArgumentFilter()) {
// Slow path of building a new table by scanning all events.
newTable.rebuild(filter);
} else {
// Build the new table and place the matching entries in it.
// This is a fast path for when we can just reuse the previously calculated
// data.
var filterFn = filter.getEventTypeFilter();
for (var n = 0; n < this.list_.length; n++) {
var entry = this.list_[n];
if (filterFn(entry.eventType)) {
newTable.eventCount_ += entry.count;
newTable.table_[entry.eventType.name] = entry;
newTable.list_.push(entry);
}
}
}
return newTable;
};
goog.exportProperty(
wtf.db.EventStatistics.Table.prototype, 'getEventCount',
wtf.db.EventStatistics.Table.prototype.getEventCount);
goog.exportProperty(
wtf.db.EventStatistics.Table.prototype, 'getEntries',
wtf.db.EventStatistics.Table.prototype.getEntries);
goog.exportProperty(
wtf.db.EventStatistics.Table.prototype, 'getEventTypeEntry',
wtf.db.EventStatistics.Table.prototype.getEventTypeEntry);
goog.exportProperty(
wtf.db.EventStatistics.Table.prototype, 'getEntriesByClass',
wtf.db.EventStatistics.Table.prototype.getEntriesByClass);
goog.exportProperty(
wtf.db.EventStatistics.Table.prototype, 'forEach',
wtf.db.EventStatistics.Table.prototype.forEach);
goog.exportProperty(
wtf.db.EventStatistics.Table.prototype, 'filter',
wtf.db.EventStatistics.Table.prototype.filter);
/**
* Abstract base type for entries in the {@see wtf.db.EventStatistics}.
* @param {!wtf.db.EventType} eventType Event type.
* @constructor
*/
wtf.db.EventDataEntry = function(eventType) {
/**
* Event type.
* @type {!wtf.db.EventType}
* @protected
*/
this.eventType = eventType;
/**
* Total number of the events encountered.
* @type {number}
* @protected
*/
this.count = 0;
};
/**
* Appends an event to the entry.
* @param {!wtf.db.EventIterator} it Event.
*/
wtf.db.EventDataEntry.prototype.appendEvent = goog.abstractMethod;
/**
* Gets the event type this entry describes.
* @return {!wtf.db.EventType} Event type.
*/
wtf.db.EventDataEntry.prototype.getEventType = function() {
return this.eventType;
};
/**
* Gets the total number of events encountered.
* @return {number} Event count.
*/
wtf.db.EventDataEntry.prototype.getCount = function() {
return this.count;
};
/**
* Gets the frequency of the events as a measure of instances/sec.
* @return {number} Instances/second.
*/
wtf.db.EventDataEntry.prototype.getFrequency = function() {
// TODO(benvanik): compute frequency of events.
return 0;
};
goog.exportSymbol(
'wtf.db.EventDataEntry',
wtf.db.EventDataEntry);
goog.exportProperty(
wtf.db.EventDataEntry.prototype, 'getEventType',
wtf.db.EventDataEntry.prototype.getEventType);
goog.exportProperty(
wtf.db.EventDataEntry.prototype, 'getCount',
wtf.db.EventDataEntry.prototype.getCount);
goog.exportProperty(
wtf.db.EventDataEntry.prototype, 'getFrequency',
wtf.db.EventDataEntry.prototype.getFrequency);
/**
* An entry in the {@see wtf.db.EventStatistics} describing scope
* event types.
* @param {!wtf.db.EventType} eventType Event type.
* @constructor
* @extends {wtf.db.EventDataEntry}
*/
wtf.db.ScopeEventDataEntry = function(eventType) {
goog.base(this, eventType);
/**
* Total time taken by all scopes.
* @type {number}
* @private
*/
this.totalTime_ = 0;
/**
* Total own time taken by all scopes.
* @type {number}
* @private
*/
this.ownTime_ = 0;
/**
* Total time taken by all scopes, minus system time.
* @type {number}
* @private
*/
this.userTime_ = 0;
/**
* Buckets of time, each 1ms.
* @type {!Uint32Array}
* @private
*/
this.buckets_ = new Uint32Array(1000);
};
goog.inherits(wtf.db.ScopeEventDataEntry, wtf.db.EventDataEntry);
/**
* @override
*/
wtf.db.ScopeEventDataEntry.prototype.appendEvent = function(it) {
if (!it.getEndTime()) {
return;
}
this.count++;
var userDuration = it.getUserDuration();
this.totalTime_ += it.getTotalDuration();
this.ownTime_ += it.getOwnDuration();
this.userTime_ += userDuration;
var bucketIndex = Math.round(userDuration) | 0;
if (bucketIndex >= 1000) {
bucketIndex = 999;
}
this.buckets_[bucketIndex]++;
};
/**
* Gets the total time spent within all scopes of this type, including
* system time.
* @return {number} Total time.
*/
wtf.db.ScopeEventDataEntry.prototype.getTotalTime = function() {
return this.totalTime_;
};
/**
* Gets the total time spent within all scopes of this type, excluding children.
* @return {number} Total time.
*/
wtf.db.ScopeEventDataEntry.prototype.getOwnTime = function() {
return this.ownTime_;
};
/**
* Gets the total time spent within all scopes of this type, excluding
* system time.
* @return {number} Total time.
*/
wtf.db.ScopeEventDataEntry.prototype.getUserTime = function() {
return this.userTime_;
};
/**
* Gets the mean time of scopes of this type.
* @return {number} Average mean time.
*/
wtf.db.ScopeEventDataEntry.prototype.getMeanTime = function() {
if (this.count) {
if (this.eventType.flags & wtf.data.EventFlag.SYSTEM_TIME) {
return this.totalTime_ / this.count;
} else {
return this.userTime_ / this.count;
}
} else {
return 0;
}
};
/**
* Gets the distribution of the events over 0-1s.
* Any event that ran longer than 1s will be in the last bucket.
* @return {!Uint32Array} Distribution.
*/
wtf.db.ScopeEventDataEntry.prototype.getDistribution = function() {
return this.buckets_;
};
goog.exportSymbol(
'wtf.db.ScopeEventDataEntry',
wtf.db.ScopeEventDataEntry);
goog.exportProperty(
wtf.db.ScopeEventDataEntry.prototype, 'getTotalTime',
wtf.db.ScopeEventDataEntry.prototype.getTotalTime);
goog.exportProperty(
wtf.db.ScopeEventDataEntry.prototype, 'getOwnTime',
wtf.db.ScopeEventDataEntry.prototype.getOwnTime);
goog.exportProperty(
wtf.db.ScopeEventDataEntry.prototype, 'getUserTime',
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goog.exportProperty(
wtf.db.ScopeEventDataEntry.prototype, 'getMeanTime',
wtf.db.ScopeEventDataEntry.prototype.getMeanTime);
goog.exportProperty(
wtf.db.ScopeEventDataEntry.prototype, 'getDistribution',
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/**
* An entry in the {@see wtf.db.EventStatistics} describing instance
* event types.
* @param {!wtf.db.EventType} eventType Event type.
* @constructor
* @extends {wtf.db.EventDataEntry}
*/
wtf.db.InstanceEventDataEntry = function(eventType) {
goog.base(this, eventType);
};
goog.inherits(wtf.db.InstanceEventDataEntry, wtf.db.EventDataEntry);
/**
* @override
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wtf.db.InstanceEventDataEntry.prototype.appendEvent = function(it) {
this.count++;
};
goog.exportSymbol(
'wtf.db.InstanceEventDataEntry',
wtf.db.InstanceEventDataEntry);
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{
"redpajama_set_name": "RedPajamaGithub"
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{"url":"http:\/\/support.sas.com\/documentation\/cdl\/en\/statug\/68162\/HTML\/default\/statug_introbayes_sect011.htm","text":"# Introduction to Bayesian Analysis Procedures\n\n### Bayesian Inference\n\nSubsections:\n\nBayesian inference about is primarily based on the posterior distribution of . There are various ways in which you can summarize this distribution. For example, you can report your findings through point estimates. You can also use the posterior distribution to construct hypothesis tests or probability statements.\n\n#### Point Estimation and Estimation Error\n\nClassical methods often report the maximum likelihood estimator (MLE) or the method of moments estimator (MOME) of a parameter. In contrast, Bayesian approaches often use the posterior mean. The definition of the posterior mean is given by\n\nOther commonly used posterior estimators include the posterior median, defined as\n\nand the posterior mode, defined as the value of that maximizes .\n\nThe variance of the posterior density (simply referred to as the posterior variance) describes the uncertainty in the parameter, which is a random variable in the Bayesian paradigm. A Bayesian analysis typically uses the posterior variance, or the posterior standard deviation, to characterize the dispersion of the parameter. In multidimensional models, covariance or correlation matrices are used.\n\nIf you know the distributional form of the posterior density of interest, you can report the exact posterior point estimates. When models become too difficult to analyze analytically, you have to use simulation algorithms, such as the Markov chain Monte Carlo (MCMC) method to obtain posterior estimates (see the section Markov Chain Monte Carlo Method). All of the Bayesian procedures rely on MCMC to obtain all posterior estimates. Using only a finite number of samples, simulations introduce an additional level of uncertainty to the accuracy of the estimates. Monte Carlo standard error (MCSE), which is the standard error of the posterior mean estimate, measures the simulation accuracy. See the section Standard Error of the Mean Estimate for more information.\n\nThe posterior standard deviation and the MCSE are two completely different concepts: the posterior standard deviation describes the uncertainty in the parameter, while the MCSE describes only the uncertainty in the parameter estimate as a result of MCMC simulation. The posterior standard deviation is a function of the sample size in the data set, and the MCSE is a function of the number of iterations in the simulation.\n\n#### Hypothesis Testing\n\nSuppose you have the following null and alternative hypotheses: is and is , where is a subset of the parameter space and is its complement. Using the posterior distribution , you can compute the posterior probabilities and , or the probabilities that and are true, respectively. One way to perform a Bayesian hypothesis test is to accept the null hypothesis if and vice versa, or to accept the null hypothesis if is greater than a predefined threshold, such as 0.75, to guard against falsely accepted null distribution.\n\nIt is more difficult to carry out a point null hypothesis test in a Bayesian analysis. A point null hypothesis is a test of versus . If the prior distribution is a continuous density, then the posterior probability of the null hypothesis being true is 0, and there is no point in carrying out the test. One alternative is to restate the null to be a small interval hypothesis: , where a is a very small constant. The Bayesian paradigm can deal with an interval hypothesis more easily. Another approach is to give a mixture prior distribution to with a positive probability of on and the density on . This prior ensures a nonzero posterior probability on , and you can then make realistic probabilistic comparisons. For more detailed treatment of Bayesian hypothesis testing, see Berger (1985).\n\n#### Interval Estimation\n\nThe Bayesian set estimates are called credible sets, which are also known as credible intervals. This is analogous to the concept of confidence intervals used in classical statistics. Given a posterior distribution , A is a credible set for if\n\nFor example, you can construct a 95% credible set for by finding an interval, A, over which .\n\nYou can construct credible sets that have equal tails. A equal-tail interval corresponds to the th and th percentiles of the posterior distribution. Some statisticians prefer this interval because it is invariant under transformations. Another frequently used Bayesian credible set is called the highest posterior density (HPD) interval.\n\nA HPD interval is a region that satisfies the following two conditions:\n\n1. The posterior probability of that region is .\n\n2. The minimum density of any point within that region is equal to or larger than the density of any point outside that region.\n\nThe HPD is an interval in which most of the distribution lies. Some statisticians prefer this interval because it is the smallest interval.\n\nOne major distinction between Bayesian and classical sets is their interpretation. The Bayesian probability reflects a person\u2019s subjective beliefs. Following this approach, a statistician can make the claim that is inside a credible interval with measurable probability. This property is appealing because it enables you to make a direct probability statement about parameters. Many people find this concept to be a more natural way of understanding a probability interval, which is also easier to explain to nonstatisticians. A confidence interval, on the other hand, enables you to make a claim that the interval covers the true parameter. The interpretation reflects the uncertainty in the sampling procedure; a confidence interval of asserts that, in the long run, of the realized confidence intervals cover the true parameter.","date":"2019-08-18 13:47:31","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\": 0.9008653163909912, \"perplexity\": 331.10064932147236}, \"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-2019-35\/segments\/1566027313889.29\/warc\/CC-MAIN-20190818124516-20190818150516-00171.warc.gz\"}"}
| null | null |
\section{Introduction}
Deep neural networks (DNNs) have produced state-of-the-art results in many challenging tasks, such as image classification~\cite{resnet,guo2018double}, face recognition~\cite{SphereFace,schroff2015facenet}, medical image analysis~\cite{zhang2019collaborative, zhang2019whole}, portfolio selection~\cite{zhang2018adaptive}, and image generation~\cite{cao2019multi}, \emph{etc.}~One of the key factors behind the success of DNNs lies in the design of effective neural architectures, such as ResNet~\cite{resnet} and MobileNet~\cite{howard2017mobilenets}.
In practice, different architectures often show different performances, and the optimal architecture may vary among different tasks.
Hence, the design of effective neural architectures highly relies on substantial human expertise. However, the human-designed process cannot fully explore the whole architecture space, resulting in suboptimal architectures~\cite{zoph2016neural}.
Besides manual design, one may resort to the neural architecture search (NAS)~\cite{zoph2016neural} technique to automatically design network architectures.
Specifically, NAS seeks to find an optimal architecture in a predefined search space~\cite{zoph2016neural,zoph2018learning}.
The searched architectures often show promising performance, demonstrating tremendous potential to surpass hand-crafted architectures~\cite{cai2018efficient,cai2018proxylessnas,liu2018progressive,liu2017hierarchical,tan2019mnasnet}.
During the search process, NAS seeks to evaluate a large number of candidate architectures and gradually learns a strategy to find good architectures.
However, the exact performance evaluation requires training all the architectures from scratch, which is highly time-consuming and computationally impractical in real-world applications.
To improve the training efficiency, a weight sharing (WS)~\cite{bender2018understanding,pham2018efficient} technique has been developed for NAS. Specifically, WS-based NAS constructs a supernet,
\emph{i.e.,~} a large computational graph, where each network architecture (subgraph) shares their parameters.
In this sense, all candidate architectures share its parameters in the supernet. Based on the supernet, one can directly estimate the performance of architectures instead of training them from scratch. In this way, WS is able to effectively accelerate the search process of NAS and reduce the search cost from 1,800 GPU days to less than 1 day~\cite{cai2018proxylessnas}. Despite the efficiency, recent studies~\cite{adam2019understanding,li2019random,sciuto2019evaluating} have empirically shown that the performance estimation provided by the WS is inaccurate, which makes it hard to identify good architectures.
As a result, the search performance of NAS cannot be guaranteed and often fails to find promising architectures~\cite{chu2019fairnas,luo2019understanding}.
In this paper, we study the risk of the WS scheme and find that WS-based methods suffer from a performance disturbance (PD) issue. That is, the training of a subsequent architecture inevitably disturbs the performance of previously trained architectures due to the update of shared parameters. As a result, the performance estimation of the previous architectures is unstable and inaccurate, which makes it difficult to search for good architectures. To address this issue, we propose a new disturbance-immune update strategy to train the WS. By exploring orthogonal gradient descent, the proposed strategy trains architectures for better performance while constraining the prediction of previously trained architectures to be unchanged. In this way, we are able to alleviate the PD issue in WS and provide more stable and accurate performance estimation. Based on this update strategy, we further propose a novel disturbance-immune training scheme for NAS, which helps to search for better architectures.
The main contributions of this paper are summarized as follows.
\begin{itemize}[leftmargin=*]
\item We propose a novel disturbance-immune WS training scheme for NAS. By updating models in an orthogonal space with orthogonal gradient decent, our method exhibits more stable/accurate performance estimation for NAS. Equipped with the disturbance-immune WS, NAS is able to learn a better search strategy and find better architectures.
\item We theoretically verify the effectiveness of the proposed disturbance-immune training scheme in alleviating the performance disturbance risk of WS. We also provide an asymptotic convergence analysis of the proposed method.
\item Extensive experiments demonstrate that the proposed method is able to alleviate the performance disturbance issue and provide more accurate performance evaluation for strategy learning. As a result, the architecture searched by our method performs better than the architectures obtained by other state-of-the-art NAS methods on the CIFAR-10 and ImageNet datasets.
\end{itemize}
\section{Related Work}
\textbf{Neural architecture search.}
In the past few years, neural architecture search (NAS) has attracted increasing attention to automatically design effective architectures.
\textcolor{black}} \def\guo{\textcolor{black}{The classical NAS problem \cite{zoph2016neural} exploits the paradigms of reinforcement learning (RL) to generate the model descriptions of DNNs.}
\textcolor{black}} \def\guo{\textcolor{black}{After that, MetaQNN \cite{baker2016designing} automatically selects the architectures of DNNs through a RL-based meta-modeling procedure. NASNet \cite{zoph2018learning} designs a new search space to improve the search performance.}
\textcolor{black}} \def\guo{\textcolor{black}{Moreover, some studies \cite{real2018regularized,liu2017hierarchical} use evolutionary algorithms to find new architectures with excellent performance.}
To guide the search process, the NAS methods need to estimate the performance of candidate architectures.
The simplest way is to train candidate architectures from scratch to obtain the performance, which, however, is time-consuming and computationally expensive (\emph{e.g.,~} several thousands of GPU days).
To reduce the computational resources, recent NAS methods adopt \textcolor{black}} \def\guo{\textcolor{black}{a} weight sharing \cite{pham2018efficient,liu2018darts,cai2018efficient,wu2019fbnet} strategy to estimate the performance of candidate architectures.
\textbf{Weight sharing approaches.}
Efficient neural architecture search (ENAS)~\cite{pham2018efficient} \textcolor{black}} \def\guo{\textcolor{black}{first proposes} a NAS training scheme with weight sharing (WS), which measures the performance of an architecture with the weights inherited from the trained supernet.
Since WS can reduce the computational resources from thousands of GPU days to one \textcolor{black}} \def\guo{\textcolor{black}{GPU day}, it is widely adopted to exploit NAS \textcolor{black}} \def\guo{\textcolor{black}{in} various applications, such as objection detection~\cite{ghiasi2019fpn,chen2019detnas}, segmentation~\cite{liu2019auto} and compact architecture design~\cite{cai2018efficient,wu2019fbnet,tan2019mnasnet}.
Besides, DARTS~\cite{liu2018darts} exploits the WS scheme with a continuous relaxation of the search space to search \textcolor{black}} \def\guo{\textcolor{black}{for} promising architectures.
However, recent studies~\cite{li2019random,sciuto2019evaluating} find that the architecture performance measured by WS training is often very inaccurate, thus leading to the inferior performance of WS based NAS methods.
To address this, NAO-V2~\cite{luo2019understanding} improves WS based NAS by training candidate architectures adequately and training complex architectures more.
FairNAS~\cite{chu2019fairnas} proposes a fair WS training strategy, which can be only applied to a single-path search space instead of a cell-based search space.
Unlike existing methods, we identify a performance disturbance (PD) issue that occurred in WS training, and propose to achieve more accurate performance estimations by alleviating this PD issue.
\textbf{Continual learning.}
Continual learning (CL) aims to continuously learn a series of tasks~\cite{kirkpatrick2017overcoming,zeng2019continual,finn2017model,farajtabar2019orthogonal}.
One of the major issues in CL is catastrophic disturbance, \emph{i.e.,~} deep models often forget about previous tasks when model weights are updated for a new task.
To \textcolor{black}} \def\guo{\textcolor{black}{address this}, EWC~\cite{kirkpatrick2017overcoming} attempts to impose constraints on the updating of model weights based on measuring the importance of previous tasks.
Moreover, regularization based methods ~\cite{he2017overcoming,zeng2019continual} neglect gradients for new tasks.
\textcolor{black}} \def\guo{\textcolor{black}{In this paper, motivated by CL, we propose a novel disturbance-immune update strategy to effectively improve the performance of WS-based NAS methods.}
\section{Problem Definition and Motivation}
\noindent \textbf{Notations.}
Throughout the paper, we use the following notations.
Let $\Omega$ be the search space of the NAS. Given any architecture $ \alpha \in \Omega$, let $w(\alpha)$ be its trainable parameters and $w^*(\alpha)$ be its optimal model parameters trained on some datasets (\emph{e.g.,~} CIFAR-10 \& ImageNet). Moreover, let $||\cdot||_{\mathrm F}$ denote the Frobenius norm, and let $[n]=\{1,\dots,n\}$.
\subsection{Neural Architecture Search and Weight Sharing}
Neural architecture search (NAS) aims to \guo{search for} an optimal architecture \guo{from} a predefined search space.
In this paper, we focus on reinforcement learning (RL)-based methods~\cite{zoph2016neural,zoph2018learning}, which seek to learn a controller with parameters $\theta$ to generate architectures (\emph{i.e.,~} $\alpha\ {\sim} \pi(\alpha; \theta)$).
To \guo{find promising} architectures, these methods learn the controller by maximizing the expectation of \guo{architecture performance using} metric $\mathcal{R}(\alpha, w^*(\alpha))$ (\emph{e.g.,~} the accuracy on the validation set):
\begin{align}\label{eq:nas_theta}
\max_{\theta}~\mathbb{E}_{\alpha\sim \pi(\alpha;\theta)}\mathcal{R}\left(\alpha, w^*(\alpha)\right), ~~\mathrm{s.t.}~ w^*(\alpha)=\arg\min_{w(\alpha)}\mathcal{L}\left(\alpha,w(\alpha)\right),
\end{align}
where $\mathcal{L}\left(\alpha,w(\alpha)\right)$ is the training loss on the training data. However, \guo{we have to} train candidate architectures from scratch to obtain $w^*(\alpha)$, resulting in an unbearable computational burden. To address this, a weight sharing scheme is proposed.
Weight sharing (WS) for NAS~\cite{pham2018efficient} constructs a supernet, \emph{i.e.,~} a large computational graph, where each network architecture (subgraph) shares their parameters.
Let $w$ be the \guo{parameters} of the whole supernet and $w(\alpha)$ be the parameters of $\alpha$ inherited from the supernet.
\guo{To train the supernet, one can sample sufficiently many architectures $(\alpha_i)_{i\in[n]}$ and train them in a sequential manner~\cite{pham2018efficient}:}
\begin{align}\label{eq:ws_general}
\min_{w(\alpha_i)}{\mathcal L} (\alpha_i, w(\alpha_i)),~\forall i\in [n].
\end{align}
Despite the training efficiency, the \guo{performance estimated} by WS is often very inaccurate~\cite{adam2019understanding,li2019random,sciuto2019evaluating} due to the sequential training method of WS.
Specifically, the training of a subsequent architecture inevitably disturbs the performance of previously trained architectures. As a result, the \guo{estimated} performance of the previous architectures becomes inaccurate, \guo{making it hard to learn a good controller and search for good architectures.}
\begin{figure*}[!t]
\centering
\subfigure[Example of weight sharing (WS)]
{
\label{fig:supernet}
\includegraphics[width=0.48\linewidth]{supernet.pdf}
}
\subfigure[Performance disturbance in WS]
{
\label{fig:disturbance}
\includegraphics[width=0.48\linewidth]{jet1.pdf}
}
\caption{Exemplar illustration of WS and performance disturbance (PD).
(a) The weights of architectures $\alpha_i$ and $\alpha_j$ are inherited from the supernet.
There are some shared weights in both $\alpha_i$ and $\alpha_j$.
(b) We record the performance of 10 architectures $(\alpha_i)_{i\in[10]}$. Element $(i,j)$ denotes the accuracy of $\alpha_i$ after the training of $\alpha_j$. Note that the training of the subsequent architectures results in apparent performance disturbance of the previously trained architectures.}
\label{fig:pi}
\vspace{0.1in}
\end{figure*}
\subsection{Performance Disturbance in Weight Sharing}
During the training of WS, for any two architectures $\alpha_i$ and $\alpha_j$ with $i {<} j$, there are some shared parameters that appear in both $\alpha_i$ and $\alpha_j$ (see examples in Fig.~\ref{fig:supernet}).
Due to the sequential training strategy in Eqn.~(\ref{eq:ws_general}), once we update the parameters of $\alpha_j$, the shared parameters would also be changed and $\alpha_i$ may yield different prediction results.
As a result, the architecture $\alpha_i$ may incur severe
\textbf{performance disturbance (PD)} and the reward ${\mathcal R}(\alpha_i, w(\alpha_i))$ becomes inaccurate. To justify this, we show an illustrative example in Fig.~\ref{fig:disturbance}.
In Fig.~\ref{fig:disturbance}, we consider training 10 architectures $(\alpha_i)_{i\in[10]}$ sequentially with shared parameters on CIFAR-10.
We record their performance in terms of validation accuracy after the training of each architecture.
The element $(i,j)$ in Fig.~\ref{fig:disturbance} denotes the validation accuracy of $\alpha_i$ after the training of $\alpha_j$.
The $i$-th row illustrates the performance disturbance of $\alpha_i$ during the whole training process.
Moreover, the more subsequent architectures are trained, the more severe performance disturbance is.
Such unstable and inaccurate performance estimation would mislead learning the policy.
The performance disturbance can be measured by the predictions of an architecture, since the predictions directly determine architecture performance.
Let ${\bf X}$ be the input data and $f({\bf X}; w(\alpha))$ be the prediction of $\alpha$ based on its parameters $w(\alpha)$.
For convenience,
we use $w_o(\alpha)$ and $w(\alpha)$ to represent the original parameters and the latest parameters after the training of other architectures that have shared parameters, respectively. During the training of WS, the performance disturbance of $\alpha$ can be measured by the differences of predictions:
\begin{equation}\label{eq:predict_difference}
{\rm PD}({\bf X}, \alpha) := || f({\bf X}; w(\alpha)) - f({\bf X}; w_o(\alpha)) ||_\mathrm{F}.
\end{equation}
\subsection{Disturbance-immune Weight Sharing Problem}\label{sec:motivation}
To reduce the prediction differences, we seek to define and address a disturbance-immune problem. Similar to model compression methods~\cite{luo2017thinet,zhuang2018discrimination}, one possible way is to restrict the changes of feature maps in each layer. In this sense, if we can update the shared parameters while maintaining the same feature maps for all previously trained architectures, the performance of these architectures will not be changed.
Specifically, for the $\kappa$-th update, we seek to improve the performance of $\alpha_\kappa$ and reduce the performance disturbance of all the previous architectures $(\alpha_i)_{i\in[\kappa-1]}$.
In this way, the weight sharing scheme is able to provide a more stable and accurate reward (or performance estimation) for all candidate architectures.
To this end, we propose a disturbance-immune training strategy that makes the update term of the model parameters orthogonal to the input features.
Note that all the layers adopt the same parameter update method. For simplicity, we only investigate the training process \emph{w.r.t.~} a single layer.
Specifically, during the training of the WS, for any layer of the supernet, we record the input feature maps ${\bf X}$ of all the previously trained architectures in ${\mathcal X}$.
To ensure that we can improve performance \emph{w.r.t.~} the current architecture, we do not include its input feature in ${\mathcal X}$.
In order to reduce PD of the previously trained architectures, we make the gradient \emph{w.r.t.~} the shared parameters orthogonal to all the input feature maps in ${\mathcal X}$. In this way, there would be no change in the prediction results for the previous architectures.
Formally, let ${\bf W}$ be the parameters of a specific layer and ${\bf W}_o$ be the original parameters before the model update. The parameters ${\bf W}$ have to satisfy a constraint that restricts the changes of output feature maps and forms a feasible set
\begin{equation}
{\mathcal Q}=\{{\bf W}: ||{\bf W}^\top{\bf X}-{\bf W}_o^{\top}{\bf X}||_\mathrm{F}\leq\gamma, \forall~{\bf X} \in {\mathcal X} \},
\end{equation}
where $\gamma$ is some small positive value. When we apply such constraints to train WS, the optimization problem becomes:
\begin{equation}
\min_{w(\alpha_i)}{\mathcal L} (\alpha_i, w(\alpha_i)), ~~\forall i\in[n]
~~\mathrm{s.t.}~{\bf W}\in {\mathcal Q}.
\label{eq:new_constraint}
\end{equation}
Unlike the original problem in Eqn.~(\ref{eq:ws_general}), we seek to train the WS by keeping the output features of all the previously trained architectures unchanged.
In this sense, the performance of these architectures becomes stable and the PD of them can be greatly reduced (see results in Fig.~\ref{fig:hpnas_epoch_interval_13}).
Note that recording all the input features is infeasible in practice. To address this issue, we propose an equivalent solution that adopts the iterative training scheme and avoids data recording (see details in Section~\ref{sec:gradient}).
More critically, we theoretically prove that the proposed method is able to reduce the risk of performance disturbance (see theoretical analysis in Section~\ref{sec:theory}). We call this method Disturbance-immune Weight Sharing and show the details in Section~\ref{sec:dfws}.
\section{Disturbance-immune Weight Sharing}\label{sec:dfws}
In this section, we first propose a disturbance-immune update strategy for architecture training with weight sharing (WS). Such a strategy aims to handle performance disturbance (PD) by constraining the change of feature maps of previously trained architectures.
Based on this strategy, we propose a new disturbance-immune training scheme for neural architecture search (DI-NAS\xspace).
Lastly, we theoretically analyze the effectiveness of the proposed method.
\subsection{Disturbance-immune Update Strategy}\label{sec:gradient}
To address PD, we propose to project the update gradient towards a useful direction but avoid large changes regarding the feature maps of shared parameters. To this end, we resort to the orthogonal gradient descent method~\cite{zeng2019continual}. Such a method projects the gradient to be orthogonal to the input features, which ensures the output features to change very slightly regarding the input.
To be specific, we maintain an orthogonal projection matrix set ${\mathcal P}$ for the supernet. Each parameter in the supernet has a corresponding projection matrix ${\bf P}$, initialized as a unit matrix ${\bf I}$. These projection matrices are used to project gradients for model training.
Overall, there are two key issues: how to update model parameters via the orthogonal projection matrix, and how to update the orthogonal projection matrix.
\subsubsection{The update of model parameters}
We train architectures based on orthogonal gradient descent~\cite{zeng2019continual}, \textcolor{black}{in which we project the gradient to be orthogonal to the input. In this way, we are able to update the model towards a useful direction but avoid large changes regarding the previous output of the shared parameters.} Formally, given any architecture $\alpha_i$
and for any layer ${\bf W}\in w(\alpha_i)$, there is a corresponding projection matrix ${\bf P}$ inherited from the projection matrix set ${\mathcal P}$. We update ${\bf W}$ by the following orthogonal gradient descent:
\begin{align}\label{eq:updating_share}
{\bf W}\leftarrow{\bf W}-\eta{\bf P}\nabla_{{\bf W}}{\mathcal L}(\alpha_i,w(\alpha_i)),
\end{align}
where ${\bf P}\nabla_{{\bf W}}{\mathcal L}(\cdot)$ is often called as the projected gradient.
Moreover, when the layers in $\alpha_i$ update, the corresponding layers in the supernet also update.
\subsubsection{The update of projection matrices}
In order to project the gradient of each shared layer to avoid large changes regarding its output, we ensure the projected gradient is orthogonal to the previous input. Specifically, for any layer ${\bf W}$ of $\alpha_i$, let ${\bf X}\in{\mathbb R}^{d\times N}$ denote the input feature maps of all previously trained architectures, with a total of $N$ feature maps (whose dimension is $d$). We compute the orthogonal projection matrix ${\bf P}$ by:
\begin{align}\label{eq:compute_p}
{\bf P} =&~{\bf I}-{\bf X}(\lambda{\bf I} + {\bf X}^{\top}{\bf X})^{-1}{\bf X}^{\top}, \end{align}
where $\lambda$ is some regularization constant.
The detailed derivations will be provided in a long version.
Based on Eqn.~(\ref{eq:compute_p}), our method is able to project the gradient to be orthogonal to the input, which helps to void a large change regarding the output of the shared layers (see Theorem~\ref{prof1} for more discussion).
A potential issue of Eqn.~(\ref{eq:compute_p}) is that the computation of ${\bf P}$ requires all previous input feature maps (as described in Section~\ref{sec:motivation}). However, such a manner may be highly storage consuming as the input feature maps increase.
To handle this, we can update the projection matrix in an iterative manner~\cite{zhang2018online,zhang2019online,zhao2018adaptive}. For each coming sample ${\bf x}\in{\mathbb R}^d$, based on Woodbury identity~\cite{horn2012matrix}, we update ${\bf P}$ by:
\begin{align}\label{eq:update_p}
{\bf P} \leftarrow {\bf P} - ({\bf P}{\bf x} {\bf x}^{\top} {\bf P})/(\lambda + {\bf x}^\top {\bf P} {\bf x}),
\end{align}
Based on Eqn.~(\ref{eq:update_p}), we update ${\bf P}$ iteratively and each iteration only requires the input feature maps \emph{w.r.t.~} a single sample. By doing so, we need not to store all previous input feature maps, thus avoiding the high storage issue. Note that the update of ${\bf P}$ also means the update of the corresponding projection matrix in ${\mathcal P}$.
\subsection{Disturbance-immune Neural Architecture Search}
The proposed disturbance-immune update strategy helps to alleviate the issue of performance disturbance in the WS, and provides more stable performance estimations for candidate architectures. As a result, we can obtain more accurate reward signals for learning a good controller.
Following this, based on the disturbance-immune update strategy,
we propose a new disturbance-immune training scheme for neural architecture search (named as DI-NAS\xspace). The overall training scheme is provided in Algorithm~\ref{alg:overall}.
The main difference between the proposed DI-NAS\xspace and other standard WS based NAS is the training of the supernet. Specifically, we train the supernet via the new proposed disturbance-immune update strategy, which alleviates the performance disturbance issue in the training process. To be specific, we maintain an orthogonal projection matrix set ${\mathcal P}$ for the supernet (see line 2 in Algorithm~\ref{alg:overall}). Each parameter in the supernet has a corresponding projection matrix ${\bf P}$, initialized as a unit matrix ${\bf I}$. Based on the orthogonal projection matrix, we update the network architecture by orthogonal gradient descent (see lines 9-10 in Algorithm~\ref{alg:overall}).
\begin{algorithm}[tb]
\caption{\small{Overall training scheme of DI-NAS\xspace}}\label{alg:overall}
\begin{algorithmic}[1]\small
\REQUIRE Training data ${\mathcal D}_{train}$, validation data ${\mathcal D}_{val}$, learning rate $\eta$, parameters $T_n$,$T_c$.
\STATE {Initialize supernet parameters $w$ and controller parameters $\theta$.}
\STATE Construct and initialize a projection matrix set ${\mathcal P}$ for the supernet.\\
\WHILE {not convergent}
\STATE // \emph{Update $w$ by minimizing the training loss}
\FOR{$i=1,...,T_n$}
\STATE Sample $\alpha\sim\pi(\alpha;\theta)$.~~~\small{//} \emph{ $\pi(\cdot)$ is the policy of the controller}
\STATE Sample a batch of data
from ${\mathcal D}_{train}$.
\STATE \small{//} \emph{\textbf{Disturbance-immune Update} for $w(\alpha)$ and ${\mathcal P}(\alpha)$}
\STATE {Update $w(\alpha)$ using Eqn.~(\ref{eq:updating_share}).}~~~\small{//} \emph{$w(\alpha)$ denotes the parameters of $\alpha$}
\STATE {Update ${\mathcal P}(\alpha)$ using Eqn.~(\ref{eq:update_p}).}~~~\small{//} \emph{${\mathcal P}(\alpha)$ denotes the projection matrices of $\alpha$}
\ENDFOR
\STATE // \emph{Update $\theta$ by maximizing the reward}
\FOR{$j=1,...,T_c$}
\STATE Sample $\alpha\sim\pi(\alpha;\theta)$.
\STATE Sample a batch of data from ${\mathcal D}_{val}$.
\STATE Update
$\theta\leftarrow\theta+\eta\mathcal{R}(\alpha, w)\nabla_\theta\text{log}\pi(\alpha;\theta)$.
\ENDFOR
\ENDWHILE
\end{algorithmic}
\label{alg:training}
\vspace{-0.03in}
\end{algorithm}
\subsection{Theoretical Analysis}\label{sec:theory}
In this section, we theoretically analyze the proposed method regarding its effectiveness and convergence.
To begin with, we analyze the effectiveness of our proposed method in alleviating the issue of performance disturbance as follows.
\begin{theorem}\label{prof1}
Given a model with parameter ${\bf W}\small{\in}{\mathbb R}^{d\times m}$ and any input matrix ${\bf X}\small{\in}{\mathbb R}^{d\times N}$, let ${\bf P}={\bf I}-{\bf X}(\lambda{\bf I}+{\bf X}^\top{\bf X})^{-1}{\bf X}^\top$ be the projection matrix and ${\bf G}$ be the gradient \emph{w.r.t.~} ${\bf W}$. For the update of ${\bf W}$ in the direction ${\bf P}{\bf G}$ and $\eta\in{\mathbb R}^+$, let $\Delta({\bf X})=||({\bf W}\small{-}\eta {\bf P}{\bf G})^\top{\bf X}\small{-}{\bf W}^\top{\bf X} ||_\mathrm{F}$ denote the output change of the original model and the updated model. When $\lambda\geq 0$, the following inequality holds:
\begin{align}\label{eq:theorem1}
\Delta({\bf X})
\leq \eta\lambda\sqrt{d} ||{\bf G}^\top||_\mathrm{F}D({\bf X}),
\end{align}
where $D({\bf X})=||{\bf X}||_\mathrm{F}R({\bf X})(1\small{+}\lambda R({\bf X}))$ and $R({\bf X})=||({\bf X}\bX^\top)^{\small{-1}}||_2$.
\end{theorem}
Theorem~\ref{prof1} indicates that for any input feature map ${\bf X}$, the distance between the original output ${\bf W}^\top{\bf X}$ and the one after the update $({\bf W}\small{-}\eta {\bf P}{\bf G})^\top{\bf X}$ is controlled by the regularization factor $\lambda$. Specifically, the lower $\lambda$, the lower distance upper bound. Therefore, our proposed orthogonal gradient descent method is able to satisfy the constraint in Eqn.~(\ref{eq:new_constraint}). In other words, our method can avoid the large change regarding the previous output of shared layers when training the supernet, thus alleviating the issue of performance disturbance in WS.
We further provide an asymptotic analysis regarding the convergence of the update for ${\bf W}$ as follows.
\begin{theorem} \label{theorem:convergence}
Given a loss function $\mathcal{L}({\bf W})$ that is $L$-smooth and convex \emph{\emph{w.r.t.~}}~${\bf W}$. Let ${\bf W}^*$ and ${\bf W}_0$ be the optimal and initial solution of ${\mathcal L}(\cdot)$. By setting $\eta = 1/L$, at the $t$-th update step, the proposed update strategy satisfies:
\begin{equation}\label{eq:convergence}
{\mathcal L}({\bf W}_t) - {\mathcal L}({\bf W}^*) \leq\frac{2L}{t}||{\bf W}_0-{\bf W}^*||_\mathrm{F}^2.
\end{equation}
\end{theorem}
This theorem illustrates that our proposed update strategy has a sublinear convergence rate of $O(1/t)$, which guarantees the effectiveness of the proposed method.
\section{Experimental Results}
We evaluate the proposed DI-NAS\xspace in two main aspects: (1) the superiority of the searched architecture by DI-NAS\xspace on CIFAR-10 and ImageNet, respectively; (2) the effectiveness of our proposed disturbance-immune update strategy. The source code will be publicly available.
\subsection{Evaluation on CIFAR-10}
In this section, we evaluate the proposed DI-NAS\xspace method on CIFAR-10~\cite{krizhevsky2009learning}. To be specific, we first use the proposed method to train a controller on CIFAR-10, and use it to search for a convolutional neural architecture. By comparing the searched architecture with other state-of-the-art architectures, we can evaluate the effectiveness of our method. To this end, we first describe the search space, training details and evaluation details.
\textbf{Search space.} Following the settings in DARTS~\cite{liu2018darts}, we aim to search for two types of convolutional cells, namely the normal cell and the reduction cell. Each cell contains 7 nodes, including 2 input nodes, 4 intermediate nodes and 1 output node. Between any two nodes, there are 8 available operations, including $3\small{\times} 3$ depthwise separable convolution, $3\small{\times} 3$ dilated convolution, $3\small{\times} 3$ max pooling, $3\small{\times} 3$ average pooling, $5\small{\times} 5$ depthwise separable convolution, $5\small{\times} 5$ dilated convolution, identity and none. After obtaining the convolutional cells, we stack them to build the final convolutional network.
\textbf{Training details.}
In the search phase, we divide the standard training set of CIFAR-10 into two parts. Specifically, we randomly select 40\% of the training set to train sampled architectures (as training data) and use the rest 60\% to learn controllers (as validation data). Moreover, we train DI-NAS\xspace for $240$ epochs in total. We first train the supernet without learning the controller, and start training the controller from epoch 90. In addition, we set $\lambda=1$ by default, where the sensitivity analysis can be found in Section~\ref{sec_discussions}. For training the supernet, we use an SGD optimizer with a weight decay of $3 \times 10^{-4}$ and a momentum of $0.9$. The learning rate is set to $0.1$.
For training the controller, we use ADAM with a learning rate of $3 \times 10^{-4}$ and a weight decay of $5 \times 10^{-4}$.
We add the controller's sample entropy to the reward, which is weighted by $0.005$.
\textbf{Evaluation details.}
In the evaluation phase, we first use the learned controller to search for a normal cell and a reduction cell. Then, we construct the final convolutional network with 17 normal cells and 2 reduction cells. Following~\cite{nayman2019xnas}, we put the two reduction cells at the $1/3$ and $2/3$ depth of the network, respectively. The initial number of the channels is set to 43. Following DARTS~\cite{liu2018darts}, we train the convolutional network for $600$ epochs with a batch size of $128$. We apply \textcolor{black}} \def\guo{\textcolor{black}{an} SGD optimizer with a weight decay of $3 \times 10^{-4}$ and a momentum of $0.9$. Moreover, we set the initial learning rate as $0.05$ and use the cosine annealing strategy~\cite{SGDR} to adjust it. We also use the cutout scheme~\cite{devries2017improved} with length 16 for data augmentation.
\begin{table*}[t]
\vspace{0.15in}
\centering
\caption{Comparisons with state-of-the-art NAS methods on CIFAR-10. Moreover, ``-" means unavailable results.
}
\label{tab:cifar-10}
{
\begin{tabular*}{0.95\textwidth}{@{}@{\extracolsep{\fill}}cccccc@{}}
\hline
\multicolumn{1}{c}{\multirow{2}[0]{*}{Architecture}} &
\multicolumn{1}{c}{\multirow{2}[0]{*}{Test Accuracy (\%)}} & &
\multicolumn{1}{c}{\multirow{2}[0]{*}{\# Params (M)}} & &
\multicolumn{1}{c}{Search Cost} \\
& & & & & (GPU days) \\
\hline
DenseNet-BC~\cite{huang2017densely}&96.54& &25.6 & & -- \\
PyramidNet-BC~\cite{han2017deep}&96.69& &26.0 & & -- \\
\hline
Random search baseline &96.71 $\pm$ 0.15 & &3.2 & & -- \\
NASNet-A + cutout~\cite{zoph2018learning}&97.35 & &3.3 & & 1,800 \\
NASNet-B~\cite{zoph2018learning}&96.27& &2.6 & & 1,800 \\
NASNet-C~\cite{zoph2018learning}&96.41& &3.1 & & 1,800 \\
AmoebaNet-A + cutout~\cite{real2018regularized}&96.66 $\pm$ 0.06& &3.2 & & 3,150\\
AmoebaNet-B + cutout~\cite{real2018regularized}&96.63 $\pm$ 0.04& &2.8 & & 3,150\\
Hierarchical Evo~\cite{liu2017hierarchical}&96.25 $\pm$ 0.12 & &15.7 & & 300 \\
SNAS~\cite{xie2018snas}&97.02& &2.9 & & 1.5\\
GHN~\cite{zhang2018graph} & 97.16 $\pm$ 0.07 & & 5.7 & & 0.8 \\
ENAS + cutout~\cite{pham2018efficient}&97.11& &4.6 & & 0.5 \\
DARTS + cutout~\cite{liu2018darts} &97.24 $\pm$ 0.09 & &3.4& & 4\\
NAT-DARTS~\cite{NAT} &97.28 & &2.7 & & --\\
NAONet~\cite{luo2018neural}&97.02& &28.6 & & 200\\
NAONet-WS~\cite{luo2018neural}&96.47& &2.5 & & 0.3\\
\hline
DI-NAS\xspace + cutout & \textbf{97.38 $\pm$ 0.04} & & 3.7 & & 1.5 \\
\hline
\end{tabular*}
}
\end{table*}
\begin{figure*}[!h]
\centering
\subfigure[Normal cell.]
{
\label{fig:normal}
\includegraphics[height=4.5cm]{normal_cell_2.pdf}
}
\subfigure[Reduction cell.]
{
\label{fig:reduction}
\includegraphics[height = 5.2cm]{reduction_cell_2.pdf}
}
\caption{The searched convolutional cells by DI-NAS\xspace on the CIFAR-10 dataset.}
\label{fig:cnn_arch}
\end{figure*}
\textbf{Comparison with state-of-the-art methods.}
We show the searched normal cells and reduction cells in Fig.~\ref{fig:cnn_arch} and report the detailed results of all methods in Table~\ref{tab:cifar-10}. For our searched architectures, we run 5 experiments with the different random initialization, and report the average performance with standard deviation. Experimental results show that the architecture searched by our DI-NAS\xspace achieves 97.38\% accuracy, outperforming all other state-of-the-art architectures, including human-designed ones and NAS ones. Note that ENAS, NAONet-WS, DARTS, and NAT also use the scheme of weight sharing (WS).
Such a result demonstrates that it is necessary to alleviate the issue of performance disturbance(PD) in WS-based NAS. This helps us to provide more accurate rewards for controller learning, and search for better neural architectures.
\subsection{Evaluation on ImageNet}
To verify the generalization of the convolutional cells searched on CIFAR-10 (as shown in Fig.~\ref{fig:cnn_arch}), we further evaluate them on a large-scale image classification dataset, namely ImageNet~\cite{deng2009imagenet}.
To begin with, we describe the evaluation details.
\textbf{Evaluation details.}
For the ImageNet dataset, we construct the convolutional network with 12 normal cells and 2 reduction cells. We put the two reduction cells at the $1/3$ and $2/3$ depth of the network, respectively. We set the number of the initial channels to 48. Following~\cite{liu2018darts}, we train the network by \textcolor{black}} \def\guo{\textcolor{black}{an} SGD optimizer with $250$ epochs and use a weight decay of $3 \times 10^{-5}$ and a momentum of $0.9$. We initialize the learning rate as $0.1$ and decrease it by the cosine annealing~\cite{SGDR}. We follow the ImageNet mobile setting~\cite{liu2018darts}, where the size of input images is set to $224 \times 224$ and the number of multiply-adds (Madds) is less than 600M.
\begin{table*}[t]
\centering
\caption{Comparison results on ImageNet, where we use the evaluation code of DARTS~\cite{liu2018darts}. Moreover, ``-'' means unavailable results.
}
\label{tab:imagenet}
{
\begin{tabular*}{0.99\textwidth}{@{}@{\extracolsep{\fill}}ccccccccc@{}}
\hline
\multicolumn{1}{c}{\multirow{2}[0]{*}{Architecture}} &
\multicolumn{2}{c}{Test Accuracy (\%)} & &
\multicolumn{1}{c}{\# Params} & &
\multicolumn{1}{c}{\# MAdds} & &
\multicolumn{1}{c}{Search Cost} \\
\cline{2-3}
& \multicolumn{1}{c}{~~Top-1} & \multicolumn{1}{c}{~~Top-5} & & (M) & & (M) & & (GPU days) \\
\hline
ResNet-18~\cite{resnet} & ~~69.8 & ~~89.1 & & 11.7 & & 1,814 & &--\\
Inception-v1~\cite{szegedy2015going} & ~~69.8 & ~~89.9 & & 6.6 & & 1,448 & &--\\
MobileNet v1 ($1\times$)~\cite{howard2017mobilenets} & ~~70.6 & ~~89.5 & & 4.2 & & 569 & &-- \\
ShuffleNet v1 ($2\times$)~\cite{Zhang2018ShuffleNetAE} & ~~70.9 & ~~89.2 & & 5.0 & & 524 & &--\\
\hline
NASNet-A~\cite{zoph2018learning} & ~~74.0 & ~~91.6 & & 5.3 & & 564 & & 3,150\\
NASNet-B~\cite{zoph2018learning} & ~~72.8 & ~~91.3 & & 5.3 & & 488 & & 3,150\\
NASNet-C~\cite{zoph2018learning} & ~~72.5 & ~~91.0 & & 4.9 & & 558 & & 3,150\\
AmoebaNet-A~\cite{real2018regularized} & ~~74.5 & ~~92.0 & & 5.1 & & 555 & & 1,800\\
AmoebaNet-B~\cite{real2018regularized} & ~~74.0 & ~~91.5 & & 5.3 & & 555 & & 1,800\\
GHN~\cite{zhang2018graph} & ~~73.0 & ~~91.3 & & 6.1 & & 569 & & 0.8\\
PNAS~\cite{liu2018progressive} & ~~74.2 & ~~91.9 & & 5.1 & & 588 & & 255\\
BayesNAS~\cite{BayesNAS} & ~~73.5 & ~~91.1 & & 3.9 & & - & & 0.2\\
DARTS~\cite{liu2018darts} & ~~73.1 & ~~91.0 & & 4.9 & & 595 & & 4\\
NAT-DARTS~\cite{NAT} & ~~73.7 & ~~91.4 & & 4.0 & & 441 & & --\\
SNAS~\cite{xie2018snas} & ~~72.7 & ~~90.8 & & 4.3 & & 522 & & 1.5\\
\hline
DI-NAS\xspace & ~~\textbf{74.7} & ~~\textbf{92.1} & & 5.2 & & 587 & & 1.5\\
\hline
\end{tabular*}
}\vspace{-0.15in}
\end{table*}
\textbf{Comparison with state-of-the-art methods.}
We compare our searched architecture with several state-of-the-art models on ImageNet. As shown in Table~\ref{tab:imagenet}, our architecture achieves 74.7\% top-1 accuracy and 92.1\% top-5 accuracy. To be specific, our architecture outperforms human-designed architectures (\emph{e.g.,~}{ResNet-18~\cite{resnet}}) by about 5\%, and outperforms most of NAS models by $0.5\% \backsim 2\%$ in terms of {top-1 accuracy}. Moreover, our architecture achieves excellent performance only using 1.5 GPU days, while AmoebaNet~\cite{real2018regularized} and NASNet~\cite{zoph2018learning} spend 3,150 and 1,800 GPU days, respectively. These results demonstrate the generalization of the searched convolutional cells and the effectiveness/efficiency of the proposed method.
\subsection{Effectiveness of Disturbance-immune Update Strategy}
In previous experiments, we have demonstrated the superiority of the proposed method. One important reason for superiority is the ability to alleviate performance disturbance (PD). In this section, we further verify the effectiveness of our proposed update strategy in dealing with PD. To this end, we use the following two metrics to measure the degree of PD and use them to evaluate the proposed method.
\textbf{\textcolor{black}} \def\guo{\textcolor{black}{Metrics for PD.}}
(1) Performance change: for any architecture $\alpha$ inherited from the supernet, performance change is defined as ($acc_j\small{-}acc_i$), where $acc_j$ and $acc_i$ denote the validation accuracy of $\alpha$ at the $j$-th and $i$-th training epoch of the supernet\textcolor{black}} \def\guo{\textcolor{black}{, respectively}. Note that $j>i$ and $(j\small{-}i)$ indicates the \textbf{epoch interval}. Overall, the larger performance change indicates the more severe performance disturbance.
(2) Kendalls Tau (KTau): we use KTau~\cite{sen1968estimates} to measure the correlation of performance ranks between two architecture sets. The range of KTau belongs to $[-1,1]$, where larger KTau means that the performance ranks of two architecture sets are more consistent.
\begin{figure*}[ht]
\centering
\begin{subfigure}[absolute performance change]{
\centering
\includegraphics[width=6.4cm]{mean_abs_pds_mean_vs_mean_box_2.pdf}}
\end{subfigure}
\begin{subfigure}[KTau at different training epochs]{
\centering
\includegraphics[width=6.4cm]{kendall_tau_mean_vs_mean_box.pdf}}
\end{subfigure}
\vspace{-0.1in}
\caption{Comparisons between our proposed disturbance-immune weight sharing (WS) scheme and the standard WS scheme. The larger the performance change is, the more severe the PD; the higher KTau is, the more accurate the performance estimation.
}
\label{fig:hpnas_epoch_interval_13}
\end{figure*}
\textbf{Evaluation in terms of performance change.}
To evaluate our method, we randomly sample 64 architectures based on the supernet. We record their performance at different training epochs of the supernet, and report the average performance change of these 64 architectures regarding epoch interval 13.
Fig.~\ref{fig:hpnas_epoch_interval_13}~(a) shows that our disturbance-immune WS scheme is able to reduce the performance change of architectures and thus alleviate the PD issue.
\textbf{Evaluation in terms of KTau.}
Based on the above architectures, we compute the KTau between their current performance rank and the one after 13 epochs (\emph{i.e.,~} epoch interval is 13). Fig.~\ref{fig:hpnas_epoch_interval_13} (b) verifies the effectiveness of our method in improving KTau. Note that the higher KTau indicates a more consistent performance estimation in the training process, which means more stable/accurate rewards for controller learning. Moreover, we compute the ground-truth KTau (GT-Tau)~\cite{sciuto2019evaluating}, which measures the performance correlation between a set of architectures inherited from the supernet and exactly trained from scratch. Specificcally, our method achieves higher GT-Tau (0.48) than the standard WS (0.16).
\subsection{More Discussions}\label{sec_discussions}
We further discuss PD for the different numbers of shared layers and parameter sensitivity $\emph{w.r.t.~} \lambda$ through some self-designed experiments.
We provide our main observations \textcolor{black}} \def\guo{\textcolor{black}{and analyses} as follows.
\textbf{Number of shared layers.}
We find that as the number of shared layers \textcolor{black}} \def\guo{\textcolor{black}{between two} architectures increases, the issue of PD becomes more severe.
Since there often exist many shared layers in WS-based NAS, this result demonstrates that it is necessary to alleviate the issue of PD for NAS.
\textbf{Regularization parameter $\lambda$.}
Generally, the optimal value of $\lambda$ varies regarding different data.
A large $\lambda$ may make the method fail to satisfy the problem constraint (see Eqn.~(\ref{eq:new_constraint})), while a small $\lambda$ may result in the irreversible issue when computing projection matrices (see Eqn.~(\ref{eq:compute_p})).
Nevertheless, the default setting $\lambda=1$ helps to achieve the best or relatively good performance in most cases.
\section{Conclusions}
In this paper, we have proposed a novel disturbance-immune training scheme for NAS to conquer performance disturbance (PD) in weight sharing (WS). Specifically, by developing a new update strategy to train sampled architectures, our method provides more stable/accurate performance estimation for architectures. As a result, the proposed method is able to learn a good controller for searching good architectures.
We theoretically and empirically verify the effectiveness of the proposed method in alleviating PD, and also provide an asymptotic analysis for its convergence. Extensive experiments demonstrate that the architecture found by our proposed method outperforms the architectures obtained by considered state-of-the-art NAS methods.
\clearpage
\bibliographystyle{abbrv}
{
\small
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
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100 Thieves, LLC is an American lifestyle brand and gaming organization based in Los Angeles, California, founded in 2017 by Matthew "Nadeshot" Haag. The organization competes in several video games, including Apex Legends, Call of Duty, Fortnite Battle Royale, League of Legends and Valorant. They currently own three franchise teams, in the League of Legends Championship Series (LCS), Valorant Americas League and Call of Duty League (branded as Los Angeles Thieves).
History
100 Thieves was founded in 2017 by Matthew "Nadeshot" Haag, a former OpTic Gaming Call of Duty team captain, (MLG) X Games gold medalist, and 2014 Esports Athlete of the year. In November 2017, 100 Thieves received a multimillion-dollar investment from Cleveland Cavaliers and Quicken Loans owner Dan Gilbert allowing the company to expand into a full-fledged esports organization.
On October 23, 2018, 100 Thieves announced that they finished their Series A funding round co-led by Scooter Braun and Drake, who both became co-owners when they raised $25 million. Included in this round was Sequoia Capital, Ludlow Ventures, Courtside Ventures, WndrCo, Marc Benioff, Drew Houston, Green Bay Ventures, Tao Capital and Advancit Capital. Their Series B funding round on July 16, 2019, was led by Artist Capital Management which raised $35 million. Included in this round is Aglae Ventures, Groupe Arnault (controlling shareholder of Louis Vuitton Moet Hennessy).
In March 2021, 100 Thieves signed a sponsorship deal with Truly Seltzer and Twisted Tea. They acquired gaming peripheral company Higround in October 2021, marking their first acquisition. On December 2, 2021, 100 Thieves announced their Series C funding round, which raised $60 million.
Divisions
League of Legends
Inaugural season
On November 20, 2017, 100 Thieves was accepted as a franchise organization for the 2018 NA LCS season. The team signed Prolly as its head coach and Ssumday, Meteos, Ryu, Cody Sun, and Aphromoo for its starting roster. The team placed first in the spring split regular season with a 12–6 record, securing a bye into the semifinals. Additionally, Aphromoo was voted as MVP of the spring split. The first team they faced in the playoffs was Clutch Gaming, and 100 Thieves won 3–2, moving on to the finals, where they were swept 0–3 by Team Liquid in the finals.
100 Thieves' second-place finish qualified them for Rift Rivals 2018, an international tournament between the top three spring teams from Europe and North America. The team elected to use substitute player Levi in place of Meteos for the tournament. Team Liquid, 100 Thieves, and Echo Fox competed against Europe's Fnatic, G2 Esports, and Splyce, going a combined 4–5 in the double round-robin group stage with 100 Thieves with a 1–2 record after a single win against Splyce. In the best-of-five "relay race" finals, the team lost their game against Fnatic, contributing to North America's combined 1–3 loss to Europe.
Before the summer split, 100 Thieves traded Meteos to Flyquest, in exchange for their jungler, AnDa. Following this roster move, the team placed third in the regular season with a 10–8 record. The team won their first match 3–0 over FlyQuest, before losing 1–3 to Team Liquid in the semi-finals, then losing 2–3 in the third place match to Team SoloMid.
The team's performance across both splits allowed them to qualify for the 2018 League of Legends World Championship in South Korea as North America's second seed. The team was drawn to Group D with Europe's Fnatic, China's Invictus Gaming, and Hong Kong's G-Rex. Shortly after qualifying for the world championship, they replaced Cody Sun with substitute player Rikara. 100 Thieves finished third in their group with a 2–4 record, and 12th overall, not qualifying for the knockout stage.
2019–2022
In 2019, the team extended top laner Ssumday's contract, while Ryu moved to an assistant position. Cody Sun and Rikara opted to leave the team, and they were replaced by Huhi and Bang. Partway through the spring split, Huhi was benched in favor or Soligo, but despite attempts at change, the team finished the spring split in dead last, with a 4–14 record, and elected to make additional roster changes for summer. Huhi left the team, and Amazing joined to replace AnDa, who was moved to the academy roster along with Ryu. A month into the split, the team replaced Soligo with Ryu from the academy roster, and also swapped Ssumday with FakeGod, as LCS rules required at least 3 North American residents on the starting roster. The team finished the summer split in eighth place and did not qualify for the post–season.
100 Thieves began the 2020 season with the announcement that PapaSmithy would be joining the team as the new general manager. Zikz was the next addition to the roster, replacing Prolly as head coach, and both Meteos and Cody Sun rejoined the team for the second time, alongside new additions Stunt and Ry0ma. Ryu, Amazing, Bang and Aphromoo all left the team, with FakeGod rejoining Academy and Ssumday rejoining the main roster. In spring, the team finished third in the regular season with a 10–8 record, and qualified for post–season for the first time since 2018. However once in the playoffs, they were swept 0–3 by Cloud9, and then lost 2–3 to Team SoloMid in the losers' bracket. After an 1–5 start in the summer split, the team parted ways with players Meteos and Stunt, and called up academy players Contractz and Poome to replace them. The team finished in seventh at the end of the summer split with a 7–11 record, and were seeded into the loser's bracket of the playoffs, where they would be swept 0–3 by Evil Geniuses to finish their 2020 season.
The 2021 season started with the team signing Closer, FBI, Damonte, and Huhi from Golden Guardians. Ry0ma and Poome were moved to the academy roster, while Cody Sun and Contractz both left the team. The team also added Freeze and Lustboy to the coaching staff. The season started off with the preseason Lock−In tournament, and the team starting strong, placing first in their group, before sweeping Immortals 2–0 in the first round of the knock−out stage. In the semifinals they faced Cloud9, and despite winning the first two games, were reverse swept to lose 2–3 in the series, and finish 3rd/4th in the tournament. In week 5 of the spring split, the team decided to bring back Ry0ma as their starter and sent Damonte to academy. They finished the spring split in third place, and in the Mid-Season Showdown, were swept 0–3 by Cloud9, before picking up a 3–0 win of their own against Dignitas in the losers bracket. The team faced Team SoloMid next, and fell 1–3, and were eliminated from playoffs.
After a fourth place finish in the spring split, the team parted ways with head coach Zikz and hired Reapered as his replacement. Prior to the start of the summer split, 100 Thieves signed Abbedagge from the LEC, and Ry0ma was subsequently sent back to academy, while Damonte was dropped by the organization. The revamped roster would go on to finish second in the regular season, with a record of 29–16. The team won their first playoff match 3–2 against Evil Geniuses, before falling 2–3 to Team Liquid. In the losers' bracket, 100 Thieves defeated Cloud9 3–1 to set up a finals rematch against Team Liquid. This time around, 100 Thieves defeated Team Liquid in a 3–0 sweep, to win the 2021 LCS Championship and qualify for the 2021 League of Legends World Championship. The team's performance secured them a bye into Group B alongside China's Edward Gaming, Korea's T1, and Japan's DetonatioN FocusMe. The team finished third in their group, with a 3–3 record, and were eliminated from the tournament, placing 9th–12th overall and ending their season.
In 2022, the team announced all five players would be returning, with top laner Tenacity as a sixth man. Mithy also joined the coaching staff in the off−season. Once again, 100 Thieves were atop their group in the Lock−In tournament, however they were upset 0–2 by Dignitas in the quarterfinals of the Knock−Out stage, and eliminated early. Through the first round robin of the spring split, 100 Thieves accumulated a 5–4 record, putting them in a three way tie for third place. 100 Thieves ultimately ended the spring split with a record of 12–6, securing themselves third place. In the first round of Playoffs, they swept Cloud 9 3–0 to advance to the winner bracket finals, where they came back from a 0–2 deficit to reverse sweep Team Liquid and advance to Grand Finals. There, they lost 0–3 to Evil Geniuses, denying 100 Thieves back to back championships.
Following a group stage exit at the 2022 World Championship, General manager PapaSmithy parted ways with the team.
2023–
For the 2023 season, 100 Thieves made an overhaul by releasing the entire roster except Closer, signing veterans Doublelift and Bjergsen and promoting Busio from the academy, and Tenacity to full-time top laner.
Call of Duty
2018–19 season
The team started the season off signing the team of Kenny, Fero, Octane, Slasher, and Enable. They placed 9th–12th at CWL Las Vegas 2019, qualifying for the Pro League Qualifiers and for the Pro League. After this, they had to re-evaluate the roster, and loan Priestahh and head coach Crowder from FaZe Clan, with Fero getting benched. At CWL Fort Worth 2019, the team finished 4th losing to Team Reciprocity, citing medical issues with player Priestahh and being replaced by Fero. At CWL London 2019, the team won the organization's first trophy. At CWL Anaheim 2019, the team won the organization's second trophy. After the Pro League, 100 Thieves finished second in Division B, qualifying for Pro League playoffs. At the Pro League Playoffs, the team placed 5th–6th, losing to Gen.G esports. In the 2019 Call of Duty World Championship, 100 Thieves finished second, with a loss to .
Los Angeles Thieves
After initially announcing their intention to not join the Call of Duty League in 2019, 100 Thieves announced in November 2020 that they would enter the league as the "Los Angeles Thieves" after acquiring OpTic Gaming's slot. They inherited the roster of Kenny, TJHaly, Slasher, and Drazah as a substitute, and later added Temp as their fourth and JKap as their coach. Following their top 6 placing at the CDL Stage 1 major, Temp was benched and the team picked up top amateur Venom. Before the Stage 2 major, Slasher was benched in favor of substitute Drazah. Before the start of the Dallas home series, TJHaly was benched and the team bought out Huke from Dallas Empire. Before the start of Stage 4, Huke was moved to the bench as TJHaly was moved back to the starting roster. Before the start of the LA Thieves home series, both Drazah and Venom were benched and Huke and Slasher were moved back to the starting roster. Before the start of Stage 5, the team added John and called up Drazah and benched Huke and TJHaly for a second time. At the 2021 Call of Duty League Championship, the team placed top 8 after being eliminated by the Minnesota ROKKR, effectively ending their season.
The team started their 2021–22 offseason by parting ways with players TJHaly, John, Venom, and Slasher. In September 2021, Huke's contract was extended and then bought out by the Los Angeles Guerillas. In the same month, the team signed Octane, reuniting him with Kenny. The day after announcing Octane, the team announced Envoy as their fourth finalizing the starting roster as Drazah, Envoy, Kenny, and Octane for the 2021–22 season with Pentagrxm joining later on as a substitute. The team finished 1st in the fourth major of the season and captured the first major championship under the LA Thieves brand. They ended the year by winning the 2022 CDL Championship in a dominant 5-2 final over Atlanta FaZe.
Valorant
On June 4, 2020, 100 Thieves Esports announced their entrance into the Valorant competitive scene through the signing of Hiko. Within three weeks of signing Hiko, the team was finalized. The original team consisted of Valliate, YaBoiDre, Venerated, and Pride. This team, however, did not last long as on August 14, 2020, all members except for Hiko were released. On the same day, the team went on the sign CS:GO veteran Nitr0. 3 weeks later, the team brought in another CS:GO veteran in Steel. 2 months later, the team was once again finalized after signing Asuna and Dicey from Immortals. The team went on to win the very first Valorant First Strike tournament defeating TSM in the grand finals.
On January 26, 2021, the team brought in Silenx as a substitute in place of Nitr0 due to Nitr0's wife going into labor. On February 28, 2021, the team benched Dicey in favor of Ethan from CS:GO. After the Masters 3 tourney, the team decided to call up their substitute B0i to the starting roster and moved Steel to the bench. The team also added Seven as a substitute. Their 2021 season ended after falling to Cloud9 in the NA Last Chance Qualifier for Valorant Champions. On November 23, Dicey and B0i were dropped from the team. Soon after, both Steel and Nitr0 departed the team.
They started 2022 with the signing of Ec1s and BabyJ and promoting analyst Jovi to head coach. After two losses in the 2022 NA VCT Challengers Stage 1 group stage, Ec1s and BabyJ were released and the team acquired Bang and JcStani on loans. After being eliminated in the 2022 NA VCT Challengers group stage, the team parted ways with Jovi and hired Sgares as head coach with Mikes joining him as an assistant, and hiring DDK as general manager. On March 27, JcStani announced his free agency after his loan period ended with the team. On April 12, Hiko announced his retirement from competitive Valorant but remained with the organization as a content creator. A couple days after, Bang was fully bought out and the team signed Derrek, Stellar, and Will while Ethan was moved to the bench, and later released. The team was able to qualify for 2022 Valorant Champions after beating The Guard in the grand finals of the NA Last Chance Qualifier. The team was then placed Group D alongside DRX, FURIA Esports, and Fnatic. The team finished the tournament 9-12th. In September and October 2022 respectively, 100 Thieves released Will and signed Cryocells, and Mikes was promoted to head coach after Sgares left the team. 100 Thieves ended the year by winning the Red Bull Home Ground tournament in Manchester with a 3–0 win over Cloud9 in December 2022.
In September 2022, 100 Thieves was selected as one of the thirty teams to be part of Riot Games' partnership program for the 2023 season.
Rosters
Former divisions
Counter-Strike: Global Offensive
In December 2017, 100 Thieves announced that they signed the former roster of Immortals. The organization had issues with visas, resulting in the team being disbanded.
In late October 2019, 100 Thieves announced the signing of the former Renegades roster. At IEM Beijing 2019, the team placed second in the tournament, losing to Astralis. The team placed 7th–8th at the ESL Season 10 Pro League Finals, losing to Fnatic. On October 12, 2020, 100 Thieves announced their departure from the competitive CS:GO scene, citing complications with travel, a focus on European events, and COVID-19 as the main reasons behind the move.
Cash App Compound
The 100 Thieves Cash App Compound is a 15,000 square foot esports, entertainment, and apparel hub, located in Culver City, California. It has four sports training rooms: the Rocket Mortgage League of Legends training room, the AT&T Valorant training room, the Totino's Fortnite training room, and the League of Legends Academy training room. It also has a content studio, which is worth around half a million dollars. Other areas include the Cash App Lounge, the Totino's basketball court, many business operation areas and four streaming pods. Chairs, catering, and PCs are provided by Secretlab, Chipotle and NZXT respectively.
The compound served as an LA County vote center for the 2020 United States presidential election.
Awards and nominations
References
External links
The Story of 100 Thieves by theScore esports (YouTube)
2016 establishments in California
American Internet groups
Esports teams based in the United States
League of Legends Championship Series teams
Red Bull
Team Razer
Valorant teams
|
{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 6,436
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Bibliography in Bits
Centre for the Study of the Book
Audio Embed
Adam Smyth talks to Professor Will Noel about the potentials of digital technology for the study of manuscripts.
Early modern plays in bits and pieces
Professor Tiffany Stern joins Dr Adam Smyth to discuss her current research on the materiality of the early modern play text. What happens to our thinking about plays when prologues, epilogues and songs become mobile pieces, detached from the whole?
The History of Oxford University Press
Adam Smyth is joined by Professor Ian Gadd to discuss his just-published collection on the history of OUP.
Creative Commons Attribution-Non-Commercial-Share Alike 2.0 UK: England & Wales; http://creativecommons.org/licenses/by-nc-sa/2.0/uk/
Will Noel
Adam Smyth
archimedes project
Department: Bodleian Libraries
Apple Podcast Audio Audio RSS Feed
|
{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
| 9,489
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Air Atlantique, Douglas DC-3
air_atlantique_dc3s-1.jpg
The Air Atlantique Group started as an air taxi operation in 1969 under the name of General Aviation Services, based in Jersey, Channel Islands. The Air Atlantique name was adopted in June 1977 when the company launched into freight charter work with a fleet of Douglas DC-3s, with most activity being transferred to the UK mainland in the early 1980s. The Group moved to its present base in Coventry in December 1985. In 1987 they were awarded an oil pollution control and spraying contract for the Department of Trade. This contract had previously been operated by Harvest Air which had gone into liquidation. Air Atlantique also acquired the two DC-3's that had been operated in the role by Harvest Air. Today, Air Atlantique still operates two DC-3 aircraft as part of the Air Atlantique Classic Flight. The Air Atlantique Classic Flight has just re-branded itself to ClassicFlight.com, but still retains the Air Atlantique heritage by using the Air Atlantique name in its logo. The DC3's in Harvest Air livery will be available as a separate download package 'harvest_dc3s.zip' .Repaints by John Payne. Original DC-3 base textures courtesy of Mark "Dark Moment" Beaumont. Requires the default Douglas DC-3.
Filename: air_atlantique_dc3s.zip
|
{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
| 1,862
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\section{Introduction}
In the fastest mixing Markov chain problem,
we choose the transition probabilities on the edges of a graph to
minimize the second-largest eigenvalue modulus of the
transition probability matrix.
In~\cite{BDX:04} we formulated this problem as a convex
optimization problem, in particular as a semidefinite program.
Thus it can be solved, up to any given precision, in polynomial time
by interior-point methods.
In this paper, we show how to exploit symmetries of a graph to make
the computation more efficient.
\subsection{The fastest mixing Markov chain problem}
We consider an undirected graph $\mathcal{G}=(\mathcal{V},\mathcal{E})$
with vertex set $\mathcal{V}=\{1,\ldots,n\}$ and edge
set~$\mathcal{E}$ and assume that $\mathcal{G}$ is connected.
We define a discrete-time Markov chain on the vertices as follows.
The state at time $t$ will be denoted $X(t) \in \mathcal V$, for
$t=0,1, \ldots $.
Each edge in the graph is associated with a transition
probability with which $X$ makes a transition between the two adjacent
vertices.
This Markov chain can be described via its transition probability
matrix $P\in{\mbox{\bf R}}^{n\times n}$, where
\[
P_{ij} = \mbox{\bf Prob}\; (\;X(t+1)=j \;|\; X(t)=i\;), \qquad
i,~j=1, \ldots, n.
\]
Note that $P_{ii}$ is the probability that $X(t)$ stays at
vertex~$i$, and $P_{ij}=0$ for $\{i,j\}\notin\mathcal{E}$
(transitions are allowed only between vertices that are linked by
an edge).
We assume that the transition probabilities are symmetric, {\it i.e.},
$P=P^T$, where the superscript $T$ denotes the transpose of a matrix.
Of course this transition probability matrix must also be stochastic:
\[
P\geq 0, \qquad P\mathbf 1=\mathbf 1,
\]
where the inequality $P \geq 0$ means elementwise, and $\mathbf 1$ denotes
the vector of all ones.
Since $P$ is symmetric and stochastic,
the uniform distribution $(1/n)\mathbf 1^T$ is stationary.
In addition, the eigenvalues of~$P$ are real, and no more than one in
modulus.
We denote them in non-increasing order
\[
1=\lambda_1(P)\geq\lambda_2(P)\geq\cdots\geq\lambda_n(P)\geq-1.
\]
We denote by $\mu(P)$ the second-largest eigenvalue modulus (SLEM) of
$P$, {\it i.e.},
\[
\mu(P) = \max_{i=2,\ldots,n} |\lambda_i(P)| = \max \; \{ \lambda_2(P), \; -\lambda_n(P) \}.
\]
This quantity is widely used to bound the asymptotic convergence rate
of the distribution of the Markov chain to its stationary
distribution, in the total variation distance or chi-squared distance
({\it e.g.}, \cite{DiS:91,DSC:93}).
In general the smaller $\mu(P)$ is, the faster the Markov chain
converges.
For more background on Markov chains, eigenvalues and rapid mixing,
see, {\it e.g.}, the text~\cite{Bre:99}.
In \cite{BDX:04}, we addressed the following problem:
What choice of $P$ minimizes $\mu(P)$?
In other words, what is the fastest mixing (symmetric) Markov chain
on the graph?
This can be posed as the following optimization problem:
\begin{equation}\label{e-fmmc}
\begin{array}{ll}
\mbox{minimize} & \mu(P)\\
\mbox{subject to} &
P\geq 0, \quad P\mathbf 1 =\mathbf 1, \quad P = P^T\\
&P_{ij} = 0, \quad \{i,j\}\notin \mathcal{E}.
\end{array}
\end{equation}
Here $P$ is the optimization variable, and the graph is the
problem data.
We call this problem the \emph{fastest mixing Markov chain} (FMMC)
problem.
This is a convex optimization
problem, in particular, the objective function can be
explicitly written in a convex form
$\mu(P) = \|P-(1/n)\mathbf 1\ones^T\|_2$,
where $\|\cdot\|_2$ denotes the spectral norm of a matrix.
Moreover, this problem can be readily transformed into a semidefinite
program (SDP):
\begin{equation}\label{e-fmmc-sdp}
\begin{array}{ll}
\mbox{minimize} & s\\
\mbox{subject to} &
-sI \preceq P - (1/n)\mathbf 1\ones^T \preceq sI\\
&P\geq 0, \quad P\mathbf 1 =\mathbf 1, \quad P = P^T\\
&P_{ij} = 0, \quad \{i,j\}\notin \mathcal{E}.
\end{array}
\end{equation}
Here $I$ denote the identity matrix, and the variables are the
matrix $P$ and the scalar $s$.
The symbol $\preceq$ denotes matrix inequality, {\it i.e.}, $X \preceq Y$
means $Y-X$ is positive semidefinite.
There has been some follow-up work on this problem.
Boyd, Diaconis, Sun, Xiao (\cite{BDSX:06}) proved analytically that on an $n$-path the fastest mixing chain can be obtained by assigning the same transition probability half at the $n-1$ edges and two loops at the two ends.
Roch (\cite{Roc:05}) used standard mixing-time analysis techniques (variational characterizations, conductance, canonical paths) to bound the fastest mixing time.
Gade and Overton (\cite{GaO:06}) have considered the fastest mixing problem for a nonreversible Markov chain. Here, the problem is non-convex and much remains to be done.
Finally, closed form solutions of fastest mixing problems have recently been applid in statistics to give a generalization of the usual spectral analysis of time series for more general discrete data. see \cite{Sal:06}.
\subsection{Exploiting problem structure}
\label{s-structure}
The SDP formulation~(\ref{e-fmmc-sdp}) means that the FMMC problem can
be efficiently solved using standard SDP solvers, at least for small
or medium size problems (with number of edges up to a thousand or so).
General background on convex optimization and SDP can be found in,
{\it e.g.}, \cite{NeN:94,VaB:96,WSV:00,BTN:01,BoV:04}.
The current SDP solvers ({\it e.g.}, \cite{Stu:99,SDPT3,SDPA6}) mostly use
interior-point methods which have polynomial time worst-case
complexity.
When solving the SDP~(\ref{e-fmmc-sdp}) by interior-point methods,
in each iteration we need to compute the first and second derivatives
of the logarithmic barrier functions (or potential functions) for
the matrix inequalities, and assemble and solve a linear system of
equations (the Newton system).
Let~$n$ be the number of vertices and~$m$ be the number of edges in
the graph (equivalently $m$ is the number of variables in the
problem).
The Newton system is a set of~$m$ linear equations with~$m$ unknowns.
Without exploiting any structure, the number of flops per iteration in
a typical barrier method is on the order $\max\{m n^3,m^2 n^2,m^3\}$,
where the first two terms come from computing and assembling the
Newton system, and the third term amounts to solving it (see, {\it e.g.},
\cite[\S11.8.3]{BoV:04}).
(Other variants of interior-point methods have similar orders of flop
count.)
Exploiting problem structure can lead to significant improvement of
solution efficiency.
As for many other problems defined on a graph, sparsity is the most
obvious structure to consider here.
In fact, many current SDP solvers already exploit sparsity.
However, as a well-known fact, exploiting sparsity alone in
interior-point methods for SDP has limited effectiveness.
The sparsity of $P$, and the sparsity plus rank-one structure of
$P-(1/n)\mathbf 1\ones^T$, can be exploited to significantly reduce the
complexity of assembling the Newton system,
but typically the Newton system itself is dense.
The computational cost per iteration can be reduced to order
$O(m^3)$, dominated by solving the dense linear system
(see analysis for similar problems in, {\it e.g.}, \cite{BYZ:00,XiB:04,XBK:07}).
In addition to using interior-point methods for the SDP
formulation~(\ref{e-fmmc-sdp}),
we can also solve the FMMC problem in the form~(\ref{e-fmmc})
by subgradient-type (first-order) methods.
The subgradients of $\mu(P)$ can be obtained by computing the extreme
eigenvalues and associated eigenvectors of the matrix~$P$.
This can be done very efficiently by iterative methods, specifically
the Lanczos method, for large sparse symmetric matrices ({\it e.g.},
\cite{GoV:96,Saa:92}).
Compared with interior-point methods, subgradient-type methods
can solve much larger problems but only to a moderate accuracy
(they don't have polynomial-time worst-case complexity).
In~\cite{BDX:04}, we used a simple subgradient method to solve the
FMMC problem on graphs with up to a few hundred thousand edges.
More sophisticated first-order methods for solving large-scale
eigenvalue optimization problems and SDPs have been reported in, {\it e.g.},
\cite{HeR:00,BuM:03,Nem:04,LNM:04,Nes:05}.
A successive partial linear programming method was developed in
\cite{Ove:92}.
In this paper, we focus on the FMMC problem on graphs with large
symmetry groups, and show how to exploit symmetries of the graph to
make the computation more efficient.
A result by Erd{\H o}s and R\'enyi \cite{ErR:63} states that with
probability one,
the symmetry group of a (suitably defined) random graph is trivial,
{\it i.e.}, it contains only the identity element.
Nevertheless, many of the graphs of theoretical and practical
interest, particularly in engineering applications have very
interesting, and sometimes very large, symmetry groups.
Symmetry reduction techniques have been explored in several different
contexts, {\it e.g.}, dynamical systems and bifurcation theory \cite{GSS:88},
polynomial system solving \cite{Gat:00,Wor:94}, numerical solution of
partial differential equations \cite{FaS:92}, and Lie symmetry
analysis in geometric mechanics \cite{MaR:99}.
In the context of optimization, a class of SDPs with symmetry has been
defined in \cite{KOMK:01}, where the authors study the invariance
properties of the search directions of primal-dual interior-point methods.
In addition, symmetry has been exploited to prune the enumeration
tree in branch-and-cut algorithms for integer programming \cite{Mar:03},
and to reduce matrix size in a spectral radius optimization problem
\cite{HOY:03}.
Closely related to our approach in this paper, the recent work
\cite{dPS:07} considers general SDPs that are invariant under the
action of a permutation group, and developed a technique based on
matrix $*$-representation to reduce problem size.
This technique has been applied to simplify computations in SDP
relaxations for graph coloring and maximal clique problems
\cite{DuR:07},
and to strengthen SDP bounds for some coding problems \cite{Lau:07}.
For the FMMC problem, we show that exploiting symmetry allows
significant reduction in both number of optimization variables and
size of matrices. Effectively, they correspond to reducing~$m$
and~$n$, respectively, in the flop counts for interior-point methods
mentioned above.
The problem can be considerably simplified and is often solvable
analytically by only exploiting symmetry.
We present two general approaches for symmetry exploitation,
based on orbit theory \cite{BDPX:05} and block-diagonalization
\cite{GatermannParrilo}, respectively.
We also establish the connection between these two approaches.
\subsection{Outline}
In~\S\ref{s-analysis}, we explain the concepts of graph automorphisms
and the automorphism group (symmetry group) of a graph. We show that
the FMMC problem always attains its optimum in the fixed-point subset
of the feasible set under the automorphism group. This allows us to
only consider a number of distinct transition probabilities that
equals the number of orbits of the edges. We then give a formulation
of the FMMC problem with reduced number of variables (transition
probabilities), which appears to be very convenient in subsequent
sections.
In~\S\ref{s-analytic}, we give closed-form solutions for
the FMMC problem on some special classes of graphs, namely
edge-transitive graphs and distance-transitive graphs.
Along the way we also discuss FMMC on graphs formed by taking
Cartesian products of simple graphs.
In~\S\ref{s-fmmc-orbit}, we first review the orbit theory for
reversible Markov chains, and give sufficient conditions on
constructing an orbit chain that contain all distinct eigenvalues of
the original chain.
This orbit chain is usually no longer symmetric but always
reversible.
We then solve the fastest reversible Markov chain problem on the orbit
graph, from which we immediately obtain optimal solution to the
original FMMC problem.
In~\S\ref{s-block-diag}, we review some group representation theory
and show how to block diagonalize the linear matrix inequalities in
the FMMC problem by constructing a symmetry-adapted basis.
The resulting blocks usually have much smaller sizes and repeated
blocked can be discarded in computation.
Extensive examples in~\S\ref{s-fmmc-orbit} and~\S\ref{s-block-diag}
reveal interesting connections between these two general symmetry
reduction methods.
In~\S\ref{s-conclusions}, we conclude the paper by pointing out some
possible future work.
\section{Symmetry analysis}
\label{s-analysis}
In this section we explain the basic concepts that are essential in
exploiting graph symmetry, and derive our result on reducing the
number of optimization variables in the FMMC problem.
\subsection{Graph automorphisms and classes}
The study of graphs that possess particular kinds of symmetry
properties has a long history.
The basic object of study is the \emph{automorphism group} of a graph,
and different classes can be defined depending on the specific form in
which the group acts on the vertices and edges.
An \emph{automorphism} of a graph
$\mathcal{G}=(\mathcal{V},\mathcal{E})$ is a permutation $\sigma$ of
$\mathcal{V}$ such that $\{i,j\} \in \mathcal{E}$ if and only if
$\{\sigma(i),\sigma(j)\} \in \mathcal{E}$. The (full)
\emph{automorphism group} of the graph, denoted by
$\mbox{Aut}(\mathcal{G})$, is the set of all such permutations, with
the group operation being composition. For instance, for the graph on
the left in Figure~\ref{fig:etvt}, the corresponding automorphism
group is generated by all permutations of the vertices $\{1,2,3\}$.
This group, isomorphic to the symmetric group $S_3$, has six elements,
namely the permutations $123\to123$ (the identity), $123\to213$,
$123\to132$, $123\to321$, $123\to231$, and $123\to312$. Note that
vertex~$4$ cannot be permuted with any other vertex.
\begin{figure}
\begin{center}
\psfrag{1}[cl]{$1$}
\psfrag{2}[tc]{$2$}
\psfrag{3}[tc]{$3$}
\psfrag{4}[bl]{$4$}
\includegraphics[width=0.55\textwidth]{figures/etvt.eps}
\caption{The graph on the left side is edge-transitive, but not
vertex-transitive. The one on the right side is vertex-transitive,
but not edge-transitive.}
\label{fig:etvt}
\end{center}
\end{figure}
Recall that an \emph{action} of a group~$G$ on a set~$\mathcal X$ is a
homomorphism from~$G$ to the set of all permutations of the elements
in~$\mathcal X$ ({\it i.e.}, the symmetric group of degree $|\mathcal X|$).
For an element $x\in\mathcal X$, the set of all images $g(x)$, as
$g$ varies through $G$, is called the \emph{orbit} of~$x$.
Distinct orbits form equivalent classes and they partition the
set~$\mathcal X$.
The action is \emph{transitive} if for every pair of elements $x, y \in
\mathcal{X}$, there is a group element $g \in G$ such that $g(x)=y$.
In other words, the action is transitive if there is only one single
orbit in~$\mathcal{X}$.
A graph $\mathcal{G} = (\mathcal{V}, \mathcal{E})$ is said to be
\emph{vertex-transitive} if $\mbox{Aut}(\mathcal G)$ acts transitively
on $\mathcal{V}$.
The action of a permutation $\sigma$ on $\mathcal V$ induces an
action on $\mathcal E$ with the rule
$\sigma(\{i,j\})=\{\sigma(i),\sigma(j)\}$.
A graph $\mathcal{G}$ is \emph{edge-transitive} if
$\mbox{Aut}(\mathcal G)$ acts transitively on $\mathcal{E}$.
Graphs can be edge-transitive without being vertex-transitive and
vice versa; simple examples are shown in Figure~\ref{fig:etvt}.
A graph is \emph{1-arc-transitive} if given any four vertices
$u,v,x,y$ with $\{u,v\},\{x,y\} \in \mathcal{E}$, there exists an
automorphism $g \in \mbox{Aut}(\mathcal G)$ such that $g(u) = x$ and
$g(v) = y$.
Notice that, as opposed to edge-transitivity, here the ordering of
the vertices is important, even for undirected graphs.
In fact, a 1-arc transitive graph must be both vertex-transitive and
edge-transitive, and the reverse may not be true.
The 1-arc-transitive graphs are called \emph{symmetric graphs} in
\cite{Biggs}, but the modern use extends this term to all graphs that
are simultaneously edge- and vertex-transitive.
Finally, let $\delta(u,v)$ denote the distance between two vertices
$u,v \in \mathcal{V}$.
A graph is called \emph{distance-transitive} if, for any four vertices
$u,v,x,y$ with $\delta(u,v) = \delta(x,y)$, there is an automorphism
$g \in \mbox{Aut}(\mathcal G)$ such that $g(u) = x$ and $g(v) = y$.
The containment relationship among the four classes of graphs
described above is illustrated in Figure~\ref{fig:graphclasses}.
Explicit counterexamples are known for each of the non-inclusions.
It is generally believed that distance-transitive graphs have been completely classified. This work has been done by classifying the distance-regular graphs.
It would take us too far afield to give a complete discussion.
See the survey in \cite[Section 7]{DSC:06}.
\begin{figure}
\begin{center}
\setlength{\unitlength}{.02cm}
\begin{picture}(350,200)(0,30)
\put(80,120){\framebox(230,40){\textbf{1-arc transitive}}}
\put(80,200){\framebox(230,40){\textbf{Distance-transitive}}}
\put(220,40){\framebox(190,40){\textbf{Vertex-transitive}}}
\put(-30,40){\framebox(190,40){\textbf{Edge-transitive}}}
\put(185,200){\vector(0,-1){40}}
\put(220,120){\vector(1,-1){40}}
\put(150,120){\vector(-1,-1){40}}
\end{picture}
\end{center}
\caption{Classes of symmetric graphs, and their inclusion relationship.}
\label{fig:graphclasses}
\end{figure}
The concept of graph automorphism can be naturally extended to
weighted graphs, by requiring that the permutation must also preserve
the weights on the edges ({\it e.g.}, \cite{BDPX:05}).
This extension allows us to exploit symmetry in more general
reversible Markov chains, where the transition probability matrix is
not necessarily symmetric.
\subsection{FMMC with symmetry constraints}
\label{s-fmmc-invariant}
A permutation $\sigma\in\mbox{Aut}(\mathcal G)$ can be represented by
a permutation matrix $Q$, where $Q_{ij}=1$ if $i=\sigma(j)$ and
$Q_{ij}=0$ otherwise.
The permutation $\sigma$ induces an action on the transition
probability matrix by $\sigma(P)=Q P Q^T$.
We denote the feasible set of the FMMC problem~(\ref{e-fmmc}) by
$\mathcal C$, {\it i.e.},
\[
\mathcal C = \{ P\in{\mbox{\bf R}}^{n\times n} ~|~ P\geq 0, ~P\mathbf 1=\mathbf 1,
~P=P^T, P_{ij}=0 ~\mbox{for}~\{i,j\}\notin\mathcal E\}.
\]
This set is invariant under the action of graph automorphism.
To see this, let $h=\sigma(i)$ and $k=\sigma(j)$.
Then we have
\[
(\sigma(P))_{hk} = (QPQ^T)_{hk} = \sum_l (QP)_{hl}Q_{kl} = (QP)_{hj}
= \sum_l Q_{hl}P_{lj}=P_{ij}.
\]
Since $\sigma$ is a graph automorphism, we have $\{h,k\}\in\mathcal E$
if and only if $\{i,j\}\in\mathcal E$, so the sparsity pattern of the
probability transition matrix is preserved.
It is straightforward to verify that the conditions $P\geq0$,
$P\mathbf 1=\mathbf 1$, and $P=P^T$, are also preserved under this action.
Let~$\mathcal F$ denote the fixed-point subset
of~$\mathcal C$ under the action of $\mbox{Aut}(\mathcal G)$; {\it i.e.},
\[
\mathcal F = \{ P\in\mathcal C ~|~ \sigma(P)=P,
~\sigma\in\mbox{Aut}(\mathcal G)\}.
\]
We have the following theorem (see also
\cite[Theorem~3.3]{GatermannParrilo}).
\begin{theorem}\label{t-fmmc-fixed-point}
The FMMC problem always has an optimal solution in
the fixed-point subset $\mathcal F$.
\end{theorem}
\begin{proof}
Let $\mu^\star$ denote the optimal value of the FMMC
problem~(\ref{e-fmmc}), {\it i.e.},
$\mu^\star=\inf\{\mu(P)|P\in\mathcal C\}$.
Since the objective function~$\mu$ is continuous and the feasible
set~$\mathcal C$ is compact, there is at least one optimal transition
matrix $P^\star$ such that $\mu(P^\star)=\mu^\star$.
Let $\overline{P}$ denote the average over the orbit of $P^\star$
under $\mbox{Aut}(\mathcal G)$
\[
\overline{P}= \frac{1}{|\mbox{Aut}(\mathcal G)|}
\sum_{\sigma\in\mbox{Aut}(\mathcal G)} \sigma(P^\star).
\]
This matrix is feasible because each $\sigma(P^\star)$ is feasible and
the feasible set is convex.
By construction, it is also invariant under the actions of
$\mbox{Aut}(\mathcal G)$.
Moreover, using the convexity of~$\mu$, we have
$\mu(\overline{P})\leq\mu(P^\star)$.
It follows that $\overline{P}\in\mathcal F$ and
$\mu(\overline{P})=\mu^\star$.
\end{proof}
As a result of Theorem~\ref{t-fmmc-fixed-point}, we can replace the
constraint $P\in\mathcal C$ by $P\in\mathcal F$ in the FMMC problem
and get the same optimal value.
In the fixed-point subset $\mathcal F$, the transition
probabilities on the edges within an orbit must be the same.
So we have the following corollaries:
\begin{corollary}\label{c-numprobs}
The number of distinct edge transition probabilities we need to
consider in the FMMC problem is at most equal to the number of
orbits of $\,\mathcal{E}$ under $\mbox{Aut}(\mathcal G)$.
\end{corollary}
\begin{corollary}\label{c-edge-transitive}
If $\mathcal{G}$ is edge-transitive, then all the edge transition
probabilities can be assigned the same value.
\end{corollary}
Note that the holding probability at the vertices can always be
eliminated using $P_{ii} = 1- \sum_j P_{ij}$.
So it suffices to only consider the edge transition probabilities.
\subsection{Formulation with reduced number of variables}
\label{s-reduce-variables}
From the results of the previous section, we can reduce the number of
optimization variables in the FMMC problem from the number of edges to
the number of edge orbits under the automorphism group.
Here we give an explicit parametrization of the FMMC problem with
the reduced number of variables.
This parametrization is also the precise characterization of the
fixed-point subset $\mathcal F$.
Recall that the \emph{adjacency matrix} of a graph with~$n$ vertices
is a $n\times n$ matrix~$A$ whose entries are given by $A_{ij}=1$ if
$\{i,j\}\in\mathcal{E}$ and $A_{ij}=0$ otherwise.
Let $\nu_i$ be the valency (degree) of vertex~$i$.
The \emph{Laplacian matrix} of the graph is given by
$L=\mathop{\bf Diag}(\nu_1,\nu_2,\ldots,\nu_n)-A$, where $\mathop{\bf Diag}(\nu)$ denotes a
diagonal matrix with the vector $\nu$ as its diagonal.
Extensive account of the Laplacian matrix and its use in algebraic
graph theory are provided in, {\it e.g.}, \cite{Mer:94,FanChung,GoR:01}.
Suppose that there are $N$ orbits of edges under the action of
$\mbox{Aut}(\mathcal G)$.
For each orbit, we define an orbit graph
$\mathcal{G}_k=(\mathcal{V},\mathcal{E}_k)$, where $\mathcal{E}_k$ is
the set of edges in the $k$th orbit.
Note that the orbit graphs are disconnected (there are disconnected
vertices) if the original graph is not edge-transitive.
Let $L_k$ be the Laplacian matrix of $\mathcal{G}_k$.
Note that the diagonal entries $(L_k)_{ii}$ equals the valency of
node~$i$ in $\mathcal{G}_k$ (which is zero if vertex~$i$ is
disconnected with all other vertices in $\mathcal{G}_k$).
By Corollary~\ref{c-numprobs}, we can assign the same transition
probability on all the edges in the $k$-th orbit.
Denote this transition probability by $p_k$ and let
$p=(p_1,\ldots,p_N)$.
Then the transition probability matrix can be written as
\begin{equation}\label{e-prob-laplacian}
P(p) = I - \sum_{k=1}^N p_k L_k.
\end{equation}
This parametrization of the transition probability matrix
automatically satisfies the constraints $P=P^T$, $P\mathbf 1=\mathbf 1$,
and $P_{ij}=0$ for $\{i,j\}\in\mathcal E$.
The entry-wise nonnegative constraint $P\geq 0$ now translates into
\begin{eqnarray*}
&& p_k \geq 0, \qquad k=1,\ldots,N \\
&& \sum_{k=1}^N (L_k)_{ii}\; p_k \leq 1, \qquad i=1,\ldots,n
\end{eqnarray*}
where the first set of constraints are for the off-diagonal entries
of~$P$, and the second set of constraints are for the diagonal entries
of~$P$.
It can be verified that the parametrization~(\ref{e-prob-laplacian}),
together with the above inequality constraints, is the precise
characterization of the fixed-point subset~$\mathcal F$.
Therefore we can explicitly write the FMMC problem restricted to the
fixed-point subset as
\begin{equation}\label{e-fmmc-orbits}
\begin{array}{ll}
\mbox{minimize} & \mu\left(I-\sum_{k=1}^N p_k L_k\right) \\[1ex]
\mbox{subject to} & p_k \geq 0, \quad k=1,\ldots,N\\[1ex]
& \sum_{k=1}^N (L_k)_{ii}\, p_k \leq 1, \quad i=1,\ldots,n.
\end{array}
\end{equation}
Later in this paper, we will also need the corresponding SDP
formulation
\begin{equation}\label{e-fmmc-orbits-sdp}
\begin{array}{ll}
\mbox{minimize} & s \\[0.5ex]
\mbox{subject to} & -s I \preceq I - \sum_{k=1}^N p_k L_k
- (1/n)\mathbf 1\ones^T \preceq s I\\[1ex]
& p_k \geq 0, \quad k=1,\ldots,N\\[1ex]
& \sum_{k=1}^N (L_k)_{ii}\, p_k \leq 1, \quad i=1,\ldots,n.
\end{array}
\end{equation}
\section{Some analytic results}
\label{s-analytic}
For some special classes of graphs, the FMMC problem can be
considerably simplified and often solved by only exploiting symmetry.
In this section, we give some analytic results for the FMMC problem on
edge-transitive graphs, Cartesian product of simple graphs, and
distance-transitive graphs (a subclass of
edge-transitive graphs).
The optimal solution is often expressed in terms of the eigenvalues
of the adjacency matrix or the Laplacian matrix of the graph.
It is interesting to notice that even for such highly structured class
of graphs, neither the maximum-degree nor the
Metropolis-Hastings heuristics discussed in~\cite{BDX:04} give the
optimal solution.
Throughout, we use $\alpha^\star$ to denote the common edge weight of the fastest mixing chain and $\mu^\star$ to denote the optimal SLEM.
\subsection{FMMC on edge-transitive graphs}
\begin{theorem}\label{t-edge-transitive}
Suppose the graph $\mathcal G$ is edge-transitive, and let~$\alpha$ be
the transition probability assigned on all the edges.
Then the optimal solution of the FMMC problem is
\begin{eqnarray}
\alpha^\star
&=& \min \left\{ \frac{1}{\nu_\mathrm{max}},\;
\frac{2}{\lambda_1(L)+\lambda_{n-1}(L)} \right\}
\label{e-et-alpha}\\[1ex]
\mu^\star
&=& \max \left\{ 1-\frac{\lambda_{n-1}(L)}{\nu_\mathrm{max}},\;
\frac{\lambda_1(L)-\lambda_{n-1}(L)}{\lambda_1(L)+\lambda_{n-1}(L)}
\right\},
\label{e-et-mu}
\end{eqnarray}
where $\nu_\mathrm{max}=\max_{i\in\mathcal{V}} \nu_i$
is the maximum valency of the vertices in the graph, and $L$ is the Laplacian matrix defined in~\S\ref{s-reduce-variables}.
\end{theorem}
\begin{proof}
By definition of an edge-transitive graph, there is a single orbit of
edges under the actions of its automorphism group.
Therefore we can assign the same transition probability $\alpha$ on
all the edges in the graph (Corollary~\ref{c-edge-transitive}),
and the parametrization~(\ref{e-prob-laplacian})
becomes $P = I - \alpha L$.
So we have
\[
\lambda_i(P) = 1-\alpha \lambda_{n+1-i}(L), \qquad i=1,\ldots,n
\]
and the SLEM
\begin{eqnarray*}
\mu(P) &=& \max\{\lambda_2(P),\;-\lambda_{n}(P)\}\\
&=&\max\{ 1-\alpha \lambda_{n-1}(L), \; \alpha \lambda_1(L)-1 \}.
\end{eqnarray*}
To minimize $\mu(P)$, we let
$1-\alpha \lambda_{n-1}(L)=\alpha \lambda_1(L)-1 $ and
get $\alpha=2/(\lambda_{n-1}(L)+\lambda_{n-1}(L))$.
But the nonnegativity constraint $P\geq 0$ requires that
the transition probability must also satisfy
$0<\alpha\leq 1/\nu_\mathrm{max}$.
Combining these two conditions gives the optimal
solution~(\ref{e-et-alpha}) and~(\ref{e-et-mu}).
\end{proof}
We give two examples of FMMC on edge-transitive graphs.
\subsubsection{Cycles}
\begin{figure}
\centering
\includegraphics[width=0.2\textwidth]{figures/cycle.eps}
\caption{The cycle graph $C_n$ with $n=9$.}
\label{f-cycle}
\end{figure}
The first example is the cycle graph $C_n$; see Figure~\ref{f-cycle}.
The Laplacian matrix is
\[
L=\left[\begin{array}{rrrrrr}
2 & -1 & 0 & \cdots & 0 & -1 \\
-1 & 2 & -1 & \cdots & 0 & 0 \\
0 & -1 & 2 & \cdots & 0 & 0 \\
\vdots & \vdots & \vdots & \ddots &\vdots & \vdots \\
0 & 0 & 0 & \cdots & 2 & -1 \\
-1 & 0 & 0 & \cdots & -1 & 2 \end{array} \right]
\]
which has eigenvalues
\[
2 - 2\cos\frac{2k\pi}{n}, \qquad k=1,\ldots,n.
\]
The two extreme eigenvalues are
\[
\lambda_1(L)= 2 - 2 \cos \frac{2\lfloor n/2\rfloor\pi}{n}, \qquad
\lambda_{n-1}(L) = 2 - 2 \cos \frac{2\pi}{n}
\]
where $\lfloor n/2\rfloor$ denotes the largest integer that is
no larger than $n/2$, which is $n/2$ for $n$ even
or $(n-1)/2$ for $n$ odd.
By Theorem~\ref{t-edge-transitive},
the optimal solution to the FMMC problem is
\begin{eqnarray}
\alpha^{\star}
&=&\frac{1}{2-\cos\frac{2\pi}{n}-\cos\frac{2\lfloor
n/2\rfloor\pi}{n}} \label{e-cycle-prob}\\[2ex]
\mu^{\star}
&=&\frac{\cos\frac{2\pi}{n}-\cos\frac{2\lfloor n/2\rfloor\pi}{n}}
{2-\cos\frac{2\pi}{n}-\cos\frac{2\lfloor n/2\rfloor\pi}{n}}.
\label{e-cycle-slem}
\end{eqnarray}
When $n\to\infty$, the transition probability $\alpha^{\star}\to 1/2$
and the SLEM $\mu^{\star}\to 1-2\pi^2/n^2$.
\subsubsection{Complete bipartite graphs}
The complete bipartite graph, denoted $K_{m,n}$, has two subsets of
vertices with cardinalities~$m$ and~$n$ respectively.
Each vertex in a subset is connected to all the vertices in the other
subset, and is not connected to any of the vertices in its own
subset; see Figure~\ref{f-bipartite}.
Without loss of generality, assume $m\leq n$.
So the maximum degree is~$\nu_\mathrm{max}=n$.
This graph is edge-transitive but not vertex-transitive.
The Laplacian matrix of this graph is
\[
L=\left[\begin{array}{cc} n I_m & -\mathbf 1_{m\times n} \\
-\mathbf 1_{n\times m} & m I_n \end{array}\right]
\]
where $I_m$ denotes the~$m$ by~$m$ identity matrix, and
$\mathbf 1_{m\times n}$ denotes the~$m$ by~$n$ matrix whose components are
all ones.
For $n\geq m\geq 2$, this matrix has four distinct eigenvalues
$m+n$, $n$, $m$ and $0$, with multiplicities $1$, $m-1$, $n-1$ and
$1$, respectively (see~\S\ref{s-bipartite-diag}).
By Theorem~\ref{t-edge-transitive}, the optimal transition probability
on the edges and the corresponding SLEM are
\begin{eqnarray}
\alpha^{\star} &=& \min \left\{ \frac{1}{n}, \;\frac{2}{n+2m} \right\}
\label{e-bipartite-alpha} \\[1ex]
\mu^{\star} &=& \max \left\{ \frac{n-m}{n},\;\frac{n}{n+2m} \right\}.
\label{e-bipartite-mu}
\end{eqnarray}
\begin{figure}
\centering
\psfrag{u}[cr]{$u\;$}
\psfrag{v}[cl]{$\;v$}
\psfrag{x}[cr]{$x\;$}
\psfrag{y}[cl]{$\;y$}
\includegraphics[width=0.22\textwidth]{figures/bipartite.eps}
\caption{The complete bipartite graph $K_{m,n}$ with $m=3$ and $n=4$.}
\label{f-bipartite}
\end{figure}
\subsection{Cartesian product of graphs}
Many graphs we consider can be constructed
by taking \emph{Cartesian product} of simpler graphs.
The Cartesian product of two graphs
$\mathcal{G}_1=(\mathcal{V}_1,\mathcal{E}_1)$ and
$\mathcal{G}_2=(\mathcal{V}_2,\mathcal{E}_2)$ is a graph with vertex
set $\mathcal{V}_1 \times \mathcal{V}_2$, where two vertices
$(u_1,u_2)$ and $(v_1, v_2)$ are connected by
an edge if and only if $u_1=v_1$ and $\{u_2,v_2\}\in \mathcal{E}_2$,
or $u_2=v_2$ and $\{u_1,v_1\}\in \mathcal{E}_1$.
Let $\mathcal G_1\oplus\mathcal G_2$ denote this Cartesian product.
Its Laplacian matrix is given by
\begin{equation}\label{e-Cartesian-laplacian}
L_{\mathcal G_1\oplus\mathcal G_2} = L_{\mathcal G_1}\otimes
I_{|\mathcal V_1|} + I_{|\mathcal V_2|}\otimes L_{\mathcal G_2}
\end{equation}
where
$\otimes$ denotes the matrix Kronecker product (\cite{Gra:81}).
The eigenvalues of $L_{\mathcal G_1\oplus\mathcal G_2}$ are given by
\begin{equation}\label{e-Cartesian-eigs}
\lambda_i(L_{\mathcal G_1}) + \lambda_j(L_{\mathcal G_2}),
\qquad i=1,\ldots,|\mathcal{V}_1|, \qquad j=1,\ldots,|\mathcal{V}_2|
\end{equation}
where each eigenvalue is obtained as many times as its multiplicity
({\it e.g.}, \cite{Moh:97}).
The adjacency matrix of the Cartesian product of graphs also has
similar properties, which we will use later for distance-transitive
graphs.
Detailed background on spectral graph theory can be found in, {\it e.g.},
\cite{Biggs,DCS:80,FanChung,GoR:01}.
Combining Theorem~\ref{t-edge-transitive} and the above expression for
eigenvalues, we can easily obtain solutions to the FMMC problem on
graphs formed by taking Cartesian product of simple graphs.
\subsubsection{Two-dimensional meshes}
\label{s-twodimgrid}
\begin{figure}
\centering
\includegraphics[width=0.22\textwidth]{figures/mesh.eps}
\caption{The two-dimensional mesh with wraparounds $M_n$ with $n=4$.}
\label{f-mesh}
\end{figure}
Here we consider the two-dimensional mesh with wraparounds at two ends
of each row and column, see Figure~\ref{f-mesh}.
It is the Cartesian product of two copies of $C_n$.
We write it as $M_n=C_n\oplus C_n$.
By equation~(\ref{e-Cartesian-eigs}), its Laplacian matrix has eigenvalues
\[
4-2\cos\frac{2i\pi}{n}-2\cos\frac{2j\pi}{n},
\qquad i,~j=1,\ldots,n.
\]
By Theorem~\ref{t-edge-transitive}, we obtain
the optimal transition probability
\[
\alpha^{\star}=\frac{1}{3-2\cos\frac{2\lfloor n/2\rfloor\pi}{n}
-\cos\frac{2\pi}{n}}
\]
and the smallest SLEM
\[
\mu^{\star}=\frac{1-2\cos\frac{2\lfloor n/2\rfloor\pi}{n}+
\cos\frac{2\pi}{n}}{3-2\cos\frac{2\lfloor n/2\rfloor\pi}{n}
-\cos\frac{2\pi}{n}}
\]
When $n\to\infty$, the transition probability $\alpha^{\star}\to 1/4$
and the SLEM $\mu^{\star}\to 1-\pi^2/n^2$.
\subsubsection{Hypercubes}
\label{s-hypercubes}
The $d$-dimensional hypercube, denoted $Q_d$, has $2^d$ vertices, each
labeled with a binary word with length~$d$.
Two vertices are connected by an edge if their words differ in exactly
one component (see Figure~\ref{f-hypercube}).
This graph is isomorphic to the Cartesian product of $d$ copies of
$K_2$, the complete graph with two vertices.
The Laplacian of $K_2$ is
\[
L_{K_2} = \left[ \begin{array}{rr} 1 & -1\\ -1 & 1\end{array}\right],
\]
whose two eigenvalues are~$0$ and~$2$.
The one-dimensional hypercube $Q_1$ is just $K_2$.
Higher dimensional hypercubes are defined recursively:
\[
Q_{k+1} = Q_k \oplus K_2, \qquad k=1,2,\ldots.
\]
By equation~(\ref{e-Cartesian-laplacian}), their Laplacian matrices are
\[
L_{Q_{k+1}} = L_{Q_k} \otimes I_2 + I_{2^k} \otimes L_{K_2}, \qquad
k=1,2,\ldots .
\]
Using equation~(\ref{e-Cartesian-eigs}) recursively,
the Laplacian of $Q_d$ has eigenvalues $2k$, $k=0,1,\ldots,d$, each
with multiplicity $\left({d\atop k}\right)$.
The FMMC is achieved for:
\[
\alpha^{\star}=\frac{1}{d+1}, \qquad
\mu^{\star}=\frac{d-1}{d+1}.
\]
This solution has also been worked out, for example, in~\cite{Moh:97}.
\begin{figure}
\centering
\includegraphics[width=0.45\textwidth]{figures/hypercube.eps}
\caption{The hypercubes $Q_1,~Q_2$ and
$Q_3$.}
\label{f-hypercube}
\end{figure}
\subsection{FMMC on distance-transitive graphs}
Distance-transitive graphs have been studied extensively in the
literature (see, {\it e.g.}, \cite{BCN:89}). In particular, they are both
edge- and vertex-transitive. In previous examples, the cycles and the
hypercubes are actually distance-transitive graphs; so are the
bipartite graphs when the two parties have equal number of vertices.
In a distance-transitive graph, all vertices have the same valency,
which we denote by~$\nu$.
The Laplacian matrix can be written as $L = \nu I - A$, with $A$ being
the adjacency matrix.
Therefore
\[
\lambda_i(L) = \nu - \lambda_{n+1-i}(A), \qquad i=1,\ldots,n.
\]
We can substitute the above equation in
equations~(\ref{e-et-alpha}) and~(\ref{e-et-mu}) to obtain the optimal
solution in terms of $\lambda_2(A)$ and $\lambda_n(A)$.
Since distance-transitive graphs usually have very large automorphism
groups, the eigenvalues of the adjacency matrix~$A$ (and the
Laplacian~$L$) often have very high multiplicities.
But to solve the FMMC problem, we only need to know the distinct
eigenvalues; actually, only $\lambda_2(A)$ and $\lambda_n(A)$ would
suffice.
In this regard, it is more convenient to use a much smaller matrix,
the \emph{intersection matrix}, which has all the distinct eigenvalues
of the adjacency matrix.
Let $D$ be the \emph{diameter} of the graph.
For a nonnegative integer~$k\leq D$, choose any two vertices~$u$
and~$v$ such that their distance satisfies $\delta(u,v)=k$.
Let $a_k$, $b_k$ and $c_k$ be the number of vertices that
are adjacent to~$u$ and whose distance from~$v$ are $k$, $k+1$ and
$k-1$, respectively. That is,
\begin{eqnarray*}
a_k &=& |\{ w\in\mathcal V ~|~ \delta(u,w)=1, \; \delta(w,v)=k\}|\\
b_k &=& |\{ w\in\mathcal V ~|~ \delta(u,w)=1, \; \delta(w,v)=k+1\}|\\
c_k &=& |\{ w\in\mathcal V ~|~ \delta(u,w)=1, \; \delta(w,v)=k-1\}|.
\end{eqnarray*}
For distance-transitive graphs, these numbers are independent
of the particular pair of vertices~$u$ and~$v$ chosen.
Clearly, we have $a_0=0$, $b_0=\nu$ and $c_1=1$.
The \emph{intersection matrix} $B$ is the following tridiagonal
$(D+1)\times(D+1)$ matrix
\[
B = \left[ \begin{array}{ccccc}
a_0 & b_0 & & & \\
c_1 & a_1 & b_1 & & \\
& c_2 & a_2 & \ddots & \\
& & \ddots & \ddots & b_{D-1}\\
& & & c_D & a_D
\end{array} \right].
\]
Denote the eigenvalues of the intersection matrix, in decreasing
order, as $\eta_0, ~\eta_1, ~\ldots, ~\eta_D$. These are precisely
the $(D+1)$ distinct eigenvalues of the adjacency matrix~$A$ (see,
{\it e.g.}, \cite{Biggs}). In particular, we have
\[
\lambda_1(A) = \eta_0 = \nu, \qquad \lambda_2(A) = \eta_1, \qquad
\lambda_n(A) = \eta_D.
\]
The following corollary is a direct consequence of
Theorem~\ref{t-edge-transitive}.
\begin{corollary}\label{c-distance-transitive}
The optimal solution of the FMMC problem on a distance-transitive
graph is
\begin{eqnarray}
\alpha^\star &=& \min\left\{ \frac{1}{\nu},\;
\frac{2}{2\nu -(\eta_1 + \eta_D)} \right\} \label{e-dist-tran-alpha}\\
\mu^\star &=& \max\left\{ \frac{\eta_1}{\nu}, \;
\frac{\eta_1 - \eta_D}{2\nu - (\eta_1 + \eta_D)} \right\} .
\label{e-dist-tran-mu}
\end{eqnarray}
\end{corollary}
Next we give solutions for the FMMC problem on several families of
distance-transitive graphs.
\subsubsection{Complete graphs}
The case of the complete graph with $n$ vertices, usually called
$K_n$, is very simple.
It is distance-transitive, with diameter $D=1$ and valency $\nu=n-1$.
The intersection matrix is
\[
B=\left[
\begin{array}{cc}
0 & n-1 \\ 1 & n-2
\end{array}
\right],
\]
with eigenvalues $\eta_0=n-1$, $\eta_1=-1$.
Using equations~(\ref{e-dist-tran-alpha}) and~(\ref{e-dist-tran-mu}),
the optimal parameters are
\[
\alpha^{\star}=\frac{1}{n}, \qquad \mu^{\star}=0.
\]
The associated matrix $P=(1/n)\mathbf 1\ones^T$ has one
eigenvalue equal to 1, and the remaining $n-1$ eigenvalues vanish.
Such Markov chains achieve perfect mixing after just one step,
regardless of the value of $n$.
\subsubsection{Petersen graph}
The Petersen graph, shown in Figure~\ref{fig:petersen}, is a well-known
distance-transitive graph with 10 vertices and 15 edges. The diameter
of the graph is $D=2$, and the intersection matrix is
\[
B=\left[
\begin{array}{ccc}
0 & 3 & 0 \\
1 & 0 & 2 \\
0 & 1 & 2
\end{array}
\right]
\]
with eigenvalues $\eta_0 =3$, $\eta_1 = 1$ and $\eta_2 = -2$.
Applying the formula~(\ref{e-dist-tran-alpha})
and~(\ref{e-dist-tran-mu}), we obtain
\[
\alpha^\star = \frac{2}{7}, \qquad \mu^\star = \frac{3}{7}.
\]
\begin{figure}
\begin{center}
\includegraphics[width=3.5cm]{figures/petersen.eps}
\caption{The Petersen graph.}
\label{fig:petersen}
\end{center}
\end{figure}
\subsubsection{Hamming graphs}
The Hamming graphs, denoted $H(d,n)$, have vertices labeled by
elements in the Cartesian product $\{1,\ldots,n\}^d$, with two
vertices being adjacent if they differ in exactly one component.
By the definition, it is clear that Hamming graphs are isomorphic to
the Cartesian product of $d$ copies of the complete graph $K_n$.
Hamming graphs are distance-transitive, with diameter $D=d$ and
valency $\nu=d\,(n-1)$. Their eigenvalues are given by
$\eta_k = d \, (n-1) - k n$ for $k = 0, \ldots, d$.
These can be obtained using an equation for eigenvalues of adjacency
matrices, similar to~(\ref{e-Cartesian-eigs}),
with the eigenvalues of~$K_n$ being $n-1$ and~$-1$.
Therefore the FMMC has parameters:
\begin{eqnarray*}
\alpha^\star
&=& \min\left\{ \frac{1}{d \,(n-1)}, \; \frac{2}{n \, (d+1)}
\right\}\\[1ex]
\mu^\star
&=& \max\left\{ 1-\frac{n}{d(n-1)}, \; \frac{d-1}{d+1} \right\}.
\end{eqnarray*}
We note that hypercubes (see \S\ref{s-hypercubes}) are special Hamming
graphs with $n=2$.
\subsubsection{Johnson graphs}
The Johnson graph $J(n,q)$ (for $1 \leq q \leq n/2$) is defined as
follows: the vertices are the $q$-element subsets of $\{1,\ldots,n\}$,
with two vertices being connected with an edge if and only if the
subsets differ exactly by one element. It is a distance-transitive
graph, with $n \choose q$ vertices and $\frac{1}{2} q \,(n-q) {n
\choose q}$ edges. It has valency $\nu=q\,(n-q)$ and diameter $D=q$.
The eigenvalues of the intersection matrix can be computed
analytically and they are:
\[
\eta_k = q\,(n-q) + k\,(k-n-1), \qquad k = 0, \ldots, q.
\]
Therefore, by Corollary~\ref{c-distance-transitive}, we obtain
the optimal transition probability
\[
\alpha^\star =
\min\left\{\frac{1}{q \,(n-q)}, ~\frac{2}{q n+n+q-q^2}\right\}
\]
and the smallest SLEM
\[
\mu^\star=\max\left\{1-\frac{n}{q(n-q)},~1-\frac{2n}{q n+n+q-q^2}\right\}.
\]
\section{FMMC on orbit graphs}
\label{s-fmmc-orbit}
For graphs with large automorphism groups, the eigenvalues of the
transition probability matrix often have very high multiplicities. To
solve the FMMC problem, it suffices to work with only the distinct
eigenvalues without consideration of their multiplicities. This is
exactly what the intersection matrix does for distance-transitive
graphs. In this section we develop similar tools for more general
graphs. More specifically, we show how to construct an orbit chain
which is much smaller in size than the original Markov chain, but
contains all its distinct eigenvalues (with much fewer
multiplicities). The FMMC on the original graph can be found by
solving a much smaller problem on the orbit chain.
\subsection{Orbit theory}
\label{s-orbit-theory}
Here we review the orbit theory developed in \cite{BDPX:05}.
Let $P$ be a symmetric Markov chain on the graph
$\mathcal G=(\mathcal V, \mathcal E)$, and $H$ be a group of automorphisms of the graph.
Often, it is a subgroup of the full automorphism group
$\mbox{Aut}(\mathcal G)$.
The vertex set $\mathcal V$ partitions into orbits
$O_v = \{hv:h\in H\}$.
For notational convenience, in this section we use $P(v,u)$,
for $v,u\in\mathcal V$,
to denote entries of the transition probability matrix.
We define the \emph{orbit chain} by
specifying the transition probabilities between orbits
\begin{equation}\label{e-orbit-transition}
P_H(O_v, O_u) = P(v,O_u) = \sum_{u'\in O_u} P(v,u').
\end{equation}
This transition probability is independent of which
$v\in O(v)$ is chosen, so it is well defined
and the lumped orbit chain is indeed Markov.
The orbit chain is in general no longer symmetric,
but it is always reversible.
Let $\pi(i)$, $i\in\mathcal V$, be the stationary distribution of the
original Markov chain.
Then the stationary distribution on the orbit chain is obtained as
\begin{equation}\label{e-orbit-distribution}
\pi_H(O_v)=\sum_{i\in O_v} \pi(i) .
\end{equation}
It can be verified that
\begin{equation}\label{e-detailed-balance}
\pi_H(O_v) P_H(O_v,O_u) = \pi_H(O_u) P_H(O_u,O_v),
\end{equation}
which is the detailed balance condition to test reversibility.
The following is a summary of the orbit theory we developed in
\cite{BDPX:05}, which relate the eigenvalues and eigenvectors of
the orbit chain $P_H$ to the eigenvalues and eigenvectors of the
original chain $P$.
\begin{itemize}
\item \emph{Lifting} (\cite[\S3.1]{BDPX:05}).
If $\bar \lambda$ is an eigenvalue of $P_H$ with associated
eigenvector $\bar f$, then $\bar \lambda$ is an eigenvalue of $P$ with
$H$-invariant eigenfunction $f(v)=\bar f(O_v)$.
Conversely, every $H$-invariant eigenfunction appears uniquely from
this construction.
\item \emph{Projection} (\cite[\S3.2]{BDPX:05}).
Let $\lambda$ be an eigenvalue of $P$ with eigenvector $f$.
Define $\bar f(O_v)=\sum_{h\in H} f(h^{-1}(v))$.
Then $\lambda$ appears as an eigenvalue of $P_H$, with eigenvector
$\bar f$, if either of the following
two conditions holds:
\begin{itemize}
\item[(a)] $H$ has a fixed point $v^*$ and $f(v^*)\neq 0$.
\item[(b)] $f$ is nonzero at a vertex $v^*$ in an
$\mbox{Aut}(\mathcal G)$-orbit which contains a fixed point of~$H$.
\end{itemize}
\end{itemize}
Equipped with this orbit theory, we would like to construct one or
multiple orbit chains that retain all the eigenvalues of the original
chain. Ideally the orbit chains are much smaller in size than the
original chain, with the eigenvalues having much fewer multiplicities.
The following theorem (Theorem 3.7 in \cite{BDPX:05}) gives sufficient
conditions that guarantee that the orbit chain(s) attain all the
eigenvalues of the original chain.
\begin{theorem}\label{t-full-eig-proj}
Suppose that $\mathcal V=O_1\cup\ldots\cup O_K$ is a disjoint union of
the orbits under $\mbox{Aut}(\mathcal G)$.
Let $H_i$ be the subgroup of $\mbox{Aut}(\mathcal G)$ that has a fixed
point in $O_i$.
Then all eigenvalues of $P$ occur among the eigenvalues of
$\{P_{H_i}\}_{i=1}^K$.
Further, every eigenvector of $P$ occurs by lifting an eigenvector of
some $P_{H_i}$.
\end{theorem}
Observe that if $H\subseteq G\subseteq \mbox{Aut}(\mathcal G)$, then
the eigenvalues of $P_H$ contain all eigenvalues of $P_G$.
This allows disregarding some of the $H_i$ in
Theorem~\ref{t-full-eig-proj}.
In particular, it is possible to construct a single orbit chain that
contains all eigenvalues of the original chain.
Therefore we have
\begin{corollary} \label{c-full-eig-proj}
Suppose that $\mathcal V=O_1\cup\ldots\cup O_k$ is a disjoint union of
the orbits under $\mbox{Aut}(\mathcal G)$, and
$H$ is a subgroup of $\mbox{Aut}(\mathcal G)$.
If $H$ has a fixed point in every $O_i$,
then all distinct eigenvalues of $P$ occur among the eigenvalues of
$P_H$.
\end{corollary}
\paragraph{Remarks.}
To find $H$ in the above corollary, we can just compute the corresponding
stabilizer, {\it i.e.}, compute the largest subgroup of $\mbox{Aut}(\mathcal G)$
that fixes one point in each orbit.
Note that the $H$ promised by the corollary may be trivial in some cases;
see the example in~\S\ref{s-graham-example}.
\vspace{2ex}
\begin{figure}[t]
\centering
\subfigure[\normalsize Orbit chain under $S_m\times S_n$.]{
\label{f-bipartite-orbit:a}
\begin{minipage}[t]{0.49\textwidth}
\centering
\psfrag{mp}[tc]{$mp$}
\psfrag{np}[bc]{$np$}
\psfrag{u}[cr]{$O_u$}
\psfrag{v}[cl]{$O_v$}
\includegraphics[width=0.5\textwidth]{figures/bipartite_orbit2.eps}
\end{minipage}}%
\subfigure[\normalsize Orbit chain under $S_{m-1}\times S_n$.]{
\label{f-bipartite-orbit:b}
\begin{minipage}[t]{0.49\textwidth}
\centering
\psfrag{np}[bc]{$np$}
\psfrag{pbr}[br]{$p$}
\psfrag{n-1p}[tl]{$np$}
\psfrag{m-1pc}[bc]{$(m\!-\!1)p$}
\psfrag{x}[cr]{$x$}
\psfrag{u}[cr]{$O_u$}
\psfrag{v}[cl]{$O_v$}
\includegraphics[width=0.5\textwidth]{figures/bipartite_orbit3m.eps}
\end{minipage}}
\newline\\[2ex]
\subfigure[\normalsize Orbit chain under $S_m\times S_{n-1}$.]{
\label{f-bipartite-orbit:c}
\begin{minipage}[t]{0.49\textwidth}
\centering
\psfrag{mp}[bc]{$mp$}
\psfrag{pbl}[bl]{$\!~p$}
\psfrag{m-1p}[tr]{$mp$}
\psfrag{n-1pc}[bc]{$(n\!-\!1)p$}
\psfrag{y}[cl]{$y$}
\psfrag{u}[cr]{$O_u$}
\psfrag{v}[cl]{$O_v$}
\includegraphics[width=0.5\textwidth]{figures/bipartite_orbit3n.eps}
\end{minipage}}%
\subfigure[\normalsize Orbit chain under $S_{m-1}\times S_{n-1}$.]{
\label{f-bipartite-orbit:d}
\begin{minipage}[t]{0.49\textwidth}
\centering
\psfrag{p}[bc]{$p$}
\psfrag{pbl}[bl]{$\!~p$}
\psfrag{pbr}[br]{$p$}
\psfrag{n-1p}[tl]{$\!(n\!-\!1)p$}
\psfrag{m-1p}[tr]{$(m\!-\!1)p\!$}
\psfrag{n-1pc}[bc]{$(n\!-\!1)p$}
\psfrag{m-1pc}[tc]{$(m\!-\!1)p$}
\psfrag{x}[cr]{$x$}
\psfrag{y}[cl]{$y$}
\psfrag{u}[cr]{$O_u$}
\psfrag{v}[cl]{$O_v$}
\includegraphics[width=0.5\textwidth]{figures/bipartite_orbit4.eps}
\end{minipage}}
\caption{Orbit chains of $K_{m,n}$ under different automorphism groups.
The vertices labeled $O_u$ and $O_v$ are orbits of vertices $u$ and
$v$ (labeled in Figure~\ref{f-bipartite}) under corresponding
actions. The vertices labeled~$x$ and~$y$ are fixed points.}
\label{f-bipartite-orbit}
\end{figure}
We illustrate the orbit theory with the bipartite graph $K_{m,n}$
shown in Figure~\ref{f-bipartite}.
It is easy to see that $\mbox{Aut}(K_{m,n})$ is the direct product of
two symmetric groups, namely $S_m \times S_n$, with each symmetric
group permuting one of the two subsets of vertices.
This graph is edge-transitive.
So we assign the same transition probability~$p$ on all the edges.
The orbit chains under four different subgroups of
$\mbox{Aut}(K_{m,n})$ are shown in Figure~\ref{f-bipartite-orbit}.
The transition probabilities between orbits are calculated using
equation~(\ref{e-orbit-transition}).
Since the transition probabilities are not symmetric, we represent
the orbit chains by directed graphs, with different transition
probabilities labeled on opposite directions between two adjacent
vertices.
The full automorphism group $\mbox{Aut}(K_{m,n})$ has two orbits of
vertices; see Figure~\ref{f-bipartite-orbit:a}.
The orbit graphs under the subgroups $S_{m-1}\times S_n$
(Figure~\ref{f-bipartite-orbit:b}) and $S_m\times S_{n-1}$
(Figure~\ref{f-bipartite-orbit:c}) each contains a fixed point of the
two orbits under $\mbox{Aut}(K_{m,n})$.
By Theorem~\ref{t-full-eig-proj}, these two orbit chains contain
all the distinct eigenvalues of the original chain on $K_{m,n}$.
Alternatively, we can construct the orbit chain under the subgroup
$S_{m-1}\times S_{n-1}$, shown in Figure~\ref{f-bipartite-orbit:d}.
This orbit chain contain a fixed point in both orbits under
$\mbox{Aut}(K_{m,n})$.
By Corollary~\ref{t-full-eig-proj}, all distinct eigenvalues of
$K_{m,n}$ appear in this orbit chain.
In particular, this shows that there are at most four distinct
eigenvalues in the original chain.
If we order the vertices in Figure~\ref{f-bipartite-orbit:d} as
$(x,y,O_u, O_v)$, then the transition probability matrix for this
orbit chain is
\[
P_H = \left[ \begin{array}{cccc}
1-np & p & 0 & (n-1)p \\
p & 1-mp & (m-1)p & 0\\
0 & p & 1-np & (n-1)p \\
p & 0 & (m-1)p & 1-mp \end{array} \right]
\]
where $H=S_{m-1}\times S_{n-1}$.
By equation~(\ref{e-orbit-distribution}),
its stationary distribution is
\[
\pi_H = \left( \frac{1}{m+n},~\frac{1}{m+n},~\frac{m-1}{m+n},
~\frac{n-1}{m+n} \right).
\]
\subsection{Fastest mixing reversible Markov chain on orbit graph}
\label{s-fmmc-reversible}
Since in general the orbit chain is no longer symmetric, we cannot
directly use the convex optimization formulation~(\ref{e-fmmc})
or~(\ref{e-fmmc-sdp}) to minimize $\mu(P_H)$. Fortunately, the
detailed balance condition~(\ref{e-detailed-balance}) leads to a
simple transformation that allow us to formulate the problem of
finding the fastest reversible Markov chain as a convex program
\cite{BDX:04}.
Suppose the orbit chain $P_H$ contains all distinct eigenvalues of the
original chain.
Let $\pi_H$ be the stationary distribution of the orbits, and
let $\Pi=\mathop{\bf Diag}(\pi_H)$.
The detailed balance condition~(\ref{e-detailed-balance}) can be
written as $\Pi P_H = P_H^T \Pi$,
which implies that the matrix $\Pi^{1/2}P_H\Pi^{-1/2}$ is symmetric
(and of course, has the same eigenvalues as $P_H$).
The eigenvector of $\Pi^{1/2}P_H\Pi^{-1/2}$ associated with the
maximum eigenvalue~$1$ is
$q=(\sqrt{\pi_H(O_1)},\ldots,\sqrt{\pi_H(O_k)})$.
The SLEM $\mu(P_H)$ equals the spectral norm of
$\Pi^{1/2}P_H\Pi^{-1/2}$ restricted to the orthogonal complement of
the subspace spanned by $q$.
This can be written as
\[
\mu(P_H) = \| (I-qq^T) \Pi^{1/2} P_H \Pi^{-1/2} (I-qq^T) \|_2
= \| \Pi^{1/2} P_H \Pi^{-1/2} - qq^T \|_2.
\]
Introducing a scalar variable $s$ to bound the above spectral norm,
we can formulate the fastest mixing reversible Markov chain problem as
an SDP
\begin{equation}\label{e-fmmc-rev}
\begin{array}{ll}
\mbox{minimize} & s \\[0.5ex]
\mbox{subject to}
& -sI \preceq \Pi^{1/2} P_H \Pi^{-1/2} - qq^T \preceq sI\\[0.5ex]
& P_H\geq 0,\quad P_H\mathbf 1=\mathbf 1,\quad\Pi P_H=P_H^T\Pi\\[0.5ex]
& P_H(O,O') = 0, \quad (O,O')\notin \mathcal{E}_H.
\end{array}
\end{equation}
The optimization variables are the matrix $P_H$ and scalar~$s$, and
problem data is given by the orbit graph and the stationary
distribution $\pi_H$. Note that the reversibility constraint $\Pi
P_H=P_H^T\Pi$ can be dropped since it is always satisfied by the
construction of the orbit chain; see
equation~(\ref{e-detailed-balance}). By pre- and post-multiplying the
matrix inequality by $\Pi^{1/2}$, we can write then another
equivalent formulation:
\begin{equation}\label{e-fmmc-rev2}
\begin{array}{ll}
\mbox{minimize} & s \\[0.5ex]
\mbox{subject to}
& -s \Pi \preceq \Pi P_H - \pi_H \pi_H^T \preceq s \Pi\\[0.5ex]
& P_H\geq 0,\quad P_H\mathbf 1=\mathbf 1, \\[0.5ex]
& P_H(O,O') = 0, \quad (O,O')\notin \mathcal{E}_H.
\end{array}
\end{equation}
To solve the fastest mixing reversible Markov chain problem on the
orbit graph, we need the following three steps.
\begin{enumerate}
\item
Conduct symmetry analysis on the original graph: identify the
automorphism graph $\mbox{Aut}(\mathcal G)$ and determine the number
of orbits of edges $N$.
By Corollary~\ref{c-numprobs},
this is the number of transition probabilities we need to consider.
\item
Find a group of automorphisms $H$ that satisfies the conditions in
Corollary~\ref{c-full-eig-proj}.
Construct its orbit chain by computing the transition probabilities
using equation~(\ref{e-orbit-transition}), and compute the stationary
distribution using equation~(\ref{e-orbit-distribution}).
Note that the entries of $P_H$ are multiples of the transition
probabilities on the original graph.
\item
Solve the fastest mixing reversible Markov chain
problem~(\ref{e-fmmc-rev}).
The optimal SLEM $\mu(P_H^\star)$ is also the optimal SLEM for the
original chain, and the optimal transition probabilities on the
original chain can be obtained by simple scaling of the optimal orbit
transition probabilities.
\end{enumerate}
We have assumed a single orbit chain that contains all distinct
eigenvalues of the original chain.
Sometimes it is more convenient to use multiple orbit chains.
Let $P_{H_i}$, $i=1,\ldots,K$, be the collection of orbit chains
in Theorem~\ref{t-full-eig-proj}.
In this case we need to minimize $\max_i \mu(P_{H_i})$.
This can be done by simply adding the set of constraints
in~(\ref{e-fmmc-rev}) for every matrix $P_{H_i}$.
For example, for the complete bipartite graph $K_{m,n}$, instead of
using the single orbit chain in Figure~\ref{f-bipartite-orbit:d},
we can use the two orbit chains in Figure~\ref{f-bipartite-orbit:b}
and Figure~\ref{f-bipartite-orbit:c} together, with two sets of
constraints in the SDP~(\ref{e-fmmc-rev}).
\subsection{Examples}
We demonstrate the above computational procedure on orbit graphs
with two examples: the graph $K_n$-$K_n$ and the complete binary
tree.
Both examples will be revisited in~\S\ref{s-block-diag} using the
method of block diagonalization.
\subsubsection{The graph $K_n$-$K_n$}
\label{s-Kn-Kn}
\begin{figure}
\centering
\subfigure[\normalsize The graph $K_n$-$K_n$.]{
\label{f-KnKn:a}
\begin{minipage}[t]{0.99\textwidth}
\centering
\psfrag{p0}[bc]{$p_0$}
\psfrag{p1}[bl]{$p_1$}
\psfrag{p3}[bc]{$p_2$}
\psfrag{x}[tl]{$x$}
\psfrag{y}[tr]{$y$}
\psfrag{z}[bl]{$z$}
\psfrag{u}[tl]{$u$}
\psfrag{v}[tr]{$v$}
\includegraphics[width=0.55\textwidth]{figures/KnKn_graph.eps}
\vspace{5ex}
\end{minipage}}
\vspace{7ex}
\subfigure[\normalsize Orbit chain under
$C_2\ltimes(S_{n-1}\times S_{n-1})$.]{
\label{f-KnKn:b}
\begin{minipage}[t]{0.99\textwidth}
\centering
\psfrag{n-1p1}[bc]{$(n-1)p_1$}
\psfrag{p1}[tc]{$p_1$}
\psfrag{p0}[bc]{$p_0$}
\psfrag{Ox}[cl]{$O_x$}
\psfrag{Oz}[cr]{$O_z$}
\psfrag{x}[tl]{$x$}
\psfrag{y}[tr]{$y$}
\psfrag{Ou}[cr]{$O_u$}
\psfrag{Ov}[cl]{$O_v$}
\includegraphics[width=0.17\textwidth]{figures/KnKn_orbit2.eps}
\vspace{5ex}
\end{minipage}}
\vspace{7ex}
\subfigure[\normalsize Orbit chain under $S_{n-1}\times S_{n-1}$.]{
\label{f-KnKn:c}
\begin{minipage}[t]{0.99\textwidth}
\centering
\psfrag{n-1p1}[bc]{$(n-1)p_1$}
\psfrag{p1}[tc]{$p_1$}
\psfrag{p0}[bc]{$p_0$}
\psfrag{Ox}[cl]{$O_x$}
\psfrag{Oz}[cr]{$O_z$}
\psfrag{x}[tl]{$x$}
\psfrag{y}[tr]{$y$}
\psfrag{Ou}[cr]{$O_u$}
\psfrag{Ov}[cl]{$O_v$}
\includegraphics[width=0.4\textwidth]{figures/KnKn_orbit4.eps}
\vspace{5ex}
\end{minipage}}
\vspace{7ex}
\subfigure[\normalsize Orbit chain under $S_{n-2}\times S_{n-1}$.]{
\label{f-KnKn:d}
\begin{minipage}[t]{0.99\textwidth}
\centering
\psfrag{p0}[bc]{$p_0$}
\psfrag{p1}[bl]{$p_1$}
\psfrag{np1}[tc]{$(n\!-\!2)p_1$}
\psfrag{np11}[tc]{$(n\!-\!1)p_1$}
\psfrag{p2}[tl]{$p_2$}
\psfrag{np2}[br]{$(n\!-\!2)p_2$}
\psfrag{x}[tl]{$x$}
\psfrag{y}[tr]{$y$}
\psfrag{z}[bl]{$z$}
\psfrag{Ou}[cr]{$O_u$}
\psfrag{Ov}[cl]{$O_v$}
\includegraphics[width=0.45\textwidth]{figures/KnKn_orbit5.eps}
\vspace{5ex}
\end{minipage}}
\vspace{7ex}
\caption{The graph $K_n$-$K_n$ and its orbit chains under
different automorphism groups.
Here $O_x,O_z,O_u,O_v$ represent orbits of the vertices
$x,z,u,v$ (labeled in Figure~\ref{f-KnKn:a}), respectively,
under the corresponding automorphism groups in each subgraph.}
\label{f-KnKn}
\end{figure}
The graph $K_n$-$K_n$ consists of two copies of the complete
graph $K_n$ joined by a bridge (see Figure~\ref{f-KnKn:a}).
We follow the three steps described in~\S\ref{s-fmmc-reversible}.
First, it is clear by inspection that the full automorphism group
is $C_2\ltimes(S_{n-1}\times S_{n-1})$.
The actions of $S_{n-1}\times S_{n-1}$ are all possible
permutations of the two set of $n-1$ vertices, distinct from the two
center vertices~$x$ and~$y$, among themselves.
The group $C_2$ acts on the graph by switching the two halves.
The semi-direct product symbol $\ltimes$ means that the actions of
$S_{n-1}\times S_{n-1}$ and $C_2$ do not commute.
By symmetry analysis in~\S\ref{s-analysis}, there are three
edge orbits under the full automorphism group: the bridging edge
between vertices~$x$ and~$y$, the edges connecting~$x$ and~$y$ to all
other vertices, and the edges connecting all other vertices.
Thus it suffices to consider just three transition probabilities
$p_0$, $p_1$, and $p_2$, each labeled in Figure~\ref{f-KnKn:a}
on one representative of the three edge orbits.
As the second step, we construct the orbit chains.
The orbit chain of $K_n$-$K_n$ under the full automorphism group is
depicted in Figure~\ref{f-KnKn:b}.
The orbit $O_x$ includes vertices~$x$ and~$y$, and the orbit $O_z$
consists of all other $2(n-1)$ vertices.
The transition probabilities of this orbit chain are calculated from
equation~(\ref{e-orbit-transition}) and are labeled on the directed
edges in Figure~\ref{f-KnKn:b}.
Similarly, the orbit chain under the subgroup $S_{n-1}\times S_{n-1}$
is depicted in Figure~\ref{f-KnKn:c}.
While these two orbit chains are the most obvious to construct,
none of them contains all eigenvalues of the original chain, nor
does their combination.
For the one in Figure~\ref{f-KnKn:b}, the full automorphism
group does not have a fixed point either of its orbit $O_x$ or $O_z$.
For the one in~\ref{f-KnKn:c}, the automorphism group
$S_{n-1}\times S_{n-1}$ has a fixed point in $O_x$ (either $x$ or
$y$), but does not have a fixed point in $O_z$
(note here $O_z$ is the orbit of~$z$ under the full automorphism
group).
To fix the problem, we consider the orbit chain under the group
$S_{n-2}\times S_{n-1}$, which leave the vertex~$x$, $y$, and $z$
fixed, while permuting the rest $n-2$ vertices on the left
and the $n-1$ points on the right, respectively.
The corresponding orbit chain is shown in
Figure~\ref{f-KnKn:d}.
By Corollary~\ref{c-full-eig-proj}, all distinct eigenvalues of the
original Markov chain on $K_n$-$K_n$ appear as eigenvalues of this
orbit chain.
Thus there are at most five distinct eigenvalues in the
original chain no matter how large~$n$ is.
To finish the second step, we calculate the transition probabilities
of the orbit chain under $H=S_{n-2}\times S_{n-1}$ using
equation~(\ref{e-orbit-transition}) and label them in Figure~\ref{f-KnKn:d}.
If we order the vertices of this orbit chain as
$(x,y,z,O_u,O_v)$, then the transition probability
matrix on the orbit chain is
\[
P_H=\left[ \begin{array}{ccccc}
1-p_0-(n-1)p_1 & p_0 & p_1 & (n-2)p_1 & 0 \\
p_0 & 1-p_0-(n-1)p_1 & 0 & 0 & (n-1)p_1 \\
p_1 & 0 & 1-p_1-(n-2)p_2 & (n-2)p_2 & 0 \\
p_1 & 0 & p_2 & 1-p_1-p_2 & 0 \\
0 & p_1 & 0 & 0 & 1-p_1 \end{array} \right] .
\]
By equation~(\ref{e-orbit-distribution}), the stationary
distribution of the orbit chain is
\[
\pi_H=\left(\frac{1}{2n},~\frac{1}{2n},~\frac{1}{2n},
~\frac{n-2}{2n},~\frac{n-1}{2n} \right) .
\]
As the third step, we solve the SDP~(\ref{e-fmmc-rev})
with the above parametrization.
It is remarkable to see that we only need to solve an SDP with~$4$
variables (three transition probabilities $p_0$, $p_1$, $p_2$,
and the extra scalar~$s$) and $5\times 5$ matrices
no matter how large the graph (the number~$n$) is.
We will revisit this example in \S\ref{s-Kn-Kn-2} using the block
diagonalization method, where we present an analytic
expression for the exact optimal SLEM and corresponding transition
probabilities.
\subsubsection{Complete binary tree}
\label{s-tree-orbit}
\begin{figure}[t]
\centering
\subfigure[\normalsize Orbit graph and chain under
$S_2 \wr S_2 \wr S_2$.]{
\label{f-binary-tree:a}
\begin{minipage}[t]{0.49\textwidth}
\centering
\psfrag{2p1}[br]{$2p_1$}
\psfrag{p1}[tl]{$p_1$}
\psfrag{2p2}[br]{$2p_2$}
\psfrag{p2}[tl]{$p_2$}
\psfrag{2p3}[br]{$2p_3$}
\psfrag{p3}[tl]{$p_3$}
\includegraphics[width=0.6\textwidth]{figures/binary_tree1.eps}
\end{minipage}}%
\subfigure[\normalsize Orbit graph under
$(S_2 \wr S_2) \times (S_2 \wr S_2)$.]{
\label{f-binary-tree:b}
\begin{minipage}[t]{0.49\textwidth}
\centering
\includegraphics[width=0.6\textwidth]{figures/binary_tree2.eps}
\end{minipage}}
\newline\\[1ex]
\subfigure[\normalsize Orbit graph under
$(S_2 \times S_2) \times (S_2 \wr S_2)$.]{
\label{f-binary-tree:c}
\begin{minipage}[t]{0.49\textwidth}
\centering
\includegraphics[width=0.6\textwidth]{figures/binary_tree3.eps}
\end{minipage}}%
\subfigure[\normalsize Orbit graph under
$S_2 \times (S_2 \wr S_2)$.]{
\label{f-binary-tree:d}
\begin{minipage}[t]{0.49\textwidth}
\centering
\includegraphics[width=0.6\textwidth]{figures/binary_tree4.eps}
\end{minipage}}
\caption{Orbit graphs of the complete binary tree $\mathcal T_n$
($n=3$) under different automorphism groups.
The vertices surrounded by a circle are fixed
points of the corresponding automorphism group.}
\label{f-binary-tree}
\end{figure}
We consider a complete binary tree with~$n$ levels of branches,
denoted as $\mathcal T_n$. The total number of nodes is
$|\mathcal{V}|= 2^{n+1}-1$. The matrix inequalities in the
corresponding SDP have size $|\mathcal{V}|\times|\mathcal{V}|$, which
is clearly exponential in $n$. However, the binary tree has a very
large automorphism group, of size $2^{(2^n-1)}$. This automorphism
group is best described recursively. Plainly, for $n=1$, we have
$\mbox{Aut}(\mathcal T_1) = S_2$. For $n>1$, it can be obtained by
the recursion
\[
\mbox{Aut}(\mathcal T_{k+1})=\mbox{Aut}(\mathcal T_k) \wr S_2,
\qquad k=1,\ldots,n-1,
\]
where $\wr$ represents the \emph{wreath product} of two groups
({\it e.g.}, \cite{JaK:81}).
More specifically, let
$g=(g_1,g_2)$ and $h=(h_1,h_2)$ be elements of the product group
$\mbox{Aut}(\mathcal T_k)\times\mbox{Aut}(\mathcal T_k)$,
and $\sigma$ and $\pi$ be in $S_2$.
The multiplication rule of the wreath product is
\[
(g,\sigma)(h,\pi) = \left( (g_1 h_{\sigma^{-1}(1)},
g_2 h_{\sigma^{-1}(2)}), \sigma\pi \right).
\]
This is a semi-direct product $\mbox{Aut}(\mathcal T_k)^2\rtimes S_2$
({\it cf.}~the automorphism group of $K_n$-$K_n$).
From the above recursion, the automorphism group of $\mathcal T_n$ is
\[
\mbox{Aut}(\mathcal T_n) = S_2 \wr S_2 \wr \cdots \wr S_2
\quad \mbox{($n$ times)}.
\]
(The wreath product is associative, but not commutative.)
The representation theory of the automorphism group of the binary tree has been thoroughly studied as this group is the Sylow 2-subgroup of a symmetric group; see \cite{OOR:04,AbV:05}.
The orbit graph of $\mathcal T_n$ under its full automorphism group is
a path with $n+1$ nodes (Figure~\ref{f-binary-tree:a}, left).
Since there are $n$ orbits of edges, there are $n$ different
transition probabilities we need to consider.
We label them as $p_k$, $k=1,\ldots,n$, from top to bottom of the
tree.
The corresponding orbit chain, represented by a directed graph labeled
with transition probabilities between orbits, is shown on the right of
Figure~\ref{f-binary-tree:a}.
To simplify presentation, only the orbit graphs are shown in other
subfigures of Figure~\ref{f-binary-tree}.
The corresponding orbit chains should be straightforward to
construct.
The largest subgroup of $\mbox{Aut}(\mathcal T_n)$ that has a fixed
point in every orbit under $\mbox{Aut}(\mathcal T_n)$ is
\[
W_n = \prod_{k=1}^{n-1} (S_2 \wr \cdots \wr S_2)~(k~\mbox{times})
\]
where $\prod$ denotes direct product of groups.
The corresponding orbit graph is shown in Figure~\ref{f-binary-tree:d}
for $n=3$.
The number of vertices in this orbit graph is
\[
1+2+\cdots+n+(n+1) = {n+1 \choose 2} = \frac{1}{2}(n+1)(n+2),
\]
which is much smaller than $2^{n+1}-1$, the size of $\mathcal T_n$.
From the above analysis, we only need to solve the fastest reversible
Markov chain problem on the orbit graph of size ${n+1 \choose 2}$ with
$n$ variables $p_1,\ldots,p_n$. In next section, using the technique
of block diagonalization, we will see that the transition probability
matrix of size ${n+1 \choose 2}$ can be further decomposed into smaller
matrices with sizes $1,2,\ldots,n+1$. Due to an eigenvalue
interlacing result, we only need to consider the orbit chain with
$2n+1$ vertices in Figure~\ref{f-binary-tree:b}.
\section{Symmetry reduction by block diagonalization}
\label{s-block-diag}
By definition of the fixed-point subspace $\mathcal F$
(in~\S\ref{s-fmmc-invariant}), any transition probability matrix
$P\in\mathcal F$ is invariant under the actions of
$\mbox{Aut}(\mathcal G)$.
More specifically, for any permutation matrix $Q$ given by
$\sigma\in\mbox{Aut}(\mathcal G)$, we have $Q P Q^T=P$, equivalently
$Q P = P Q$.
In this section we show that this property allows the construction
of a coordinate transformation matrix that can block diagonalize
every $P\in\mathcal F$.
The resulting blocks usually have much smaller sizes and repeated
blocks can be discarded in computation.
The method we use in this section is based on classical group
representation theory ({\it e.g.}, \cite{Ser:77}).
It was developed for more general SDPs in~\cite{GatermannParrilo},
and has found applications in sum-of-squares decomposition for
minimizing polynomial
functions~\cite{PhD:Parrilo,Par:03,ParriloSturmfels} and controller
design for symmetric dynamical systems~\cite{CLP:03}.
A closely related approach is developed in \cite{dPS:07}, which is
based on a low-order representation of the commutant
(collection of invariant matrices) of the matrix algebra generated
by the permutation matrices.
\subsection{Some group representation theory}
\label{s-representation}
Let $G$ be a group.
A \emph{representation} $\rho$ of $G$ assigns an invertible matrix
$\rho(g)$ to each $g\in G$ in such a way that the matrix assigned to
the product of two elements in $G$ is the product of the matrices
assigned to each element: $\rho(gh)=\rho(g)\rho(h)$.
The matrices we work with are all invertible and are considered over
the real or complex numbers.
We thus regard $\rho$ as a homomorphism from $g$ to the linear maps on
a vector space $V$.
The dimension of $\rho$ is the dimension of $V$.
Two representations are \emph{equivalent} if they are related by a
fixed similarity transformation.
If $W$ is a subspace of $V$ invariant under $G$, then $\rho$
restricted to $W$ gives a \emph{subrepresentation}.
Of course the zero subspace and the subspace $W=V$ are trivial
subrepresentations.
If the representation $\rho$ admits no non-trivial subrepresentation,
then $\rho$ is called \emph{irreducible}.
We consider first complex representations, as the theory is
considerably simpler in this case.
For a finite group $G$ there are only finitely many inequivalent
irreducible representations $\vartheta_1,\ldots,\vartheta_h$ of
dimensions $n_1,\ldots,n_h$, respectively.
The degrees $n_i$ divide the group order $|G|$, and satisfy the
condition $\sum_{i=1}^h n_i^2=|G|$.
Every linear representation of $G$ has a canonical decomposition as a
direct sum of irreducible representations
\[
\rho = m_1\vartheta_1 \oplus m_2\vartheta_2 \oplus \cdots \oplus
m_h\vartheta_h,
\]
where $m_1,\ldots,m_h$ are the multiplicities.
Accordingly, the representation space $\mathbf{C}^n$ has an
\emph{isotypic decomposition}
\begin{equation}\label{e-isotypic}
\mathbf{C}^n = V_1 \oplus \cdots \oplus V_h
\end{equation}
where each isotypic components consists of $m_i$ invariant subspaces
\begin{equation}\label{e-equivalent}
V_i = V_i^1 \oplus \cdots \oplus V_i^{m_i},
\end{equation}
each of which has dimension $n_i$ and transforms after the manner of
$\vartheta_i$.
A basis of this decomposition transforming with respect to the
matrices $\vartheta_i(g)$ is called \emph{symmetry-adapted} and can be
computed using the algorithm presented in \cite[\S2.6-2.7]{Ser:77}
or \cite[\S5.2]{FaS:92}.
This basis defines a change of coordinates by a matrix $T$ collecting
the basis as columns.
By Schur's lemma, if a matrix~$P$ satisfies
\begin{equation}\label{e-invariance}
\rho(g)P=P\rho(g), \qquad \forall g\in G,
\end{equation}
then $T^{-1}PT$ has block diagonal form with one block
$P_i$ for each isotypic component of dimension $m_i n_i$, which
further decomposes into $n_i$ equal blocks $B_i$ of dimension $m_i$.
That is
\begin{equation}\label{e-block-diag}
T^{-1} P T = \left[ \begin{array}{ccc}
P_1 & & 0\\ & \ddots & \\ 0 & & P_h \end{array} \right], \qquad
P_i = \left[ \begin{array}{ccc}
B_i & & 0\\ & \ddots & \\ 0 & & B_i \end{array} \right].
\end{equation}
For our application of semidefinite programs, the problems are
presented in terms of real matrices, and therefore we would like to
use real coordinate transformations.
In fact a generalization of the classical theory to the real case is
presented in \cite[\S13.2]{Ser:77}.
If all $\vartheta_i(g)$ are real matrices the irreducible
representation is called \emph{absolutely irreducible}.
Otherwise, for each $\vartheta_i$ with complex character its complex
conjugate will also appear in the canonical decomposition.
Since $\rho$ is real both will have the same multiplicity and real
bases of $V_i+\bar V_i$ can be constructed.
So two complex conjugate irreducible representations form one real
irreducible representation of \emph{complex type}.
There is a third case, real irreducible representations of
\emph{quaternonian type}, rarely seen in practical examples.
In this paper, we assume that the representation $\rho$ is orthogonal,
{\it i.e.}, $\rho(g)^T\rho(g)=\rho(g)\rho(g)^T=I$ for all $g\in G$.
As a result, the transformation matrix~$T$ can also be chosen to be
orthogonal.
Thus $T^{-1}=T^T$ (for complex matrices, it is the conjugate
transpose).
For symmetric matrices the block corresponding to a representation of
complex type or quaternonian type simplifies to a collection of equal
subblocks.
For the special case of circulant matrices, complete
diagonalization reveals all the eigenvalues \cite[page 50]{Dia:88}.
\subsection{Block diagonalization of SDP constraint}
\label{s-diag-sdp}
As in~\S\ref{s-fmmc-invariant}, for every
$\sigma\in\mbox{Aut}(\mathcal G)$ we assign a permutation matrix
$Q(\sigma)$ by letting $Q_{ij}(\sigma)=1$ if
$i=\sigma(j)$ and $Q_{ij}(\sigma)=0$ otherwise.
This is an $n$-dimensional representation of $\mbox{Aut}(\mathcal G)$,
which is often called the \emph{natural representation}.
As mentioned in the beginning of this section, every matrix~$P$
in the fixed-point subset $\mathcal F$ has the symmetry of
$\mbox{Aut}(\mathcal G)$; {\it i.e.}, it satisfies the
condition~(\ref{e-invariance}) with $\rho=Q$.
Thus a coordinate transformation matrix~$T$ can be constructed such
that $P$ can be block diagonalized into the
form~(\ref{e-block-diag}).
Now we consider the SDP~(\ref{e-fmmc-orbits-sdp}), which is the
FMMC problem formulated in the fixed-point subset~$\mathcal F$.
In \S\ref{s-reduce-variables}, we have derived the expression
$P(p)=I-\sum_{k=1}^N p_k L_k$,
where $L_k$ is the Laplacian matrix for the $k$th orbit graph and $p_k$ is the common transition probability assigned on all edges in the $k$th orbit graph.
Note the matrix $P(p)$ has the symmetry of $\mbox{Aut}(\mathcal G)$.
Applying the coordinate transformation~$T$ to the linear matrix
inequalities, we obtain the following equivalent problem
\begin{equation}\label{e-fmmc-block-diag}
\begin{array}{ll}
\mbox{minimize} & s \\[1ex]
\mbox{subject to} & -s I_{m_i} \preceq B_i(p) - J_i \preceq s I_{m_i},
\quad i=1,\ldots,h\\[1ex]
& p_k \geq 0, \quad k=1,\ldots,N\\[1ex]
& \sum_{k=1}^N (L_k)_{ii} \;p_k \leq 1, \quad i=1,\ldots,n
\end{array}
\end{equation}
where $B_i(p)$ correspond to the small blocks $B_i$
in~(\ref{e-block-diag}) of the transformed matrix $T^TP(p)T$,
and~$J_i$ are the corresponding diagonal blocks of
$T^T(1/n)\mathbf 1\ones^T T$.
The number of matrix inequalities~$h$ is the number of inequivalent
irreducible representations, and the size of each matrix
inequality~$m_i$ is the multiplicity of the corresponding
irreducible representation.
Note that we only need one out of~$n_i$ copies of each~$B_i$ in the
decomposition~(\ref{e-block-diag}).
Since $m_i$ can be much smaller than~$n$ (the number of vertices in
the graph), the improvement in computational complexity over the SDP
formulation~(\ref{e-fmmc-orbits-sdp}) can be significant (see the flop
counts discussed in~\S\ref{s-structure}).
This is especially the case when there are high-dimensional
irreducible representations
({\it i.e.}, when $n_i$ is large; see, {\it e.g.}, $K_n$-$K_n$ defined in~\S\ref{s-Kn-Kn}).
The transformed SDP formulation~(\ref{e-fmmc-block-diag}) needs some
further justification.
Namely, all the off-diagonal blocks of the matrix
$T^T(1/n)\mathbf 1\ones^T T$ have to be zero.
This is in fact the case.
Moreover, the following theorem reveals an interesting connection
between the block diagonalization approach and the orbit theory
in~\S\ref{s-fmmc-orbit}.
\begin{theorem}\label{t-orbit-diag-connection}
Let $H$ be a subgroup of $\mathrm{Aut}(\mathcal G)$, and $T$ be the
coordinate transformation matrix whose columns are a symmetry-adapted
basis for the natural representation of~$H$. Suppose a Markov
chain~$P$ defined on the graph has the symmetry of~$H$. Then the
matrix $T^T(1/n)\mathbf 1\ones^T T$ has the same block diagonal form as
$T^TPT$. Moreover, there is only one nonzero block. Without loss of
generality, let this nonzero block be $J_1$ and the corresponding
block of $T^TPT$ be $B_1$. These two blocks relate to the orbit chain
$P_H$ by
\begin{eqnarray}
B_1 &=& \Pi^{1/2}P_H\Pi^{-1/2}
\label{e-orbit-diag-connection} \\
J_1 &=& qq^T \label{e-orbit-diag-J1}
\end{eqnarray}
where $\Pi=\mathop{\bf Diag}(\pi_H)$, $q=\sqrt{\pi_H}$,
and $\pi_H$ is the stationary distribution of $P_H$.
\end{theorem}
\begin{proof}
First we note that $P$ always has a single eigenvalue~$1$ with
associated eigenvector~$\mathbf 1$. Thus~$\mathbf 1$ spans an invariant
subspace of the natural representation, which is obviously
irreducible. The corresponding irreducible representation is
isomorphic to the trivial representation (which assigns the scalar~$1$
to every element in the group). Without loss of generality, let $V_1$
be the isotypic component that contains the vector~$\mathbf 1$. Thus
$V_1$ is a direct product of $H$-fixed vectors (each corresponds to a
copy of the trivial representation), and~$\mathbf 1$ is a linear
combination of these vectors.
Let $m_1$ be the dimension of $V_1$, which is the number of $H$-fixed
vectors. We can calculate~$m_1$ by Frobenius reciprocity, or
``Burnside's Lemma''; see, {\it e.g.}, \cite{Ser:77}. To do so, we note that
the character~$\chi$ of the natural representation~$Q(g)$,
$g\in H$, is the number of fixed points of~$g$, {\it i.e.},
\[
\chi(g) = \mathop{\bf Tr} Q(g) = \mathrm{FP}(g)=\#\{v\in \mathcal V: g(v)=v \}.
\]
``Burnside's Lemma'' says that
\[
\frac{1}{|H|} \sum_{g\in H} \mathrm{FP}(g) = \# \mathrm{orbits}.
\]
The left-hand side is the inner product of~$\chi$ with the trivial
representation. It thus counts the number of $H$-fixed vectors
in~$V$. So $m_1$ equals the number of orbits under $H$.
Suppose that $\mathcal V=O_1\cup\ldots\cup O_{m_1}$ as a disjoint
union of $H$-orbits.
Let $b_i(v)=1/\sqrt{|O_i|}$ if $v\in O_i$ and zero otherwise.
Then $b_1,\ldots,b_{m_1}$ are $H$-fixed
vectors, and they form an orthonormal symmetry-adapted basis
for $V_1$ (these are not unique).
Let $T_1=[b_1 \cdots b_{m_1}]$ be the first $m_1$ columns
of~$T$.
They are orthogonal to all other columns of~$T$.
Since~$\mathbf 1$ is a linear combination of $b_1,\ldots,b_{m_1}$, it is
also orthogonal to other columns of~$T$.
Therefore the matrix $T^T(1/n)\mathbf 1\ones^T T$ has all its elements
zero except for the first $m_1\times m_1$ diagonal block, which we
denote as $J_1$.
More specifically, $J_1=qq^T$ where
\begin{eqnarray*}
q &=&\frac{1}{\sqrt{n}} T_1^T \mathbf 1 = \frac{1}{\sqrt{n}}
\left[ b_1^T\mathbf 1 ~\cdots~ b_{m_1}^T\mathbf 1 \right]^T\\
&=& \frac{1}{\sqrt{n}} \left[ \frac{|O_1|}{\sqrt{|O_1|}} ~\cdots~
\frac{|O_{m_1}|}{\sqrt{|O_{m_1}|}} \; \right]^T
= \left[ \sqrt{\frac{|O_1|}{n}} ~\ldots~
\sqrt{\frac{|O_{m_1}|}{n}} \; \right]^T.
\end{eqnarray*}
Note that by~(\ref{e-orbit-distribution}) the stationary distribution
of the orbit chain $P_{H}$ is
\[
\pi_{H}
=\left[ \; \frac{|O_1|}{n} ~\cdots~ \frac{|O_{m_1}|}{n} \; \right]^T.
\]
Thus we have $q=\sqrt{\pi_{H}}$.
This proves~(\ref{e-orbit-diag-J1}).
Finally we consider the relationship between $B_1=T_1^T P T_1$ and
$P_{H}$.
We prove~(\ref{e-orbit-diag-connection}) by showing
\[
\Pi^{-1/2} B_1 \Pi^{1/2} = \Pi^{-1/2} T_1^T P T_1 \Pi^{1/2} = P_H.
\]
It is straightforward to verify that
\begin{eqnarray*}
&&
\Pi^{-1/2} T_1^T = \sqrt{n} \left[ \begin{array}{c}
b'^T_1\\ \vdots\\ b'^T_{m_1} \end{array} \right], \qquad
b'_i(v) = \left\{ \begin{array}{lll}
\displaystyle \frac{1}{|O_i|} && \mbox{if}~v\in O_i\\[2ex]
0 && \mbox{if}~v\notin O_i \end{array} \right. \\[2ex]
&&
T_1 \Pi^{1/2} = \frac{1}{\sqrt{n}}
\left[ b''_1 ~\cdots~ b''_{m_1}\right], \qquad
b''_i(v) = \left\{ \begin{array}{lll}
1 && \mbox{if}~v\in O_i\\[1ex]
0 && \mbox{if}~v\notin O_i \end{array} \right.
\end{eqnarray*}
The entry at the $i$-th row and $j$-th column of the matrix
$\Pi^{-1/2} T_1^T P T_1 \Pi^{1/2}$ are given by
\[
b'^T_i P b''_j
= \frac{1}{|O_i|} \sum_{v\in O_i} \sum_{u\in O_j} P(v,u)
= \frac{1}{|O_i|} \sum_{v\in O_i} P_H(O_i, O_j)
= P_H( O_i, O_j ).
\]
In the last equation, we have used the fact that $P_H(O_i, O_j)$ is
independent of which $v\in O_i$ is chosen.
This completes the proof.
\end{proof}
From Theorem~\ref{t-orbit-diag-connection}, we know that
$B_1$ contains the eigenvalues of the orbit chain under $H$.
Other blocks $B_i$ contain additional eigenvalues (not including those
of $P_{H}$) of the orbit
chains under various subgroups of $H$.
(Note that the eigenvalues of the orbit chain under
$H$ are always contained in the orbit chain
under its subgroups).
With this observation, it is possible to identify the multiplicities
of eigenvalues in orbit chains under various subgroups of
$\mbox{Aut}(\mathcal{G})$ by relating to the
decompositions~(\ref{e-isotypic}), (\ref{e-equivalent}) and
(\ref{e-block-diag})
(some preliminary results are discussed in \cite{BDPX:05}).
\subsubsection{A running example}
\begin{figure}
\begin{center}
\psfrag{1}{$1$}
\psfrag{2}{$2$}
\psfrag{3}{$3$}
\psfrag{4}{$4$}
\psfrag{5}{$5$}
\psfrag{6}{$6$}
\psfrag{7}{$7$}
\psfrag{8}{$8$}
\psfrag{9}{$9$}
\includegraphics[width=0.2\textwidth]{figures/grid3.eps}
\caption{A $3 \times 3$ grid graph.}
\label{fig:grid9}
\end{center}
\end{figure}
As a running example for this section, we consider a
Markov chain on a $3\times 3$ grid $\mathcal{G}$, with a total of 9
nodes (see Figure~\ref{fig:grid9}). The automorphism group
$\mbox{Aut}(\mathcal{G})$ is isomorphic to the 8-element dihedral group
$D_4$, and corresponds to flips and 90-degree rotations of the graph.
The orbits of $\mbox{Aut}(\mathcal{G})$ acting on the vertices and
edges are
\[
\{1,3,7,9\}, \qquad \{5\}, \qquad \{2,4,6,8\}
\]
and
\[
\{ \{1,2\}, \{1,4\}, \{2,3\}, \{3,6\}, \{4,7\}, \{7,8\},
\{6,9\}, \{8,9\} \}, \qquad
\{ \{2,5\}, \{4,5\}, \{5,6\}, \{5,8\} \},
\]
respectively.
So $\mathcal{G}$ is neither vertex- nor edge-transitive.
By Corollary~\ref{c-numprobs}, we associate transition
probabilities~$a$ and~$b$ to the two edge orbits, respectively.
The transition probability matrix has the form
\[
P=\left[\begin{array}{ccccccccc}
1\!-\!2a & a & 0 & a & 0 & 0 & 0 & 0 & 0 \\
a & 1\!-\!2a\!-\!b & a & 0 & b & 0 & 0 & 0 & 0 \\
0 & a & 1\!-\!2a & 0 & 0 & a & 0 & 0 & 0 \\
a & 0 & 0 & 1\!-\!2a\!-\!b & b & 0 & a & 0 & 0\\
0 & b & 0 & b & 1\!-\!4b & b & 0 & b & 0 \\
0 & 0 & a & 0 & b & 1\!-\!2a\!-\!b & 0 & 0 & a \\
0 & 0 & 0 & a & 0 & 0 & 1\!-\!2 a & a & 0 \\
0 & 0 & 0 & 0 & b & 0 & a & 1\!-\!2a\!-\!b& a \\
0 & 0 & 0 & 0 & 0 & a & 0 & a & 1\!-\!2a
\end{array}
\right].
\]
The matrix $P$ satisfies $Q(\sigma)P=P Q(\sigma)$ for every
$\sigma\in\mbox{Aut}(\mathcal G)$. Using the algorithm in
\cite[\S5.2]{FaS:92}, we found a symmetry-adapted basis for the
representation $Q$, which we take as columns to form
\[
T = \frac{1}{2}
\left[\begin{array}{rrrrrrrrr}
0 & 1 & 0 & 1 & 0 & \sqrt{2} & 0 & 0 & 0 \\
0 & 0 & 1 & 0 &-1 & 0 & 1 & 0 & 1 \\
0 & 1 & 0 &-1 & 0 & 0 & 0 & \sqrt{2} & 0 \\
0 & 0 & 1 & 0 & 1 & 0 & 1 & 0 & -1\\
2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 1 & 0 & 1 & 0 &-1 & 0 & 1 \\
0 & 1 & 0 &-1 & 0 & 0 & 0 & -\sqrt{2} & 0 \\
0 & 0 & 1 & 0 &-1 & 0 &-1 & 0 & -1 \\
0 & 1 & 0 & 1 & 0 & -\sqrt{2} & 0 & 0 & 0
\end{array}
\right].
\]
With this coordinate transformation matrix, we obtain
\[
T^TPT=\left[\begin{array}{ccccccccc}
1\!-\!4 b & 0 & 2b & & & & & & \\
0 & 1\!-\!2a & 2a & & & & & & \\
2b & 2 a & 1\!-\!2a\!-\!b & & & & & & \\
& & & 1\!-\!2a & & & & & \\
& & & & 1\!-\!2a\!-\!b & & & & \\
& & & & & 1\!-\!2a & \sqrt{2} a & & \\
& & & & & \sqrt{2} a & 1\!-\!2 a \!-\!b & & \\
& & & & & & & 1\!-\!2a & \sqrt{2} a \\
& & & & & & & \sqrt{2} a & 1\!-\!2a\!-\!b
\end{array}
\right].
\]
The 3-dimensional block $B_1$ contains the single eigenvalue~$1$, and
it is related to the orbit chain in Figure~\ref{f-grid-orbit} by the
equation~(\ref{e-orbit-diag-connection}).
The corresponding nonzero block of $T^T(1/n)\mathbf 1\ones^T T$ is
\[
J_1 = \frac{1}{9} \left[ \begin{array}{ccc}
1 & 2 & 2\\ 2 & 4 & 4\\ 2 & 4 & 4 \end{array} \right].
\]
\begin{figure}
\begin{center}
\psfrag{o1}[cr]{$O_1$}
\psfrag{o2}[bc]{$O_2$}
\psfrag{o5}[cl]{$O_5$}
\psfrag{2aa}[bc]{$2a$}
\psfrag{2ab}[tc]{$2a$}
\psfrag{b}[bc]{$b$}
\psfrag{4b}[tc]{$4b$}
\includegraphics[width=0.35\textwidth]{figures/grid3_orbit.eps}
\caption{The orbit chain of the $3 \times 3$ grid graph.}
\label{f-grid-orbit}
\end{center}
\end{figure}
Next, we substitute the above expressions into the
SDP~(\ref{e-fmmc-block-diag}) and solve it numerically.
Since there are repeated $2\times 2$ blocks, the original $9 \times 9$
matrix is replaced by four smaller blocks, of dimension 3,1,1,2.
The optimal solutions are
\[
a^\star \approx 0.363, \qquad b^\star \approx 0.2111, \qquad
\mu^\star \approx 0.6926.
\]
Interestingly, it can be shown that these optimal values are not
rational, but instead algebraic numbers with defining minimal
polynomials:
\begin{eqnarray*}
18157\, a^5-17020\, a^4+6060\, a^3-1200\, a^2+180\, a-16 &=& 0 \\
1252833\, b^5-1625651\, b^4+791936\, b^3-173536\, b^2+15360\, b-256 &=& 0 \\
54471\, \mu^5-121430\, \mu^4+88474\, \mu^3-18216\, \mu^2-2393\, \mu+262 &=& 0.
\end{eqnarray*}
\subsection{Examples}
We revisit some previous examples with the block diagonalization
method, and draw connections to the method based on orbit theory
in~\S\ref{s-fmmc-orbit}.
We also discuss some additional examples that are difficult if one uses
the orbit theory, but are nicely handled by block diagonalization.
In many of the examples, the coordinate transformation matrix~$T$
can be constructed directly by inspection.
\subsubsection{Complete bipartite graphs}
\label{s-bipartite-diag}
For the complete bipartite graph $K_{m,n}$ (see
Figure~\ref{f-bipartite}),
This graph is edge-transitive, so we can assign the same transition
probability~$p$ on all the edges.
The transition probability matrix has the form
\[
P(p)=\left[
\begin{array}{cc}
(1 - n p) I_m & p \, \mathbf 1_{m\times n} \\
p \, \mathbf 1_{n\times m} & (1- m p) I_n
\end{array}
\right]
\]
We can easily find a decomposition of the associated matrix
algebra. It will have three blocks, and an orthogonal
block-diagonalizing change of basis is given by
\[
T = \left[ \begin{array}{cccc}
(1/\sqrt{m})\mathbf 1_{m\times 1} & 0 & F_m & 0 \\
0 & (1/\sqrt{n}) \mathbf 1_{n\times 1} & 0 & F_m
\end{array} \right]
\]
where $F_n$ is an $n \times (n-1)$ matrix whose columns are an
orthogonal basis of the subspace complementary to that generated by
$\mathbf 1_{n \times 1}$.
In the new coordinates, the matrix $T^T P(p) T$ has the following
diagonal blocks
\[
\left[
\begin{array}{cc}
1-m p & p \sqrt{nm} \\
p \sqrt{nm} & 1-n p
\end{array}
\right],
\qquad
I_{n-1} \otimes (1-m p), \qquad
I_{m-1} \otimes (1-n p).
\]
The $2\times 2$ block has eigenvalues~$1$ and~$1-(m+n)p$.
The other diagonals reveal the eigenvalue $1-mp$ and $1-np$, with
multiplicities $n-1$ and $m-1$, respectively.
The optimal solution to the FMMC problem can be easily obtained as
in~(\ref{e-bipartite-alpha}) and~(\ref{e-bipartite-mu}).
To draw connections to the orbit theory, we note that the above
$2\times 2$ block is precisely $B_1$ in the
equation~(\ref{e-orbit-diag-connection}), and the corresponding
$P_H$ is the orbit chain shown in Figure~\ref{f-bipartite-orbit:a}.
In addition to the two eigenvalues in~$B_1$, the extra eigenvalue in
the orbit chain of Figure~\ref{f-bipartite-orbit:b} is $1-np$, and the
extra eigenvalue in Figure~\ref{f-bipartite-orbit:c} is $1-mp$.
All these eigenvalues appear in the orbit chain in
Figure~\ref{f-bipartite-orbit:d}.
As we have seen, the block diagonalization technique reveals the
multiplicities in the original chain of the eigenvalues from various
orbit chains.
\subsubsection{Complete $k$-partite graphs}
The previous example generalizes nicely to the complete $k$-partite
graph $K_{n_1,\ldots,n_k}$.
In this case, the fixed-point reduced matrix will have
dimensions $\sum _i n_i$, and the structure
\[
P(p)=\left[
\begin{array}{cccc}
(1 - \sum_{j \not = 1} n_j p_{1j} ) I_{n_1} &
p_{12} \mathbf 1_{n_1 \times n_2} & \cdots &
p_{1k} \mathbf 1_{n_1 \times n_k}\\
p_{21} \mathbf 1_{n_2 \times n_1} &
(1 - \sum_{j \not = 2} n_j p_{2j} ) I_{n_2} &
\cdots &
p_{2k} \mathbf 1_{n_2 \times n_k}\\
\vdots & \vdots & \ddots & \vdots \\
p_{k1} \mathbf 1_{n_k \times n_1} &
p_{k2} \mathbf 1_{n_k \times n_2} &
\cdots &
(1 - \sum_{j \not = k} n_j p_{kj} ) I_{n_k}
\end{array}
\right]
\]
where the probabilities satisfy $p_{ij}=p_{ji}$.
There are total ${k \choose 2}$ independent variables.
In a very similar fashion to the bipartite case, we can explicitly
write the orthogonal coordinate transformation matrix
\[
T = \left[ \begin{array}{cccccc}
(1/\sqrt{n_1}) \mathbf 1_{n_1 \times 1} & \ldots & \mathbf{0} &
F_{n_1} & \ldots & \mathbf{0} \\
\vdots & \ddots & \vdots & \vdots & \ddots & \vdots \\
\mathbf{0} & \ldots & (1/\sqrt{n_{k}}) \mathbf 1_{n_{k} \times 1} &
\mathbf{0} & \ldots & F_{n_k}
\end{array} \right].
\]
The matrix $T^T P(p) T$ decomposes into $k+1$ blocks: one of
dimension $k$, with the remaining $k$ blocks each having dimension
$n_i-1$. The decomposition is:
\begin{eqnarray*}
&& \left[ \begin{array}{cccc}
(1 - \sum_{j \not = 1} n_j p_{1j} ) &
p_{12} \sqrt{n_1 n_2}& \cdots &
p_{1k} \sqrt{n_1 n_k} \\
p_{21} \sqrt{n_2 n_1}&
(1 - \sum_{j \not = 2} n_j p_{2j} ) &
\cdots &
p_{2k} \sqrt{n_2 n_k} \\
\vdots & \vdots & \ddots & \vdots \\
p_{k1} \sqrt{n_k n_1} &
p_{k2} \sqrt{n_k n_2} &
\cdots &
(1 - \sum_{j \not = k} n_j p_{kj} )
\end{array} \right], \\[2ex]
&& I_{n_i-1} \otimes (1 -\sum_{j \not = i} n_j p_{i j} ),
\qquad i = 1,\ldots,k.
\end{eqnarray*}
These blocks can be substituted into the SDP~(\ref{e-fmmc-block-diag})
to solve the FMMC problem.
\subsubsection{Wheel graph}
\begin{figure}
\begin{center}
\includegraphics[width=0.22\textwidth]{figures/wheel2.eps}
\caption{The wheel graph with $n=9$ (total 10 nodes).}
\label{f-wheel}
\end{center}
\end{figure}
The wheel graph consists of a center vertex (the \emph{hub}) and a
ring of $n$ peripheral vertices, each connected to the hub; see
Figure~\ref{f-wheel}. It has total $n+1$ nodes. Its automorphism
group is isomorphic to the dihedral group $D_n$ with order $2n$. The
transition probability matrix has the structure
\begin{equation}
P=\left[\begin{array}{cccccc}
1-n p & p & p & \ldots & p & p \\
p & 1-p-2q & q & \ldots & 0 & q \\
p & q & 1-p-2q & \ldots & 0 & 0 \\
\vdots & \vdots & \vdots & \ddots & \vdots & \vdots \\
p & 0 & 0 & \ldots & 1-p-2q & q \\
p & q & 0 & \ldots & q & 1 -p-2q
\end{array}\right],
\end{equation}
where $p$ and $q$ are the transition probabilities between the hub and
each peripheral vertex, and between adjacent peripheral vertices,
respectively.
For this structure, the block-diagonalizing transformation is given by
\[
T = \mathop{\bf Diag}(1,\mathcal{F}_n), \qquad
[\mathcal{F}_n]_{jk} = \frac{1}{\sqrt{n}}
e^\frac{2 \pi \imath (j-1)(k-1)}{n}
\]
where $\mathcal{F}_n$ is the unitary Fourier matrix of size
$n\times n$.
As a consequence, the matrix $T^{-1} P T$ is block diagonal with
a $2\times 2$ matrix and $n-1$ scalars on its diagonal, given by
\[
\left[\begin{array}{cc}
1-np & \sqrt{n} p \\
\sqrt{n} p & 1 - p
\end{array}\right] \\
\]
and
\[
1 - p + (\omega_n^k + \omega_n^{-k} - 2 ) \cdot q,
\qquad k=1,\ldots,n-1
\]
where $\omega_n= e^\frac{2 \pi \imath}{n}$ is an elementary $n$-th
root of unity.
The $2\times 2$ block is $B_1$, which contains eigenvalues of the
orbit chain under $D_n$ (it has only two orbits).
With the above decomposition, we obtain the optimal solution to the
FMMC problem in closed form
\[
p^\star = \frac{1}{n}, \qquad
q_\star = \frac{1-\frac{1}{n}}{2-\cos\frac{2\pi}{n}
-\cos\frac{2\lfloor n/2\rfloor\pi}{n}}.
\]
The optimal value of the SLEM is
\[
\mu^{\star} = \left(1-\frac{1}{n}\right)
\frac{\cos\frac{2\pi}{n}-\cos\frac{2\lfloor n/2\rfloor\pi}{n}}
{2-\cos\frac{2\pi}{n}-\cos\frac{2\lfloor n/2\rfloor\pi}{n}}.
\]
Compared with the optimal solution for the cycle graph
in~(\ref{e-cycle-prob}) and~(\ref{e-cycle-slem}), we see an extra
factor of $1-1/n$ in both the SLEM and the transition probability
between peripheral vertices.
This is exactly the factor improved by adding the central hub
over the pure $n$-cycle case.
The wheel graph is an example for which the block diagonalization
technique works out nicely, while the orbit theory leads to much less
reduction.
Although there are only two orbits under the full automorphism group,
any orbit graph that has a fixed peripheral vertex
will have at least $(n+1)/2$ orbits (the corresponding symmetry is
the reflection through that vertex).
\subsubsection{$K_n$-$K_n$}
\label{s-Kn-Kn-2}
We did careful symmetry analysis for the graph $K_n$-$K_n$
in~\S\ref{s-Kn-Kn}; see Figure~\ref{f-KnKn}.
The transition probability matrix on this graph has the structure
\[
P = \left[ \begin{array}{cccc}
C & p_1 \mathbf 1 & 0 & 0 \\
p_1 \mathbf 1^T & 1-p_0-(n-1)p_1 & p_0 & 0 \\
0 & p_0 & 1-p_0-(n-1)p_1 & p_1 \mathbf 1^T \\
0 & 0 & p_1 \mathbf 1 & C \end{array} \right]
\]
where $C$ is a circulant matrix
\[
C = (1-p_1-(n-3)p_2)I_{n-1} + p_2 \mathbf 1_{(n-1)\times(n-1)}.
\]
Since circulant matrices are diagonalized by Fourier matrices, we
first use the transformation matrix
\[
T_1 = \left[ \begin{array}{cccc}
\mathcal F_{n-1} & 0 & 0 & 0\\
0 & 1 & 0 & 0 \\
0 & 0 & 1 & 0 \\
0 & 0 & 0 & \mathcal F_{n-1} \end{array} \right]
\]
where $\mathcal F_{n-1}$ is the unitary Fourier matrix of
dimension $n-1$.
This corresponds to block diagonalization using the symmetry group
$S_{n-1}\times S_{n-1}$, which is a subgroup of
$\mbox{Aut}(\mbox{$K_n$-$K_n$})$.
The matrix $T_1^{-1} P T_1$ has diagonal blocks
\[
B_1'=\left[ \begin{array}{cccc}
1-p_1 & \sqrt{n-1}p_1 & 0 & 0\\
\sqrt{n-1}p_1 & 1-p_0-(n-1)p_1 & p_0 & 0\\
0 & p_0 & 1-p_0-(n-1)p_1 & \sqrt{n-1}p_1\\
0 & 0 & \sqrt{n-1}p_1 & 1-p_1 \end{array} \right]
\]
and
\begin{equation}\label{e-KnKn-2n-4}
I_{2n-4} \otimes (1-p_1-(n-1)p_2).
\end{equation}
From this we know that $P$ has an eigenvalue $1-p_1-(n-1)p_2$ with
multiplicity $2n-4$, and the remaining four eigenvalues are the
eigenvalues of the above $4\times 4$ block $B_1'$. The block $B_1'$
corresponds to the orbit chain under the symmetry group
$H=S_{n-1}\times S_{n-1}$. More precisely, $B_1'=\Pi^{1/2} P_H
\Pi^{-1/2}$, where $\Pi=\mathop{\bf Diag}(\pi_H)$, $P_H$ and $\pi_H$ are the
transition probability matrix and stationary distribution of the orbit
chain shown in Figure~\ref{f-KnKn:c}, respectively.
Exploring the full automorphism group of $K_n$-$K_n$, we can further
block diagonalize $B_1'$.
Let
\[
T = T_1 \left[ \begin{array}{ccc}
I_{n-2} & & \\ & T_2 & \\ & & I_{n-2} \end{array} \right], \qquad
T_2 = \frac{1}{\sqrt{2}} \left[ \begin{array}{ccrr}
1 & 0 & 0 & 1\\ 0 & 1 & 1 & 0 \\ 0 & 1 & -1 & 0\\ 1 & 0 & 0 & -1
\end{array} \right].
\]
The $4\times 4$ block $B_1'$ is decomposed into
\[
\left[ \begin{array}{cc}
1-p_1 & \sqrt{n-1}p_1 \\
\sqrt{n-1}p_1 & 1-(n-1)p_1
\end{array} \right],
\qquad
\left[ \begin{array}{cc}
1-2p_0-(n-1)p_1 & \sqrt{n-1}p_1\\
\sqrt{n-1}p_1 & 1-p_1 \end{array} \right]
\]
The first block is $B_1$, which has eigenvalues~$1$ and $1-np_1$.
By Theorem~\ref{t-orbit-diag-connection}, $B_1$ is related to the
orbit chain under $\mbox{Aut}(\mbox{$K_n$-$K_n$})$
(see Figure~\ref{f-KnKn:b}) by the
equation~(\ref{e-orbit-diag-connection}).
The second $2\times 2$ block has eigenvalues
\[
1-p_0-(1/2)np_1\pm\sqrt{(p_0+(1/2)np_1)^2-2p_0p_1}.
\]
These are the eigenvalues contained in the orbit chain of
Figure~\ref{f-KnKn:c} but not in Figure~\ref{f-KnKn:b}.
In summary, the distinct eigenvalues of the Markov chain on
$K_n$-$K_n$ are
\[
1,\quad 1-np_1, \quad
1-p_0-(1/2)np_1\pm\sqrt{(p_0+(1/2)np_1)^2-2p_0p_1},
\quad 1-p_1-(n-1)p_2
\]
where the last one has multiplicity $2n-4$, and all the rest have
multiplicity~1. To solve the FMMC problem, we still need to solve the
SDP~(\ref{e-fmmc-block-diag}). There are three blocks of matrix
inequality constraints, with sizes $2$, $2$, $1$, respectively. Note
that the total size is $5$, which is exactly the size of the single
matrix inequality in the SDP~(\ref{e-fmmc-rev}) when we used the orbit
theory to do symmetry reduction. As we mentioned before, the huge
reduction for $K_n$-$K_n$ is due to the fact that it has an
irreducible representation with high dimension $2n-4$ and
multiplicity~$1$ (see \cite[Proposition 2.4]{BDPX:05}). In the
decomposition~(\ref{e-block-diag}), this means a block of size~$1$
repeated $2n-4$ times; see equation~(\ref{e-KnKn-2n-4}).
Since now the problem has been reduced to something much more
tractable, we can even obtain an analytic expression for the optimal
transition probabilities. The optimal solution for the $K_n$-$K_n$
graph (for $n \geq 2$) is given by:
\[
p_0^\star = (\sqrt{2}-1) \frac{n+\sqrt{2}-2}{n+2-2\sqrt{2}}, \qquad
p_1^\star = \frac{2 - \sqrt{2}}{n+2-2\sqrt{2}}, \qquad
p_2^\star = \frac{n-\sqrt{2}}{(n-1)(n+2-2\sqrt{2})}.
\]
The corresponding optimal convergence rate is
\[
\mu^\star = \frac{n-4+2\sqrt{2}}{n+2-2\sqrt{2}}.
\]
For large $n$, we have
$\mu^\star=1-\frac{6 - 4 \sqrt{2}}{n} + O\left(\frac{1}{n^2}\right)$.
This is quite close to the SLEM of a suboptimal construction with
transition probabilities
\begin{equation}\label{e-sub-optimal}
p_0 = \frac{1}{2}, \qquad p_1=p_2 = \frac{1}{2(n-1)}.
\end{equation}
As shown in \cite{BDPX:05}, the corresponding SLEM is of the order
$\mu = 1 - \frac{1}{3n} + O\left(\frac{1}{n^2}\right)$;
here we have $6 - 4 \sqrt{2} \approx 0.3431$.
The limiting value of the optimal
transition probability between the two clusters is $\sqrt{2}-1 \approx
0.4142$.
\subsubsection{Complete binary trees}
Since the automorphism groups of the complete binary trees $\mathcal
T_n$ are given recursively (see~\S\ref{s-tree-orbit}), it is also
convenient to write the transition probability matrices in a recursive
form.
We start from the bottom by considering the last level of branches.
If we cut-off the rest of the tree, the last level has three nodes and
two edges with the transition probability matrix
\begin{equation}
P_n =
\left[\begin{array}{ccc}
1-2 p_n & p_n & p_n \\
p_n & 1-p_n & 0 \\
p_n & 0 & 1-p_n
\end{array}\right].
\end{equation}
For the tree with $n$ levels $\mathcal T_n$, the transition matrix $P_1$
can be computed from the recursion
\begin{equation}
P_{k-1} =
\left[\begin{array}{ccc}
1-2 p_{k-1} & p_{k-1}e_k^T & p_{k-1}e_k^T\\
p_{k-1}e_k & P_k - p_{k-1} e_k e_k^T & 0\\
p_{k-1}e_k & 0 & P_k - p_{k-1} e_k e_k^T
\end{array}\right], \qquad k=n,n-1\ldots,2
\end{equation}
where $e_k=[1~0\ldots~0]$, a unit vector in ${\mbox{\bf R}}^{t_k}$ with
$t_k = 2^{k+1}-1$.
The coordinate transformations are also best written in recursive
form.
Let
\[
T_n \mathop{\bf Diag}(1,\mathcal{F}_2), \qquad
\mathcal{F}_2 =
\frac{1}{\sqrt{2}}
\left[\begin{array}{cc}
1 & 1 \\
1 & -1
\end{array}
\right],
\]
and define the matrices
\[
T_{k-1} = \mathop{\bf Diag}(1,\mathcal{F}_2 \otimes T_k),
\qquad k=n,n-1,\ldots,2.
\]
It is clear that all the $T_k$ are orthogonal.
It is easy to verify that $T_n$ block-diagonalizes $P_n$
\[
T_n^T P_n T_n = \left[ \begin{array}{ccc}
1-2p_n & \sqrt{2}p_n & 0 \\
\sqrt{2}p_n & 1-p_n & 0\\
0 & 0 & 1-p_n \end{array} \right] .
\]
In fact $T_k$ block-diagonalizes $P_k$, and the transformed matrices
can be obtained recursively
\[
T_{k-1}^T P_{k-1} T_{k-1} = \left[ \begin{array}{ccc}
1-2p_{k-1} & \sqrt{2}p_{k-1} e_k^T & 0\\ \sqrt{2}p_{k-1} e_k &
T_k^T P_k T_k - p_{k-1} e_k e_k^T & 0\\
0 & 0 & T_k^T P_k T_k - p_{k-1} e_k e_k^T
\end{array} \right]
\]
for $k=n,n-1,\ldots,2$.
The matrix $T_1^T P_1 T_1$ has a very special structure.
It has $n+1$ distinct blocks, each with size $1,\ldots,n+1$,
respectively.
Order these blocks with increasing sizes as $B_1,B_2,\ldots,B_{n+1}$.
The largest block of size $n+1$ is
\[
B_{n+1} = \left[ \begin{array}{cccccc}
1\!-\!2p_1 & \sqrt{2}p_1 & & & & \\
\sqrt{2}p_1 & 1\!-\!p_1\!-\!2p_2 & \sqrt{2}p_2 & & & \\
& \sqrt{2}p_2 & 1\!-\!p_2\!-\!2p_3 & \sqrt{2} p_3 & & \\[2ex]
& & \ddots & \ddots & \ddots & \\[2ex]
& & & \sqrt{2}p_{n\!-\!1} & 1\!-\!p_{n\!-\!1}\!-\!2p_n & \sqrt{2}p_n \\
& & & & \sqrt{2}p_n & 1\!-\!p_n
\end{array} \right].
\]
The matrix $B_n$ is the submatrix of~$B_{n+1}$ by removing its first
row and column.
The matrix $B_{n-1}$ is the submatrix of~$B_{n+1}$ by removing its
first two rows and first two columns, and so on.
The matrix $B_1$ is just the scalar $1-p_n$.
The matrix $B_{n+1}$ only appears once and it is related
by~(\ref{e-orbit-diag-connection}) to the orbit chain in
Figure~\ref{f-binary-tree:a} (for this example we use
$B_{n+1}$ instead of $B_1$ for notational convenience).
The eigenvalues of $B_{n+1}$ appear in $\mathcal T_n$ with
multiplicity one.
For $k=1,\ldots,n$, the block $B_k$ is repeated $2^{n-k}$
times.
These blocks, in a recursive form, contain additional eigenvalues of
$\mathcal T_n$, and the numbers of their occurrences reveal the
multiplicities of the eigenvalues.
More specifically, we note that the orbit chain under the full
automorphism group has only one fixed point --- the root vertex
(see Figure~\ref{f-binary-tree:a}).
We consider next the orbit chain that has a fixed point in the first
level of child vertices (the other child vertex in the same level
is also fixed).
This is the orbit graph in Figure~\ref{f-binary-tree:b}, which has
$2n+1$ vertices.
The matrix $B_n$ contains exactly the~$n$
eigenvalues that appear in this orbit chain but not in the one of
Figure~\ref{f-binary-tree:a}.
These~$n$ eigenvalues each has multiplicity $2^{n-n}=1$
in $\mathcal T_n$.
Then we consider the orbit chain that has a fixed point in the second
level of child vertices (it also must have a fixed point in the
previous level).
This is the orbit graph in Figure~\ref{f-binary-tree:c}, which has
$3n$ vertices.
The matrix $B_{n-1}$ contains exactly the $n-1$
eigenvalues that appear in this orbit chain but not in the previous
one.
These $n-1$ eigenvalues each has multiplicity $2^{n-(n-1)}=2$.
In general, for $k=1,\ldots,n$,
the size of the orbit chain that has a fixed point in the
$k$-th level of child vertices is
\[
(n+1) + n + \cdots + (n+1-k)
\]
(it must have a fixed point in all previous levels).
Compared with the orbit chain of $(k-1)$-th level,
the orbit chain of $k$-th level contains additional $n+1-k$
eigenvalues.
These are precisely the eigenvalues of the matrix $B_{n+1-k}$,
and they all appear in $\mathcal T_n$ with multiplicity
$2^{n-(n+1-k)}=2^{k-1}$.
Because of the special structure of $B_1,\ldots,B_{n+1}$, we have the
following eigenvalue interlacing result
({\it e.g.}, \cite[Theorem 4.3.8]{HoJ:85})
\[
\lambda_{k+1}(B_{k+1}) \leq \lambda_k(B_k) \leq \lambda_k(B_{k+1})
\leq \lambda_{k-1}(B_k) \leq \cdots \leq \lambda_2(B_k) \leq
\lambda_2(B_{k+1}) \leq \lambda_1(B_k) \leq \lambda_1(B_{k+1})
\]
for $k=1,\ldots,n$.
Thus for the FMMC problem, we only need to consider the
two blocks $B_{n+1}$ and $B_n$ (note that $\lambda_1(B_{n+1})=1$).
In other words, we only need to consider the orbit chain with $2n+1$
vertices in Figure~\ref{f-binary-tree:b}.
This is a further simplification over the method based on orbit theory.
We conjecture that the optimal transition probabilities are
\[
p^\star_k = \frac{1}{3} \left( 1 - \left(- \frac{1}{2} \right)^k \right),
\qquad k = 1,\ldots,n.
\]
Notice that these probabilities \emph{do not} depend explicitly
on $n$, and so they coincide for any two binary trees, regardless of
the height. With increasing $k$, the limiting optimal values
oscillate around and converge to $1/3$.
\subsubsection{An example of Ron Graham}
\label{s-graham-example}
\begin{figure}
\begin{center}
\includegraphics[width=0.8\textwidth]{figures/graham_graph.eps}
\caption{Left: the simplest graph with no symmetry.
Right: two copies joined head-to-tail.}
\label{f-graham-graph}
\end{center}
\end{figure}
\begin{figure}
\begin{center}
\psfrag{p1}[bc]{$p_1$}
\psfrag{p2}[bc]{$p_2$}
\psfrag{p3}[br]{$p_3$}
\psfrag{p3r}[bl]{$p_3$}
\psfrag{p4}[bl]{$p_4$}
\psfrag{2p3}[tr]{$2p_3$}
\psfrag{2p4}[tc]{$2p_4$}
\includegraphics[width=0.8\textwidth]{figures/graham_orbit.eps}
\caption{Left: orbit graph with $C_n$ symmetry.
Right: orbit graph with $D_n$ symmetry.}
\label{f-graham-orbits}
\end{center}
\end{figure}
We finish this section with an example raised by Ron Graham.
Consider the simplest graph with no symmetry
(Figure~\ref{f-graham-graph}, left).
Take~$n$ copies of this six vertex graph and join them, head to tail,
in a cycle.
By construction, this $6n$ vertex graph certainly has $C_n$ symmetry.
Careful examination reveals that the automorphism group is isomorphic to the dihedral group $D_n$ (with order $2n$).
The construction actually brings symmetry under reflections in addition to rotations (Figure~\ref{f-graham-graph}, right).
The orbit graphs under $C_n$ and $D_n$ are shown in Figure~\ref{f-graham-orbits}.
Although the automorphism group of this graph (with $6n$ vertices) is isomorphic to the ones of $n$-cycles (Figure~\ref{f-cycle}) and wheels (Figure~\ref{f-wheel}), finding the symmetry-adapted basis for block-diagonalization is a bit more involved.
This is due to the different types of orbits we have for this graph.
The details of block-diagonalizing this type of graphs is described in \cite[\S3.1]{FaS:92}.
The diagonal blocks of the resulting matrix all have sizes no larger than $6\times 6$.
Numerical experiments show that for $n\geq 3$, the fastest mixing chain seems to satisfy
\[
p^\star_1=p^\star_4=\frac{1}{2}, \qquad p^\star_2+p^\star_3=\frac{1}{2}.
\]
Intuitively, this $6n$ vertex graph is the same as modifying a $5n$ vertex cycle by adding a triangular bump (with an additional vertex) for every $5$ vertices.
Recall that for a pure cycle, we have to use a transition probability that is slightly less than $1/2$ to achieve fastest mixing; see equation~(\ref{e-cycle-prob}).
Here because of the added bumps, it seems optimal to assign transition probability $1/2$ to every edge on the cycle ($p^\star_1$ and $p^\star_4$), except for edges being part of a bump.
For the bumps, the probability $1/2$ is shared between the original edge on the cycle ($p^\star_2$) and the edge connecting to the bump points ($p^\star_3$).
Moreover, we observe that as $n$ increases, $p^\star_3$ gets smaller and $p^\star_2$ gets closer to $1/2$.
So for large $n$, the added bump vertices seem to be ignored, with very small probability to be reached; but once it is reached, it will staying there with high probability.
\iffalse
For this example, the orbit theory doesn't work (see Remark~3.10 in
\cite{BDPX:05}): if we want an orbit graph that fixes any vertex,
we must fix the whole large graph with $6n$ vertices.
However, the block diagonalization approach can still exploit the
$C_n$ symmetry.
There are~$7$ orbits of edges under $C_n$, each labeled with
a transition probability $p_1,\ldots,p_7$ as shown in
Figure~\ref{f-graham-graph}, right.
Without writing explicitly the transition probability matrix $P$
for the large graph,
we directly give the coordinate transformation matrix
\[
T = \mathcal F_n \otimes I_6.
\]
The matrix $T^* P T$ has $n$ diagonal blocks,
each with size $6\times 6$.
For $k=1,\ldots,n$, these blocks are
\[
\left[
\begin{array}{cccccc}
1-p_1-p_7 & p_1 & 0 & 0 & 0 & \omega_n^{-k}\;p_7\\
p1 & 1-p_1-p_2-p_3 & p_2 & p_3 & 0 & 0\\
0 & p_2 & 1-p_2-p_4-p_5 & p_4 & p_5 & 0\\
0 & p_3 & p_4 & 1-p_3-p_4 & 0 & 0\\
0 & 0 & p_5 & 0 & 1-p_5-p_6 & p_6\\
\omega_n^k\;p_7 & 0 & 0 & 0 & p_6 & 1-p_6-p_7
\end{array}
\right]
\]
where $\omega_n=e^{2\pi i/n}$ is the $n$-th root of unity.
The only differences among these blocks are the two entries at the
bottom-left and top-right corners.
With the above block diagonalization, in solving the
SDP~(\ref{e-fmmc-block-diag}), we need to work with $n$
matrix inequalities each with size $6\times 6$.
For interior-point methods, this makes the number of flops per
Newton step depend linearly on~$n$ (it would be $n^3$ if we do not
exploit any structure).
However, we note that this reduction can also be achieved by exploiting
sparsity alone, given the structure of the $P$ matrix.
\fi
\section{Conclusions}
\label{s-conclusions}
We have shown that exploiting graph symmetry can lead to significant
reduction in both the number of variables and the size of matrices, in
solving the FMMC problem. For special classes of graphs such as
edge-transitive and distance-transitive graphs, symmetry reduction
leads to closed form solutions in terms of the eigenvalues of the
Laplacian matrix or the intersection matrix. For more general graphs,
we gave two symmetry reduction methods, based on orbit theory and
block diagonalization, respectively.
The method based on orbit theory is very intuitive, but the
construction of ``good'' orbit chains can be of more art than
technique. The method of block diagonalization can be mostly
automated once the irreducible representations of the automorphism
groups are generated (for small graphs, they can be generated using
software for computational discrete algebra such as GAP \cite{GAP}).
These two approaches have an interesting connection: orbit theory
gives nice interpretation of the diagonal blocks, while the block
diagonalization approach offers theoretical insights about the
construction of the orbit chains.
The symmetry reduction method developed in this paper can be very
useful in many combinatorial optimization problems where the graph has
rich symmetry properties, in particular, problems that can be
formulated as or approximated by SDP or eigenvalue optimization
problems involving weighted Laplacian matrices ({\it e.g.},
\cite{MoP:93,Goe:97}). In addition to the reduction of problem size,
other advantages of symmetry exploitation includes degeneracy removal,
better conditioning and reliability \cite{GatermannParrilo}.
There is still much to do in understanding how to exploit symmetry in
semidefinite programming. The techniques presented in this paper
requires a good understanding of orbit theory, group representation
theory and interior-point methods for SDP. It is of practical
importance to develop general purpose methods that can automatically
detect symmetries ({\it e.g.}, the code \texttt{nauty} \cite{nauty} for
graph automorphisms), and then exploit them in computations. A good
model here is general purpose (but heuristic) methods for exploiting
sparsity in numerical linear algebra, where symbolic operations on
graphs ({\it e.g.}, minimum degree permutation) reduce fill-ins in numerical
factorization ({\it e.g.}, \cite{GeL:81}). As a result of this work, even
very large sparse optimization problems are now routinely solved by
users who are not experts in sparse matrix methods. For exploiting
symmetry in SDP, the challenges include the development of fast
methods to detect large symmetry groups (for computational purposes,
it often suffices to recognize parts of the symmetries), and the
integration of algebraic methods ({\it e.g.}, orbit theory and group
representations) and numerical algorithms ({\it e.g.}, interior-point
methods).
\bibliographystyle{alpha}
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 4,871
|
\section{Introduction}
The task of Weakly-supervised Temporal Action Localization (W-TAL) aims at simultaneously localizing and classifying all action instances in a long untrimmed video given only video-level categorical labels in the learning phase. Compared to its fully-supervised counterpart, which requires frame-level annotations of all action instances during training, W-TAL greatly simplifies the procedure of data collection and avoids annotation bias of human annotators, therefore has been widely studied \cite{kumar2017hide,wang2017untrimmednets,shou2018autoloc,nguyen2018weakly,paul2018w,alwassel2019refineloc,liu2019completeness,zhai2019action,liu2019weakly,nguyen2019weakly,narayan20193c,yu2019temporal,lee2020background} in recent years.
Several W-TAL methods \cite{wang2017untrimmednets,paul2018w,nguyen2018weakly,liu2019completeness,nguyen2019weakly,narayan20193c,lee2020background} adopt a Multiple Instance Learning (MIL) framework, where a video is treated as a bag of frames/snippets to perform the video-level action classification. During testing, the trained model slides over time and generates a Temporal-Class Activation Map (T-CAM) \cite{zhou2016learning,nguyen2018weakly} (\textit{i.e.}, a sequence of probability distributions over action classes at each time step) and an attention sequence that measures the relative importance of each snippet.
The action proposals are generated by thresholding the attention value and/or the T-CAM.
This MIL framework is usually built on two feature modalities, \textit{i.e.,} RGB frames and optical flow, which are fused in two possible ways. \textit{Early fusion} methods \cite{paul2018w,shou2018autoloc,alwassel2019refineloc,liu2019completeness,liu2019weakly,lee2020background} concatenate the RGB and optical flow features before they are fed into the network, and \textit{late fusion} methods \cite{nguyen2018weakly,liu2019completeness,nguyen2019weakly,narayan20193c} compute a weighted sum of their respective outputs before generating action proposals. An example of late fusion is shown in Fig.~\ref{fig:twoStream}.
\begin{figure}[t]
\centering
\includegraphics[width=\linewidth]{images/illustration.pdf}
\caption{Visualization of two-stream outputs and their late fusion result. The first two rows are an input video and the ground truth action instances, respectively. The last three rows are attention sequences (scaled from $0$ to $1$) predicted by the RGB stream, the flow stream and their weighted sum (\textit{i.e.}, the fusion result), respectively, and the horizontal and vertical axes denote the time and the intensity of attention values, respectively.
The green boxes denote the localization results generated by thresholding the attention at the value of $0.5$. By properly combining the two different attention distributions predicted by the RGB and flow streams, the late fusion result achieves a higher true positive rate and a lower false positive rate, and thus has better localization performance}
\label{fig:twoStream}
\end{figure}
Despite these recent development, two major challenges still persist.
One of the most critical problems that prior W-TAL methods suffer from is the lack of ability to rule out false positive action proposals.
Without frame-level annotations, they localize action instances that do not necessarily correspond to the video-level labels.
For example, a model may falsely localize the action ``swimming" by only checking the existence of water in the scene. Therefore, it is necessary to exploit more fine-grained supervision to guide the learning process.
Another problem lies in the generation of action proposals.
In previous methods, action proposals are generated by thresholding the activation sequence with a fixed threshold, which is preset empirically.
It has a significant impact on the quality of action proposals: a high threshold may result in incomplete action proposals while a low threshold can bring more false positives. But how to get out of this dilemma was rarely studied.
In this paper, we introduce a Two-Stream Consensus Network (TSCN) to address the two aforementioned problems.
To eliminate false positive action proposals, we design an iterative refinement training scheme, where a frame-level pseudo ground truth is generated from late fusion attention sequence, and serves as a more precise frame-level supervision to iteratively update two-stream models.
Our intuition is simple: late fusion is essentially a voting ensemble of the RGB and flow streams, and if a proper fusion parameter (\textit{i.e.}, the hyperparameter to control the relative importance of two streams) is selected, late fusion can provide more accurate result compared with each individual stream.
The advantage of combining these two streams has been demonstrated by the Two-Stream Convolutional Networks \cite{simonyan2014two} for action recognition.
As shown in Fig.~\ref{fig:twoStream}, the two streams produce different activation distributions, which lead to different false positives and false negatives.
However, when they are combined, the false positive action proposals that only exist in one stream can be largely eliminated, and a high activation value occurs only when both streams are confident that an action instance exists.
Since the late fusion result is of higher quality than single stream result, it can in turn serve as a frame-level pseudo ground truth to supervise and refine both streams.
To generate high-quality action proposals, we introduce a new attention normalization loss. It pushes the predicted attention to approach extreme values, \emph{i.e.}, 0 and 1, so as to avoid ambiguity. As a result, simply setting the threshold to 0.5 yields high-quality action proposals.
Formally, given an input video, RGB and optical flow features are first extracted from pre-trained deep networks.
Then two-stream base models are trained with video-level labels on RGB and optical flow features, respectively, where the attention normalization loss is used to learn the attention distribution.
After obtaining two-stream attention sequences, a frame-level pseudo ground truth is generated based on their weighted sum (\textit{i.e.}, the late fusion attention sequence), and in turn provides frame-level supervision to improve the two-stream models.
We iteratively update the pseudo ground truth and refine the two-stream base models, and the normalization term at the same time forces the predicted attention to approach a binary selection.
The final localization result is obtained by thresholding the late fusion attention sequence.
To summarize, our contribution is threefold:
\begin{itemize}
\item We introduce a Two-Stream Consensus Network (TSCN) for W-TAL. The proposed TSCN uses an iterative refinement training method, where a pseudo ground truth generated from late fusion attention sequence at previous iteration can provide more precise frame-level supervision to current iteration.
\item We propose an attention normalization loss function, which forces the attention to act like a binary selection, and thus improves the quality of action proposals generated by the thresholding method.
\item Extensive experiments are conducted on two standard benchmarks (\textit{i.e.}, THUMOS14 and ActivityNet) to demonstrate the effectiveness of the proposed method. Our TSCN significantly outperforms previous state-of-the-art W-TAL methods, and even achieves comparable results to some recent fully-supervised TAL methods.
\end{itemize}
\section{Related Work}
\textbf{Action Recognition.} Traditional methods \cite{laptev2005space,dalal2005histograms,dalal2006human,wang2011action} aim to model spatio-temporal information via hand-crafted features.
Two-Stream Convolutional Networks \cite{simonyan2014two} use two separate Convolutional Neural Networks (CNNs) to exploit appearance and motion clues from RGB frames and optical flow, respectively, and use a late fusion method to reconcile the two-stream outputs. \cite{feichtenhofer2016convolutional} focuses on studying different ways to fuse the two streams.
The Inflated 3D ConvNet (I3D) \cite{carreira2017quo} expands the 2D CNNs in two-stream networks to 3D CNNs.
Several recent methods \cite{zhao2018recognize,crasto2019mars,shou2019dmc,wang2019hallucinating,repflow2019} focus on directly learning motion clues from RGB frames instead of calculating optical flow.
\noindent\textbf{Fully-supervised Temporal Action Localization.} Fully-supervised TAL requires frame-level annotations of all action instances during training. Several large-scale datasets have been created for this task, such as THUMOS \cite{jiang2014thumos,gorban2015thumos}, ActivityNet \cite{caba2015activitynet}, and Charades \cite{sigurdsson2016hollywood}.
Many methods \cite{shou2017cdc,zhao2017temporal,gao2017cascaded,heilbron2017scc,dai2017temporal,xu2017r,lin2018bsn,chao2018rethinking} adopt a two-stage pipeline, \emph{i.e.}, action proposal generation followed by action classification.
Several methods \cite{xu2017r,dai2017temporal,gao2017turn,chao2018rethinking} adopt the Faster R-CNN \cite{ren2015faster} framework to TAL.
Most recently, some methods \cite{lin2018bsn,long2019gaussian,lin2019bmn} try to generate action proposals with more flexible durations.
Zeng \textit{et al.} \cite{zeng2019graph} apply the Graph Convolutional Networks (GCN) \cite{kipf2016semi,tan2015learning} to TAL to exploit proposal-proposal relations.
\noindent\textbf{Weakly-supervised Temporal Action Localization.} W-TAL, which only requires video-level supervision during training, greatly relieves the data annotation efforts, and draws more and more attention from the community recently. Hide-and-Seek \cite{kumar2017hide} randomly hides part of the input video to guide the network to discover other relevant parts. UntrimmedNet \cite{wang2017untrimmednets} consists of a selection module to select the important snippets and a classification module to perform per snippet classification. Sparse Temporal Pooling Network (STPN) \cite{nguyen2018weakly} improves UntrimmedNet by adding a sparse loss to enforce the sparsity of selected segments. W-TALC \cite{paul2018w} jointly optimizes a co-activity similarity loss and a multiple instance learning loss to train the network. AutoLoc \cite{shou2018autoloc} is one of the first two-stage methods in W-TAL, and it first generates initial action proposals and then regresses the boundaries of the action proposals with an Outer-Inner-Contrastive loss. CleanNet \cite{liu2019weakly} improves AutoLoc by leveraging the temporal contrast in snippet-level action classification predictions.
Liu \textit{et al}. \cite{liu2019completeness} propose a multi-branch network to model different stages of action.
Besides, several methods~\cite{nguyen2019weakly,lee2020background} focus on modeling the background and achieve state-of-the-art performances.
Recently, RefineLoc \cite{alwassel2019refineloc} uses an iterative refinement method to help the model capture a \textit{complete} action instance.
And our method is distinct from RefineLoc in three main aspects.
(1) We adopt a late fusion framework, while RefineLoc adopts an early fusion framework.
(2) Our pseudo ground truth is generated from two-stream late fusion attention sequences, which provides better localization performance than each single stream, while RefineLoc generates the pseudo ground truth by expanding previous localization results, which might result in coarser and over-complete action proposals.
(3) We introduce a new attention normalization loss to explicitly avoid the ambiguity of attention, while RefineLoc has no explicit constraints on attention values.
\begin{figure}[t]
\centering
\includegraphics[width=\linewidth]{images/framework.pdf}
\caption{An overview of the proposed Two-Stream Consensus Network, which consists of three parts: (1) RGB and optical flow snippet-level features are extracted with pre-trained models; (2) two-stream base models are separately trained using these RGB and optical flow features; (3) frame-level pseudo ground truth is generated from the two-stream late fusion attention sequence, and in turn provides frame-level supervision to two-stream base models}
\label{fig:framework}
\end{figure}
\section{Two-Stream Consensus Network}
In this section, we first formulate the task of Weakly-supervised Temporal Action Localization (W-TAL), and then describe the proposed Two-Stream Consensus Network (TSCN) in detail. The overall architecture is shown in Fig.~\ref{fig:framework}.
\subsection{Problem Formulation}
Assume we are given a set of training videos. For each video $v$, we only have its video-level categorical label $\mathbf{y}$, where $\mathbf{y} \in \mathbb{R}^C$ is a normalized multi-hot vector, and $C$ is the number of action categories. The goal of temporal action localization is to generate a set of action proposals $\{ (t_{s}, t_{e}, c, \psi) \}$ for each testing video, where $t_{s}, t_{e}, c, \psi$ denote the start time, the end time, the predicted action category and the confidence score of the action proposal, respectively.
\subsection{Feature Extraction}
Following recent W-TAL methods \cite{shou2018autoloc,nguyen2018weakly,paul2018w,liu2019completeness,liu2019weakly,nguyen2019weakly,narayan20193c,yu2019temporal,lee2020background}, we construct TSCN upon snippet-level feature sequences extracted from the raw video volume. The RGB and optical flow features are extracted with pre-trained deep networks (\textit{e.g.}, I3D \cite{carreira2017quo}) from non-overlapping fixed-length RGB frame snippets and optical flow snippets, respectively. They provide high-level appearance and motion information of the corresponding snippets.
Formally, given a video with $T$ non-overlapping snippets, we denote the RGB features and optical flow features as $\{ \mathbf{f}_{\text{RGB},i} \}_{i=1}^T$ and $\{ \mathbf{f}_{\text{flow},i} \}_{i=1}^T$, respectively, where $\mathbf{f}_{\text{RGB},i}, \mathbf{f}_{\text{flow},i} \in \mathbb{R}^{D}$ are the feature representations of the $i$-th RGB frame and optical flow snippet, respectively, and $D$ denotes the channel dimension.
\subsection{Two-Stream Base Models}
After obtaining the RGB and optical flow features, we first use two-stream base models to perform the video-level action classification, and then iteratively refine the base models with a frame-level pseudo ground truth. The features of two modalities are fed into two separate base models, respectively, and the two base models use the same architecture but do not share parameters. Therefore, in this subsection, we omit the subscript $\text{RGB}$ and $\text{flow}$ for conciseness.
Since the features are not originally trained for the W-TAL task, we concatenate the $T$ input features $\{ \textbf{f}_i \}_{i=1}^{T}$, and use a set of temporal convolutional layers to generate a set of new features $\{ \textbf{x}_i \}_{i=1}^{T}$, where $\textbf{x}_i \in \mathbb{R}^{D'}$, and $D'$ denotes the output feature dimension.
As a video may contain background snippets, to perform video-level classification, we need to select snippets that are likely to contain action instances and meanwhile filter out snippets that are likely to contain background. To this end, an attention value $A_i \in (0, 1)$ to measure the likelihood of the $i$-th snippet containing an action is given by a fully-connected (FC) layer:
\begin{equation}
A_i = \sigma \left( \mathbf{w}_A \cdot \mathbf{x}_i + b_A \right ),
\end{equation}
where $\sigma (\cdot)$, $\mathbf{w}_A$, and $b_A$ are the sigmoid function, weight vector and bias of the attention layer. We then perform attention-weighted pooling over the feature sequence to generate a single foreground feature $\mathbf{x}_{\text{fg}}$, and feed it to an FC softmax layer to get the video-level prediction:
\begin{equation}
\mathbf{x}_{\text{fg}} = \frac{1}{\sum_{i=1}^{T} A_{i}} \sum_{i=1}^{T}A_{i} \mathbf{x}_i,
\end{equation}
\begin{equation}
\hat{y}_{c} = \frac{e^{\mathbf{w}_c \cdot \mathbf{x}_{\text{fg}} + b_c}}{\sum_{i=1}^{C}e^{\mathbf{w}_i \cdot \mathbf{x}_{\text{fg}} + b_i}},
\end{equation}
where $\hat{y}_{c}$ is the probability that the video contains the $c$-th action, and $\mathbf{w}_c$ and $b_c$ are the weight and bias of the FC layer for category $c$. The classification loss function $\mathcal{L}_{\text{cls}}$ is defined as the standard cross entropy loss:
\begin{equation}
\mathcal{L}_{\text{cls}} = -\sum_{c=1}^{C} y_{c}\log (\hat{y}_{c}),
\end{equation}
where $y_{c}$ denotes the value of label vector $\textbf{y}$ at index $c$.
Ideally, an attention value is expected to be binary, where $1$ indicates the presence of action while $0$ indicates background. Recently, several methods \cite{nguyen2019weakly,lee2020background} introduce a background category, and use the background classification to guide the learning of attention. In this work, instead of using background classification, we introduce an attention normalization term to force the attention to approach extreme values:
\begin{equation}
\mathcal{L}_{\text{att}} = \frac{1}{l}\min_{\substack{A \subset \{A_i\} \\|A|=l}} \sum_{a \in A}a - \frac{1}{l} \max_{\substack{A \subset \{A_i\} \\|A|=l}} \sum_{a \in A}a,
\label{eq:lNorm}
\end{equation}
where $l=\max \left(1, \lfloor\frac{T}{s}\rfloor \right)$ and $s$ is a hyperparameter to control the selected snippets. This normalization loss aims to maximize the difference between the average top-$l$ attention values and the average bottom-$l$ attention values, and force the foreground attention to be $1$ and background attention to be $0$.
Therefore, the overall loss for the base model training is the weighted sum of the classification loss and the attention normalization term:
\begin{equation}
\mathcal{L}_{\text{base}} = \mathcal{L}_{\text{cls}} + \alpha \mathcal{L}_{\text{att}},
\end{equation}
where $\alpha$ is a hyperparameter to control the weight of the normalization loss.
In addition, the temporal-class activation map (T-CAM) $\{ \textbf{s}_i \}_{i=1}^{T}$, $\textbf{s}_i \in \mathbb{R}^{C}$ is also generated by sliding the classification FC softmax layer over all snippets:
\begin{equation}
s_{i,c} = \frac{e^{\mathbf{w}_c \cdot \mathbf{x}_{i} + b_c}}{\sum_{j=1}^{C}e^{\mathbf{w}_j \cdot \mathbf{x}_{i} + b_j}},
\end{equation}
where $s_{i,c}$ is the T-CAM value of $i$-th snippet for category $c$.
\subsection{Pseudo Ground Truth Generation}
We iteratively refine the two-stream base models with a frame-level pseudo ground truth.
Specifically, we divide the whole training process into several refinement iterations.
At refinement iteration $0$, only video-level labels are used for training.
And at refinement iteration $n+1$, a frame-level pseudo ground truth is generated at refinement iteration $n$, and provides frame-level supervision for the current refinement iteration.
However, without \textit{true} ground truth annotations, we can neither measure the quality of the pseudo ground truth, nor guarantee the pseudo ground truth can help the base models achieve higher performance.
Inspired by two-stream late fusion, we introduce a simple yet effective method to generate the pseudo ground truth. Intuitively, locations at which both streams have high activations are likely to contain ground truth action instances; locations at which only one stream has high activations are likely to be either false positive action proposals or true action instances that only one stream can detect; locations at which both streams both have low activations are likely to be the background.
Following this intuition, we use the fusion attention sequence $\{ A_{\text{fuse},i}^{(n)} \}_{i=1}^T$ at refinement iteration $n$ to generate pseudo ground truth $\{ \mathcal{G}_{i}^{(n+1)} \}_{i=1}^T$ for refinement iteration $n + 1$, where $A_{\text{fuse},i}^{(n)}=\beta A_{\text{RGB},i}^{(n)} + (1-\beta)A_{\text{flow},i}^{(n)}$, and $\beta \in [0, 1]$ is a hyperparameter to control the relative importance of RGB and flow attentions.
We introduce two pseudo ground truth generation methods.
\noindent\textbf{Soft pseudo ground truth} means to directly use the fusion attention values as pseudo labels: $\mathcal{G}_{i}^{(n+1)} = A_{\text{fuse},i}^{(n)}$. The soft pseudo labels contain the probability of a snippet being the foreground action, but also add uncertainty to the model.
\noindent\textbf{Hard pseudo ground truth} thresholds the attention sequence to generate a binary sequence:
\begin{equation}
\mathcal{G}_{i}^{(n+1)} =
\left \{
\begin{array}{lr}
1, \quad A_{\text{fuse},i}^{(n)} > \theta; & \\
0, \quad A_{\text{fuse},i}^{(n)} \leq \theta, & \\
\end{array}
\right.
\end{equation}
where $\theta$ is the threshold value. Setting a large value of $\theta$ will eliminate the action proposals that only one stream has high activations, and therefore reduces the false positive rate. In contrast, setting a small value of $\theta$ will help models to generate more action proposals and achieve a higher recall. Hard pseudo labels remove the uncertainty and provide stronger supervision, but introduce a hyperparameter.
After generating the frame-level pseudo ground truth, we force the attention sequence generated by \textit{each} stream to be similar to the pseudo ground truth with a mean square error (MSE) loss\footnote{Although it is straightforward to use a cross entropy loss for hard pseudo ground truth, we found in practice that the cross entropy loss and the MSE loss achieve similar performance. To simplify training, we use the MSE loss for both kinds of pseudo ground truth.}:
\begin{equation}
\mathcal{L}_{\mathcal{G}}^{(n + 1)} = \frac{1}{T} \sum_{i=1}^{T} \left( A_{i}^{(n + 1)} - \mathcal{G}_{i}^{(n+1)} \right)^2.
\end{equation}
At refinement iteration $n + 1$, the total loss for each stream is
\begin{equation}
\mathcal{L}_{\text{total}}^{(n + 1)} = \mathcal{L}_{\text{base}} + \gamma \mathcal{L}_{\mathcal{G}}^{(n + 1)},
\end{equation}
where $\gamma$ is a hyperparameter to control the relative importance of two losses.
\subsection{Action Localization}
During testing, following BaS-Net \cite{lee2020background}, we first temporally upsample the attention sequence and T-CAM by a factor of $8$ via linear interpolation. Then, we select top-$k$ action categories from the fusion video-level prediction $\hat{\textbf{y}}_{\text{fuse}}$ to perform action localization, where $\hat{\textbf{y}}_{\text{fuse}}=\beta \hat{\textbf{y}}_{\text{RGB}} + (1-\beta) \hat{\textbf{y}}_{\text{flow}}$.
For each of these categories, following our intention that the attention performs a binary selection, we generate action proposals by directly thresholding the attention value at $0.5$ and concatenating consecutive snippets.
The action proposals are scored via a variant of the Outer-Inner-Constrastive score \cite{shou2018autoloc}: instead of using average T-CAM, we use attention weighted T-CAM to measure the outer and inner temporal contrast.
Formally, given action proposal $(t_s, t_e, c)$, fusion attention $\{A_{\text{fuse},i} \}_{i=1}^{T}$ and T-CAM $\{ \textbf{s}_{\text{fuse},i} \}_{i=1}^{T}$, where $\textbf{s}_{\text{fuse},i} = \beta \textbf{s}_{\text{RGB},i} + (1 - \beta)\textbf{s}_{\text{flow},i} $, the score $\psi$ is computed as
\begin{equation}
\psi = \frac{\sum_{i=t_s}^{t_e} A_{\text{fuse},i} s_{\text{fuse},i,c}}{t_e - t_s}
- \frac{\sum_{i=T_s}^{T_e} A_{\text{fuse},i} s_{\text{fuse},i,c} - \sum_{i=t_s}^{t_e} A_{\text{fuse},i} s_{\text{fuse},i,c}}{T_e - T_s - (t_e - t_s)},
\end{equation}
where $T_s=t_s-\frac{L}{4}$, $T_e=t_e+\frac{L}{4}$, and $L=t_e-t_s$. We discard action proposals with confidence scores lower than $0$.
\section{Experiments}
\subsection{Dataset and Evaluation}
\textbf{THUMOS14 dataset} \cite{jiang2014thumos} contains $200$ validation videos and $213$ testing videos within $20$ categories for the TAL task. We use the $200$ validation videos for training, and use the $213$ testing videos for evaluation.
\noindent\textbf{ActivityNet dataset} \cite{caba2015activitynet} has two release versions, \textit{i.e.}, ActivityNet v1.3 and ActivityNet v1.2. ActivityNet v1.3 covers $200$ action categories, with a training set of $10,024$ videos and a validation set of $4,926$ videos.
ActivityNet v1.2 is a subset of ActivityNet v1.3, and covers $100$ action categories, with $4,819$ and $2,383$ videos in the training and validation set, respectively.\footnote{In our experiments, there are $9,937$ and $4,575$ videos in training and validation set of ActivityNet v1.3, respectively, and $4,471$ and $2,211$ videos in training and validation set of ActivityNet v1.2, respectively, because the rest of the videos are unaccessible from YouTube.}
We use the training set and the validation set for training and testing, respectively.
\noindent\textbf{Evaluation Metrics.} Following the standard protocol on temporal action localization, we evaluate our method with mean Average Precision (mAP) under different Intersection-over-Union (IoU) thresholds. We use the evaluation code provided by ActivityNet\footnote{https://github.com/activitynet/ActivityNet/tree/master/Evaluation} to perform the experiments.
\subsection{Implementation Details}
Two off-the-shelf feature extraction backbones are used in our experiments, \textit{i.e.}, UntrimmedNet \cite{wang2017untrimmednets} and I3D \cite{carreira2017quo}, with snippet lengths of $15$ frames and $16$ frames, respectively.
The two backbones are pre-trained on ImageNet~\cite{Deng2009ImageNet} and Kinetics~\cite{carreira2017quo}, respectively, and are not fine-tuned for fair comparison.
The RGB and flow snippet-level features are extracted at the \texttt{global\_pool} layer as $1024$-D vectors.
The networks are implemented in PyTorch \cite{paszke2017automatic}.
We use the Adam \cite{kingma2014adam} optimizer with a fixed learning rate $0.0001$.
We train the base models $200$ and $80$ epochs at refinement iteration $0$, and $100$ and $40$ epochs for later refinement iterations for ActivityNet and THUMOS14, respectively.
We set the maximal number of refinement iterations to $4$ for the THUMOS14 dataset, and $24$ for the ActivityNet datasets, and choose base models that achieve the lowest loss at the previous refinement iteration to generate the pseudo ground truth.
To eliminate fragmentary action proposals, temporal max pooling of kernel size $5$ and stride $1$ is used on the fusion attention sequence before pseudo ground truth generation on ActivityNet dataset.
We use a whole video as a batch.
All hyperparameters are determined via grid search: $s=8$, $\alpha=0.1$, $\beta=0.4$, $\gamma=2$. We set $\theta$ to $0.55$ and $0.5$ for THUMOS14 and ActivityNet, respectively. We choose top-$2$ action categories and also reject categories whose fusion classification prediction scores are lower than $0.1$ to perform action localization.
\begin{table}[t]
\centering
\caption{Comparison of our method with state-of-the-art TAL methods on the THUMOS14 testing set. UNT and I3D are abbreviations for UntrimmedNet feature and I3D feature, respectively}
\label{tab:comparisonOnTHUMOS14}
\begin{tabular}{c|c|C{0.7cm}C{0.7cm}C{0.7cm}C{0.7cm}C{0.7cm}C{0.7cm}C{0.7cm}C{0.7cm}C{0.7cm}}
\hline
\multirow{2}{*}{} & \multirow{2}{*}{Method} & \multicolumn{9}{c}{mAP@IoU (\%)}\\
& & 0.1 & 0.2 &0.3 & 0.4 & 0.5 & 0.6 & 0.7 & 0.8 & 0.9 \\
\hline
\multirow{8}{*}{\rotatebox{90}{Fully-supervised}} & Yuan \textit{et al.} \cite{yuan2017temporal} & 51.0 & 45.2 & 36.5 & 27.8 & 17.8 & - & - & - & - \\
& CDC \cite{shou2017cdc} & - & - & 40.1 & 29.4 & 23.3 & 13.1 & 7.9 & - & - \\
& R-C3D \cite{xu2017r} & 54.5 & 51.5 & 44.8 & 35.6 & 28.9 & - & - & - & - \\
& SSN \cite{zhao2017temporal} & 66.0 & 59.4 & 51.9 & 41.0 & 29.8 & - & - & - & - \\
& BSN \cite{lin2018bsn} & - & - & 53.5 & 45.0 & 36.9 & 28.4 & 20.0 & - & - \\
& TAL-Net \cite{chao2018rethinking} & 59.8 & 57.1 & 53.2 & 48.5 & 42.8 & 33.8 & 20.8 & - & - \\
& GTAN \cite{long2019gaussian} & 69.1 & 63.7 & 57.8 & 47.2 & 38.8 & - & - & - & - \\
& BMN \cite{lin2019bmn} & - & - & 56.0 & 47.4 & 38.8 & 29.7 & 20.5 & - & - \\
\hline
\multirow{16}{*}{\rotatebox{90}{Weakly-supervised}} & UntrimmedNet \cite{wang2017untrimmednets} & 44.4 & 37.7 & 28.2 & 21.1 & 13.7 & - & - & - & - \\
& STPN (UNT) \cite{nguyen2018weakly} & 45.3 & 38.8 & 31.1 & 23.5 & 16.2 & 9.8 & 5.1 & 2.0 & 0.3 \\
& AutoLoc (UNT) \cite{shou2018autoloc} & - & - & 35.8 & 29.0 & 21.2 & 13.4 & 5.8 & - & - \\
& W-TALC (UNT) \cite{paul2018w} & 49.0 & 42.8 & 32.0 & 26.0 & 18.8 & - & 6.2 & - & - \\
& Liu \textit{et al.} (UNT) \cite{liu2019completeness} & 53.5 & 46.8 & 37.5 & 29.1 & 19.9 & 12.3 & 6.0 & - & - \\
& RefineLoc (UNT) \cite{alwassel2019refineloc} & - & - & 36.1 & - & 22.6 & - & 5.8 & - & - \\
& CleanNet (UNT) \cite{liu2019weakly} & - & - & 37.0 & 30.9 & 23.9 & 13.9 & 7.1 & - & - \\
& BaS-Net (UNT) \cite{lee2020background} & 56.2 & 50.3 & 42.8 & 34.7 & 25.1 & 17.1 & 9.3 & 3.7 & \textbf{0.5} \\
& Ours (UNT) & \textbf{58.9} & \textbf{52.9} & \textbf{45.0} & \textbf{36.6} & \textbf{27.6} & \textbf{18.8} & \textbf{10.2} & \textbf{4.0} & \textbf{0.5} \\
\cline{2-11}
& STPN (I3D) \cite{nguyen2018weakly} & 52.0 & 44.7 & 35.5 & 25.8 & 16.9 & 9.9 & 4.3 & 1.2 & 0.1 \\
& W-TALC (I3D) \cite{paul2018w} & 55.2 & 49.6 & 40.1 & 31.1 & 22.8 & - & 7.6 & - & - \\
& Liu \textit{et al.} (I3D) \cite{liu2019completeness} & 57.4 & 50.8 & 41.2 & 32.1 & 23.1 & 15.0 & 7.0 & - & - \\
& RefineLoc (I3D) \cite{alwassel2019refineloc} & - & - & 40.8 & - & 23.1 & - & 5.3 & - & - \\
& Nguyen \textit{et al.} (I3D) \cite{nguyen2019weakly} & 60.4 & 56.0 & 46.6 & 37.5 & 26.8 & 17.6 & 9.0 & 3.3 & 0.4 \\
& BaS-Net (I3D) \cite{lee2020background} & 58.2 & 52.3 & 44.6 & 36.0 & 27.0 & 18.6 & \textbf{10.4} & \textbf{3.9} & 0.5 \\
& Ours (I3D) & \textbf{63.4} & \textbf{57.6} & \textbf{47.8} & \textbf{37.7} & \textbf{28.7} & \textbf{19.4} & 10.2 & \textbf{3.9} & \textbf{0.7} \\
\hline
\end{tabular}
\end{table}
\subsection{Comparison with the State-of-the-art}
\textbf{Experiments on THUMOS14}. Table~\ref{tab:comparisonOnTHUMOS14} summarizes the performance comparison between the proposed TSCN and state-of-the-art fully-supervised and weakly-supervised TAL methods on the THUMOS14 testing set. With UntrimmedNet features, TSCN outperforms other W-TAL methods by a large margin, and even achieves comparable results to some recent W-TAL methods with I3D features (\textit{e.g.}, Nguyen \textit{et al}. \cite{nguyen2019weakly} and BaS-Net \cite{lee2020background}) at high IoU thresholds.
With I3D features, our performance boosts significantly, and outperforms previous W-TAL methods at most IoU thresholds. We note the proposed TSCN can achieve a comparable performance to some recent fully-supervised methods (\textit{e.g.}, R-C3D \cite{xu2017r}). TSCN even outperforms TAL-net \cite{chao2018rethinking} at IoU thresholds $0.1$ and $0.2$. However, as the IoU threshold increases, the performance of TSCN drops significantly, because generating more precise action boundaries need true frame-level ground truth supervision.
\noindent\textbf{Experiments on ActivityNet}. The performance comparisons on ActivityNet v1.2 and v1.3 are shown in Table~\ref{tab:comparsionOnActivityNet1.2} and Table~\ref{tab:comparsionOnActivityNet1.3}, respectively, where our models are trained with I3D features. The proposed TSCN outperforms previous W-TAL methods at the average mAP at IoU threshold $0.5:0.05:0.95$ on both release versions of ActivityNet, verifying the efficacy of our design intuition.
\begin{table}[t]
\hfill
\begin{minipage}[h]{0.48\linewidth}
\centering
\caption{Comparison of our method with state-of-the-art W-TAL methods on the ActivityNet v1.2 validation set. The Avg column indicates the average mAP at IoU thresholds 0.5:0.05:0.95}
\label{tab:comparsionOnActivityNet1.2}
\begin{tabular}{c|C{0.65cm}C{0.65cm}C{0.65cm}|C{0.65cm}}
\hline
\multirow{2}{*}{Method} & \multicolumn{3}{c|}{mAP@IoU (\%)} & \multirow{2}{*}{Avg}\\
& 0.5 & 0.75 & 0.95 & \\
\hline
UntrimmedNet \cite{wang2017untrimmednets} & 7.4 & 3.2 & 0.7 & 3.6 \\
AutoLoc \cite{shou2018autoloc} & 27.3 & 15.1 & 3.3 & 16.0 \\
W-TALC \cite{paul2018w} & 37.0 & - & - & 18.0 \\
Liu \textit{et al.} \cite{liu2019completeness} & 36.8 & 22.0 & 5.6 & 22.4 \\
Ours & \textbf{37.6} & \textbf{23.7} & \textbf{5.7} & \textbf{23.6} \\
\hline
\end{tabular}
\end{minipage}
\hfill
\begin{minipage}[h]{0.48\linewidth}
\centering
\caption{Comparison of our method with state-of-the-art W-TAL methods on the ActivityNet v1.3 validation set. The Avg column indicates the average mAP at IoU thresholds 0.5:0.05:0.95}
\label{tab:comparsionOnActivityNet1.3}
\begin{tabular}{c|C{0.65cm}C{0.65cm}C{0.65cm}|C{0.65cm}}
\hline
\multirow{2}{*}{Method} & \multicolumn{3}{c|}{mAP@IoU (\%)} & \multirow{2}{*}{Avg}\\
& 0.5 & 0.75 & 0.95 & \\
\hline
STPN \cite{nguyen2018weakly} & 29.3 & 16.9 & 2.7 & - \\
Liu \textit{et al.} \cite{liu2019completeness} & 34.0 & 20.9 & \textbf{5.7} & 21.2 \\
Nguyen \textit{et al.} \cite{nguyen2019weakly} & \textbf{36.4} & 19.2 & 2.9 & - \\
Ours & 35.3 & \textbf{21.4} & 5.3 & \textbf{21.7} \\
\hline
\end{tabular}
\end{minipage}
\hfill
\end{table}
\subsection{Ablation Study}
\label{subsec:ablationStudy}
In this subsection, a set of ablation studies is conducted on the THUMOS14 testing set with UntrimmedNet feature to analyze the efficacy of each component in the proposed TSCN.
\noindent\textbf{Ablation study on $\mathcal{L}_{\text{att}}$}.
The goal of $\mathcal{L}_{\text{att}}$ in Equation~\eqref{eq:lNorm} is to force the attention values to approach extreme values, and therefore generate a clean foreground feature $\textbf{x}_{\text{fg}}$ and improve action proposal quality.
Some recent methods~\cite{nguyen2019weakly,lee2020background} introduce background classification to W-TAL.
Particularly, background classification loss $\mathcal{L}_{bg}$~\cite{nguyen2019weakly} is introduced to classify the background, where a background attention is defined as $1-A_i$, and a background feature is generated via background attention-weighted pooling over all snippets to perform the background classification.
Therefore, $\mathcal{L}_{bg}$ is in essence an implicit attention normalization loss.
However, one drawback of such background loss is that assigning background labels to all videos will make the value of the background category in the T-CAM increase.
We reproduce $\mathcal{L}_{bg}$ in our model, compare it with our proposed $\mathcal{L}_{\text{att}}$, and list the results in Table~\ref{tab:comparisonWithDifferentNormalization}.
The results reveal that both $\mathcal{L}_{bg}$ and $\mathcal{L}_{\text{att}}$ help improve the performance.
And the proposed $\mathcal{L}_{\text{att}}$ achieves higher attention variance and better localization performance than $\mathcal{L}_{bg}$, demonstrating that the our attention normalization term $\mathcal{L}_{\text{att}}$ can better avoid the ambiguity of attention.
Surprisingly, with both $\mathcal{L}_{bg}$ and $\mathcal{L}_{\text{att}}$, the localization performance is still lower than that with only $\mathcal{L}_{\text{att}}$, and we think this is because the noise of background classification reduces the accuracy of action proposal scores.
\begin{figure}[t]
\begin{minipage}[h]{0.5\linewidth}
\centering
\captionof{table}{Comparison of our method with different attention normalization functions on the THUMOS14 testing set. $\mathcal{L}_{bg}$ is the background classification loss introduced in \cite{nguyen2019weakly}, and $\mathcal{L}_{att}$ is defined in Equation~\eqref{eq:lNorm}. The var column denotes the average attention variance over the whole testing set}
\label{tab:comparisonWithDifferentNormalization}
\begin{tabular}{ccc|C{0.8cm}C{0.8cm}C{0.8cm}|c}
\hline
\multirow{2}{*}{$\mathcal{L}_{\text{cls}}$} & \multirow{2}{*}{$\mathcal{L}_{bg}$} & \multirow{2}{*}{$\mathcal{L}_{att}$} & \multicolumn{3}{c|}{mAP@IoU (\%)} & \multirow{2}{*}{Var} \\
& & & 0.3 & 0.5 & 0.7 & \\
\hline
\checkmark & - & - & 29.6 & 16.1 & 4.1 & 0.0440 \\
\checkmark & \checkmark & - & 34.3 & 19.3& 6.7 & 0.0599 \\
\checkmark & - & \checkmark & \textbf{40.9} & \textbf{24.0} & \textbf{8.2} & \textbf{0.0937} \\
\checkmark & \checkmark & \checkmark & 40.6 & 23.6 & 7.8 & 0.0886 \\
\hline
\end{tabular}
\end{minipage}
\begin{minipage}[h]{0.5\linewidth}
\centering
\includegraphics[width=0.83\linewidth]{images/curve.pdf}
\caption{Comparison between models trained with different pseudo ground truth on the THUMOS14 testing set. The upper bounds denote models trained with ground truth actionness sequence}
\label{fig:comparisonWithDifferentPseudoGT}
\end{minipage}
\end{figure}
\noindent\textbf{Ablation study on Pseudo Ground Truth}.
Fig.~\ref{fig:comparisonWithDifferentPseudoGT} plots performance comparison between different pseudo ground truth methods at different refinement iterations.
Both soft and hard pseudo ground truth help improve the localization performance.
The hard pseudo ground truth removes uncertainty to the model, and thus achieves higher performance improvement.
However, with the same frame-level supervision, the flow stream outperforms the RGB stream by a large margin. We think this is because of the nature of two modalities: the RGB modality is less sensitive to actions than the optical flow modality.
To demonstrate this, we generate a \textit{true} frame-level ground truth actionness sequence (action categories are not used), train our model in the same way as the pseudo ground truth. The results are plotted in Fig.~\ref{fig:comparisonWithDifferentPseudoGT} as an upper bound. The results verify our hypothesis and demonstrate that the optical flow modality is more suitable for the action localization task than the RGB modality.
\begin{table}[t]
\centering
\caption{Comparison between the model trained with only video-level labels and the model trained with hard pseudo ground truth on the THUMOS14 testing set. The label column denotes the supervision used in training, where ``video" indicates only video-level labels are leveraged, and ``frame" indicates the hard pseudo ground truth is also leveraged during training. Precision, recall and F-measure are calculated under IoU threshold $0.5$}
\label{tab:}
\begin{tabular}{c|C{1cm}|C{0.7cm}C{0.7cm}C{0.7cm}C{0.7cm}C{0.7cm}|c|c|c}
\hline
\multirow{2}{*}{Modality} & \multirow{2}{*}{Label} & \multicolumn{5}{c|}{mAP@IoU (\%)} & \multirow{2}{*}{Precision (\%)} & \multirow{2}{*}{Recall (\%)} & \multirow{2}{*}{F-measure} \\
& & 0.3 & 0.4 & 0.5 & 0.6 & 0.7 & & & \\
\hline
RGB & video & 19.8 & 13.2 & 8.2 & 4.5 & 1.9 & 10.2 & 20.9 & 0.1371 \\
RGB & frame & 31.4 & 22.1 & 14.4 & 8.9 & 5.2 & 20.9 & 30.8 & 0.2489 \\
\hline
Flow & video & 40.2 & 32.0 & 23.2 & 15.4 & 7.2 & 25.5 & 43.3 & 0.3207 \\
Flow & frame & 40.8 & 32.7 & 24.1 & 16.8 & 8.7 & 30.9 & 42.4 & 0.3573 \\
\hline
Fusion & video & 40.9 & 32.4 & 24.0 & 15.9 & 8.2 & 23.6 & 44.4 & 0.3078 \\
Fusion & frame & 45.0 & 36.5 & 27.6 & 18.8 & 10.2 & 31.3 & 44.6 & 0.3680 \\
\hline
\end{tabular}
\end{table}
Table~\ref{eq:lNorm} lists the detailed performance comparison between the model trained with only video-level labels and that trained with the hard pseudo ground truth. The results show that pseudo ground truth improves the localization performance for both modalities at all IoU thresholds, and thus improves the performance of the fusion result. Also, the pseudo ground truth greatly improves the precision and recall for the RGB stream and the fusion result, and improves the precision for the flow stream with a minor loss of recall (the overall F-measure improves significantly), which demonstrates that the pseudo ground truth can help eliminate false positive action proposals.
\noindent\textbf{Qualitative Analysis.} Three representative examples of TAL results are plotted in Fig.~\ref{fig:qualitativeResult} to illustrate the efficacy of the proposed pseudo supervision.
In the first example of diving and cliff diving, with only video-level labels, the RGB stream provides worse localization result than the flow stream, and thus leads to a noisy fusion attention sequence. The pseudo ground truth guides the RGB stream to identify false positive action proposals and discover true action instances, and further leads to a cleaner fusion attention sequence, where high activations correspond better to the ground truth.
In the second example of cricket shot, with only video-level supervision, the RGB stream can only distinguish certain scenes, and fails to separate proximate action instances. In contrast, the flow stream can precisely detect action instances. Therefore, the pseudo ground truth helps the RGB stream to separate consecutive action instances.
In the last example of soccer penalty, both streams have high activations on certain false positive temporal locations. Under this circumstance, the false positive action proposals will have higher activations under frame-level pseudo supervision. To eliminate such false positive action proposals, however, need true ground truth supervision.
To summarize, the two modalities have their own strengths and limitations: the RGB stream is sensitive to appearance, thus it fails in scenes shot from unusual angles or separating proximate action instances in the same scene; the flow stream is sensitive to motion, and provides more accurate results, but it fails in slow or occluded motion.
Qualitative results reveal that the pseudo ground truth helps two streams reach a consensus at most temporal locations.
Therefore, the fusion attention sequence becomes cleaner and helps generate more precise action proposals and more reliable confidence scores.
\begin{figure}[t]
\centering
\includegraphics[width=0.99\linewidth]{images/qualitative_result.pdf}
\caption{Qualitative results on the THUMOS14 testing set. The eight rows in each example are input video, ground truth action instance, RGB stream, flow stream, and fusion attention sequences from the model trained with only video-level labels and frame-level pseudo ground truth, respectively. Action proposals are represented by green boxes. The horizontal and vertical axes are time and intensity of attention, respectively}
\label{fig:qualitativeResult}
\end{figure}
\section{Conclusions}
In this paper, we propose a Two-Stream Consensus Network (TSCN) for W-TAL, which benefits from an iterative refinement training method and a new attention normalization loss.
The iterative refinement training uses a novel frame-level pseudo ground truth as fine-grained supervision, and iteratively improves the two-stream base models.
The attention normalization loss function reduces the ambiguity of attention values, and thus leads to more precise action proposals.
Experiments on two benchmarks demonstrate the proposed TSCN outperforms current state-of-the-art methods, and verify our design intuition.
\section*{Acknowledgement}
This work was supported partly by National Key R\&D Program of China Grant 2018AAA0101400, NSFC Grants 61629301, 61773312, and 61976171, China Postdoctoral Science Foundation Grant 2019M653642, Young Elite Scientists Sponsorship Program by CAST Grant 2018QNRC001, and Natural Science Foundation of Shaanxi Grant 2020JQ-069.
\clearpage
\bibliographystyle{splncs04}
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 6,487
|
// ======================================================================== //
// Copyright 2009-2017 Intel Corporation //
// //
// Licensed under the Apache License, Version 2.0 (the "License"); //
// you may not use this file except in compliance with the License. //
// You may obtain a copy of the License at //
// //
// http://www.apache.org/licenses/LICENSE-2.0 //
// //
// Unless required by applicable law or agreed to in writing, software //
// distributed under the License is distributed on an "AS IS" BASIS, //
// WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied. //
// See the License for the specific language governing permissions and //
// limitations under the License. //
// ======================================================================== //
#pragma once
#include <ospcommon/vec.h>
#include <ospcommon/box.h>
#include <stdexcept>
/*! _everything_ in the ospray core universe should _always_ be in the
'ospray' namespace. */
namespace ospray {
/*! though not required, it is good practice to put any module into
its own namespace (isnide of ospray:: ). Unlike for the naming of
library and init function, the naming for this namespace doesn't
particularlly matter. E.g., 'bilinearPatch', 'module_blp',
'bilinar_patch' etc would all work equally well. */
namespace bilinearPatch {
// use ospcommon for vec3f etc
using namespace ospcommon;
/*! helper class to parse command-line arguments */
struct CommandLine {
CommandLine(int ac, const char **av);
std::vector<std::string> inputFiles;
};
inline CommandLine::CommandLine(int ac, const char **av)
{
for (int i=1;i<ac;i++) {
const std::string arg = av[i];
if (arg[0] == '-') {
throw std::runtime_error("un-handled cmdline argument '"+arg+"'");
} else {
// no arg: must be an input file
inputFiles.push_back(arg);
}
}
}
} // ::ospray::bilinearPatch
} // ::ospray
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 9,813
|
// Copyright 1998-2016 Epic Games, Inc. All Rights Reserved.
#pragma once
#include "Curves/RichCurve.h"
#include "MovieSceneSection.h"
#include "IKeyframeSection.h"
#include "MovieSceneMpcFloatSection.generated.h"
/**
* A single floating point section
*/
UCLASS( MinimalAPI )
class UMovieSceneMpcFloatSection
: public UMovieSceneSection
, public IKeyframeSection<float>
{
GENERATED_UCLASS_BODY()
public:
/**
* Updates this section
*
* @param Position The position in time within the movie scene
*/
virtual float Eval( float Position ) const;
/**
* @return The float curve on this section
*/
FRichCurve& GetFloatCurve() { return FloatCurve; }
const FRichCurve& GetFloatCurve() const { return FloatCurve; }
public:
//~ IKeyframeSection interface
void AddKey( float Time, const float& Value, EMovieSceneKeyInterpolation KeyInterpolation );
bool NewKeyIsNewData(float Time, const float& Value) const;
bool HasKeys( const float& Value ) const;
void SetDefault( const float& Value );
public:
//~ UMovieSceneSection interface
virtual void MoveSection(float DeltaPosition, TSet<FKeyHandle>& KeyHandles) override;
virtual void DilateSection(float DilationFactor, float Origin, TSet<FKeyHandle>& KeyHandles) override;
virtual void GetKeyHandles(TSet<FKeyHandle>& OutKeyHandles, TRange<float> TimeRange) const override;
virtual TOptional<float> GetKeyTime( FKeyHandle KeyHandle ) const override;
virtual void SetKeyTime( FKeyHandle KeyHandle, float Time ) override;
private:
/** Curve data */
UPROPERTY()
FRichCurve FloatCurve;
};
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 795
|
It's springtime and Easter is here! We have created lots of Easter printables for your kids to enjoy. Including our new Goody Bag Toppers. And for a some giggles - Visit our Easter Egg Fun Page - It will Make You Smile!
Scroll below to enjoy our Easter coloring pages, mazes, bingo cards, stickers and more fun printables.
|
{
"redpajama_set_name": "RedPajamaC4"
}
| 4,767
|
Q: SQL to django ORM I have the model:
class DeviceVals(Model):
device_id = UUIDField()
dt = DateTimeField()
value = FloatField()
I need calculate sum of changes from this table, here is sql which works fine:
SELECT
dt,
SUM(cons)
FROM
(SELECT
device_id,
dt,
value,
("value" - LAG("value", 1) OVER (PARTITION BY "device_id"
ORDER BY "dt" ASC)) AS "cons"
FROM "devicevals") AS ss
GROUP BY "dt"
ORDER BY "dt";
I need to convert this sql to django ORM. How can I do that?
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 3,723
|
WKLH (96.5 FM) is a classic rock-formatted radio station in Milwaukee, Wisconsin. The station is owned by Saga Communications. Its studios (which are shared with the other four sister stations) are located on Milwaukee's West Side and the transmitter is the MPTV tower in Shorewood.
WKLH broadcasts in the digital hybrid HD Radio format.
History
Classical (1956-1983)
The station received its Federal Communications Commission (FCC) license on November 30, 1955. They signed on the air as WFMR on June 26, 1956 with a classical music format from the Bayshore Shopping Center in Glendale, with an effective radiated power (ERP) of 24.5 kilowatts and an antenna height of 35 feet above average terrain (HAAT). They began broadcasting in stereo in 1962. In June 1969, it boosted its power to 40 kW, becoming the most powerful FM station in Milwaukee at that time. They got another power upgrade in July 1974, this time to 50 kW.
Adult contemporary (1983-1986)
In January 1983, the station's owners switched the format to adult contemporary, picking up the call sign WMGF and named the station Magic 96.5. Three months later, on April 27, 1983, the owner of (WXJY) picked up the WFMR call sign, its classical music format, and several of its air talents.
Classic hits (1986-2005)
WMGF carried on for three years until flipping formats to classic hits as WKLH on January 27, 1986. WKLH was an immediate success, almost tripling its ratings and landing in the top five stations in the market in its first ratings book.
Classic rock (2005-present)
The station tweaked its format to classic rock in early 2005, as the classic hits format itself in the industry drifted towards a more pop-leaning sound with 80s and early 90s music. WKLH has long been a ratings success in the Milwaukee area, and has also won many awards, including Radio & Records "Classic Rock Station of the Year" in 2001. On April 23, 2015, WKLH rebranded as "Hometown Rock 96.5."
The WKLH call letters previously belonged to AM 1580 and FM 92.1 (K-92) in Saint Johns, Michigan, from October 1983 to September 1985. But before that, the WKLH call letters belonged to a Montgomery, Alabama radio station operating at 92.3 MHz from 1973 through 1977 and was known as "Stereo 92."
Dave and Carole
The Dave and Carole morning show dominated the Milwaukee radio ratings since WKLH debuted in 1986. Based on a yearly 4 rating book average, this show has finished either #1 or #2 in Adults 25-54 for 22 consecutive years.
The show was hosted by Dave Luczak and Carole Caine, who did the news and added the "female" perspective. The other regular was Kevin (KB) Brandt. KB is a former stand-up comedian and a musician who also performs in several local bands. Since early 2002, the show has been produced by Marcus Allen.
Over the past 20+ years, the show has featured a rotating cast. Regulars have included many current or former athletes and coaches. These include former Milwaukee Brewer Paul Molitor (86-92), former Brewer manager Phil Garner (92-98), then-Green Bay Packer coach Mike Holmgren (92-98), QB Brett Favre (97-00), defensive lineman Kabeer Gbaja-Biamila (02-06), and former Brewer pitching coach Mike Maddux (05-08). Other sports guests include LeRoy Butler, a member of the Packer Hall of Fame and Jason Wilde of the Wisconsin State Journal, twice Wisconsin sports writer of the year and famed glee club aficionado.
Other current or former regular cast members include:;
Mike Gousha from WISN 12 in Milwaukee, improv master Dylan Bolin "Out Of The Box", Segway King Angelo "Mort" Snotlocker, Rudy "Itchy" Itchkowski, Gino (The Judge) Salomone, "MR. Angry",
Dutch "Wandering Wisconsin", Joyce Garbaciak from WISN-12, Katrina Cravy from FOX 6, Greg "The Gristleman" Koch, John McGivern, Willy Porter, Genesis touring guitarist Daryl Stuermer and more.
Dave and Carole have also hosted hundreds of thousands of comedians in their studios. Since 1988 they have featured the weekly headliner at the Comedy Cafe in Milwaukee. Other comedy "regulars" include: Fred Klett, Jim Gaffigan, Kyle Cease, Kathleen Madigan and Stephen Lynch and Wisconsin native Frank Caliendo.
Dave and Carole also produce and host many live shows in the Milwaukee area. These include their yearly "Comedy 4 Kids" at the Riverside Theater, "Dave and Carole-ing" for Christmas, their annual Christmas show also at the Riverside Theater, and the "Music of the Morning 'KLH" featuring the talented musicians of the morning show. All of these shows are fund raisers for their various charity endeavors and all shows typically sell out within seconds.
On June 29, 2015, Carole Caine was let go by Saga after her contract was not renewed, ending a 29-year career at WKLH. Mornings are currently hosted by Dave Luczak.
Miracle Marathon
Once every year starting in 1998, WKLH's morning show, Dave & Carole, host the "Miracle Marathon" to raise money for Children's Hospital of Wisconsin. Over the course of three days in 2008, they raised over $1.4 million. Dave and Carole's 11-year total is over $11 million.
References
External links
WKLH official website
KLH
Classic rock radio stations in the United States
Radio stations established in 1956
1956 establishments in Wisconsin
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Using ELO ratings to analyse football championships
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{{Infobox tennistoernooi
| Naam = ATP-toernooi van Kitzbühel
| Officiële naam = Interwetten Austrian Open
| Plaats = Kitzbühel
| Land = Oostenrijk
| Locatie = Kitzbüheler Tennis Club
| Auspiciën = ATP
| Deelnemers = 32 enkel, 32 kwal. / 16 dubbel
| Toernooidirecteur = Ronnie Leitgeb
| Categorie = ATP World Tour 250
| Ondergrond = Gravel, buiten
| Prijzengeld = € 450.000
| Jaar = 2009
| Datum = 17 - 23 mei
| Editie = 65
| Laatst direct toegelaten = Nicolás Massú (84)
| Vorige = 2008
| Volgende = 2011
| Winnaar enkelspel =
Het ATP-toernooi van Kitzbühel 2009 (met de officiële naam Interwetten Austrian Open 2009) werd van 17 tot en met 23 mei gespeeld.
Er werd gespeeld op de gravelbanen van de Kitzbüheler Tennis Club in de Oostenrijkse plaats Kitzbühel.
Enkelspel
Geplaatste spelers
Prijzengeld en ATP-punten
Finale
Bovenste helft
Onderste helft
Dubbelspel
Geplaatste speelsters
Prijzengeld en ATP-punten
Toernooischema
Externe links
Kitzbühel
ATP-toernooi van Kitzbühel
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Outro dia liguei na Rádio Nacional e só ouvi música brasileira. Parecia que eu estava em Nova York.
The other day I turned on Radio Nacional and I only heard Brazilian music. It felt like I was in New York.
Hello and thanks for reading! My name is Victoria Broadus. I first came to Brazil in 2008, when I was studying Portuguese at Georgetown University, and now I've lived here since early 2012. I started this site as a way to share more Brazilian music with friends and family and any other readers out there who wonder what the songs they're listening to are saying, and why. I try to keep a balance between popular songs (e.g. bossa nova) and songs that aren't as well known outside Brazil, and even in Brazil. Please let me know if there is a song you would like translated or would like to hear more about. You can leave a comment here or anywhere else on the site, or on the Facebook page. For requests, if I can't write a full post about the song, I can usually offer the translation at least.
You can follow the blog by clicking "Follow" at the bottom of the main page, and follow blog posts and other updates on Facebook: http://www.facebook.com/lyricalbrazil.
Hello, already seen this clip of CRIOLO?
Hi Evandro, I hadn't seen it yet- thanks!!!
Obrigado pela atenção Victoria, eu sou brasileiro e não falo inglês ainda, você fala português?.
Thank you for your Victoria, I'm Brazilian and not speak English yet, you speak Portuguese?. The video is not official, is my own, but who saw in Brazil, like much, hope you like it too, and if possible, publish the site 🙂 Thank you!
Great site! I've been obsessed with Brazilian music for years, and this is the perfect way to learn Portuguese while discovering more great tunes. Just a suggestion – would it be possible to post the original Portuguese lyrics alongside your translations?
I'm glad you're enjoying the site, and thanks for your suggestion! I had been thinking about posting the Portuguese lyrics — I'm going to try to find the best way to include them.
I'm admiring your work here. I too have translated some MPB songs and am currently working on Chico Buarque's lyrics. I wrote a book about him (http://www.amazon.co.uk/Simple-Brazilian-Song-Journeys-Through/dp/0349108498). I'd be interested to discuss your work and perhaps tell you about mine. I'm giving a talk on Chico in Cambridge (UK) in October. You can find me on Facebook. If my e-address shows up where you are, do mail me.
Hi Vitoria, a question regarding the live recordings of Gilberto Gil's Domingo No Parque as seen on this longer version recording on Youtube http://youtu.be/nrstmBhpZts.
Why is Gil being booed practically all through the song? Is it because it was to innovative?
and many thanks for your answer! keep up the good work!
Booing was really common throughout the festivals. In this case, it was particularly bad because Gilberto Gil was playing such an innovative song and using electric guitars — part of what he and Caetano called their "universal sound" — which a lot of people thought had no place in Brazilian popular music.
Awesome! Thank you very much for the info and your rapid response!
The video's show a really 'charged' and electrifying atmosphere by the way, with the crowd wholeheartedly booing anyone that they didn't like, and religiously chanting the lyrics with others (Chico Buarque's Roda Viva). Was that a 'selected' audience? (as national television was state-run) or does it reflect the general attitude of the people towards their MBP in those days?
I searched for Dorival Caymmi's song "Marina" here, but I could not find it. Could you please provide a translation of that song?
Hi! Thanks so much for reading and for your comments. I will plan to translate Marina soon. And Lapinha is on the blog: https://lyricalbrazil.com/2012/06/19/lapinha/. Besouro was legendary capoeirista; he was killed when he was 24. Hope this helps! Let me know if you have more questions.
I really appreciate your help. I never considered the possibility that "besouro" is the name of a person. I'll look that up, thanks.
Marina is such a great song — I didn't even know who Caymmi was until this week! I noticed he's the author of a Santana song on their "Borboletta" album, "Promise of a Fisherman" and I looked him up.
PS: I just read your explanation — thanks for all the details. I always wondered how a Brazilian got the name "Baden Powell"….!
— is probably untranslatable, but I like your choice of "indifference" for "desamor". Perhaps, though, "so disheartens" might work for "tão desesperador" better than "so maddening"…? I hesitate to even suggest anything, since you obviously know a million times more than me about Portuguese and Brazil!
Translation of any song or interview by Criolo would be very much appreciated. I am studying an MSW degree right now and am quite interested in the LIberation Theology movement in Brazil, Paulo Freire in particular, and what some contemporary artists are saying/rapping about social justice. Word is Mr. Criolo has much to say and I would love to hear(read) more!
Thanks for "Marina", Victoria! What a charmingly odd little song!
Hey, glad you liked the post! I´ll try to do the Toquinho song soon, too.
Hi Victoria, I'd love to know the meaning of another song — "Alô Alô" by Toquinho. If you can find the time, I would be so grateful.
Victoria, I was looking for reference material for an article about Gozanga because he would turn 100 years this week and then I was redirected by google to your website, and wow good job!Well done for you! All the cultural explanations before the lyrics are so well presented, I will take a look into the references that you recommend, keep up the great job!
Hi can you translate "voce e linda" by caetano Veloso? that would be awesome. I love that song!
I don't know if anyone could help me. Does anyone know of a master's degree focused on Brazilian music? I'm looking for a study programme along those lines, and I'm not sure where to look and where to start from. I'm more into the music and lyrical content, however I might be interested in delving into the social link between the music and it's people too. I'm considering other countries too, although I'm based in France.
Thanks in advance. Any help would be appreciated.
Hi Frangale – the best programs I know of along these lines in the United States would be at Tulane University in New Orleans (with Christopher Dunn) and University of Florida (with Charles Perrone), but both would be Portuguese programs. I'm not sure of programs in France.
I am very impressed with your work. I worked in Brazil 1965-7 and 1969-70 and have periodically looked for the wonderful carnival songs I heard in those days, in Salvador and Curitiba.
5 years ago on Youtube one would only find some old guy plucking out the melodies on his verandah–or, more often: nothing. When I last visited, in 1995, you couldn't buy the old songs in Rio and most young people didn't seem to value Samba even. Pagode was all the rage. I had to do my own halting translations of the songs I could remember, eg: A banda, Tristezam, Quem nao gosta… A felicidade.
Ate hoje, tenho saudades do Brasil! Parabens para o website.
It seems that "how insensitive" translates "Assim tão desalmado"?
Hi Gurkski, I'm glad you liked it. Yeah – no part of the song in Portuguese literally translates to "how insensitive", but "tão desalmado" comes the closest (so soulless). Hope this helps!
Cobra criado (the lyrics Elis Regina sings.. there are 2 different lyrics. I absolute adore elis and to understand her's.
We love the song Lapinha by Sergio Mendez and Brazil 66. Is there an English translation?
Hello Victoria , do you have any medium through which i can contact you directly ?
Hi, you can email me at Torysmail@gmail.com.
Have you heard the version by US-born singer Karrin Allyson of the song "Faltando Um Pedaco" ? It's amazingly beautiful. How did she learn Brazilian Portuguese so well!?
Could you please translate the lyrics to that song?
I don't delete comments on here, I just realized I hadn't clicked "approve" yet.
Anyway, you may be interested in this English version of Tom Jobim's "Derradeira primavera," "The Final Springtime." I recorded it with some of my musician friends.
Hello from Austin Texas, Victoria! And on behalf of the Austin Samba School, thank you for this fabulous resource! I stumbled upon your site today while searching for a translation of Aquarela Brasileira, which is going to be our samba-enredo for our next Carnaval show. We are already being asked to learn the song, and when I sent out the link to your version I got a lot of enthusiastic replies about it and about your site in general. A number of us are also studying the music and the language of Brazil in more depth, and your blog will be a great help.
Thanks so much, Dell! Let me know any requests you have, as well. I hope to be able to update more soon.
Thanks for your comment! I'll add that to my list!
Thank you so much for this site! I am in love with brazilian music and have recently started learning bossa nova and samba on guitar and find that a lot of the translations done in the 60's are pretty hokey and don't get what i believe to be the full meaning across. I want to play covers in english (as i don't speak portuguese). Glad i stumbled on this site, I was looking for "Luz Negra", Nara Leao's version being one of my favorite songs. Valuable! Thanks!
Oh, and a few songs i would love to have translations for: "E Preciso Perdoar" by Joao Gilberto, "Pois E" by Nara Leao, and "A Tamba" by Jorge Ben. And also, I live in Brooklyn, NY. Do you know anywhere to go see Brazillian music performed in the city? Thanks again, and hope all is well!
Hi Victoria. Thanks for a fantastic site, and such a great resource for me. I am a jazz singer and am in love with Brazilian music. I love to know what I am singing about, but its difficult to fid decent translations of songs. Could you add Djavan's "A Rota Individuo" to your list please? I've searched and searched for a translation, but no luck, and Google translate is unintelligible!
Hi Amanda, Thanks so much for your kind comments, I'm glad you're finding the site useful. I've added "A rota do individuo" to my list of songs to translate! Let me know any other requests.
Olá, Victoria, tudo bem? Eu gostaria que, se possível, você traduzisse algumas das músicas de Los Hermanos. Eles são uma banda popular entre os jovens brasileiros e têm letras bonitas. Eu traduziria (na realidade já tentei) Mas sempre que as traduzo sinto que escrevi algo errado. Se possível, gostaria que traduzisse estas quatro em especial: "A flor", "De onde vem a calma", "Último Romance" e "Conversa de Botas Batidas".
Hi Victoria… would it be possible to get your email address.
Thanks Victoria. Love your website! Would love to see your translations for Noites Cariocas, Berimbau and Canto de Ossanha. Hope you got my email. All the best – Anna.
Gostei muito do teu site.
A seleção musical é de primeira e a tradução com interpretação é ótima.
Tomei a liberdade de fazer um post sobre ele no google+ http://bit.ly/1gpHoAT.
Obrigada, fico muito contente que gostou do site!
My favorite songs (without the accents that I don't know how to make) are E De Oxum and Aguas De Marches. The second is widely sung in English since Tom made his own translation of it. But I would love to see a professional translation of E De Oxum!
Thanks Larry! I will work on É D'Oxum.
Hi I met a very nice man from Brazil on a trip to Mexico and we have become good friends. He has asked me to listen to the song "Roza Azul".. But I can't seem to find the English lyrics anywhere.. Please help, thanks!
Hi Sandra, do you have a link to the song on YouTube by any chance?
I do..I think..let me try and find it..thanks!
I'm not sure if these are the same song..?
Hi I sent you two links, can you tell me if their the same song.. I'd really appreciate thank you!!
Hi I sent you the song and link from YouTube..thank you!
Please translate or let me know where I can find these lyrics in English or Spanish.. Thanks!
What a beautiful site you have! I found it looking for a translation/more about Mar Grande (Paulinho da Viola). I absolutely love that song, and I find it quite meaningful and poetic. Nevertheless, there are a couple of parts that aren't quite clear, and I can't make sense of them with just the dictionary… I'm sure there are some contexts or some uses I'm missing. I would love it if you find the time to zrite about it. Thanks a bunch!
I love your blog! What a beautiful space you have here. I found it looking for Paulinho da Viola's Mar Grande lyrics. I wonder if you could write about that song here. I understand almost everything, but I would love to read more about it and discuss the meaning….
Thank you for this intelligent introduction into a new cosmos.
Hello! This blog is so amazing!!!!! Thank you! It's so nice be able to share my favorite brazilian songs with my friends. Could you translate some songs by Los Hermanos?
I of course meant, the English translations!
I need to know the song meaning of Acai by Djavan. …I realy love that song. ..music. .o love jazz…voice… I love Brazilian song also….can u help me to know ….
Hi Victoria, I hope you are doing well.
I have started a project on teaching Brazilian guitar and Brazilian music for English speakers and I'd like to tell you that I put a link to your blog in my website. Translation of lyrics is always something hard to do due the intrinsic subjectivity of poetry and the culture of a people and you are doing a great job.
Thanks so much Renato! Where are you based?
I'm an acoustic guitar teacher from Brazil. I have been teaching acoustic guitar here since 1991 and I'm used to teach Bossa Nova, Samba, Choro and the most traditional styles of Brazilian music. I have started this project because I think that many people around the world would like to learn Brazilian music on guitar and that there is a lack of available online resources that could really help them.
If you want, take a look at my website, although it is a site for members anyone can subscribe and get 7 days of full access without credit card or whatever.
Thanks for creating this terrific site. It's addictive.
I'd love a translation/backstory for "Roda-Viva" by Chico Buarque.
As a side note, your blog is great — I took a class on Brazilian music and globalization in college, and I'm rediscovering everything (and learning Portuguese) now. Thanks for sharing with us.
Thanks for the breakdown of Chico Buarque's "Tanto Mar." I'm Portuguese-American and have always loved the song and was looking for a way to adequately explain it and its political/cultural significance to folks. I thin you mostly hit the nail on the head. Great blog, keep up the great work!
Hi! As someone who loves Brazilian music but doesn't speak a bit of the language, this is a godsend. Having the lyrics translated by someone who knows the language and the history beats trying to gather the meaning behind the lyrics from google translate. Especially your interpretations and explanations of the historical context – I have loved going through this blog and discovering new meanings to songs I thought I knew well. Hope you carry on this blog, I will be reading!
I am currently living in Rio and learning to play the Mandolim. This is my first song I am learning, and I am wondering if you would enjoy helping me translate it to english.
I find it really beautiful.
So lovely to happen upon your website! I look forward to listening to and looking at lyrics of the many songs you are sharing. Thank you!
splendid, is only missing more alternative sounds, o rappa, nação zumbi … congratulations!
I only today found your site! I'm very excited! I've been listening to Bassa Nova for years and have been wishing there was somewhere like this. Have no idea how I missed you! I'm nearly 60 now and while I've always enjoyed songs like Girl from Ipanima I got heavily into Brazilian music when I heard Rosa Passos sing 'Voce Vai Ver' on an album of women singers from around the world. I searched and found more of her music. I fell in love with the language. I picked up Pimsleur's Brazilian Portuguese and learned enough that when I visited Rio I could just about converse. I have work colleagues in Brazil who have helped me over the years learn some but without being immersed it's too hard to really learn well. But I continue to find Brazilian music. I've even gotten to see Rosa live in New York!! 🙂 Some singers I have in my playlist include Ceu (she sings a great song with Herbie Hancock), Joao Gilberto, Vanessa Da Mata, Tania Maria and others. Even tho I grew up with my parents speaking Spanish and think it's one of the most elegant languages around, there's just something about Portuguese that I just love. Previously, when I listen to a song and want to know the translation, I used both google translate and some of my works friends. Needless to say, it's not a perfect way. So I appreciate your site which helps me both enjoy the music more and at the same time learn the language a bit.
I only today found your site! I'm very excited! I've been listening to Bassa Nova for years and have been wishing there was somewhere like this. Have no idea how I missed you! I'm nearly 60 now and while I've always enjoyed songs like Girl from Ipanima I got heavily into Brazilian music when I heard Rosa Passos sing 'Voce Vai Ver' on an album of women singers from around the world. I searched and found more of her music. I fell in love with the language. I picked up Pimsleur's Brazilian Portuguese and learned enough that when I visited Rio I could just about converse. I have work colleagues in Brazil who have helped me over the years learn some but without being immersed it's too hard to really learn well. But I continue to find Brazilian music. I've even gotten to see Rosa live in New York!!🙂 Some singers I have in my playlist include Ceu (she sings a great song with Herbie Hancock), Joao Gilberto, Vanessa Da Mata, Tania Maria and others. Even tho I grew up with my parents speaking Spanish and think it's one of the most elegant languages around, there's just something about Portuguese that I just love. Previously, when I listen to a song and want to know the translation, I used both google translate and some of my works friends. Needless to say, it's not a perfect way. So I appreciate your site which helps me both enjoy the music more and at the same time learn the language a bit.
Here's another song which I really didn't know what she was singing but loved it. I was told by one of my friends that it's difficult to translate. But it would be great if you would.
Hi! I can't remember if I've posted here before, but I've been following this blog for a while now and felt the need to comment. This page is absolutely amazing for a clueless foreigner! I have a huge love for Brazilian music, how varied and deep it is, with so many incredible song writers and musicians. I love that this site makes it easy to understand the lyrics too! That is definitely something I've been missing while listening, so you're doing something amazing here.
I have always been fascinated by this performance of Partido Alto from the Caetano Veloso and Chico Buarque live CD from 1972. Not only is it a really livewire performance with this amazing creeping tension to it, but I've also heard that it's a protest song and somehow significant to the anti-military movement of the time? Obviously I'm having a lot of trouble trying to discern the meaning behind the words. Is this something you might be able to explain? I would love to just get a good English translation to be honest!
This website pretty much sums up what the internet is all about. Thanks and keep up the good work!
An entry on "Trilha de Sumé/Culto à Terra/Bailado das Muscarias" from the Paebiru record would be greatly appreciated.
What a great site. Thank you for making it. Do you think you can add "Agua No Feijao" to your list of translated songs? Much appreciated.
Some years ago I wrote a blog post about Chico's 'A Foto da Capa' and am embarrassed now with my translation. Would you please comment on my version?
This immediately brings to mind the album "Paratodos" (RCA / BMG Ariola, 1993) by the great Brazilian singer-songwriter-playwright-poet-novelist and soccer fiend Chico Buarque. The booklet accompanying the CD is sprinkled throughout with head shots of people an anthropologist might recognize as Brazilian, black and white and of all ages. Most of these are frontal photos of people who were obviously relaxed at the instant of the shot. And then there is this pair of shots of Chico himself, front and profile, that stands in sharp contrast to the others. The frontal shot has an identifying number below it, and the circumstances are revealed inside by a reproduction of the document that places the portraits in context. "SECÇÃO DE IDENTIFICAÇÃO J.V.P.M." the card says on top, and if I take the last two letters to signify "Policial Militar", this is definitely an arrest record. The obverse carries the two photographs stapled to the right half of the card and bears a thumb print on the lower left. The document is dated 29-12-61; the 17 year old Chico had apparently been up to some felonious post-Noël hi-jinks. The eyes stare out and the mood is somber. It is unlike any other face of Chico that I have seen over the years.
The lyrics on the facing page are of the last song of the CD, "A Foto Da Capa" — the photo from the cover. Here is my translation from the Portuguese, made unreliable by my meagre grasp of the literary and the colloquial.
Cujo foco toda lírica solapa // Whose focus disguises all that's lyrical.
E o fotógrafo frontal batendo a chapa // And the photographer in front taking the shot.
Chico is an acknowledged master of wordplay and it is impossible to convey the multiple meanings of his lyrics in translation. Is he singing about the cover photo or of the cover-up he presents to the camera? His focus or the camera's? Is it the future he is not seeing or it's not his fiancee that he is looking at? Is it the cop who is snarling or is he? Merely the obverse, the face covered? A face to the shot or a shot to the face? Chico has the upper hand on me, and I claim the license of a journeyman translator.
Still, in the space of sixteen lines the erudite Chico manages to personalize the experience and address the ambiguities of the painful situation, echoing the sentiments of Michaelson and Ruchira. To quote from a famous book on contemporary Brazilian song "Buarque's mature songs are notable for ingenious structurings, for variety of personae, and for ambiguity and plurisignification, especially as strategies for making social commentaries". I couldn't have said it worse; I hope I said it better!
You have to hear the original for the unfailing rhymes and the tongue-twister of a final line which brings an abrupt end to the song.
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{"url":"http:\/\/zims-en.kiwix.campusafrica.gos.orange.com\/wikipedia_en_all_nopic\/A\/Tangent_lines_to_circles","text":"# Tangent lines to circles\n\nIn Euclidean plane geometry, a tangent line to a circle is a line that touches the circle at exactly one point, never entering the circle's interior. Tangent lines to circles form the subject of several theorems, and play an important role in many geometrical constructions and proofs. Since the tangent line to a circle at a point P is perpendicular to the radius to that point, theorems involving tangent lines often involve radial lines and orthogonal circles.\n\n## Tangent lines to one circle\n\nA tangent line t to a circle C intersects the circle at a single point T. For comparison, secant lines intersect a circle at two points, whereas another line may not intersect a circle at all. This property of tangent lines is preserved under many geometrical transformations, such as scalings, rotation, translations, inversions, and map projections. In technical language, these transformations do not change the incidence structure of the tangent line and circle, even though the line and circle may be deformed.\n\nThe radius of a circle is perpendicular to the tangent line through its endpoint on the circle's circumference. Conversely, the perpendicular to a radius through the same endpoint is a tangent line. The resulting geometrical figure of circle and tangent line has a reflection symmetry about the axis of the radius.\n\nNo tangent line can be drawn through a point within a circle, since any such line must be a secant line. However, two tangent lines can be drawn to a circle from a point P outside of the circle. The geometrical figure of a circle and both tangent lines likewise has a reflection symmetry about the radial axis joining P to the center point O of the circle. Thus the lengths of the segments from P to the two tangent points are equal. By the secant-tangent theorem, the square of this tangent length equals the power of the point P in the circle C. This power equals the product of distances from P to any two intersection points of the circle with a secant line passing through P.\n\nThe tangent line t and the tangent point T have a conjugate relationship to one another, which has been generalized into the idea of pole points and polar lines. The same reciprocal relation exists between a point P outside the circle and the secant line joining its two points of tangency.\n\nIf a point P is exterior to a circle with center O, and if the tangent lines from P touch the circle at points T and S, then \u2220TPS and \u2220TOS are supplementary (sum to 180\u00b0).\n\nIf a chord TM is drawn from the tangency point T of exterior point P and \u2220PTM \u2264 90\u00b0 then \u2220PTM = (1\/2)\u2220TOM.\n\n### Compass and straightedge constructions\n\nIt is relatively straightforward to construct a line t tangent to a circle at a point T on the circumference of the circle:\n\n\u2022 A line a is drawn from O, the center of the circle, through the radial point T;\n\u2022 The line t is the perpendicular line to a.\n\nThales' theorem may be used to construct the tangent lines to a point P external to the circle C:\n\n\u2022 A circle is drawn centered on the midpoint of the line segment OP, having diameter OP, where O is again the center of the circle C.\n\u2022 The intersection points T1 and T2 of the circle C and the new circle are the tangent points for lines passing through P, by the following argument.\n\nThe line segments OT1 and OT2 are radii of the circle C; since both are inscribed in a semicircle, they are perpendicular to the line segments PT1 and PT2, respectively. But only a tangent line is perpendicular to the radial line. Hence, the two lines from P and passing through T1 and T2 are tangent to the circle C.\n\nAnother method to construct the tangent lines to a point P external to the circle using only a straightedge:\n\n\u2022 Draw any three different lines through the given point P that intersect the circle twice.\n\u2022 Let ${\\displaystyle A_{1},A_{2},B_{1},B_{2},C_{1},C_{2}}$ be the six intersection points, with the same letter corresponding to the same line and the index 1 corresponding to the point closer to P.\n\u2022 Let D be the point where the lines ${\\displaystyle A_{1}B_{2}}$ and ${\\displaystyle A_{2}B_{1}}$ intersect,\n\u2022 Similarly E for the lines ${\\displaystyle B_{1}C_{2}}$ and ${\\displaystyle B_{2}C_{1}}$.\n\u2022 Draw a line through D and E.\n\u2022 This line meets the circle at two points, F and G.\n\u2022 The tangents are the lines PF and PG.[1]\n\n### Tangential polygons\n\nA tangential polygon is a polygon each of whose sides is tangent to a particular circle, called its incircle. Every triangle is a tangential polygon, as is every regular polygon of any number of sides; in addition, for every number of polygon sides there are an infinite number of non-congruent tangential polygons.\n\n#### Tangent quadrilateral theorem and inscribed circles\n\nA tangential quadrilateral ABCD is a closed figure of four straight sides that are tangent to a given circle C. Equivalently, the circle C is inscribed in the quadrilateral ABCD. By the Pitot theorem, the sums of opposite sides of any such quadrilateral are equal, i.e.,\n\n${\\displaystyle {\\overline {AB}}+{\\overline {CD}}={\\overline {BC}}+{\\overline {DA}}.}$\n\nThis conclusion follows from the equality of the tangent segments from the four vertices of the quadrilateral. Let the tangent points be denoted as P (on segment AB), Q (on segment BC), R (on segment CD) and S (on segment DA). The symmetric tangent segments about each point of ABCD are equal, e.g., BP=BQ=b, CQ=CR=c, DR=DS=d, and AS=AP=a. But each side of the quadrilateral is composed of two such tangent segments\n\n${\\displaystyle {\\overline {AB}}+{\\overline {CD}}=(a+b)+(c+d)={\\overline {BC}}+{\\overline {DA}}=(b+c)+(d+a)}$\n\nproving the theorem.\n\nThe converse is also true: a circle can be inscribed into every quadrilateral in which the lengths of opposite sides sum to the same value.[2]\n\nThis theorem and its converse have various uses. For example, they show immediately that no rectangle can have an inscribed circle unless it is a square, and that every rhombus has an inscribed circle, whereas a general parallelogram does not.\n\n## Tangent lines to two circles\n\nFor two circles, there are generally four distinct lines that are tangent to both (bitangent) \u2013 if the two circles are outside each other \u2013 but in degenerate cases there may be any number between zero and four bitangent lines; these are addressed below. For two of these, the external tangent lines, the circles fall on the same side of the line; for the two others, the internal tangent lines, the circles fall on opposite sides of the line. The external tangent lines intersect in the external homothetic center, whereas the internal tangent lines intersect at the internal homothetic center. Both the external and internal homothetic centers lie on the line of centers (the line connecting the centers of the two circles), closer to the center of the smaller circle: the internal center is in the segment between the two circles, while the external center is not between the points, but rather outside, on the side of the center of the smaller circle. If the two circles have equal radius, there are still four bitangents, but the external tangent lines are parallel and there is no external center in the affine plane; in the projective plane, the external homothetic center lies at the point at infinity corresponding to the slope of these lines.[3]\n\n### Outer tangent\n\nThe red line joining the points ${\\displaystyle (x_{3},y_{3})}$ and ${\\displaystyle (x_{4},y_{4})}$ is the outer tangent between the two circles. Given points ${\\displaystyle (x_{1},y_{1})}$, ${\\displaystyle (x_{2},y_{2})}$ the points ${\\displaystyle (x_{3},y_{3})}$, ${\\displaystyle (x_{4},y_{4})}$ can easily be calculated with help of the angle ${\\displaystyle \\alpha }$:\n\n{\\displaystyle {\\begin{aligned}x_{3}&=x_{1}+r\\cos({\\tfrac {\\pi }{2}}-\\alpha )\\\\y_{3}&=y_{1}+r\\sin({\\tfrac {\\pi }{2}}-\\alpha )\\\\x_{4}&=x_{2}+R\\cos({\\tfrac {\\pi }{2}}-\\alpha )\\\\y_{4}&=y_{2}+R\\sin({\\tfrac {\\pi }{2}}-\\alpha )\\\\\\end{aligned}}}\n\nHere R and r notate the radii of the two circles and the angle ${\\displaystyle \\alpha }$ can be computed using basic trigonometry. You have ${\\displaystyle \\alpha =\\gamma -\\beta }$ with ${\\displaystyle \\gamma =-\\arctan \\left({\\tfrac {y_{2}-y_{1}}{x_{2}-x_{1}}}\\right)}$ and ${\\displaystyle \\beta =\\arcsin \\left({\\tfrac {R-r}{\\sqrt {(x_{2}-x_{1})^{2}+(y_{2}-y_{1})^{2}}}}\\right)}$. [4]\n\n### Inner tangent\n\nAn inner tangent is a tangent that intersects the segment joining two circles' centers. Note that the inner tangent will not be defined for cases when the two circles overlap.\n\n### Construction\n\nThe bitangent lines can be constructed either by constructing the homothetic centers, as described at that article, and then constructing the tangent lines through the homothetic center that is tangent to one circle, by one of the methods described above. The resulting line will then be tangent to the other circle as well. Alternatively, the tangent lines and tangent points can be constructed more directly, as detailed below. Note that in degenerate cases these constructions break down; to simplify exposition this is not discussed in this section, but a form of the construction can work in limit cases (e.g., two circles tangent at one point).\n\n#### Synthetic geometry\n\nLet O1 and O2 be the centers of the two circles, C1 and C2 and let r1 and r2 be their radii, with r1\u00a0>\u00a0r2; in other words, circle C1 is defined as the larger of the two circles. Two different methods may be used to construct the external and internal tangent lines.\n\nExternal tangents\n\nA new circle C3 of radius r1\u00a0\u00a0r2 is drawn centered on O1. Using the method above, two lines are drawn from O2 that are tangent to this new circle. These lines are parallel to the desired tangent lines, because the situation corresponds to shrinking both circles C1 and C2 by a constant amount, r2, which shrinks C2 to a point. Two radial lines may be drawn from the center O1 through the tangent points on C3; these intersect C1 at the desired tangent points. The desired external tangent lines are the lines perpendicular to these radial lines at those tangent points, which may be constructed as described above.\n\nInternal tangents\n\nA new circle C3 of radius r1\u00a0+\u00a0r2 is drawn centered on O1. Using the method above, two lines are drawn from O2 that are tangent to this new circle. These lines are parallel to the desired tangent lines, because the situation corresponds to shrinking C2 to a point while expanding C1 by a constant amount, r2. Two radial lines may be drawn from the center O1 through the tangent points on C3; these intersect C1 at the desired tangent points. The desired internal tangent lines are the lines perpendicular to these radial lines at those tangent points, which may be constructed as described above.\n\n#### Analytic geometry\n\nLet the circles have centres c1 = (x1,y1) and c2 = (x2,y2) with radius r1 and r2 respectively. Expressing a line by the equation ${\\displaystyle ax+by+c=0,}$ with the normalization a2\u00a0+\u00a0b2\u00a0=\u00a01, then a bitangent line satisfies:\n\nax1\u00a0+\u00a0by1\u00a0+\u00a0c\u00a0=\u00a0r1 and\nax2\u00a0+\u00a0by2\u00a0+\u00a0c\u00a0=\u00a0r2.\n\nSolving for ${\\displaystyle (a,b,c)}$ by subtracting the first from the second yields\n\na\u0394x\u00a0+\u00a0b\u0394y\u00a0=\u00a0\u0394r\n\nwhere \u0394x\u00a0=\u00a0x2\u00a0\u00a0x1, \u0394y\u00a0=\u00a0y2\u00a0\u00a0y1 and \u0394r\u00a0=\u00a0r2\u00a0\u00a0r1.\n\nIf ${\\displaystyle d={\\sqrt {(\\Delta x)^{2}+(\\Delta y)^{2}}}}$ is the distance from c1 to c2 we can normalize by X\u00a0=\u00a0\u0394x\/d, Y\u00a0=\u00a0\u0394y\/d and R\u00a0=\u00a0\u0394r\/d to simplify equations, yielding the equations aX\u00a0+\u00a0bY\u00a0=\u00a0R and a2\u00a0+\u00a0b2\u00a0=\u00a01, solve these to get two solutions (k\u00a0=\u00a0\u00b11) for the two external tangent lines:\n\na\u00a0=\u00a0RX\u00a0\u00a0kY\u221a(1\u00a0\u00a0R2)\nb\u00a0=\u00a0RY\u00a0+\u00a0kX\u221a(1\u00a0\u00a0R2)\nc\u00a0=\u00a0r1\u00a0\u00a0(ax1\u00a0+\u00a0by1)\n\nGeometrically this corresponds to computing the angle formed by the tangent lines and the line of centers, and then using that to rotate the equation for the line of centers to yield an equation for the tangent line. The angle is computed by computing the trigonometric functions of a right triangle whose vertices are the (external) homothetic center, a center of a circle, and a tangent point; the hypotenuse lies on the tangent line, the radius is opposite the angle, and the adjacent side lies on the line of centers.\n\n(X,\u00a0Y) is the unit vector pointing from c1 to c2, while R is ${\\displaystyle \\cos \\theta }$ where ${\\displaystyle \\theta }$ is the angle between the line of centers and a tangent line. ${\\displaystyle \\sin \\theta }$ is then ${\\displaystyle \\pm {\\sqrt {1-R^{2}}}}$ (depending on the sign of ${\\displaystyle \\theta }$, equivalently the direction of rotation), and the above equations are rotation of (X,\u00a0Y) by ${\\displaystyle \\pm \\theta ,}$ using the rotation matrix:\n\n${\\displaystyle {\\begin{pmatrix}R&\\mp {\\sqrt {1-R^{2}}}\\\\\\pm {\\sqrt {1-R^{2}}}&R\\end{pmatrix}}}$\nk\u00a0=\u00a01 is the tangent line to the right of the circles looking from c1 to c2.\nk\u00a0=\u00a01 is the tangent line to the right of the circles looking from c2 to c1.\n\nThe above assumes each circle has positive radius. If r1 is positive and r2 negative then c1 will lie to the left of each line and c2 to the right, and the two tangent lines will cross. In this way all four solutions are obtained. Switching signs of both radii switches k\u00a0=\u00a01 and k\u00a0=\u00a01.\n\n#### Vectors\n\nIn general the points of tangency t1 and t2 for the four lines tangent to two circles with centers v1 and v2 and radii r1 and r2 are given by solving the simultaneous equations:\n\n{\\displaystyle {\\begin{aligned}(t_{2}-v_{2})\\cdot (t_{2}-t_{1})&=0\\\\(t_{1}-v_{1})\\cdot (t_{2}-t_{1})&=0\\\\(t_{1}-v_{1})\\cdot (t_{1}-v_{1})&=r_{1}^{2}\\\\(t_{2}-v_{2})\\cdot (t_{2}-v_{2})&=r_{2}^{2}\\\\\\end{aligned}}}\n\nThese equations express that the tangent line, which is parallel to ${\\displaystyle t_{2}-t_{1},}$ is perpendicular to the radii, and that the tangent points lie on their respective circles.\n\nThese are four quadratic equations in two two-dimensional vector variables, and in general position will have four pairs of solutions.\n\n### Degenerate cases\n\nTwo distinct circles may have between zero and four bitangent lines, depending on configuration; these can be classified in terms of the distance between the centers and the radii. If counted with multiplicity (counting a common tangent twice) there are zero, two, or four bitangent lines. Bitangent lines can also be generalized to circles with negative or zero radius. The degenerate cases and the multiplicities can also be understood in terms of limits of other configurations \u2013 e.g., a limit of two circles that almost touch, and moving one so that they touch, or a circle with small radius shrinking to a circle of zero radius.\n\n\u2022 If the circles are outside each other (${\\displaystyle d>r_{1}+r_{2}}$), which is general position, there are four bitangents.\n\u2022 If they touch externally at one point (${\\displaystyle d=r_{1}+r_{2}}$) \u2013 have one point of external tangency \u2013 then they have two external bitangents and one internal bitangent, namely the common tangent line. This common tangent line has multiplicity two, as it separates the circles (one on the left, one on the right) for either orientation (direction).\n\u2022 If the circles intersect in two points (${\\displaystyle |r_{1}-r_{2}|), then they have no internal bitangents and two external bitangents (they cannot be separated, because they intersect, hence no internal bitangents).\n\u2022 If the circles touch internally at one point (${\\displaystyle d=|r_{1}-r_{2}|}$) \u2013 have one point of internal tangency \u2013 then they have no internal bitangents and one external bitangent, namely the common tangent line, which has multiplicity two, as above.\n\u2022 If one circle is completely inside the other (${\\displaystyle d<|r_{1}-r_{2}|}$) then they have no bitangents, as a tangent line to the outer circle does not intersect the inner circle, or conversely a tangent line to the inner circle is a secant line to the outer circle.\n\nFinally, if the two circles are identical, any tangent to the circle is a common tangent and hence (external) bitangent, so there is a circle's worth of bitangents.\n\nFurther, the notion of bitangent lines can be extended to circles with negative radius (the same locus of points, ${\\displaystyle x^{2}+y^{2}=(-r)^{2},}$ but considered \"inside out\"), in which case if the radii have opposite sign (one circle has negative radius and the other has positive radius) the external and internal homothetic centers and external and internal bitangents are switched, while if the radii have the same sign (both positive radii or both negative radii) \"external\" and \"internal\" have the same usual sense (switching one sign switches them, so switching both switches them back).\n\nBitangent lines can also be defined when one or both of the circles has radius zero. In this case the circle with radius zero is a double point, and thus any line passing through it intersects the point with multiplicity two, hence is \"tangent\". If one circle has radius zero, a bitangent line is simply a line tangent to the circle and passing through the point, and is counted with multiplicity two. If both circles have radius zero, then the bitangent line is the line they define, and is counted with multiplicity four.\n\nNote that in these degenerate cases the external and internal homothetic center do generally still exist (the external center is at infinity if the radii are equal), except if the circles coincide, in which case the external center is not defined, or if both circles have radius zero, in which case the internal center is not defined.\n\n### Applications\n\n#### Belt problem\n\nThe internal and external tangent lines are useful in solving the belt problem, which is to calculate the length of a belt or rope needed to fit snugly over two pulleys. If the belt is considered to be a mathematical line of negligible thickness, and if both pulleys are assumed to lie in exactly the same plane, the problem devolves to summing the lengths of the relevant tangent line segments with the lengths of circular arcs subtended by the belt. If the belt is wrapped about the wheels so as to cross, the interior tangent line segments are relevant. Conversely, if the belt is wrapped exteriorly around the pulleys, the exterior tangent line segments are relevant; this case is sometimes called the pulley problem.\n\n## Tangent lines to three circles: Monge's theorem\n\nFor three circles denoted by C1, C2, and C3, there are three pairs of circles (C1C2, C2C3, and C1C3). Since each pair of circles has two homothetic centers, there are six homothetic centers altogether. Gaspard Monge showed in the early 19th century that these six points lie on four lines, each line having three collinear points.\n\n## Problem of Apollonius\n\nMany special cases of Apollonius's problem involve finding a circle that is tangent to one or more lines. The simplest of these is to construct circles that are tangent to three given lines (the LLL problem). To solve this problem, the center of any such circle must lie on an angle bisector of any pair of the lines; there are two angle-bisecting lines for every intersection of two lines. The intersections of these angle bisectors give the centers of solution circles. There are four such circles in general, the inscribed circle of the triangle formed by the intersection of the three lines, and the three exscribed circles.\n\nA general Apollonius problem can be transformed into the simpler problem of circle tangent to one circle and two parallel lines (itself a special case of the LLC special case). To accomplish this, it suffices to scale two of the three given circles until they just touch, i.e., are tangent. An inversion in their tangent point with respect to a circle of appropriate radius transforms the two touching given circles into two parallel lines, and the third given circle into another circle. Thus, the solutions may be found by sliding a circle of constant radius between two parallel lines until it contacts the transformed third circle. Re-inversion produces the corresponding solutions to the original problem.\n\n## Generalizations\n\nThe concept of a tangent line to one or more circles can be generalized in several ways. First, the conjugate relationship between tangent points and tangent lines can be generalized to pole points and polar lines, in which the pole points may be anywhere, not only on the circumference of the circle. Second, the union of two circles is a special (reducible) case of a quartic plane curve, and the external and internal tangent lines are the bitangents to this quartic curve. A generic quartic curve has 28 bitangents.\n\nA third generalization considers tangent circles, rather than tangent lines; a tangent line can be considered as a tangent circle of infinite radius. In particular, the external tangent lines to two circles are limiting cases of a family of circles which are internally or externally tangent to both circles, while the internal tangent lines are limiting cases of a family of circles which are internally tangent to one and externally tangent to the other of the two circles.[5]\n\nIn M\u00f6bius or inversive geometry, lines are viewed as circles through a point \"at infinity\" and for any line and any circle, there is a M\u00f6bius transformation which maps one to the other. In M\u00f6bius geometry, tangency between a line and a circle becomes a special case of tangency between two circles. This equivalence is extended further in Lie sphere geometry.\n\nRadius and tangent line are perpendicular at a point of a circle, and hyperbolic-orthogonal at a point of the unit hyperbola. The parametric representation of the unit hyperbola via radius vector is ${\\displaystyle p(a)\\ =\\ (\\cosh a,\\sinh a).}$ The derivative of p(a) points in the direction of tangent line at p(a), and is ${\\displaystyle {\\frac {dp}{da}}\\ =\\ (\\sinh a,\\cosh a).}$ The radius and tangent are hyperbolic orthogonal at a since ${\\displaystyle p(a)\\ {\\text{and}}\\ {\\frac {dp}{da}}}$ are reflections of each other in the asymptote y=x of the unit hyperbola. When interpreted as split-complex numbers (where j j = +1), the two numbers satisfy ${\\displaystyle jp(a)\\ =\\ {\\frac {dp}{da}}.}$\n\n## References\n\n1. \"Finding tangents to a circle with a straightedge\". Stack Exchange. August 15, 2015.\n2. Paul Kunkel. \"Tangent circles\". Whistleralley.com. Retrieved 2008-09-29.\n3. Libeskind, Shlomo (2007), Euclidean and Transformational Geometry: A Deductive Inquiry, pp.\u00a0110\u2013112 (online copy, p. 110, at Google Books)\n4. Kunkel, Paul (2007), \"The tangency problem of Apollonius: three looks\" (PDF), BSHM Bulletin: Journal of the British Society for the History of Mathematics, 22 (1): 34\u201346, doi:10.1080\/17498430601148911","date":"2020-10-24 21:53:37","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 41, \"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\": 0.6780163645744324, \"perplexity\": 352.6955564166394}, \"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\/1603107884755.46\/warc\/CC-MAIN-20201024194049-20201024224049-00479.warc.gz\"}"}
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Q: SecTrustEvaluate always returns kSecTrustResultRecoverableTrustFailure with SecPolicyCreateSSL My application tries to evaluate a server trust certificate for a self signed certificate. This is working fine with SecPolicyCreateBasicX509 but not working for SecPolicyCreateSSL
Here is my code:
if (challenge.protectionSpace.authenticationMethod == NSURLAuthenticationMethodServerTrust) {
// create trust from protection space
SecTrustRef trustRef;
int trustCertificateCount = SecTrustGetCertificateCount(challenge.protectionSpace.serverTrust);
NSMutableArray* trustCertificates = [[NSMutableArray alloc] initWithCapacity:trustCertificateCount];
for (int i = 0; i < trustCertificateCount; i++) {
SecCertificateRef trustCertificate = SecTrustGetCertificateAtIndex(challenge.protectionSpace.serverTrust, i);
[trustCertificates addObject:(id) trustCertificate];
}
// set evaluation policy
SecPolicyRef policyRef;
// policyRef = SecPolicyCreateBasicX509(); this is working
policyRef = SecPolicyCreateSSL(NO, (CFStringRef)
SecTrustCreateWithCertificates((CFArrayRef) trustCertificates, policyRef, &trustRef);
[trustCertificates release];
// load known certificates from keychain and set as anchor certificates
NSMutableDictionary* secItemCopyCertificatesParams = [[NSMutableDictionary alloc] init];
[secItemCopyCertificatesParams setObject:(id)kSecClassCertificate forKey:(id)kSecClass];
[secItemCopyCertificatesParams setObject:@"Server_Cert_Label" forKey:(id)kSecAttrLabel];
[secItemCopyCertificatesParams setObject:(id)kCFBooleanTrue forKey:(id)kSecReturnRef];
[secItemCopyCertificatesParams setObject:(id)kSecMatchLimitAll forKey:(id)kSecMatchLimit];
CFArrayRef certificates;
certificates = nil;
SecItemCopyMatching((CFDictionaryRef) secItemCopyCertificatesParams, (CFTypeRef*) &certificates);
if (certificates != nil && CFGetTypeID(certificates) == CFArrayGetTypeID()) {
SecTrustSetAnchorCertificates(trustRef, certificates);
SecTrustSetAnchorCertificatesOnly(trustRef, NO);
}
SecTrustResultType result;
OSStatus trustEvalStatus = SecTrustEvaluate(trustRef, &result);
if (trustEvalStatus == errSecSuccess) {
if (result == kSecTrustResultConfirm || result == kSecTrustResultProceed || result == kSecTrustResultUnspecified) {
// evaluation OK
[challenge.sender useCredential:[NSURLCredential credentialForTrust: challenge.protectionSpace.serverTrust] forAuthenticationChallenge:challenge];
} else {
// evaluation failed
// ask user to add certificate to keychain
} else {
// evaluation failed - cancel authentication
[[challenge sender] cancelAuthenticationChallenge:challenge];
}
}
After a lot of research i have already made changes to the self-signed certificate by adding extension like mentioned in this post: Unable to trust a self signed certificate on iphone
Does anyone have another hint what might be missing here?
A: After a lot of testing I have worked out this problem. The following has been changed.
*
*The policy is set to NO for server evaluation. This means the certificate is checked for client authentication. Obviously the server certificate will not have this! Setting this to YES will actually check if extendedKeyUsage is set to serverAuth for the server certificate.
*SecTrustSetAnchorCertificates and SecTrustSetAnchorCertificatesOnly should always be called before evaluation and not only if you are providing your own anchor certificates. You need to call this with an empty array, otherwise the system known anchor certificates are not used for evaluation. Even installed trusted root certificates from MDM are working then.
Here is a working sample based on the first code:
if (challenge.protectionSpace.authenticationMethod == NSURLAuthenticationMethodServerTrust) {
// create trust from protection space
SecTrustRef trustRef;
int trustCertificateCount = SecTrustGetCertificateCount(challenge.protectionSpace.serverTrust);
NSMutableArray* trustCertificates = [[NSMutableArray alloc] initWithCapacity:trustCertificateCount];
for (int i = 0; i < trustCertificateCount; i++) {
SecCertificateRef trustCertificate = SecTrustGetCertificateAtIndex(challenge.protectionSpace.serverTrust, i);
[trustCertificates addObject:(id) trustCertificate];
}
// set evaluation policy
SecPolicyRef policyRef;
// set to YES to verify certificate extendedKeyUsage is set to serverAuth
policyRef = SecPolicyCreateSSL(YES, (CFStringRef) challenge.protectionSpace.host);
SecTrustCreateWithCertificates((CFArrayRef) trustCertificates, policyRef, &trustRef);
[trustCertificates release];
// load known certificates from keychain and set as anchor certificates
NSMutableDictionary* secItemCopyCertificatesParams = [[NSMutableDictionary alloc] init];
[secItemCopyCertificatesParams setObject:(id)kSecClassCertificate forKey:(id)kSecClass];
[secItemCopyCertificatesParams setObject:@"Server_Cert_Label" forKey:(id)kSecAttrLabel];
[secItemCopyCertificatesParams setObject:(id)kCFBooleanTrue forKey:(id)kSecReturnRef];
[secItemCopyCertificatesParams setObject:(id)kSecMatchLimitAll forKey:(id)kSecMatchLimit];
CFArrayRef certificates;
certificates = nil;
SecItemCopyMatching((CFDictionaryRef) secItemCopyCertificatesParams, (CFTypeRef*) &certificates);
if (certificates != nil && CFGetTypeID(certificates) == CFArrayGetTypeID()) {
SecTrustSetAnchorCertificates(trustRef, certificates);
SecTrustSetAnchorCertificatesOnly(trustRef, NO);
} else {
// set empty array as own anchor certificate so system anchos certificates are used too!
SecTrustSetAnchorCertificates(trustRef, (CFArrayRef) [NSArray array]);
SecTrustSetAnchorCertificatesOnly(trustRef, NO);
}
SecTrustResultType result;
OSStatus trustEvalStatus = SecTrustEvaluate(trustRef, &result);
if (trustEvalStatus == errSecSuccess) {
if (result == kSecTrustResultConfirm || result == kSecTrustResultProceed || result == kSecTrustResultUnspecified) {
// evaluation OK
[challenge.sender useCredential:[NSURLCredential credentialForTrust: challenge.protectionSpace.serverTrust] forAuthenticationChallenge:challenge];
}
else {
// evaluation failed
// ask user to add certificate to keychain
}
}
else {
// evaluation failed - cancel authentication
[[challenge sender] cancelAuthenticationChallenge:challenge];
}
}
Hope this will help someone.
A: It may be a server certificate problem....
Check here, I solved my kSecTrustResultRecoverableTrustFailure problem, adding subjectAltName = DNS:example.com into openssl config file, specifically in server key generation...
If you are not using openssl to generate it, I'm sorry but I can help you.. Anyway if you want to use openssl, here is a good tutorial to generate those keys and sign then with your own root certificate authority.
From this tutorial, I just changed my openssl server config file to:
[ server ]
basicConstraints = critical,CA:FALSE
keyUsage = digitalSignature, keyEncipherment, dataEncipherment
extendedKeyUsage = serverAuth
nsCertType = server
subjectAltName = IP:10.0.1.5,DNS:office.totendev.com
Hope it helps !
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 9,610
|
By: John L. Turner for CRESLI
At first glance Long Island's coastal environment might seem to be a fairly homogeneous place - simply the interface between land and water. Yet, a closer, more detailed look reveals a great diversity of habitats. These range from salt marshes fringing the shore of sheltered embayments to intertidal mud- and sandflats. They include subtidal bay bottoms, both muddy and sandy, some of which provides a substrate for ecologically important submerged aquatic vegetation such as eelgrass. There are the sandy beaches that take the brunt of the Atlantic Ocean's energy.
Even the ocean itself, an environment that might seem to be pretty homogeneous, turns out to be remarkably varied, given to differences in depth, salinity, temperature, and surface and subsurface currents. These differences often result in local concentration of food resources that are readily exploited by pelagic (ocean loving) birds. It is not surprising, therefore, that this diverse coastal environment provides suitable habitat to a surprising diversity of birds.
Following is a list of the birds that might be seen and studied in Long Island's coastal environment. The list is, of course, somewhat arbitrary. Birds that are occasional or are vagrants were excluded since the likelihood of seeing them is very slim. Also, birds that frequent habitats near the coast were excluded. These include, for example, waterfowl that exclusively frequent freshwater ponds near the coast but don't typically occur on brackish or salt water. The criteria used as to whether a species was included or not can be summed up in a question "Is there a reasonable chance I could see this bird if I made the effort at the right time of year?" If the answer is a definite or probable "no" then the bird was left off.
Family Procellariidae (Shearwaters and Petrels)
Northern Fulmar (Fulmarus glacialis) - a common pelagic in the winter; rarely seen from land. Tubenose on bill separates it from similar appearing gulls.
Sooty Shearwater (Ardenna grisea) - a common pelagic seabird especially in late spring; sightings are almost guaranteed during ffshore birding trips. Occasionally seen from land.
Great Shearwater (Puffinus gravis) - a very common pelagic seabird seen throughout the warmer months, when, like the preceding species, it is "overwintering", away from its southern hemisphere breeding grounds.
Cory's Shearwater (Calonectris borealis) - a common pelagic seabird in late summer. Can sometimes be seen from land.
Scopoli's Shearwater (Calonectric diomedea) - relative of Cory's, smaller, and with a lighter underwing pattern than Cory's.
Manx Shearwater (Puffinus puffinus) - an uncommon to occasional pelagic seabird during the summer months; probably found offshore during the winter too.
Audubon Shearwater (Puffinus Iherminieri) - regularly seen in small numbers offshore during the summer months; it is the smallest of the shearwater species that occur in coastal Long Island.
Family Hydrobatidae (Storm-Petrels)
Wilson's Storm-Petrel (Oceanites oceanicus) - probably the most common pelagic species seen off Long Island; flocks have been known to number in the thousands. Has unique flight behavior, often dipping its feet on the water, lending the appearance that it is walking on the water. Old-time mariners called the bird Mother Carey's chicken, a derivation from mater cara, the Virgin Mary, a reference to its habit of walking on the water surface. The word petrel is derived from "Little Peter", a reference to St. Peter who was able to walk on water with Christ's help. Breeds in Southern hemisphere
Leach's Storm-Petrel (Oceanodroma leucorhoa) - less abundant pelagic than the ubiquitous Wilson's Storm-Petrel; fairly common through the warmer months. Nests throughout northern Atlantic as far south as Massachusetts. Has distinctive forked tail.
Family Sulidae (Boobies & Gannets)
Northern Gannet (Morus bassanus) - a member of the Booby family; Common pelagic bird during spring and fall migration and as an overwintering species. Can often be seen from land especially at Montauk Point and along the south shore. Regular in small numbers in Long Island Sound. Spectacular plunges into schools of bait fish sending up plumes of spray. All age classes can be seen.
Family Pelecanidae (Pelicans)
Brown Pelican (Pelecanus occidentalis) - twenty years ago the sighting of a brown pelican in Long Island's coastal waters would have been cause for celebration. Now they are seen with some regularity in the summer along the south shore, often flying in small groups in typical flight behavior - low to the water in single file. They have expanded their breeding range northward over the past few decades and now breed on the DELMARVA peninsula.
Family Phalacrocoracidae (Cormorants)
Great Cormorant (Phalacrocorax carbo) - a regular winter visitor to Long Island. Is larger than, and has a distinctive white flank patch from, the following species.
Double-crested Cormorant (Phalacrocorax auritus) - an abundant coastal bird. It is unlikely not to be seen on any trip to the coast. Has recovered from past persecution to the point that contend it has become a pest. Has returned as breeding bird on Long Island.
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Sound::Sound(QObject *parent) : sm({"carState", "controlsState", "deviceState"}) {
qInfo() << "default audio device: " << QAudioDeviceInfo::defaultOutputDevice().deviceName();
for (auto &[alert, fn, loops] : sound_list) {
QSoundEffect *s = new QSoundEffect(this);
QObject::connect(s, &QSoundEffect::statusChanged, [=]() {
assert(s->status() != QSoundEffect::Error);
});
s->setVolume(Hardware::MIN_VOLUME);
s->setSource(QUrl::fromLocalFile("../../assets/sounds/" + fn));
sounds[alert] = {s, loops};
}
QTimer *timer = new QTimer(this);
QObject::connect(timer, &QTimer::timeout, this, &Sound::update);
timer->start(1000 / UI_FREQ);
};
void Sound::update() {
const bool started_prev = sm["deviceState"].getDeviceState().getStarted();
sm.update(0);
const bool started = sm["deviceState"].getDeviceState().getStarted();
if (started && !started_prev) {
started_frame = sm.frame;
}
// no sounds while offroad
// also no sounds if nothing is alive in case thermald crashes while offroad
const bool crashed = (sm.frame - std::max(sm.rcv_frame("deviceState"), sm.rcv_frame("controlsState"))) > 10*UI_FREQ;
if (!started || crashed) {
setAlert({});
return;
}
// scale volume with speed
if (sm.updated("carState")) {
float volume = util::map_val(sm["carState"].getCarState().getVEgo(), 11.f, 20.f, 0.f, 1.0f);
volume = QAudio::convertVolume(volume, QAudio::LogarithmicVolumeScale, QAudio::LinearVolumeScale);
volume = util::map_val(volume, 0.f, 1.f, Hardware::MIN_VOLUME, Hardware::MAX_VOLUME);
for (auto &[s, loops] : sounds) {
s->setVolume(std::round(100 * volume) / 100);
}
}
setAlert(Alert::get(sm, started_frame));
}
void Sound::setAlert(const Alert &alert) {
if (!current_alert.equal(alert)) {
current_alert = alert;
// stop sounds
for (auto &[s, loops] : sounds) {
// Only stop repeating sounds
if (s->loopsRemaining() > 1 || s->loopsRemaining() == QSoundEffect::Infinite) {
s->stop();
}
}
// play sound
if (alert.sound != AudibleAlert::NONE) {
auto &[s, loops] = sounds[alert.sound];
s->setLoopCount(loops);
s->play();
}
}
}
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<?xml version="1.0" encoding="UTF-8"?>
<beans xmlns="http://www.springframework.org/schema/beans"
xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance"
xmlns:context="http://www.springframework.org/schema/context"
xsi:schemaLocation="http://www.springframework.org/schema/beans
http://www.springframework.org/schema/beans/spring-beans-4.0.xsd
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Defines a pluggable datasource which allow varying connection pool type
and database type without having to recompile the application.
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ignore-unresolvable="true" order="0"/>
<import resource="classpath:com/github/rbuck/dash/services/spring/spring-${dash.db.pool.type:bonecp}-ds.xml"/>
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{"url":"http:\/\/physics.aps.org\/story\/v26\/st15","text":"# Focus: Vibrations Skip Across Vacuum\n\nPhys. Rev. Focus 26, 15\nHeat-carrying vibrational waves can be transmitted across a small vacuum barrier between a metal point and a flat surface, according to a new experiment.\n\nThis story was slightly revised on 11 October in the description of the 2005 experiment.\n\nThe Sun warms the Earth by radiating heat across the emptiness of space, but the physics of heat transfer is not well understood for very short distances. An experiment reported in the 15 October Physical Review Letters demonstrates that a sharp metal point will emit heat at a rate many times greater than normal when brought within a few atom-lengths of a cold, metal surface. The warm tip transfers heat through its fluctuating electric fields, which jiggle electrons in the surface. The researchers call the effect \u201cphonon tunneling\u201d because quantized molecular vibrations called phonons seemingly hop over the forbidden vacuum gap. The phenomenon could be important for future nanoscale devices, the researchers say.\n\nHeat can flow between two objects at different temperatures in three possible ways: conduction, convection, and radiation. Convection requires a medium, such as air, to carry the heat, whereas radiation refers to electromagnetic waves that can travel through a vacuum. Conduction only occurs when the objects are touching, when faster-vibrating atoms in the hotter body jiggle the slower atoms in the colder body. Researchers describe this process as transmission of phonons (quanta of vibration), and it would seem to be impossible between separated objects. \u201cThe vacuum is a forbidden zone for phonons,\u201d says Igor Altfeder of the Air Force Research Laboratory at Wright-Patterson Air Force Base in Ohio.\n\nBut heat transfer at short range shows some surprises. In one recent experiment, a sharp point or \u201ctip\u201d was placed near a cold surface, and the heat flow did not match theoretical predictions at separations less than 10 nanometers. However, the experimental design did not permit a measurement of the temperature at the end of the tip.\n\nTo measure that temperature and better understand what was happening, Altfeder and his colleagues used a technique called inelastic electron tunneling spectroscopy (IETS). It involves applying a range of voltages between the tip and the surface and recording the current produced by electrons jumping across the gap. These electrons are affected by thermal vibrations in the materials, so measuring the current leads to the temperature, especially that of the very last atom or molecule on the tip.\n\nUnder high vacuum conditions, the team brought a platinum-iridium tip at 275 Kelvin to within 0.3 nanometers of a gold surface cooled to 90, 150 or 210 Kelvin. From the IETS data, they found the tip temperature but also discovered that a single carbon monoxide (CO) molecule had attached to the very end of their tip. This is not uncommon, Altfeder says, but it turned out to be fortuitous, because the CO molecule was thermally isolated to some extent from the rest of the tip. The team was surprised to find that the CO molecule had roughly the same temperature as the surface, implying that heat was escaping from the tip-end at an enormous rate, much faster than in the previous experiment.\n\nThe researchers explain the cooling as a result of electric fields that have influence across the narrow gap. The electric field from the CO molecule rearranges the top layer of electrons in the gold surface. From classical electrodynamics, this rearrangement can be represented more simply as a single \u201cimage charge\u201d underneath the surface. The CO molecule and its image vibrate in unison, but the shaking of the image charge\u2013or alternatively, the surface electrons\u2013induces heat-carrying phonons inside the gold, assuming a strong electron-phonon coupling.\n\nThis transmission of vibrations across a forbidden gap has previously been called phonon tunneling because it mimics quantum tunneling\u2013for example, when electrons cross this same vacuum gap, even though they lack the energy to escape the tip. But in past experiments, the gap was another material, rather than a vacuum. The authors calculate that the CO molecule dissipates heat ${10}^{10}$ times faster by generating phonons in the gold than by emitting radiation directly into the vacuum.\n\nThe proposed mechanism coincides with recent theoretical work by Mika Prunnila and Johanna Meltaus of the VTT Technical Research Centre of Finland, which predicted phonon tunneling between nearby piezoelectric materials that by definition have a strong coupling between phonons and electric field variations [2]. Prunnila thinks this research will be applicable in nanoelectronics and the development of devices that harvest energy from temperature gradients. \u201cAcoustic phonons are the major heat carriers in many solids, and in this sense it is somewhat surprising that phonon tunneling didn\u2019t receive more attention previously,\u201d he says.\n\n\u2013Michael Schirber\n\nMichael Schirber is a Corresponding Editor for Physics\u00a0based based in Lyon, France.\n\n## References\n\n1. A. Kittel, W. M\u00fcller-Hirsch, J. Parisi, S.-A. Biehs, D. Reddig, and M. Holthaus, \u201cNear-Field Heat Transfer in a Scanning Thermal Microscope,\u201d Phys. Rev. Lett. 95, 224301 (2005)\n2. M. Prunnila and J. Meltaus, \u201cAcoustic Phonon Tunneling and Heat Transport due to Evanescent Electric Fields,\u201d Phys. Rev. Lett. 105, 125501 (2010)\n\nNanophysics\n\n## Related Articles\n\nNanophysics\n\n### Synopsis: Coulomb Drag in a Double Dot\n\nElectric current passing through a quantum dot can generate current in a nearby dot through a coordinated tunneling mechanism. Read More \u00bb\n\nAtomic and Molecular Physics\n\n### Synopsis: Taking Pictures with Single Ions\n\nA new ion microscope with nanometer-scale resolution builds up images using single ions emitted one at a time from an ion trap. Read More \u00bb\n\nMagnetism\n\n### Synopsis: Measuring Spin One Atom at a Time\n\nElectron microscopy experiments have measured the spin state of individual metal atoms on a graphene layer, characterizing their potential for information storage applications. \u00a0 Read More \u00bb","date":"2016-09-30 08:24:13","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 1, \"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\": 0.5508395433425903, \"perplexity\": 1652.0677769347888}, \"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-2016-40\/segments\/1474738662133.5\/warc\/CC-MAIN-20160924173742-00185-ip-10-143-35-109.ec2.internal.warc.gz\"}"}
| null | null |
Санта Круз Папалутла има више значења:
Општина Санта Круз Папалутла (Оахака), општина у савезној држави Оахака у Мексику
Санта Круз Папалутла (Санта Круз Папалутла, Оахака), насеље у савезној држави Оахака у Мексику
|
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"redpajama_set_name": "RedPajamaWikipedia"
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\section{Introduction}
The problem of obtaining a variational description of mechanical systems subject to external forces has been present in the literature for long time. Concretely, this article is concerned with the variational nature of the dynamical equations of a Lagrangian system subject to what we call {\it fractional damping}, namely:
\begin{equation}\label{IntEqCont}
\frac{d}{dt}\left(\frac{\partial L}{\partial\dot x}\right)-\frac{\partial L}{\partial x}=-\rho D^{\alpha}_{-}D^{\alpha}_{-}x,
\end{equation}
where $x$ is the dynamical variable, $L(x,\dot x)$ is a Lagrangian function, $\rho\in\R_+$ and $D^{\alpha}_{-}$, with $\alpha\in[0,1]$, is the retarded fractional derivative defined below \eqref{FracDer} (as usual, the dot notation represents the time derivative). The right hand side is non-local in time, and therefore the previous equation represents a particular example of non-locally damped mechanical system (we shall focus on Lagrangians given by kinetic minus potential energy), which are ubiquitous in mechanical engineering applications (see \cite{AdWoo,Rudin,Sri} and references therein). In addition, \eqref{IntEqCont} also involves the linear damping case, since $D^{2\alpha}_{-}x= \dot x$ when $\alpha= 1/2$, where we have used \eqref{Aditive},\eqref{Alpha1}. This case is a pardigmatic example of non-Lagrangian/Hamiltonian system, i.e.~ its dynamics cannot be obtained from the Hamilton's principle \cite{AbMa} given a Lagrangian/Hamiltonian function, as proven in \cite{Bauer}. A remarkable approach towards the variational modelling of external forces, due to its phenomenological versatility, is the Lagrange-d'Alembert principle \cite{Bloch}, where the variation of the action is set equal to the work done by external forces under virtual displacements. As happens with the usual Hamilton's principle, the Lagrange-d'Alembert's can be performed in Lagrangian and Hamiltonian fashions, both related to each other by the Legendre transformation \cite{AbMa}. It is important to note, however, that Lagrange-d'Alembert principle is not variational in the pure sense on the word, circumstance that we try to avoid in our apprach.
There are previous approaches to generate a purely variational principle by including fractional derivatives into the state space of the considered Lagrangian functions; more concretely, we base our work on \cite{Cresson, Riewe}. These references delivered promising results, but they present some drawbacks. Say: in \cite{Riewe} Riewe did not take into account the asymmetric integration by parts of the fractional derivatives \eqref{IntegrationByParts}; Cresson and collaborators, in \cite{Cresson,CressonBook}, designed the so-called {\it asymmetric embeddings} to surpass that issue, but such objects are unclear from the point of view of calculus of variations. This last approach implies the doubling of curves in the state space; i.e. it is necessary to add an extra $y$-mirror system, which is natural when treating externally forced system in a variational way, see \cite{Bateman, Galley, DaRo} and references therein.
Taking as well into account a doubled space of curves, in this work we establish a novel approach to surpass the asymmetry issue while obtaining the correct fractional damped dynamics \eqref{IntEqCont}, embodied in a particular restriction of the usual calculus of variations. Namely, we will apply the Hamilton's principle over a properly designed (in a geometric way) Lagrangian function, but we shall restrict the class of varied curves. We denote this principle by {\it restricted Hamilton's principle}. Out of this principle, we shall obtain both \eqref{IntEqCont} and the dynamics of the $y$-mirror system. However, we shall rigorously prove, both in Lagrangian and Hamiltonian settings, that the latter is nothing but the former in reversed time, which implies that the restricted Hamilton's principle is {\it not} adding extra physics to the studied system. As it is made clear below, the price to pay for applying this new principle is that the dynamical equations are not anymore necessary and sufficient conditions for the extremals of the action (as in the classical Hamilton's principle), but only sufficient. Finally, given that the linear damping case is also included, we elucidate the connection between the restricted Hamilton's and Lagrange-d'Alembert principles.
\medskip
During the last years, the discretisation of variational principles (Hamilton's, Lagrage-d'A\-lem\-bert and others \cite{MaWe,ObJuMa}) has been of high interest in numerical integration theory, since such discretisations generate numerical integrators that approximate the systems' dynamics faithfully both from dynamical and geometrical perspectives, presenting as well a superior behaviour in the integration of the energy in the long-term. Thus, we proceed to discretise the restricted Hamilton's principle, which leads to the discrete counterpart of \eqref{IntEqDisc}, i.e.:
\begin{equation}\label{IntEqDisc}
D_1L_d(x_k,x_{k+1})+D_2L_d(x_{k-1},x_{k})=h\,\rho\,\Delta_{-}^{\alpha}\Delta_{-}^{\alpha}x_k,
\end{equation}
where $x_k$ is the discrete dynamical variable, $L_d(x_k,x_{k+1})$ the discrete Lagrangian and $\Delta_{-}^{\alpha}x_k$ is a given approximation of $D^{\alpha}_{-}x(t_k)$ for a time grid $t_k$ with time step $h$ (other approaches to discretise fractional mechanical problems can be found in \cite{Bastos, Cresson1, CreGre, CressonBook} and references therein). The elements of the discrete principle are analogous to the continuous' ones: the discrete dynamical equations are sufficient conditions for the extremals of the discrete action, and we have an extra $y$-mirror system (which again is proven to be the $x$-system in reversed discrete time). The initialisation of \eqref{IntEqDisc} as a numerical integrator of \eqref{IntEqCont} (which we will call Fractional Variational Integator, FVI) requires an initial condition which is based on the discrete Legendre transform linking the Lagrangian and Hamiltonian approaches. With that aim, we develop the restricted Hamilton's principle also in a Hamiltonian fashion (both in continuous and discrete scenarios), where, as in the usual discrete mechanics \cite{MaWe}, the momentum mathching condition will be a crucial element when constructing numerical integrators approximating the Hamiltonian version of \eqref{IntEqCont}. As in the continuous side, we elucidate the connection between the restricted Hamilton's and Lagrange-d'Alembert principles in the discrete side. Finally, we test the generated integrators with respect to well-known examples. With that aim, we use a particular benchmark approximation of the solution of inhomogeneous fractional-differential equations \cite{Vinagre}, based on a matrix discretisation of the fractional-differential operators and, afterwards, the resolution of the discrete dynamics as a matricial-algebraic equation \cite{Podlubny}.
\medskip
\hspace{-0.6cm} The paper is organised as follows: In \S\ref{Preli} we introduce the basics on fractional derivatives (since they will be used as a tool, we do not put emphasis on the mathematical specifics, and refer the interested reader to the proper literature); moreover we present the variational description of Lagrangian/Hamiltonian systems (Hamilton's principle) both in the continuous and discrete settings, as well as externally forced systems (Lagrange-d'Alembert principle). \S\ref{CRHP} is devoted to develop the continuous restricted Hamilton's principle, which is stated in its Lagrangian version in Theorem \ref{ConsFraELLag}. The relationship between the $x$-system and the $y$-system in reversed time is given in Proposition \ref{InvTime}. The preservation of a particular presymplectic form by \eqref{IntEqCont} is established in Proposition \ref{GeomPreservation}, for which the existence and uniqueness of solutions of \eqref{IntEqCont} is necessary, whose proof is developed in Appendix. The Hamiltonian version of the restricted Hamilton's principle is given in Theorem \ref{ConsFraELHam}. \S\ref{DRHP} accounts for the discrete restricted Hamilton's principle, established in Proposition \ref{FullLagProposition} and Theorem \ref{Theo:DiscEqs}. The relationship between the $x$-system and the $y$-system in reversed discrete time is given in Proposition \ref{InvDiscTime}. The connection between the discrete restricted Hamilton's principle and the discrete Lagrange-d'Alembert's is given in Corollary \ref{Corolario}, for which Lemma \ref{LemmaDisc} is essential. The definition of the discrete Legendre transform, needed for the initialisation of the FVIs, is established in Definition \ref{Def:DLT}, whereas the momentum matching condition and the Hamiltonian version of the FVIs is given in Proposition \ref{MMPropo}. Finally, in \S\ref{Simu} we display the performance of the FVIs in two examples.
\section{Preliminaries}\label{Preli}
\subsection{Fractional derivatives}\label{FracDerPreli}
Let $\alpha\in[0,1]\subset\R$ and $f:[a,b]\subset\R\rightarrow\R$, $[a,b]\subset \R$, a smooth function. The fractional derivatives are defined by
\begin{small}
\begin{subequations}\label{FracDer}
\begin{align}
& _{_{RL}}D^{\alpha}_{-}f(t)=\frac{1}{\Gamma(1-\alpha)}\frac{d}{dt}\int_a^t(t-\tau)^{-\alpha}f(\tau)d\tau,\,\,\, _{_{RL}}D^{\alpha}_{+}f(t)=-\frac{1}{\Gamma(1-\alpha)}\frac{d}{dt}\int_t^b(\tau-t)^{-\alpha}f(\tau)d\tau,\label{FracDer:RL}\\
&\,\, _{_{C}}D^{\alpha}_{-}f(t)=\frac{1}{\Gamma(1-\alpha)}\int_a^t(t-\tau)^{-\alpha}\dot f(\tau)d\tau,\,\,\,\quad\,\,\, _{_{C}}D^{\alpha}_{+}f(t)=-\frac{1}{\Gamma(1-\alpha)}\int_t^b(\tau-t)^{-\alpha}\dot f(\tau)d\tau,\label{FracDer:Ca}
\end{align}
\end{subequations}
\end{small}
for $t\in$\,[a,b], and $\Gamma(z)$ the gamma function, in their Riemann-Liouville and Caputo expressions. These two kinds of fractional derivatives are related to each other, indeed it can be shown that:
\begin{equation}\label{Relation}
_{_{RL}}D^{\alpha}_{-}f(t)=\frac{-1}{\Gamma(1-\alpha)}\frac{f(a)}{(t-a)^{\alpha}}+_{_{C}}D^{\alpha}_{-}f(t),\,\,\,\,\,\,\, _{_{RL}}D^{\alpha}_{+}f(t)=\frac{1}{\Gamma(1-\alpha)}\frac{f(b)}{(b-t)^{\alpha}}+ _{_{C}}D^{\alpha}_{+}f(t).
\end{equation}
In this work, the function $f$ will represent the dynamical variable of a mechanical system. Thus, it is interesting to remark that we can always set $f(a)=0$, i.e. the system is at the origin of coordinates at initial time, and consequently both retarded Riemman-Liouville and Caputo versions are equivalent, according to the first equation in \eqref{Relation}\footnote{This is particularly apparent for $\alpha=0$. Namely: $_{_{RL}}D^{0}_{-}f(t)=f(t)$, whereas $_{_{C}}D^{0}_{-}f(t)=f(t)-f(a)$.}, for dynamical purposes. We will assume henceforth that $D_{\pm}^{\alpha}$ is determined by the Riemann-Liouville expression \eqref{FracDer:RL}, unless otherwise stated. Further relevant properties are:
\begin{subequations}
\begin{align}
\int_a^bf(t)D^{\alpha}_{\sigma}g(t)dt&=\int_a^b\left( D^{\alpha}_{-\sigma}f(t)\right) g(t)dt,\,\, \sigma=\left\{-,+\right\}, \label{IntegrationByParts}\\
D_{\sigma}^{\alpha}D_{\sigma}^{\beta}&=D_{\sigma}^{\alpha+\beta},\quad 0\leq\alpha,\beta\leq1,\label{Aditive}\\
D_{-}^{\alpha}= d/dt,\quad & D_{+}^{\alpha}= -d/dt,\quad\mbox{when}\quad \alpha= 1. \label{Alpha1}
\end{align}
\end{subequations}
For the proof of these properties and more details on fractional derivatives, we refer to \cite{TheBook}.
\subsection{Continuous Lagrangian and Hamiltonian description of mechanics}\label{ContinuousFramework}
In this subsection we shall consider the configuration space of the studied systems as a finite dimensional smooth manifold $Q$. Moreover, $TQ$ and $T^*Q$ will denote its tangent and cotangent bundles, locally represented by coordinates $(q,\dot q)$ and $(q,p)$, respectively. For more details on the geometric formulation of mechanics we refer to \cite{AbMa}.
\subsubsection{Conservative systems}
Given a Lagrangian function $L:TQ\rightarrow\R$, the associated action functional in the time interval $t\in[a,b]\subset\R$ for a smooth curve $q:[a,b]\rightarrow Q$ is defined by $S(q)=\int_a^bL(q(t),\dot q(t))\,dt$. Through Hamilton's principle, i.e. the true evolution of the system $q(t)$ with fixed endpoints $q(a)$ and $q(b)$ will satisfy
\begin{equation}\label{HPrinciple}
\delta\int_a^bL(q(t),\dot q(t))\,dt=0,
\end{equation}
we obtain the Euler-Lagrange equations via calculus of variations:
\begin{equation}\label{ELeqs}
\frac{d}{dt}\left(\frac{\partial L}{\partial\dot q}\right)-\frac{\partial L}{\partial q}=0.
\end{equation}
Define the Legendre transformation:
\begin{equation}\label{LegendreTrans}
\mathbb{F} L:TQ\rightarrow T^*Q; \quad (q,\dot q)\mapsto \left(q,p=\frac{\partial L}{\partial\dot q}\right).
\end{equation}
If \eqref{LegendreTrans} is a global diffeomorphism we say that it is {\it hyperregular}, and we call the Lagrangian function {\it hyperregular}. Under the assumption of hyperregularity (which we will take throughout the article), via the Legendre transformation we can define the Hamiltonian function $H:T^*Q\rightarrow\R$:
\begin{equation}\label{HamFunc}
H(q,p):=\langle p,\dot q\rangle-L(q,\dot q),
\end{equation}
where $\langle\cdot,\cdot\rangle:T^*Q\times TQ\rightarrow\R$ is the natural pairing. From the definition of the Hamiltonian function \eqref{HamFunc} it follows that $L(q,\dot q)=\langle p,\dot q\rangle-H(q,p)$. Furthermore, from \eqref{HPrinciple} we can write the stationary condition of the action functional in a Hamiltonian version, i.e.
\begin{equation}\label{HHPrin}
\delta\int_a^b\,\left\{\langle p(t),\dot q(t)\rangle-H(q(t),p(t))\right\}\,dt=0.
\end{equation}
Again, using calculus of variations we obtain the Hamilton equations:
\begin{equation}\label{Heqs}
\dot q=\frac{\partial H}{\partial p},\quad\,\,\, \dot p=-\frac{\partial H}{\partial q}.
\end{equation}
We shall consider that the physical energy of the system is given by the Hamiltonian function \eqref{HamFunc}. It is easy to check that $dH(q,p)/dt=0$ under \eqref{Heqs}, showing that the system is conservative. Equivalently, the Lagrangian energy
\begin{equation}\label{LagEner}
E(q,\dot q):=\Big<\frac{\partial L}{\partial\dot q},\dot q\Big>-L(q,\dot q),
\end{equation}
is invariant under \eqref{ELeqs}, i.e. $dE(q,\dot q)/dt=0.$
\subsubsection{Forced systems} \label{ContForcedSystems}
First we model the external forces (which might include damping, dragging, etc.) through the mapping:
\begin{equation}\label{ExForces}
f_L:TQ\rightarrow T^*Q.
\end{equation}
The forced dynamics is provided by the Lagrange-d'Alembert principle \cite{Arn, Bloch}: the true evolution of the system $q(t)$ between fixed points $q(a)$ and $q(b)$ will satisfy
\begin{equation}\label{LdAPrinL}
\delta\int_a^bL(q,\dot q)\,dt+\int_a^b\langle f_L(q,\dot q),\delta q\rangle\,dt=0,
\end{equation}
where $\delta q\in TQ$, which provides the forced Euler-Lagrange equations:
\begin{equation}\label{ForcedEL}
\frac{d}{dt}\left(\frac{\partial L}{\partial\dot q}\right)-\frac{\partial L}{\partial q}=f_L(q,\dot q).
\end{equation}
Now, the Lagrangian energy of the system \eqref{LagEner} is not preserved by \eqref{ForcedEL}. In particular $dE(q,\dot q)/dt=\big< f_L(q,\dot q),\dot q\big>$, showing that this kind of systems is not conservative.
The dual version of the Lagrange-d'Alembert principle \eqref{LdAPrinL} is naturally obtained through the Legendre transformation \eqref{LegendreTrans}. The dual external forces $f_H:T^*Q\rightarrow T^*Q$ are defined by $f_H:=f_L\circ \left(\mathbb{F} L\right)^{-1}$ (we recall that we are assuming $L$ hyperregular), while the dynamics is established by
\begin{equation}\label{LdAPrinH}
\delta\int_a^b\left\{\langle p,\dot q\rangle-H(q,p)\right\}\,dt+\int_a^b\langle f_H(q, p),\delta q\rangle\,dt=0,
\end{equation}
yielding the forced Hamilton equations:
\begin{equation}\label{ForcedHam}
\dot q=\frac{\partial H}{\partial p},\quad\,\,\, \dot p=-\frac{\partial H}{\partial q}+f_H(q,p).
\end{equation}
Obviously, the Hamiltonian function \eqref{HamFunc} is not preserved under \eqref{ForcedHam}. In particular $dH(q,p)/dt=\big<f_H(q,p),\frac{\partial H}{\partial p}\big>$.
\subsection{Discrete Lagrangian and Hamiltonian description of mechanics}\label{DiscreteFramework}
\subsubsection{Conservative systems}
The construction of the discrete version of mechanics relies on the substitution of $TQ$ by the Cartesian product $Q\times Q$ (note that these two spaces contain the same amount of information at local level) \cite{MaWe,MoVe}. The continuous curves $q(t)$ will be replaced by discrete ones, say $\gamma_d=\left\{ q_k\right\}_{0:N}:=\left\{ q_0,q_1,...,q_N\right\}\in Q^{N+1}$, where $N\in \N$ and the power $N+1$ indicates the Cartesian product of $N+1$ copies of $Q$. Given an increasing sequence of times $\left\{ t_k=a+hk\,|\,k=0,...,N\right\}\subset \R$, with $h=(b-a)/N$, the points in $\gamma_d$ will be considered as an approximation of the continuous curve at time $t_k$, i.e. $q_k\simeq q(t_k).$ Defining the discrete Lagrangian $L_d:Q\times Q\rightarrow\R$ as an approximation of the action integral in one time step, say $L_d(q_k,q_{k+1},h)\simeq \int_{t_k}^{t_k+h}L(q(t),\dot q(t))\,dt$ (we shall omit the $h$ dependence of the discrete Lagrangian unless needed), we can establish the so called discrete action sum:
\begin{equation}\label{AcSum}
S_d(\gamma_d)=\sum_{k=0}^{N-1}L_d(q_k,q_{k+1}).
\end{equation}
Applying the Hamilton's principle over \eqref{AcSum}, i.e. considering variations of $\gamma_d$ with fixed endpoints $q_0=q(a),\, q_{N}=q(b)$ and extremizing $S_d$, we obtain the discrete Euler-Lagrange equations
\begin{equation}\label{DEL}
D_1L_d(q_k,q_{k+1})+D_2L_d(q_{k-1},q_{k})=0,\quad k=1,...,N-1,
\end{equation}
where $D_1$ and $D_2$ denote the partial derivative with respect to the first and second variables, respectively. If $L_d$ is regular, i.e.~the matrix $\big[D_{12}L_d\big]$ is invertible, the equations \eqref{DEL} define a discrete Lagrangian flow $F_{L_d}:Q\times Q\rightarrow Q\times Q$; $(q_{k},q_{k+1})\mapsto (q_{k+1},q_{k+2})$, which is normally called {\it variational integrator} of the continuous dynamics provided by the Euler-Lagrange equations \eqref{ELeqs} (indistinctly, we shall call the equations \eqref{DEL} also variational integrator). Moreover, \eqref{DEL} are a discretisation in finite differences of \eqref{ELeqs}.
In order to establish the Hamiltonian picture we need to introduce the discrete Legendre transforms.
From $L_d$, two of them can be defined:
\[
\mathbb{F}^{-}L_d,\mathbb{F}^{+}L_d:Q\times Q\rightarrow T^*Q,
\]
in particular
\begin{subequations}\label{DLT}
\begin{align}
\mathbb{F}^{-}L_d(q_k,q_{k+1})&=(\,\,\,q_k\,\,\,,p_k^{-}=-D_1L_d(q_k,q_{k+1})),\label{DLT:a}\\
\mathbb{F}^{+}L_d(q_k,q_{k+1})&=(q_{k+1},p_{k+1}^{+}=D_2L_d(q_k,q_{k+1})).\label{DLT:b}
\end{align}
\end{subequations}
We observe that the {\it momentum matching} condition, i.e.
\begin{equation}\label{MomMat}
p_k^{-}=p_{k}^{+},
\end{equation}
provides the discrete Euler-Lagrange equations \eqref{DEL} according to \eqref{DLT} (based on this, we shall refer indistinctly to the discrete Legendre transform as momentum matching). Under the regularity of $L_d$, both discrete Legendre transforms are invertible and the discrete Hamiltonian flow $\tilde F_{L_d}:T^*Q\rightarrow T^*Q$; $(q_k,p_{k})\mapsto (q_{k+1},p_{k+1})$ can be defined by any of the following identities:
\begin{equation}\label{DiscHamFlow}
\tilde F_{L_d}=\mathbb{F}^{+}L_d\circ (\mathbb{F}^{-}L_d)^{-1}=\mathbb{F}^{+}L_d\circ F_{L_d}\circ (\mathbb{F}^{+}L_d)^{-1}=\mathbb{F}^{-}L_d\circ F_{L_d}\circ (\mathbb{F}^{-}L_d)^{-1};
\end{equation}
see \cite{MaWe} for the proof. At the Hamiltonian level, the map $\tilde F_{L_d}$ is called {\it variational integrator} of the continuous dynamics provided by the Hamilton equations \eqref{Heqs}. Moreover, the discrete equations provided by \eqref{DLT} are a discretisation in finite differences of \eqref{Heqs}.
\begin{remark}\label{InitialFree}
Another advantage of the Hamiltonian version of variational integrators is that it provides a natural way of initialisating the numerical scheme. As showed in the previous discussion, two points $q_0,q_1$ are necessary in order to start \eqref{DEL} and establish the discrete flow $F_{L_d}$. On the other hand, a mechanical problem involves as initial data $q(a)=q_0$, $\dot q(a)=v_0$ and $p(a)=p_0$; thus an extra step, providing $q_1$, is in order. This step is naturally determined by \eqref{DLT:a}, leading to the algorithm (where the regularity of $L_d$ is assumed):
\begin{algorithm}{\rm Variational Integrator Scheme}
\label{Algorithm}
\begin{algorithmic}[1]
\State {\bf Initial data}: $N,\, h,\, q_0,\, p_0.$
\State {\bf solve for} $q_1$ {\bf from} $p_0=-D_1L_d(q_0,q_1).$
\State {\bf Initial points:} $q_0,\,q_1.$
\For {$k= 1: N-1$}
{\bf solve for} $q_{k+1}$ {\bf from} $D_1L_d(q_k,q_{k+1})+D_2L_d(q_{k-1},q_{k})=0$
\EndFor
\State {\bf Output:} $(q_2,...,q_N).$
\end{algorithmic}
\end{algorithm}
Other definitions of $q_1$, different from Step 2, can lead to the instability of the discrete flow $F_{L_d}$. \hfill $\diamond$
\end{remark}
A crucial feature of variational integrators is their {\it symplecticity}. If $\Omega_{T^*Q}$ is the canonical symplectic form on $T^*Q$ (which, according to Darboux theorem, can be locally written as $\Omega_{T^*Q}=dq\wedge dp$), define $\Omega_{Q\times Q}:=(\mathbb{F}^{-}L_d)^*\Omega_{T^*Q}=(\mathbb{F}^{+}L_d)^*\Omega_{T^*Q}$. Thus, the symplecticity of the variational integrators imply $F_{L_d}^*\Omega_{Q\times Q}=\Omega_{Q\times Q}$ \cite{MaWe, MoVe}, which furthermore imply that the energy cannot be conserved at the same time \cite{GeMa}. However, symplectic integrators have proven to present a stable energy behaviour even in long-term simulations \cite{SS}, behaviour that can be explained in terms of Backward Error Analysis \cite{BEA,HLW}.
\subsubsection{Forced systems} \label{DiscLdAlem}
As discrete version of the external forces \eqref{ExForces} we consider the maps:
\[
f_{L_d}^{-}, f_{L_d}^{+}:Q\times Q\rightarrow T^*Q
\]
such that
\[
\langle f_{L_d}^{-}(q_k,q_{k+1}),\delta q_k\rangle+\langle f_{L_d}^{+}(q_k,q_{k+1}),\delta q_{k+1}\rangle\simeq \int_{t_k}^{t_k+h}\langle f_L(q,\dot q),\delta q\rangle\,dt.
\]
Note that the previous equation implies that $f_{L_d}^{-}(q_k,q_{k+1})\in T^*_{q_k}Q$ and $f_{L_d}^{+}(q_k,q_{k+1})\in T^*_{q_{k+1}}Q$. The discrete Lagrange-d'Alembert principle \cite{MaWe,ObJuMa} provides discrete curves between fixed $q_0,q_{N}$ satisfying the critical condition
\[
\delta\sum_{k=0}^{N-1}L_d(q_k,q_{k+1})+\sum_{k=0}^{N-1}\Big[\langle f_{L_d}^{-}(q_k,q_{k+1}),\delta q_k\rangle+\langle f_{L_d}^{+}(q_k,q_{k+1}),\delta q_{k+1}\rangle\Big]=0.
\]
These curves are given by the forced discrete Euler-Lagrange equations
\begin{equation}\label{ForcedDEL}
D_1L_d(q_k,q_{k+1})+D_2L_d(q_{k-1},q_{k})+f_{L_d}^{-}(q_k,q_{k+1})+f_{L_d}^{+}(q_{k-1},q_{k})=0,\,\, k=1,...,N-1;
\end{equation}
they are a discretisation in finite differences of \eqref{ForcedEL} and, under the regularity of the matrix $\big[D_{12}L_d(q_k,q_{k+1})+D_2f_{L_d}^{-}(q_k,q_{k+1})\big]$, provide a forced discrete Lagrangian map $F_{L_d}^{\tiny \mbox{f}}:Q\times Q\rightarrow Q\times Q$ approximating their continuous solution. In the forced case, the discrete Legendre transformation is defined by
\begin{subequations}\label{DForcedLT}
\begin{align}
\mathbb{F}^{-}L_d^{\tiny \mbox{f}}(q_k,q_{k+1})&=(\,\,\,q_k\,\,\,,\,\,p_k^{-}=-D_1L_d(q_k,q_{k+1})-f_{L_d}^{-}(q_k,q_{k+1})),\label{DForcedLT:a}\\
\mathbb{F}^{+}L_d^{\tiny \mbox{f}}(q_k,q_{k+1})&=(q_{k+1},\,\,p_{k+1}^{+}=D_2L_d(q_k,q_{k+1})+f_{L_d}^{+}(q_k,q_{k+1})).\label{DForcedLT:b}
\end{align}
\end{subequations}
The momentum matching condition \eqref{MomMat} reproduces the forced discrete Euler-Lagrange equations \eqref{ForcedDEL}. Moreover, \eqref{DForcedLT} provide a discretisation in finite differences of \eqref{ForcedHam}, whereas \eqref{DiscHamFlow} (for $F_{L_d}^{\tiny \mbox{f}}$ and $\mathbb{F}^{\pm}L_d^{\tiny \mbox{f}}$) yields an approximation $\tilde F_{L_d}^{\tiny \mbox{f}}:T^*Q\rightarrow T^*Q$ of their continuous flow.
\begin{remark}
Establishing the forced variational flow $F_{L_d}^{\tiny \mbox{f}}:Q\times Q\rightarrow Q\times Q$ requires as well two initial points $q_0,q_1$ as in the free case discussed in Remark \ref{InitialFree}. According to \eqref{DForcedLT:a} the algorithm for the forced variational integrator is given by:
\begin{algorithm}{\rm Forced Variational Integrator Scheme}
\label{ForcedAlgorithm}
\begin{algorithmic}[1]
\State {\bf Initial data}: $N,\, h,\, q_0,\, p_0.$
\State {\bf solve for} $q_1$ {\bf from} $p_0=-D_1L_d(q_0,q_1)-f^{-}_{L_d}(q_0,q_1).$
\State {\bf Initial points:} $q_0,\,q_1.$
\For {$k= 1: N-1$}
\hspace{-0.6cm} {\bf solve for} $q_{k+1}$ {\bf from} $D_1L_d(q_k,q_{k+1})+D_2L_d(q_{k-1},q_{k})+f_{L_d}^{-}(q_k,q_{k+1})+f_{L_d}^{+}(q_{k-1},q_{k})=0$
\EndFor
\State {\bf Output:} $(q_2,...,q_N).$
\end{algorithmic}
\end{algorithm} \hfill $\diamond$
\end{remark}
\section{continuous restricted Hamilton's principle}\label{CRHP}
\subsection{Fractional state space}\label{FracPhSpa}
Consider a smooth curve $\gamma:[a,b]\subset\R\rightarrow\R^d$ for $d\in\N$. The local representation of the curve is given by $\gamma(t)=(x^1(t),...,x^d(t))$, $t\in[a,b]$. For the set of all smooth curves $C^{\infty}([a,b],\R^d)$, let us define the {\it fractional tangent vector} of the curve $\gamma$ by means of the following mapping
\[
\begin{split}
&X_{\sigma}^{\alpha}: C^{\infty}([a,b],\R^d) \rightarrow\quad\R^d,\\
&\quad\quad\quad\quad\quad\,\,\,\gamma\quad\quad\,\mapsto\,\,\,\, X_{\sigma}^{\alpha}\gamma,
\end{split}
\]
where the fractional tangent vector is defined by
\[
X_{\sigma}^{\alpha}\gamma:=D^{\alpha}_{\sigma}\gamma(t)=(D^{\alpha}_{\sigma}x^1(t),...,D^{\alpha}_{\sigma}x^d(t)),\, t\in[a,b],
\]
and $D^{\alpha}_{\sigma}$ represents the $\alpha$-fractional derivatives \eqref{FracDer}, $\sigma=\left\{-,+\right\}$. In the sequel we shall omit the $t$-dependence in curves and fractional tangent vectors; furthermore, we will omit as well the coordinate superindex $i$, refering to the local expression of $X_{\sigma}^{\alpha}\gamma$ just as $D^{\alpha}_{\sigma}x$.
\begin{remark}
Observe that we are choosing $\R^d$ as configuration space instead of a $d$-dimensional smooth manifold $Q$, as in \S\ref{ContinuousFramework}. We do so because the definition of $D^{\alpha}_{\sigma}\gamma(t)$ for $\gamma(t)\subset Q$ is not unique and depends on the particular set of charts employed to cover the manifold. \hfill $\diamond$
\end{remark}
\begin{definition}\label{FracVectorSpace}
{\it We define the {\rm fractional tangent space} of the curve $\gamma$ as}
{\rm
\[
V^{\alpha}_{\sigma}\R^d=\left\{ X_{\sigma}^{\alpha}\gamma\,|\, \mbox{for} \,\,\gamma\in C^{\infty}([a,b],\R^d)\,,\,t\in [a,b]\right\}.
\]
}
\end{definition}
\begin{proposition}\label{FracVectorSpace}
{\it $V^{\alpha}_{\sigma}\R^d$ is a vector space with {\rm dim} $=d$.}
\end{proposition}
\begin{proof} The vector structure for curves is defined pointwise, i.e. for $\gamma, \delta\in C^{\infty}([a,b],\R^d)$ and $\lambda\in\R$
\[
(\gamma+\delta)(t)=\gamma(t)+\delta(t), \quad (\lambda\gamma)(t)=\lambda\gamma(t).
\]
Noting that $D^{\alpha}_{\sigma}$ is a linear operator according to \eqref{FracDer}, the vector structure for $V^{\alpha}_{\sigma}\R^d$ is defined by
\[
\begin{split}
X^{\alpha}_{\sigma}\gamma+X^{\alpha}_{\sigma}\delta:=X^{\alpha}_{\sigma}(\gamma+\delta),\quad
\lambda\cdot X^{\alpha}_{\sigma}\gamma:=X^{\alpha}_{\sigma}(\lambda\gamma).
\end{split}
\]
Considering that for any curve $\gamma$ the canonical basis of $\R^d$ is a linearly independent generating system with $d$ elements for $X^{\alpha}_{\sigma}\gamma$ with coordinates $D^{\alpha}_{\sigma}x^i$, we conclude that $V^{\alpha}_{\sigma}\R^d$ is a vector space with dim $=d$.
\end{proof}
Now, we enlarge our space of cuves in the following way: given $\gamma_x,\gamma_y\in C^{\infty}([a,b],\R^d)$ it is straightforward to define the curve $\tilde\gamma\in C^{\infty}([a,b],\R^d\times\R^d)$ by $\tilde\gamma(t):=(\gamma_x(t),\gamma_y(t))$ for $t\in[a,b]$, which locally we shall denote $\tilde\gamma=(x,y)$ (which we will use indistinctly). Furthermore, for the space of curves $C^{\infty}([a,b],\R^d\times\R^d)$ we establish the vector bundle $\mathbb{T}^{\alpha}\R^d$, with $V^{\alpha}_{-}\R^d\times V^{\alpha}_{+}\R^d$ (whose vector structure is straightforward after Definiton \ref{FracVectorSpace} and the Cartesian product; see \cite{AbMa} for further details) as the fiber over the base space $\R^d\times\R^d$. More particuarly, for a given curve $\tilde\gamma$, $\Omega_{\tilde\gamma}^{\alpha}\in\mathbb{T}^{\alpha}\R^d$ is defined by $\Omega_{\tilde\gamma}^{\alpha}:=((\gamma_x,\gamma_y),(X^{\alpha}_{-}\gamma_x,X^{\alpha}_{+}\gamma_y))$, $t\in[a,b],$ and has natural coordinates $(x,y,D^{\alpha}_{-}x,D^{\alpha}_{+}y)$ (henceforth, we shall use indistinctly $\Omega_{\tilde\gamma}^{\alpha}$ and its coordinates). The bundle projection $\tau^{\alpha}:\mathbb{T}^{\alpha}\R^d\rightarrow\R^d\times\R^d$ is given by $\tau^{\alpha}(\Omega_{\tilde\gamma}^{\alpha})=(x,y)$; furthermore, it is apparent that $\mathbb{T}^{\alpha}\R^d\cong \R^d\times\R^d\times\R^d\times\R^d$. On the other hand, $\mathbb{T}\R^d$ is defined as the vector bundle with $V^{1}_{-}\R^d\times V^{1}_{-}\R^d$ as fiber over $\R^d\times\R^d$, with elements $\Xi_{\tilde\gamma}:=((\gamma_x,\gamma_y),(X^{1}_{-}\gamma_x,X^{1}_{-}\gamma_y))=((\gamma_x,\gamma_y),(D^1_{-}\gamma_x,D^1_{-}\gamma_y))=((\gamma_x,\gamma_y),(\dot\gamma_x,\dot\gamma_y))$, $t\in[a,b]$, and local coordinates $((x,y),(\dot x,\dot y)).$ The bundle projection is given by $\tau:\mathbb{T}\R^d\rightarrow\R^d\times\R^d$, $\tau(\Xi_{\tilde\gamma})=(x,y).$ With these elements we define the {\it fractional state space}.
\begin{definition}\label{FracPhSp}
{\it Consider the vector bundles $\mathbb{T}^{\alpha}\R^d$ and $\mathbb{T}\R^d$. We define the {\rm fractional state space} as the bundle product of them over $\R^d\times\R^d$, i.e.
\[
\mathfrak{T}\R^d:=\mathbb{T}\R^d\otimes_{\tiny{\R^d\times\R^d}}\mathbb{T}^{\alpha}\R^d.
\]
Thus, $\Sigma_{(x,y)}:=(\gamma_{x},\gamma_{y},X_{-}\gamma_{x}, X_{-}\gamma_{y},X_{-}^{\alpha}\gamma_{x}, X_{+}^{\alpha}\gamma_{y})\in\mathfrak{T}\R^d$, $t\in[a,b]$, is locally described by
\begin{equation}\label{TotalCoord}
\Sigma_{(x,y)}=(x,y,\dot x,\dot y,D^{\alpha}_{-}x,D^{\alpha}_{+}y).
\end{equation}
The bundle projection $\mathcal{T}:\mathfrak{T}\R^d\rightarrow\R^d\times\R^d$ is defined by $\mathcal{T}(\Sigma_{(x,y)})=(x,y)$. }
\end{definition}
For more details on bundle products we refer to \cite{AbMa}. The construction of the dual bundle
\[
\mathfrak{T}^*\R^d:=\mathbb{T}^*\R^d\otimes_{\R^d\times\R^d}\mathbb{T}^{\alpha *}\R^d,
\]
which we will denote {\it fractional phase space}, follows straightforwardly from the dual bundles $\mathbb{T}^{\alpha *}\R^d$ and $\mathbb{T}^*\R^d$.
For $P_{(x,y)}\in \mathfrak{T}^*\R^d$, we fix local coordinates
\begin{equation}\label{CoorHam}
P_{(x,y)}=(x,y,p_x,p_y,p^{\alpha}_x,p^{\alpha}_y).
\end{equation}
The bundle projection $\mathcal{P}:\mathfrak{T}^*\R^d\rightarrow\R^d\times\R^d$ is locally given by $\mathcal{P}(P_{(x,y)})=(x,y)$. It is easy to see that both $\mathfrak{T}\R^d$ and $\mathfrak{T}^*\R^d$ are locally equivalent to the Cartesian product of 6 copies of $\R^d$.
\subsection{Fractional dynamics}\label{FrDy}
In order to establish the fractional dynamics we are going to consider a subclass of curves $C^{\infty}(\tilde\gamma^{_{(a,b)}};\R^d\times\R^d)\subset C^{\infty}([a,b],\R^d\times\R^d)$, in particular those $\tilde\gamma=(\gamma_x,\gamma_y)$ such that $\gamma_x(a)=x_a,\,\gamma_x(b)=x_b$ and $\gamma_y(a)=y_a,\,\gamma_y(b)=y_b$ for $x_a,\,x_b,\,y_a,\,y_b\in\R^d$; i.e. those curves with fixed endpoints. Let us define de action sum $S:C^{\infty}(\tilde\gamma^{_{(a,b)}};\R^d\times\R^d)\rightarrow\R$ for a Lagrangian function $\mathcal{L}:\mathfrak{T}\R^d\rightarrow\R$ (which henceforth we shall consider $C^2$) by
\begin{equation}\label{ContAction}
S(\tilde\gamma)=\int_a^b\mathcal{L}\left( x(t),y(t),\dot x(t),\dot y(t),D^{\alpha}_{-}x(t),D^{\alpha}_{+}y(t)\right)\,dt,
\end{equation}
and a particular set of varied curves and variations, namely:
\begin{definition}\label{ConsVariations}
{\it Define the set of {\rm restricted varied curves} as $\Gamma_{(\eta,\epsilon)}(t):=\tilde\gamma(t)+\epsilon\eta(t)$, $\epsilon\in\R$, where $\Gamma_{(\eta,\epsilon)}(t)\in C^{\infty}(\tilde\gamma^{_{(a,b)}};\R^d\times\R^d)$ and $\eta(t):=(\delta\gamma_x(t),\delta\gamma_x(t))$, $t\in[a,b]$, is defined such that $\delta\gamma_x\in C^{\infty}([a,b],\R^d)$ with $\delta\gamma_x(a)=\delta\gamma_x(b)=0$. As it is easy to see, an unrestricted variation would be defined by $(\delta\gamma_x(t),\delta\gamma_y(t))$, with $\delta\gamma_x\neq\delta\gamma_y$. We impose $\delta\gamma_x(t)=\delta\gamma_y(t)$ to the variations, which locally is expressed by $\delta x=\delta y,$ vanishing at the endpoints.}
\end{definition}
Note that the introduction of $D^{\alpha}_{+}y(t)$ in the action \eqref{ContAction} breaks its causality, since it depends on future times $[t,b]$. Thus, apparently it is not suitable for the physical description of the $x$ and $y$ systems. We will see in the following discussion, however, that their dynamics can be decoupled thanks to the restriction of the variations presented in Definition \ref{ConsVariations}. Moreover, the choice of mechanical Lagrangians and the definition of the fractional derivatives \eqref{FracDer:RL} help to surpass the causality issue.
We have already all the ingredients to establish the restricted Hamilton's principle:
\begin{theorem}\label{ConsFraELLag}
{\it A sufficient conditions for a curve $\tilde\gamma\in C^{\infty}([a,b],\R^d\times\R^d)$, subject to the restricted variations in Definition \ref{ConsVariations}, to be an extremal of the action $S:C^{\infty}(\tilde\gamma^{_{(a,b)}};\R^d\times\R^d)\rightarrow\R$ \eqref{ContAction} is the so-called {\rm restricted fractional Euler-Lagrange equations}}:
\begin{subequations}\label{ResFracEL}
\begin{align}
\frac{d}{dt}\left(\frac{\partial \mathcal{L}}{\partial \dot x}\right)-\frac{\partial \mathcal{L}}{\partial x}-D^{\alpha}_{-}\left(\frac{\partial \mathcal{L}}{\partial D^{\alpha}_{+} y}\right)=0,\label{FracEL:-}\\
\frac{d}{dt}\left(\frac{\partial \mathcal{L}}{\partial \dot y}\right)-\frac{\partial \mathcal{L}}{\partial y}-D^{\alpha}_{+}\left(\frac{\partial \mathcal{L}}{\partial D^{\alpha}_{-} x}\right)=0.\label{FracEL:+}
\end{align}
\end{subequations}
\end{theorem}
\begin{proof}
To find the extremals of $S$ for restricted varied curves $\Gamma_{(\eta,\epsilon)}(t)$ we impose the usual critical condition, i.e. $\delta S:=\frac{d}{d\epsilon}S(\Gamma_{(\eta,\epsilon)})\big|_{\epsilon=0}=0$. Using that
\[
\frac{d}{d\epsilon}\mathcal{L}(\Sigma_{_{\Gamma_{(\eta,\epsilon)}}})\Big|_{\epsilon=0}=\frac{\partial \mathcal{L}}{\partial x}\delta x+\frac{\partial \mathcal{L}}{\partial\dot x}\delta\dot x+\frac{\partial \mathcal{L}}{\partial D_{-}^{\alpha}x} D_{-}^{\alpha}\delta x+\frac{\partial \mathcal{L}}{\partial y}\delta x+\frac{\partial \mathcal{L}}{\partial\dot y}\delta\dot x+\frac{\partial \mathcal{L}}{\partial D_{+}^{\alpha}y}D_{+}^{\alpha}\delta x,
\]
where
\[
\Sigma_{_{\Gamma_{(\eta,\epsilon)}}}=(x+\epsilon\,\delta x,y+\epsilon\,\delta x,\dot x+\epsilon\, d(\delta x)/dt,\dot y+\epsilon\, d(\delta x)/dt, D_{-}^{\alpha}x+\epsilon\,D_{-}^{\alpha}\delta x,D_{+}^{\alpha}y+\epsilon\,D_{+}^{\alpha}\delta x),
\]
according to $\Gamma_{(\eta,\epsilon)}$ in Definition \ref{ConsVariations}, we obtain:
\begin{eqnarray}
\delta S&&=\int_a^b\Big\{\frac{\partial \mathcal{L}}{\partial x}\delta x+\frac{\partial \mathcal{L}}{\partial\dot x}\delta\dot x+\frac{\partial \mathcal{L}}{\partial D_{-}^{\alpha}x} D_{-}^{\alpha}\delta x+\frac{\partial \mathcal{L}}{\partial y}\delta x+\frac{\partial \mathcal{L}}{\partial\dot y}\delta\dot x+\frac{\partial \mathcal{L}}{\partial D_{+}^{\alpha}y}D_{+}^{\alpha}\delta x\Big\}\,dt\label{Variation}\\
&&=\int_{a}^b\Big\{\frac{\partial \mathcal{L}}{\partial x}-\frac{d}{dt}\left(\frac{\partial \mathcal{L}}{\partial \dot x}\right)+D^{\alpha}_{+}\left(\frac{\partial \mathcal{L}}{\partial D^{\alpha}_{-} x}\right)+\frac{\partial \mathcal{L}}{\partial y}-\frac{d}{dt}\left(\frac{\partial \mathcal{L}}{\partial \dot y}\right)+D^{\alpha}_{-}\left(\frac{\partial \mathcal{L}}{\partial D^{\alpha}_{+} y}\right)\Big\}\delta x\,dt\nonumber\\
&&\hspace{11cm} +\frac{\partial \mathcal{L}}{\partial \dot x}\delta x\Big|_a^b+\frac{\partial \mathcal{L}}{\partial \dot y}\delta x\Big|_a^b,\nonumber
\end{eqnarray}
where in the second equality we have integration by parts with respect to the total and fractional derivatives \eqref{IntegrationByParts}. According to the vanishing endpoint conditions, all the border terms are equal to zero, leading to
\begin{eqnarray}\label{RestVar}
\delta S=-\int_{a}^b\Big\{\Big[\frac{d}{dt}\left(\frac{\partial \mathcal{L}}{\partial \dot x}\right)-\frac{\partial \mathcal{L}}{\partial x}-D^{\alpha}_{-}\left(\frac{\partial \mathcal{L}}{\partial D^{\alpha}_{+}y}\right)\Big]\,\delta x+\Big[\frac{d}{dt}\left(\frac{\partial \mathcal{L}}{\partial \dot y}\right)-\frac{\partial \mathcal{L}}{\partial y}-D^{\alpha}_{+}\left(\frac{\partial \mathcal{L}}{\partial D^{\alpha}_{-}x}\right)\Big]\delta x\Big\}\,dt.\nonumber
\end{eqnarray}
Finally, from this last expression and considering arbitrary $\delta x$, it is easy to see that the restricted fractional Euler-Lagrange equations \eqref{FracEL} are a sufficient condition for $\delta S=0$; and the claim holds.
\end{proof}
\begin{remark}
For unrestricted variations $\delta x\neq \delta y$, both arbitrary, and fixed endpoint conditions, the necessary and sufficient conditions for the extremals of \eqref{ContAction} are the (unrestricted) fractional Euler-Lagrange equations:
\begin{equation}\label{FracEL}
\frac{d}{dt}\left(\frac{\partial \mathcal{L}}{\partial \dot x}\right)-\frac{\partial \mathcal{L}}{\partial x}-D^{\alpha}_{+}\left(\frac{\partial \mathcal{L}}{\partial D^{\alpha}_{-} x}\right)=0, \quad \frac{d}{dt}\left(\frac{\partial \mathcal{L}}{\partial \dot y}\right)-\frac{\partial \mathcal{L}}{\partial y}-D^{\alpha}_{-}\left(\frac{\partial \mathcal{L}}{\partial D^{\alpha}_{+} y}\right)=0.
\end{equation}
Other derivations of fractional Euler-Lagrange equations can be found in \cite{Agra}. Moreover, from the linearity of the fractional derivatives \eqref{FracDer}, it is easy to prove that the restricted Euler-Lagrange equations \eqref{ResFracEL} are invariant under linear constant change of variables, i.e. $x=\Lambda\,z$, $y=\Lambda\,\bar z$, with $\Lambda\in \R^{d\times d}$ a regular constant matrix. If furthermore we pick the Caputo definition of the fractional derivatives \eqref{FracDer:Ca}, the equations are invariant under affine change of variables $x=\Lambda\,z +\zeta$, $y=\Lambda\,\bar z+\bar\zeta$; for constant $\zeta,\bar\zeta\in\R^d$ (see \cite{JiOb} for more details). \hfill $\diamond$
\end{remark}
Define now the Lagrangian $\mathcal{L}: \mathfrak{T}\R^d\rightarrow\R$ by
\begin{equation}\label{PartLagrangian}
\mathcal{L}(x,y,\dot x,\dot y,D^{\alpha}_{-}x,D^{\alpha}_{+}y):=L(x,\dot x)+L(y,\dot y)- D^{\alpha}_{-}x\,\rho\, D^{\alpha}_{+}y,
\end{equation}
where $L:T\R^d\rightarrow\R$ is a $C^2$ function, $\rho=\,$diag$(\rho_1,\cdots,\rho_d)\in \R^{d\times d}$ and we represent the matrix product $\mu^T\,M\,\nu$, for $\mu,\nu\in\R^d$ and $M\in\R^{d\times d}$, by $\mu\,M\,\nu$. In this case, the restricted fractional Euler-Lagrange equations \eqref{ResFracEL} read:
\begin{subequations}\label{ResFracELDamped}
\begin{align}
\frac{d}{dt}\left(\frac{\partial L(x,\dot x)}{\partial \dot x}\right)-\frac{\partial L(x,\dot x)}{\partial x}+\rho\,D^{\alpha}_{-} D^{\alpha}_{-} x&=0,\label{ResFracELDamped:a}\\
\frac{d}{dt}\left(\frac{\partial L(y,\dot y)}{\partial \dot y}\right)-\frac{\partial L(y,\dot y)}{\partial y}+\rho D^{\alpha}_{+} D^{\alpha}_{+} y&=0,\label{ResFracELDamped:b}
\end{align}
\end{subequations}
which are the usual Euler-Lagrange equations for a Lagrangian system plus a fractional damping term \eqref{IntEqCont}. The following proposition provides an interpretation of the $y$-mirror system.
\begin{proposition}\label{InvTime}
If we set $y(t):=x(a+b-t)$, which implies that $\tilde\gamma\in C^{\infty}(\tilde\gamma^{_{(a,b)}};\R^d\times\R^d)$ with $\gamma_y(a)=x_b,\,\gamma_y(b)=x_a$, and we consider an even Lagrangian function $L:T\R^d\rightarrow\R$ in the second variable, i.e. $L(z,-\dot z)=L(z,\dot z)$, then \eqref{ResFracELDamped:b} is \eqref{ResFracELDamped:a} in reversed time.
\end{proposition}
\begin{proof}
Defining the reverse time by $\tilde t:=a+b-t$ for $t\in[a,b]$, it is easy to see that $\dot y(t)=-x^{\prime}(\tilde t)$ by applying the chain rule, where we consider $x^{\prime}(\tilde t)=dx(\tilde t)/d\tilde t$. Then
\[
\begin{split}
\frac{\partial L(y(t),\dot y(t))}{\partial y}&=\frac{\partial L(x(\tilde t),-x^{\prime}(\tilde t))}{\partial x}=^{_1}\frac{\partial L(x(\tilde t),x^{\prime}(\tilde t))}{\partial x},\\
\frac{d}{dt}\left(\frac{\partial L(y(t),\dot y(t))}{\partial \dot y}\right)&=\frac{d}{dt}\left(\frac{\partial L(x(\tilde t),-x^{\prime}(\tilde t))}{\partial \dot y}\right)=^{_2}-\frac{d}{dt}\left(\frac{\partial L(x(\tilde t),-x^{\prime}(\tilde t))}{\partial x^{\prime}}\right)=^{_3}\frac{d}{d\tilde t}\left(\frac{\partial L(x(\tilde t),x^{\prime}(\tilde t))}{\partial x^{\prime}}\right),
\end{split}
\]
where in $=^{_1}$ we have used the parity $L(x,-x^{\prime})=L(x,x^{\prime})$, and in $=^{_2}$, $=^{_3}$, the chain rule and the redefinition of the time $\tilde t=a+b-t$ and parity again, respectively. On the other hand, using \eqref{Aditive} and according to \eqref{FracDer:RL} and $y(t)=x(a+b-t)$:
\[
\begin{split}
D^{2\alpha}_{+}y(t)=&-\frac{1}{\Gamma(1-2\alpha)}\frac{d}{dt}\int_t^b(\tau-t)^{-2\alpha}x(a+b-\tau)d\tau\\
&\,\,\quad=^{_1}\frac{1}{\Gamma(1-2\alpha)}\frac{d}{dt}\int_{a+b-t}^a((a+b-t)-\tilde\tau)^{-2\alpha}x(\tilde\tau)d\tilde\tau\\
&\quad\quad\quad\quad=^{_2}\frac{1}{\Gamma(1-2\alpha)}\frac{d}{d\tilde t}\int_{a}^{\tilde t}(\tilde t-\tilde\tau)^{-2\alpha}x(\tilde\tau)d\tilde\tau\,\,=^{_3}\,\,D^{2\alpha}_{-}x(\tilde t),
\end{split}
\]
where in $=^{_1}$ we have used the change of variables $\tilde\tau=a+b-\tau$, in $=^{_2}$ the redefinition of time and, in $=^{_3}$, the definition of $D^{\alpha}_{-}$ in \eqref{FracDer:RL}. Plugging all these elements in \eqref{ResFracELDamped:b} we obtain
\[
\frac{d}{d\tilde t}\left(\frac{\partial L(x(\tilde t), x^{\prime}(\tilde t))}{\partial \dot x}\right)-\frac{\partial L(x(\tilde t),x^{\prime}(\tilde t))}{\partial x}+\rho\,D^{\alpha}_{-} D^{\alpha}_{-} x(\tilde t)=0
\]
for $\tilde t:b\rightarrow a$, and the claim holds.
\end{proof}
Thus, our conclusion is that, for the (mechanical) Lagrangians of interest in this work, i.e.
\begin{equation}\label{MechLag}
L(z,\dot z):=\frac{1}{2}\dot z\,m\,\dot z-U(z),
\end{equation}
with $m=\,$diag\,$(m_1,\cdots,m_d)\in\R^{d\times d}_{+}$ and $U:\R^d\rightarrow\R$ a smooth function, we can consider the $y$-mirror system as the $x$-system in reversed time, as long as $\gamma_y(t)$ is bounded by $y(a)=x_b$ and $y(b)=x_a$. Thus, doubling the space of curves as exposed in \S\ref{FracPhSpa} allows us to surpass the issue raised by the asymetric integration by parts \eqref{IntegrationByParts} without adding extra unnecessary (physical) information to the dynamics. In addition, the non-causal terms in the action \eqref{ContAction} due to the presence of the advanced fractional derivative become causal in reversed time since $D^{\alpha}_{+}y(t)=D^{\alpha}_{-}x(\tilde t)$, as we just proved. Finally, we observe that the restricted varied curves introduced in Definition \ref{ConsVariations} acquire a particular shape when we set $y(t)=x(a+b-t)$, as we show in Figure \ref{Curves}.
\begin{figure}[!htb]
\includegraphics[scale=0.37]{Curves.PNG}
\caption{We show the varied curves $\Gamma_{(\eta,\epsilon)}(t)=(x_{\epsilon}(t),y_{\epsilon}(t))=(x(t),y(t))+\epsilon(\delta x(t),\delta x(t))$, according to Definition \ref{ConsVariations}, when we set $y(t)=x(a+b-t)$. The horizontal arrows represent the two directions of time.}
\label{Curves}
\end{figure}
\begin{remark}
Note that the (unrestricted) fractional Euler-Lagrange equations \eqref{FracEL} for the Lagrangians \eqref{PartLagrangian} read
\[
\frac{d}{dt}\left(\frac{\partial L(x,\dot x)}{\partial \dot x}\right)-\frac{\partial L(x,\dot x)}{\partial x}+\rho\,D^{\alpha}_{+} D^{\alpha}_{+} y=0,\,\,\,\,\frac{d}{dt}\left(\frac{\partial L(y,\dot y)}{\partial \dot y}\right)-\frac{\partial L(y,\dot y)}{\partial y}+\rho\,D^{\alpha}_{-} D^{\alpha}_{-} x=0,
\]
which are a coupled system of fractional differential equations with meaningless dynamics. \hfill $\diamond$
\end{remark}
In addition, $D^{\alpha}_{-}D^{\alpha}_{-}=D^{2\alpha}_{-}$ \eqref{Aditive}; therefore $\alpha= 1/2$ implies $D^{2\alpha}_{-}= d/dt$ \eqref{Alpha1}. According to this, it is easy to see that, given $\alpha=1/2$, then equation \eqref{ResFracELDamped:a} is equivalent to the forced Euler-Lagrange equations \eqref{ForcedEL} when $f_L(x,\dot x)=-\rho\,\dot x$. This establishes the relationship between the restricted Hamilton's principle and the Lagrange-d'Alembert principle, expressed in Figure \ref{Diagram}.
\begin{figure}[!htb]
\includegraphics[scale=0.52]{ContLagDiagram.PNG}
\caption{In the diagram it is reflected that the Lagrange-d'Alembert principle for a general Lagrangian and a particular set of external forces produce the same forced equations as enlarging the phase space, then applying the restricted Hamilton's principle for the action $S=\int_a^b\mathcal{L}\,dt$ \eqref{ContAction}, and then setting $\alpha=1/2$.}
\label{Diagram}
\end{figure}
Picking the mechanical Lagrangian \eqref{MechLag}, equations \eqref{ResFracELDamped} read
\begin{subequations}\label{MechDynFracDamp}
\begin{align}
m\,\ddot x+\nabla U(x)+\rho\,D^{\alpha}_{-} D^{\alpha}_{-} x&=0,\label{MechDynFracDamp:a}\\
m\,\ddot y+\nabla U(y)+\rho\,D^{\alpha}_{+} D^{\alpha}_{+} y&=0,\label{MechDynFracDamp:b}
\end{align}
\end{subequations}
which are the dynamical equations of a Lagrangian system with $2\alpha$ fractional damping, and will be considered in the examples, \S\ref{Simu}.
\subsection{Preservation of presymplectic form}
Before proving the preservation of a given presymplectic form in $\mathcal{T}\R^d$ we point out that the equations \eqref{ResFracELDamped} define a continuous flow $F_t:\mathcal{T}\R^d\rightarrow \mathcal{T}\R^d$, with $F_0=$Id. The existence of this flow is ensured by the proof of existence and uniqueness of the cited equations in Appendix. In particular, it is proven that given initial conditions $(x_0,\dot x_0)$, the existence and uniqueness of smooth $(x(t),\dot x(t))$ is ensured, ensuring furthermore the existence and uniqueness of $D_{-}^{\alpha}x(t)$ according to definition \eqref{FracDer} (equivalently for the mirror $y$-system). We shall
ignore issues related to global versus local flows, which are easily dealt with
by restricting the domains of the flows. Given initial conditions $X_0^{\alpha}:=(x_0,y_0,\dot x_0, \dot y_0,0,y^{\alpha}_0)\footnote{Observe that, according to \eqref{FracDer}, the only available value of $D^{\alpha}_{-}x(t_0)$ is 0.}$, the flow provides $F_t(X_0^{\alpha})=X^{\alpha}(t)$, with $X^{\alpha}(t):=(x(t),y(t),\dot x(t),\dot y(t),D^{\alpha}_{-}x(t),D^{\alpha}_{+}y(t)).$
Now, define the one-form and two-form on $\mathcal{T}\R^d$, with $L:T\R^d\rightarrow\R$ a $C^2$ function:
\begin{subequations}
\begin{align}
\Theta:=&\left(\frac{\partial L(x,\dot x)}{\partial\dot x^i}+\frac{\partial L(y,\dot y)}{\partial\dot y^i}\right)\,dx^i, \label{OneForm}\\
\Omega:=&\mbox{d}\Theta=\frac{\partial^2L}{\partial x^j\partial\dot x^i}dx^i\wedge dx^j+\frac{\partial^2L}{\partial\dot x^j\partial\dot x^i}dx^i\wedge d\dot x^j+\frac{\partial^2L}{\partial y^j\partial\dot y^i}dx^i\wedge dy^j+\frac{\partial^2L}{\partial \dot y^j\partial\dot y^i}dx^i\wedge d\dot y^j. \label{TwoForm}
\end{align}
\end{subequations}
It is straightforward to see that $\Omega$ is presymplectic on $\mathcal{T}\R^d$, since the fractional variables are absent, therefore it is degenerate, and $\mbox{d}\Omega=\mbox{d}\mbox{d}\Theta=0.$ We can establish the following preservation result.
\begin{proposition}\label{GeomPreservation}
The flow generated by the restricted fractional Euler-Lagrange equations when we set the Lagrangian function \eqref{PartLagrangian}, i.e.~ equations \eqref{ResFracELDamped}, preserves the presymplectic form $\Omega$ \eqref{TwoForm}. In other words, $F_t^*\Omega=\Omega.$
\end{proposition}
\begin{proof}
The action \eqref{ContAction}, in our fractional context, can be seen as a function $S:\mathcal{T}\R^d\rightarrow\R$, and its variation $\delta S=\langle \mbox{d}S\,,\,\delta X^{\alpha}_0\rangle$, where we set $\delta X_0^{\alpha}:=(\delta x_0,\delta y_0, \delta\dot x_0, \delta\dot y_0,\delta x^{\alpha}_0,\delta y^{\alpha}_0)$
$\in T\mathcal{T}\R^d$ arbitrary. According to \eqref{Variation}, when we pick the Lagrangian \eqref{PartLagrangian} and along the solutions of \eqref{ResFracELDamped}, we have that
\[
\begin{split}
\langle \mbox{d}S\,,\,\delta X^{\alpha}_0\rangle=&\,\,\,\left(\frac{\partial L(x,\dot x)}{\partial\dot x^i}+\frac{\partial L(y,\dot y)}{\partial\dot y^i}\right)\,\delta x^i\Big|_{t_0}^t=^{_1}\langle\Theta(X^{\alpha}(t)),\delta X^{\alpha}(t)\rangle-\langle\Theta(X^{\alpha}_0),\delta X^{\alpha}_{0}\rangle\\
=^{_2}&\,\,\,\,\langle\Theta(X^{\alpha}(t)),(F_t)_{*}\delta X^{\alpha}_0\rangle-\langle\Theta(X^{\alpha}_0),\delta X^{\alpha}_{0}\rangle=\langle(F_t)^{*}\Theta(X^{\alpha}_0),\delta X^{\alpha}_0\rangle-\langle\Theta(X^{\alpha}_0),\delta X^{\alpha}_{0}\rangle\\
=&\,\,\,\langle (F_t)^{*}\Theta(X^{\alpha}_0)-\Theta(X^{\alpha}_0)\,,\,\delta X^{\alpha}_0\rangle,
\end{split}
\]
where we have used arbitrary initial and final times $t_0, t$, respectively. In $=^{_1}$ we have employed the definition of $\Theta$ \eqref{OneForm}; in $=^{_2}$ we have used the flow $F_t:\mathcal{T}\R^d\rightarrow \mathcal{T}\R^d$ introduced above. It follows straightforwardly that
\[
\mbox{d}S=(F_t)^{*}\Theta-\Theta.
\]
Now, taking the differential in both sides, considering that the differential and the pull-back commute and the definition of $\Omega$ \eqref{TwoForm}, we obtain $(F_t)^{*}\mbox{d}\Theta-\mbox{d}\Theta=0$ and then $F_t^*\Omega=\Omega,$ from which the result follows.
\end{proof}
\subsection{The Legendre transformation}\label{LgTr}
Let us define the {\it fractional Legendre transform} $\mathcal{F}\mathcal{L}:\mathfrak{T}\R^d\rightarrow \mathfrak{T}^*\R^d$ as the fiber derivative for a Lagrangian function $\mathcal{L}:\mathfrak{T}\R^d\rightarrow\R$, i.e.
\begin{equation} \label{LegTrans}
\begin{split}
\mathcal{F}\mathcal{L}:\mathfrak{T}_{(x,y)}\R^d&\longrightarrow \mathfrak{T}^*_{(x,y)}\R^d\\
\Sigma_{(x,y)}&\mapsto D_{(x,y)}\mathcal{L}(\Sigma_{(x,y)}),
\end{split}
\end{equation}
where $D_{(x,y)}$ denotes the partial derivative in the fiber $\mathcal{T}^{-1}((x,y))$. Locally we have
\begin{equation}\label{FracLeg}
\mathcal{F}\mathcal{L}(\Sigma_{(x,y)})=\left(\frac{\partial \mathcal{L}}{\partial\dot x},\frac{\partial \mathcal{L}}{\partial\dot y},\frac{\partial \mathcal{L}}{\partial D^{\alpha}_{-}x},\frac{\partial \mathcal{L}}{\partial D^{\alpha}_{+}y}\right).
\end{equation}
It is easy to check that $\mathcal{F}\mathcal{L}$ is fiber preserving \cite{AbMa}. Moreover, we will say that $\mathcal{F}\mathcal{L}$ is {\it regular} if it is a diffeomorphism, and furthermore we will call $\mathcal{L}$ {\it regular} if that is the case (which we will assume throughout the article). Hence, we define the Hamiltonian function $\mathcal{H}:\mathfrak{T}^*\R^d\rightarrow\R$ by
\begin{equation}\label{HamDef}
\mathcal{H}(P_{(x,y)}):=\langle \mathcal{F}\mathcal{L}(\Sigma_{(x,y)}),\Sigma_{(x,y)}\rangle-\mathcal{L}(\Sigma_{(x,y)}),
\end{equation}
where the coordinates of $P_{(x,y)}:=\mathcal{F}\mathcal{L}(\Sigma_{(x,y)})$ are given in \eqref{CoorHam} and $\langle\cdot,\cdot\rangle:\mathfrak{T}_{(x,y)}^*\R^d\times \mathfrak{T}_{(x,y)}\R^d\rightarrow\R$ denotes the natural pairing. Employing these elements, we can establish the Hamiltonian version of the restricted Hamilton's principle.
\begin{theorem}\label{ConsFraELHam}
{\it A sufficient condition for a curve $\tilde\gamma\in C^{\infty}([a,b],\R^d\times\R^d)$, subject to the restricted variations in Definition \ref{ConsVariations}, to be an extremal of the action $S:C^{\infty}(\tilde\gamma^{_{(a,b)}};\R^d\times\R^d)\rightarrow\R$ \eqref{ContAction} is the so-called {\rm restricted fractional Hamilton equations}}:
\begin{subequations}\label{FracHam}
\begin{align}
\dot x&=\frac{\partial \mathcal{H}}{\partial p_x},\quad\quad D^{\alpha}_{-}x=\frac{\partial \mathcal{H}}{\partial p^{\alpha}_x}, \quad\quad \dot p_x=-\frac{\partial \mathcal{H}}{\partial x}+D^{\alpha}_{-}p^{\alpha}_y,\\
\dot y&=\frac{\partial \mathcal{H}}{\partial p_y}, \quad\quad
D^{\alpha}_{+}y=\frac{\partial \mathcal{H}}{\partial p^{\alpha}_y}, \quad\quad \dot p_y=-\frac{\partial \mathcal{H}}{\partial y}+D^{\alpha}_{+}p^{\alpha}_x.
\end{align}
\end{subequations}
\end{theorem}
\begin{proof}
Setting the action \eqref{ContAction} in its Hamiltonian form, i.e.
\[
S(\tilde\gamma)=\int_a^b\big\{ p_x\dot x+p_y\dot y+p_x^{\alpha}D^{\alpha}_{-}x+p_y^{\alpha}D^{\alpha}_{+}y-\mathcal{H}(x,y,p_x,p_y,p^{\alpha}_x,p^{\alpha}_y)\big\}\,dt,
\]
and imposing the critical condition with restricted variations, i.e. $\delta S=\frac{d}{d\epsilon}S(\Gamma_{(\eta,\epsilon)})\big|_{\epsilon=0}=0$, after applying fractional and total integration by parts we arrive to
\[
\begin{split}
\delta S=\int_a^b&\Big\{\delta p_x\left(\dot x-\frac{\partial \mathcal{H}}{\partial p_x}\right)+\delta p_y\left(\dot y-\frac{\partial \mathcal{H}}{\partial y}\right)+\delta p_x^{\alpha}\left( D^{\alpha}_{-}x-\frac{\partial \mathcal{H}}{\partial p_x^{\alpha}}\right)+\delta p_y^{\alpha}\left( D^{\alpha}_{+}y-\frac{\partial \mathcal{H}}{\partial p_y^{\alpha}}\right)\\
&\quad\quad\quad\,\,\,+\left(-\dot p_x-\frac{\partial \mathcal{H}}{\partial x}+D^{\alpha}_{+}p^{\alpha}_x-\dot p_y-\frac{\partial \mathcal{H}}{\partial y}+D^{\alpha}_{-}p^{\alpha}_y\right)\,\delta x\Big\}dt+p_x\delta x\Big|_a^b+p_y\delta x\Big|_a^b,
\end{split}
\]
which leads to
\[
\begin{split}
\delta S=\int_a^b\Big\{&\delta p_x\left(\dot x-\frac{\partial \mathcal{H}}{\partial p_x}\right)+\delta p_x^{\alpha}\left( D^{\alpha}_{-}x-\frac{\partial \mathcal{H}}{\partial p_x^{\alpha}}\right)+\left(-\dot p_x-\frac{\partial \mathcal{H}}{\partial x}+D^{\alpha}_{-}p^{\alpha}_y\right)\delta x\\
+&\delta p_y\left(\dot y-\frac{\partial \mathcal{H}}{\partial y}\right)+\delta p_y^{\alpha}\left( D^{\alpha}_{+}y-\frac{\partial \mathcal{H}}{\partial p_y^{\alpha}}\right)+\left(-\dot p_y-\frac{\partial \mathcal{H}}{\partial y}+D^{\alpha}_{+}p^{\alpha}_x\right)\,\delta x\Big\}dt,
\end{split}
\]
since the boundary terms vanish. From this last expression is straightforward to see that \eqref{FracHam} is a sufficient condition for the extremals given arbitrary variations $\delta x,\, \delta p_x,\,\delta p_y,\, \delta p^{\alpha}_x,\,\delta p^{\alpha}_y$.
\end{proof}
\begin{remark}
For unrestricted variations $\delta x\neq \delta y$, both arbitrary, and fixed endpoint conditions, the necessary and sufficient conditions for the extremals of \eqref{ContAction} are the (unrestricted) fractional Hamilton equations:
\[
\begin{split}
\dot x&=\frac{\partial \mathcal{H}}{\partial p_x},\quad\quad D^{\alpha}_{-}x=\frac{\partial \mathcal{H}}{\partial p^{\alpha}_x}, \quad\quad \dot p_x=-\frac{\partial \mathcal{H}}{\partial x}+D^{\alpha}_{+}p^{\alpha}_x,\\
\dot y&=\frac{\partial \mathcal{H}}{\partial p_y}, \quad\quad
D^{\alpha}_{+}y=\frac{\partial \mathcal{H}}{\partial p^{\alpha}_y}, \quad\quad \dot p_y=-\frac{\partial \mathcal{H}}{\partial y}+D^{\alpha}_{-}p^{\alpha}_y.
\end{split}
\]
\hfill $\diamond$
\end{remark}
From \eqref{PartLagrangian} and \eqref{HamDef} we get the Hamiltonian
\begin{equation}\label{PartHamiltonian}
\mathcal{H}(x,y,p_x,p_y,p_x^{\alpha},p_y^{\alpha})=H(x,p_x)+H(y,p_y)-p_x^{\alpha}\,\rho^{-1}\,p^{\alpha}_y,
\end{equation}
where $H(z,p_z):=\langle p_z,\,\dot z\rangle-L(z,\dot z)$. For the particular Hamiltonian \eqref{PartHamiltonian}, the restricted Hamilton equations \eqref{FracHam} read:
\begin{subequations}\label{FracPartiHam}
\begin{align}
\dot x&=\frac{\partial H}{\partial p_x},\quad\quad D^{\alpha}_{-}x=-\rho^{-1}\,p_y^{\alpha}, \quad\quad \dot p_x=-\frac{\partial H}{\partial x}+D^{\alpha}_{-}\,p^{\alpha}_y,\label{FracPartiHam:a}\\
\dot y&=\frac{\partial H}{\partial p_y}, \quad\quad
D^{\alpha}_{+}y=-\rho^{-1}\,p_x^{\alpha}, \quad\quad \dot p_y=-\frac{\partial H}{\partial y}+D^{\alpha}_{+}\,p^{\alpha}_x.\label{FracPartiHam:b}
\end{align}
\end{subequations}
We see that the $x$ and $y$ dynamics are also coupled in principle. On the other hand, in the next result we show that the relationship between $x$ and $y$ systems is consistent with the Lagrangian case under inversion of time.
\begin{proposition}
If we set $y(t):=x(a+b-t)$, $p_y(t):=-p_x(a+b-t)$ and $p_y^{\alpha}(t):=p_x(a+b-t)$, and we consider an even Hamiltonian function in the second variable, i.e. $H(z,-p_z)=H(z,p_z)$, then \eqref{FracPartiHam:b} is \eqref{FracPartiHam:a} in reversed time.
\end{proposition}
\begin{proof}
Given that $\dot y(t)=-x^{\prime}(\tilde t)$ and
\[
\frac{\partial H(y(t),p_y(t))}{\partial p_y}=^{_{1}}\frac{\partial H(x(\tilde t),-p_x(\tilde t))}{\partial p_y}=^{_{2}}-\frac{\partial H(x(\tilde t),-p_x(\tilde t))}{\partial p_x}=^{_{3}}-\frac{\partial H(x(\tilde t),p_x(\tilde t))}{\partial p_x},
\]
where in $=^{_{1}}$ we use the hypotheses, in $=^{_{2}}$ the chain rule and in $=^{_{3}}$ the parity of the Hamiltonian; we see that
\[
\dot y(t)=\frac{\partial H(y(t),p_y(t))}{\partial p_y} \Rightarrow -x^{\prime}(\tilde t)=-\frac{\partial H(x(\tilde t),p_x(\tilde t))}{\partial p_x}\Rightarrow x^{\prime}(\tilde t)=\frac{\partial H(x(\tilde t),p_x(\tilde t))}{\partial p_x}.
\]
From the proof of Proposition \ref{InvTime}, $D^{\beta}_{+}y(t)=D^{\beta}_{-}x(\tilde t)$, and therefore it is easy to see that
\[
D^{\alpha}_{+}y(t)=-\rho^{-1}\,p^{\alpha}_x(t)\Rightarrow D^{\alpha}_{-}x(\tilde t)=-\rho^{-1}\,p^{\alpha}_y(\tilde t).
\]
Finally, considering
\[
\frac{\partial H(y(t),p_y(t))}{\partial y}=\frac{\partial H(x(\tilde t),-p_x(\tilde t))}{\partial y}=\frac{\partial H(x(\tilde t),p_x(\tilde t))}{\partial x},
\]
where again we use the chain rule and the parity of the Hamiltonain, and that $\dot p_y(t)=p_x^{\prime}(\tilde t)$ according to the hypotheses, we get
\[
\dot p_y(t)=-\frac{\partial H(y(t),p_y(t))}{\partial y}+D^{\alpha}_{+}\,p_x(t)\Rightarrow p_x^{\prime}(\tilde t)=-\frac{\partial H(x(\tilde t),p_x(\tilde t))}{\partial x}+D^{\alpha}_{-}p_y^{\alpha}(\tilde t),
\]
and the claim holds.
\end{proof}
The mechanical Hamiltonian
\begin{equation}\label{MechHamilto}
H(z,p_z)=\frac{1}{2}p_z\,m^{-1}\,p_z+U(z)
\end{equation}
presents even parity in the second variable, and therefore the previous result applies. The restricted fractional Hamilton equations \eqref{FracPartiHam} in this case, after replacing the fractional relationship in the dynamical equation for $p_x$ and $p_y$, read
\begin{subequations}\label{FracHamEqs:Mech}
\begin{align}
\dot x&=m^{-1}\,p_x, \quad\quad \dot p_x=-\nabla U(x)-\rho\,\,D^{\alpha}_{-}D^{\alpha}_{-}\,x,\\
\dot y&=m^{-1}\,p_y, \quad\quad \dot p_y=-\nabla U(y)-\rho\,\,D^{\alpha}_{+}D^{\alpha}_{+}\,y,
\end{align}
\end{subequations}
where we recognize the usual Hamilton equations for mechanical systems plus a fractional damping term. We observe that, after getting rid of the pure fractional equation, the dynamics of $x$ and $y$ are again decoupled. Furthermore, when $\alpha = 1/2$ it is easy to see that the $x$-equations (analogous arguments can be applied to the $y$-equations) are equivalent to the forced Hamilton equations \eqref{ForcedHam} when $f_H(x,p_x)=-\rho\,m^{-1}p_x$. Indeed, when $\alpha=1/2$:
\[
\dot x=^{_{1}}m^{-1}p_x, \quad\quad \dot p_x=-\nabla U(x)-\rho\,D^{_{1/2}}_{-}D^{_{1/2}}_{-}\,x=^{_{2}}-\nabla U(x)-\rho\,\dot x=-\nabla U(x)-\rho\,m^{-1}p_x,
\]
which are the usual Hamilton equations for mechanical systems with linear damping.
\begin{remark}
We observe in the right hand side of $=^{_{2}}$ that the damping force is not actually defined in $T^*\R^d$, but in $T\R^d$, i.e. $f_L(x,\dot x)=-\rho\,\dot x$, and that we can relate it to a pure ``cotangent'' force thanks to $=^{_{1}}$, say $f_H(x,p_x)=-\rho\,m^{-1}p_x$. This is a specific phenomenon of our approach, and differs from the usual description of forced systems \S\ref{ContForcedSystems}. However, we observe as well that both dynamics, i.e.
\begin{eqnarray}
&\dot x&=m^{-1}p_x,\quad\quad \dot p_x=-\nabla U(x)-\rho\,\dot x,\nonumber\\
&\dot x&=m^{-1}p_x,\quad\quad \dot p_x=-\nabla U(x)-\rho\,m^{-1}p_x, \label{HamLinearDamp}
\end{eqnarray}
define the same subspace in $TT^*\R^d$, whose natural coordinates are $(x,p_x,\dot x,\dot p_x).$ \hfill $\diamond$
\end{remark}
\begin{remark}
Define the one-form on $\mathfrak{T}^*\R^d$ given by $\Theta_H:=(\mathcal{F}\mathcal{L})^*\Theta$, with $\Theta$ defined in \eqref{OneForm} and $\mathcal{F}\mathcal{L}$ in \eqref{FracLeg}. Using the Lagrangian \eqref{PartLagrangian} and the coordinates \eqref{CoorHam}, it is easy to see that $\Theta_H=(p_x+p_y)\,dx$. Using analogous arguments than in Proposition \ref{GeomPreservation}, it is easy to see that the flow generated by equations \eqref{FracPartiHam} preserves the presymplectic form $\Omega_H:=\mbox{d}\Theta_H=dx\wedge dp_x+dx\wedge dp_y$. \hfill $\diamond$
\end{remark}
\section{Discrete restricted Hamilton's principle}\label{DRHP}
\subsection{Discrete Lagrangian dynamics}
Let us consider the increasing sequence of times $\{ t_k=a+hk\,|\,k=0,...,N\}\subset\R$ where $h$ is the fixed time step determined by $h=(b-a)/N$. Define a discrete curve as a collection of points in $\R^d$ i.e. $\gamma_d:=\left\{ x_0,x_1,...,x_{N-1},x_N\right\}=\left\{ x_k\right\}_{0:N}\in\R^{(N+1)d}$ (here $\R^{(N+1)d}$ denotes $(\R^d)^{(N+1)}:=\R^d\times\cdots\times\R^d$, the Cartesian product of $(N+1)$ copies of $\R^d$). As usual, we will consider these points as an approximation of the continuous curve at time $t_k$, namely $x_k\simeq x(t_k).$ Given $\left\{ z_k\right\}_{0:N}$ (later on we shall particularise in $\left\{ x_k\right\}_{0:N}$ and $\left\{ y_k\right\}_{0:N}$) define the following sequences:
\begin{subequations}\label{AllSequencs}
\begin{align}
&\left\{ S^{\kappa}\,z_k\right\}_{\tiny 0:N-1},\,\,\,\quad\, S^{\kappa}\,z_k:=\kappa\,z_k+(1-\kappa)\,z_{k+1},\,\,\quad\kappa\in[0,1]\subset\R,\label{KappaRule}\\
&\left\{ \Delta_{-}^{\alpha}z_k\right\}_{\tiny 0:N},\,\,\,\,\,\,\,\,\quad\Delta_{-}^{\alpha}z_k:=\frac{1}{h^{\alpha}}\sum_{n=0}^k\alpha_nz_{k-n},\label{DiscFracDerDef:1}\\
&\left\{ \Delta_{+}^{\alpha}z_k\right\}_{\tiny 0:N},\,\,\,\,\,\,\,\,\quad\Delta_{+}^{\alpha}z_k:=\frac{1}{h^{\alpha}}\sum_{n=0}^{N-k}\alpha_nz_{k+n},\label{DiscFracDerDef:2}
\end{align}
\end{subequations}
where
\begin{equation}\label{AlphaDef}
\alpha_n:=\frac{-\alpha\,(1-\alpha)\,(2-\alpha)\cdot\cdot\cdot(n-1-\alpha)}{n!};\quad\alpha_0:=1.
\end{equation}
The discrete series $\Delta_{-}^{\alpha}x_k$ (resp. $\Delta_{+}^{\alpha}y_k$) are an approximation of $D^{\alpha}_{-} x(t_k)$ (resp. $D^{\alpha}_{+} y(t_k)$). For more details on the approximation of fractional derivatives we refer to (\cite{CressonBook}, Chapter 5).
\begin{remark}\label{SumEndpoint}
We observe that \eqref{DiscFracDerDef:1}, \eqref{DiscFracDerDef:2} are well-defined for $k=0$ and $k=N$. Namely, straightforward computations lead to $\Delta_{-}^{\alpha}x_0=\alpha_0x_0/h^{\alpha}$ and $\Delta_{+}^{\alpha}y_N=\alpha_0y_N/h^{\alpha}$. \hfill $\diamond$
\end{remark}
Given two sequences $\left\{ F_k\right\}_{\tiny 0:N}, \left\{ G_k\right\}_{\tiny 0:N}$, the discrete fractional derivatives \eqref{DiscFracDerDef:1}, \eqref{DiscFracDerDef:2}, obey the following discrete integration by parts relationships:
\begin{lemma}
Consider $\left\{ F_k\right\}_{\tiny 0:N}, \left\{ G_k\right\}_{\tiny 0:N}$, with $F_0=F_N=G_0=G_N=0$. Then
\begin{subequations}
\begin{align}
\sum_{k=0}^{N-1}(\Delta_{+}^{\alpha}G_k)F_k&=\sum_{k=1}^NG_k(\Delta_{-}^{\alpha}F_k). \label{DIParts:b}\\
\sum_{k=0}^{N-1}G_{k+1}(\Delta_{-}^{\alpha}F_{k+1})&=\sum_{k=1}^{N-1}(\Delta_{+}^{\alpha}G_{k})F_{k}. \label{DIParts:c}
\end{align}
\end{subequations}
\end{lemma}
\begin{proof}
\eqref{DIParts:b} (see \cite{Cresson1} for more details):
\[
\begin{split}
\sum_{k=1}^NG_k(\Delta_{-}^{\alpha}F_k)&=^{_1}\frac{1}{h^{\alpha}}\sum_{k=1}^N\sum_{n=0}^k\alpha_nG_kF_{k-n}=^{_2}\frac{1}{h^{\alpha}}\sum_{k=0}^N\sum_{n=0}^k\alpha_nG_kF_{k-n}=^{_3}\frac{1}{h^{\alpha}}\sum_{n=0}^N\sum_{k=n}^N\alpha_nG_kF_{k-n}\\
&=^{_4}\frac{1}{h^{\alpha}}\sum_{n=0}^N\sum_{k=0}^{N-n}\alpha_nG_{k+n}F_{k}=^{_5}\frac{1}{h^{\alpha}}\sum_{k=0}^N\sum_{n=0}^{N-k}\alpha_nG_{k+n}F_{k}=^{_6}\frac{1}{h^{\alpha}}\sum_{k=0}^{N-1}\sum_{n=0}^{N-k}\alpha_nG_{k+n}F_{k}\\
=^{_7}\sum_{k=0}^{N-1}(\Delta_{+}^{\alpha}G_k)F_k&.
\end{split}
\]
In $=^{_1}$ the definition \eqref{DiscFracDerDef:1} is used. In $=^{_2}$ $F_0=0$ is taken into account. To prove $=^{_3}$ is enough to notice that, for a fixed $j=0,...,N,$ the elements $a_{i}:=\alpha_iG_jF_{j-i}$, $i=0,...,j$, on the left hand side, disposed in columns, form an upper diagonal $(N+1)\times (N+1)$, whereas the same elements on the right hand side, for $j=0,...,N$ and $i=j,...,N$, account for the transposed matrix; therefore their total sums are equal. In $=^{_4}$ the sum index is rearranged. In $=^{_5}$ equivalent arguments to $=^{_3}$ can be used. In $=^{_6}$ $F_N=0$ is taken into account. Finally, in $=^{_7}$ the definition \eqref{DiscFracDerDef:2} is used.
\eqref{DIParts:c}:
\[
\sum_{k=0}^{N-1}G_{k+1}(\Delta_{-}^{\alpha}F_{k+1})=^{_{1}}\sum_{k=1}^{N}G_{k}(\Delta_{-}^{\alpha}F_{k})=^{_{2}}\sum_{k=0}^{N-1}(\Delta_{-}^{\alpha}G_{k})F_{k}=^{_{3}}\sum_{k=1}^{N-1}(\Delta_{+}^{\alpha}G_{k})F_{k}.
\]
In $=^{_{1}}$ we have arranged the sum index. In $=^{_{2}}$ we have used \eqref{DIParts:b} and, finally, in $=^{_{3}}$ we have used $F_0=0.$
\end{proof}
\begin{remark}\label{MixedIndices}
By simple inspection it is easy to check that the discrete asymetric integration by parts does not hold for {\it out of phase} indices, for instance $\sum_{k=0}^{N-1}(\Delta^{\alpha}_{+}G_k)F_{k+1}\neq \sum_{k=1}^{N-1}G_k(\Delta^{\alpha}_{-}F_{k+1})$ and similar cases. \hfill $\diamond$
\end{remark}
Next, according to \S\ref{DiscreteFramework}, we shall consider the discrete Lagrangian as an approximation of the continuous action \eqref{ContAction}, i.e.
\[
\mathcal{L}_d\simeq \int_{t_k}^{t_k+h}\mathcal{L}\left( x(t),y(t),\dot x(t),\dot y(t),D^{\alpha}_{-}x(t),D^{\alpha}_{+}y(t)\right)\,dt,
\]
for $a+h<t_k< b-h$. As mentioned above, we shall use \eqref{DiscFracDerDef:1},\eqref{DiscFracDerDef:2} as discrete counterparts for the fractional derivatives. We see that these discrete series imply the whole discrete past for $x$ and the whole discrete future for $y$ at time $t_k$. Thus, it is clear that the approximation of the action in the discrete time interval $[k,k+1]$ would depend on $x_{(0,k+1)}:=(x_0,...,x_{k+1})\in \R^{d(k+2)}$ and $y_{(k,N)}:=(y_k,...,y_{N})\in \R^{d(N+1-k)}$, accounting for the discrete version of the non-locality of the fractional derivatives. Moreover, it is explicit that the approximation of the action shall depend as well on the interval $[k,k+1]$ and therefore on $k$. According to this, we establish the function
\begin{equation}\label{DiscLag}
\begin{split}
\mathcal{L}_d^k:&\,\,\,\R^{d(k+2)}_X\times \R^{d(N+1-k)}_Y\quad\longrightarrow\quad\R\\
&\,\,\,\,\,(x_{(0,k+1)},y_{(k,N)})\quad\quad\quad\mapsto \mathcal{L}_d^k(x_{(0,k+1)},y_{(k,N)}),
\end{split}
\end{equation}
where we remark the correspondence of the first $k+2$ entries with the discrete $x$-path, and the last $N+1-k$ with the $y$-path. If we define the sets of discrete curves as
\[
\begin{split}
C_d^x&=\left\{ \gamma_d^x=\left\{ x_k\right\}_{\tiny 0:N}\in\R^{d(N+1)}\,|\,x_0=x_a,\,\,x_N=x_b\right\},\\
C_d^y&=\left\{ \gamma_d^y=\left\{ y_k\right\}_{\tiny 0:N}\in\R^{d(N+1)}\,|\,\,y_0=y_a,\,\,\,y_N=y_b\right\},
\end{split}
\]
then the discrete action sum is defined naturally by $S_d:C_d^x\times C_d^y\rightarrow\R$:
\begin{equation}\label{DiscAct}
S_d(\tilde\gamma_d):=\sum_{k=0}^{N-1}\mathcal{L}_d^k(x_{(0,k+1)},y_{(k,N)}),
\end{equation}
where $\tilde\gamma_d:=(\gamma_d^x,\gamma_d^y)\in C_d^x\times C_d^y$. Next, we introduce the discrete restricted variations.
\begin{definition}\label{DiscreteVariations}
{\it Given a discrete curve $\tilde\gamma_d=(\gamma_d^x,\gamma_d^y)\in C_d^x\times C_d^y$, we define the set of {\rm varied discrete curves} by
\begin{equation}\label{DiscVar}
\Gamma^x_{\epsilon}:=\gamma_d^x+\epsilon\,\delta\gamma^x_d,\quad
\Gamma^y_{\epsilon}:=\gamma_d^y+\epsilon\,\delta\gamma^y_d.
\end{equation}
where $\delta\gamma_d^x:=\left\{ \delta x_k\right\}_{0:N}$, $\delta\gamma_d^y:=\left\{ \delta y_k\right\}_{0:N}$ are the {\rm discrete variations}, defined such that
\begin{equation}\label{DiscEndpoints}
\delta x_0=\delta x_N=0,\quad\,\delta y_0=\delta y_N=0.
\end{equation}
We define the set of {\rm restricted varied discrete curves}, by
\begin{equation}\label{DiscRestVar}
\Gamma^x_{(\epsilon,\eta)}:=\gamma_d^x+\epsilon\,\eta_d,\quad
\Gamma^y_{(\epsilon,\eta)}:=\gamma_d^y+\epsilon\,\eta_d,
\end{equation}
where we establish $\eta_d=\delta\gamma_d^x=\delta\gamma_d^y$. In other words, we set $\delta x_k=\delta y_k$ for $k=1,...,N-1$.}
\end{definition}
In the following proposition we establish the necessary and sufficient condition for the extremals of \eqref{DiscAct} under restricted variations (discrete restricted Hamilton's principle).
\begin{proposition}\label{FullLagProposition}
Given the discrete action \eqref{DiscAct} subject to restricted variations \eqref{DiscRestVar} and endpoint conditions \eqref{DiscEndpoints}, the necessary and sufficient conditions for the extremals are
\begin{equation}\label{DELEqs:NS}
\sum_{i=k-1}^{N-1}D_{x_k}\mathcal{L}_d^i(x_{(0,i+1)},y_{(i,N)}) +\sum_{i=0}^{k}D_{y_k}\mathcal{L}_d^i(x_{(0,i+1)},y_{(i,N)})=0,
\end{equation}
for $k=1,...,N-1$, where $D_{x_k}:=\partial/\partial x_k$ and $D_{y_k}:=\partial/\partial y_k$.
\end{proposition}
\begin{proof}
The condition for the extremals is $\frac{d}{d\epsilon}S_d((\Gamma^x_{(\epsilon,\eta)},\Gamma^y_{(\epsilon,\eta)}))\Big|_{\epsilon=0}=0$, which, as it is easy to see, is equivalent to
\[
\begin{split}
\delta \sum_{k=0}^{N-1}&\mathcal{L}_d^k(x_{(0,k+1)},y_{(k,N)})\\
&=\delta \mathcal{L}_d^0(x_{(0,1)},y_{(1,N)}) + \delta \mathcal{L}_d^1(x_{(0,2)},y_{(1,N)})+\cdots\\
&\hspace{4cm} \cdots+\delta \mathcal{L}_d^{N-2}(x_{(0,N-1)},y_{(N-2,N)})+\delta \mathcal{L}_d^{N-1}(x_{(0,N)},y_{(N-1,N)})=0,
\end{split}
\]
given that $\delta x_k=\delta y_k$. Considering the variations with respect to $x$ and $y$ separately, we obtain
\[
\begin{split}
&\delta_x\sum_{k=0}^{N-1}\mathcal{L}_d^k(x_{(0,k+1)},y_{(k,N)})\\
=\,&D_{_{x_0}}\mathcal{L}_d^0\,\delta x_0+D_{_{x_1}}\mathcal{L}_d^0\,\delta x_1\\
+&D_{_{x_0}}\mathcal{L}_d^1\,\delta x_0+D_{_{x_1}}\mathcal{L}_d^1\,\delta x_1+D_{_{x_2}}\mathcal{L}_d^1\,\delta x_2\\
&\hspace{0.6cm} \vdots\hspace{2.4cm} \vdots\hspace{2.4cm} \vdots\\
+&D_{_{x_0}}\mathcal{L}_d^{_{N-1}}\,\delta x_0+D_{_{x_1}}\mathcal{L}_d^{_{N-1}}\,\delta x_1+D_{_{x_2}}\mathcal{L}_d^{_{N-1}}\,\delta x_2+\cdots+D_{_{x_{N-1}}}\mathcal{L}_d^{_{N-1}}\,\delta x_{N-1}+D_{_{x_N}}\mathcal{L}_d^{_{N-1}}\,\delta x_N,
\end{split}
\]
where $D_{x_i}$ represents the partial derivative with respect to $x_i$, and
\[
\begin{split}
&\delta_y\sum_{k=0}^{N-1}\mathcal{L}_d^k(x_{(0,k+1)},y_{(k,N)})\\
=\,&D_{_{y_0}}\mathcal{L}_d^{0}\,\delta x_0+D_{_{y_1}}\mathcal{L}_d^{0}\,\delta x_1+D_{_{y_2}}\mathcal{L}_d^{0}\,\delta x_2+\cdots+D_{_{y_{N-1}}}\mathcal{L}_d^{0}\,\delta x_{N-1}+D_{_{y_N}}\mathcal{L}_d^{0}\,\delta x_N\\
&\hspace{1.7cm} +D_{_{y_1}}\mathcal{L}_d^1\,\delta x_1+D_{_{y_2}}\mathcal{L}_d^1\,\delta x_2+\cdots+D_{_{y_{N-1}}}\mathcal{L}_d^{1}\,\delta x_{N-1}+D_{_{y_N}}\mathcal{L}_d^{1}\,\delta x_N\\
&\hspace{9cm} \vdots\hspace{2.2cm}\vdots\\
&\hspace{6.2cm} +D_{_{y_{N-1}}}\mathcal{L}_d^{_{N-1}}\,\delta x_{N-1}+D_{_{y_N}}\mathcal{L}_d^{_{N-1}}\,\delta x_N.
\end{split}
\]
Adding up the columns, we arrive to
\[
\begin{split}
&\delta \sum_{k=0}^{N-1}\mathcal{L}_d^k(x_{(0,k+1)},y_{(k,N)})=\Big(\sum_{i=0}^{N-1}D_{_{x_0}}\mathcal{L}_d^i+D_{_{y_0}}\mathcal{L}_d^0\Big)\,\delta x_0\\
&\hspace{3cm} +\sum_{k=1}^{N-1}\Big(\sum_{i=k-1}^{N-1}D_{x_k}\mathcal{L}_d^i +\sum_{i=0}^{k}D_{y_k}\mathcal{L}_d^i\Big)\,\delta x_k
+\Big( D_{_{x_N}}\mathcal{L}_d^{N-1}+\sum_{i=0}^{k}D_{y_k}\mathcal{L}_d^i\Big)\,\delta x_N.
\end{split}
\]
Equating the last expression to 0, considering $\delta x_0=\delta x_N=0$ and arbitrary variations $\delta x_k$, we arrive directly to \eqref{DELEqs:NS}.
\end{proof}
\begin{remark}\label{AllVariables}
Observe that the discrete Lagrangian problem established by \eqref{DiscLag} and \eqref{DiscAct} is of the higher-order type, i.e. the discrete Lagrangian depends on multiple copies (more than two) of the configuration manifold (see \cite{HO:1,HO:2} for more details). As striking difference with respect to the kind of problem in these references, we note that the number of copies of $\R^d_X$ and $\R^d_Y$ the discrete Lagrangian \eqref{DiscLag} depends on is not fixed and is determined by $k$. This circumstance
prevents in general the definition of a discrete flow in the sense expressed in \S\ref{DiscreteFramework}, where $F_{L_d}:Q\times Q\rightarrow Q\times Q$ can be obtained from \eqref{DEL} under regularity conditions. This can be seen as well noticing that all variables are present at the same time in \eqref{DELEqs:NS} for any $k$, which makes mandatory to solve them simultaneously in order to obtain the sequences $\left\{ x_k\right\}_{0:N}$, $\left\{ y_k\right\}_{0:N}$. However, we will see below that particular choices of $\mathcal{L}_d^k$ lead to actual discrete flows.
\hfill $\diamond$
\end{remark}
\begin{remark}\label{RemarkLagrangianMixing}
For unrestricted variations \eqref{DiscVar}, i.e. $\delta x_k\neq\delta y_k$ and both arbitrary, and endpoint conditions \eqref{DiscEndpoints}, the necessary and sufficient conditions for the extremals of \eqref{DiscAct} read
\[
\sum_{i=k-1}^{N-1}D_{x_k}\mathcal{L}_d^i(x_{(0,i+1)},y_{(i,N)})=0,\,\,\,\quad\sum_{i=0}^{k}D_{y_k}\mathcal{L}_d^i(x_{(0,i+1)},y_{(i,N)})=0;\,\,\, k=1,...,N-1. \quad\quad\quad \diamond
\]
\end{remark}
Next, let us pick the discrete Lagrangian
\begin{equation}\label{DiscLag:Dyn}
\mathcal{L}_d^k(x_{(0,k+1)},y_{(k,N)}):=L_d(x_k,x_{k+1})+L_d(y_k,y_{k+1})-h\,\Delta_{-}^{\alpha}x_k\,\rho\,\Delta_{+}^{\alpha}y_k,
\end{equation}
where $L_d:\R^d\times\R^d\rightarrow\R$ is a particular discretisation of the continuous action integral defined for the Lagrangian $L:T\R^d\rightarrow\R$ in \eqref{PartLagrangian}, the discrete fractional derivatives are defined in \eqref{DiscFracDerDef:1}, \eqref{DiscFracDerDef:2} and $\rho\in\R^{d\times d}$. Furthermore, we set $\mathcal{L}_d^N:=0$ since $L_d$ is not defined in such a case. Naturally, this is the discrete counterpart of the continuous Lagrangian \eqref{PartLagrangian}.
\begin{theorem}\label{Theo:DiscEqs}
Given the discrete Lagrangian \eqref{DiscLag:Dyn}, restricted variations \eqref{DiscRestVar} and endpoint conditions \eqref{DiscEndpoints}, a sufficient condition for the extremals of \eqref{DiscAct} is
\begin{subequations}\label{RestDEL:Dyn}
\begin{align}
&D_1L_d(x_k,x_{k+1})+D_2L_d(x_{k-1},x_{k})-h\,\rho\,\Delta_{-}^{\alpha}\Delta_{-}^{\alpha}x_k=0,\label{RestDEL:Dyn:a}\\
&D_1L_d(y_k,y_{k+1})\,+\,D_2L_d(y_{k-1},y_{k})-h\,\,\rho\,\Delta_{+}^{\alpha}\Delta_{+}^{\alpha}y_k=0,\label{RestDEL:Dyn:b}
\end{align}
\end{subequations}
for $k=1,...,N-1$, and $D_i$ denotes the partial derivative with respect to the $i$-th variable.
\end{theorem}
\begin{proof}
To prove the claim it is enough to show that equations \eqref{RestDEL:Dyn} are a sufficient condition of \eqref{DELEqs:NS} to be satisfied.
First, we see that
\begin{subequations}\label{Sums}
\begin{align}
\frac{\partial}{\partial x_k}\sum_{i=k-1}^{N-1}\mathcal{L}_d^i&=\frac{\partial}{\partial x_k}\Big(-\sum_{i=0}^{k-2}\mathcal{L}_d^i+\sum_{i=0}^{N-1}\mathcal{L}_d^i\Big)=\frac{\partial}{\partial x_k}\sum_{i=0}^{N-1}\mathcal{L}_d^i=\frac{\partial}{\partial x_k}\sum_{i=0}^{N}\mathcal{L}_d^i,\label{Sums:a}\\
\frac{\partial}{\partial y_k}\sum_{i=0}^{k}\mathcal{L}_d^i&=\frac{\partial}{\partial y_k}\Big(-\sum_{i=k+1}^{N-1}\mathcal{L}_d^i+\sum_{i=0}^{N-1}\mathcal{L}_d^i\Big)=\frac{\partial}{\partial y_k}\sum_{i=0}^{N-1}\mathcal{L}_d^i.\label{Sums:b}
\end{align}
\end{subequations}
In \eqref{Sums:a} we use that $\sum_{i=0}^{k-2}\partial\mathcal{L}_d^i/\partial x_k=0$ because there is no dependence of $\mathcal{L}_d^i$ on $x_k$ in the range $[0,k-2]$, and that $\mathcal{L}_d^N=0$ in the last equality (which implies that we have already picked the Lagrangian \eqref{DiscLag:Dyn}). Equivalent arguments lead to $\sum_{i=k+1}^{N-1}\partial\mathcal{L}_d^i/\partial y_k=0$ in order to prove \eqref{Sums:b}.
Recalling that $k=1,...,N-1$, from \eqref{Sums:a} we get:
\[
\begin{split}
\frac{\partial}{\partial x_k}\sum_{i=k-1}^{N-1}\mathcal{L}_d^i=\frac{\partial}{\partial x_k}\sum_{i=0}^{N}\mathcal{L}_d^i=^{_{1}}&\sum_{i=1}^{N-1}\Big(D_1L_d(x_i,x_{i+1})+D_2L_d(x_{i-1},x_{i})\Big)\delta_{ik}-h\frac{\partial}{\partial x_k}\sum_{i=1}^{N}\Delta_{-}^{\alpha}x_i\rho\Delta_{+}^{\alpha}y_i\\
=&\sum_{i=1}^{N-1}\Big(D_1L_d(x_i,x_{i+1})+D_2L_d(x_{i-1},x_{i})\Big)\delta_{ik}-h\frac{\partial}{\partial x_k}\sum_{i=1}^{N-1}x_i\rho\Delta_{+}^{\alpha}\Delta_{+}^{\alpha}y_i\\
=^{_{2}}&\sum_{i=1}^{N-1}\Big(D_1L_d(x_i,x_{i+1})+D_2L_d(x_{i-1},x_{i})-h\,\rho\,\Delta_{+}^{\alpha}\Delta_{+}^{\alpha}y_i\Big)\delta_{ik}\\
=&\quad \,\,\,\,\,\,\,\,D_1L_d(x_k,x_{k+1})+D_2L_d(x_{k-1},x_{k})-h\,\rho\,\Delta_{+}^{\alpha}\Delta_{+}^{\alpha}y_k,
\end{split}
\]
where $\delta_{ij}$ represents the Kronecker delta. In the previous computation, right hand side of $=^{_{1}}$, we have taken into account that
\[
\frac{\partial}{\partial x_k}\sum_{i=0}^{N}\Delta_{-}^{\alpha}x_i\,\rho\,\Delta_{+}^{\alpha}y_i=\frac{\partial}{\partial x_k}\sum_{i=1}^{N}\Delta_{-}^{\alpha}x_i\,\rho\,\Delta_{+}^{\alpha}y_i
\]
given that $\partial(\Delta_{-}^{\alpha}x_0\,\rho\,\Delta_{+}^{\alpha}y_0)/\partial x_k=0$ for $k=1,...,N-1$. Moreover, in the right hand side of $=^{_{2}}$ we have used \eqref{DIParts:b}, $\partial (x_0\,\rho\,\Delta_{+}^{\alpha}\Delta_{+}^{\alpha}y_0)/\partial x_k=0$ and $\partial x_i/\partial x_k=\delta_{ij}$ for $k$ in the range $[1,N-1]$.
Using equivalent arguments, it can be proven from \eqref{Sums:b} that:
\[
\begin{split}
\frac{\partial}{\partial y_k}\sum_{i=0}^{k}\mathcal{L}_d^i=\frac{\partial}{\partial y_k}\sum_{i=0}^{N-1}\mathcal{L}_d^i=&\sum_{i=1}^{N-1}\Big(D_1L_d(y_i,y_{i+1})+D_2L_d(y_{i-1},y_{i})-h\,\rho\,\Delta_{-}^{\alpha}\Delta_{-}^{\alpha}x_i\Big)\delta_{ik}\\
=&\hspace{0.5cm}\,\,\,\,\,\,\,\, D_1L_d(y_k,y_{k+1})+D_2L_d(y_{k-1},y_{k})-h\,\rho\,\Delta_{-}^{\alpha}\Delta_{-}^{\alpha}x_k,
\end{split}
\]
where again \eqref{DIParts:b} and the range $k=1,...,N-1$ are used.
Adding both terms together and equating the sum to 0, which accounts for \eqref{DELEqs:NS}, we obtain:
\[
\begin{split}
&D_1L_d(x_k,x_{k+1})+D_2L_d(x_{k-1},x_{k})-h\,\rho\,\Delta_{+}^{\alpha}\Delta_{+}^{\alpha}y_k\\
&\,\,+D_1L_d(y_k,y_{k+1})+D_2L_d(y_{k-1},y_{k})-h\,\rho\,\Delta_{-}^{\alpha}\Delta_{-}^{\alpha}x_k=0.
\end{split}
\]
Swapping the fractional terms and equating each block to 0, we directly obtain \eqref{RestDEL:Dyn} as a sufficient condition for \eqref{DELEqs:NS} to hold.
\end{proof}
We note that equations \eqref{RestDEL:Dyn} are the natural discrete version of \eqref{ResFracELDamped}.
\begin{remark}\label{Alphak+1}
Observe that a term $h\Delta_{-}^{\alpha}x_{k+1}\Delta_{+}^{\alpha}y_{k+1}$ is admissible in \eqref{DiscLag:Dyn} according to the definition \eqref{DiscLag}, leading to a term $h\sum_{k=0}^{N-1}\Delta_{-}^{\alpha}x_{k+1}\Delta_{+}^{\alpha}y_{k+1}$ in the action sum \eqref{DiscAct}. However, it provides the same discrete dynamics as $\Delta_{-}^{\alpha}x_{k}\Delta_{+}^{\alpha}y_{k}$, according to \eqref{DIParts:c}, which makes it redundant. Further terms, as those described in Remark \ref{MixedIndices}, are meaningless since the asymetric integration by parts is not defined for them. \hfill $\diamond$
\end{remark}
In the next result we prove that the $x$ and $y$ dynamics in \eqref{RestDEL:Dyn} are also related under inversion of time at a discrete level.
\begin{proposition}\label{InvDiscTime}
Given $y_k:=x_{N-k}$, then \eqref{RestDEL:Dyn:b} is \eqref{RestDEL:Dyn:a} in reversed discrete time.
\end{proposition}
\begin{proof}
We define the reversed discrete time as $\tilde k:=N-k$, such that $\tilde k=N,...,0$, for $k=0,...,N$. Given that, we observe that, under inversion of time:
\[
\begin{split}
D_1L_d(y_k,y_{k+1})\,+\,D_2L_d(y_{k-1},y_{k})&=D_1L_d(x_{N-k},x_{(N-k)+1})\,+\,D_2L_d(x_{(N-k)-1},x_{N-k})\\
&=D_1L_d(x_{\tilde k},x_{\tilde k+1})\,+\,D_2L_d(x_{\tilde k-1},x_{\tilde k}).
\end{split}
\]
On the other hand:
\[
\begin{split}
\Delta_{+}^{\alpha}\Delta_{+}^{\alpha}y_k=\frac{1}{h^{2\alpha}}\sum_{n=0}^{N-k}\alpha_n\sum_{p=0}^{N-k-n}\alpha_p\,y_{k+n+p}&=\frac{1}{h^{2\alpha}}\sum_{n=0}^{(N-k)}\alpha_n\sum_{p=0}^{(N-k)-n}\alpha_p\,x_{(N-k)-n-p}\\
&=\frac{1}{h^{2\alpha}}\sum_{n=0}^{\tilde k}\alpha_n\sum_{p=0}^{\tilde k-n}\alpha_p\,x_{\tilde k-n-p}=\Delta_{-}^{\alpha}\Delta_{-}^{\alpha}x_{\tilde k}.
\end{split}
\]
Multiplying $\Delta_{-}^{\alpha}\Delta_{-}^{\alpha}x_{\tilde k}$ by $h\,\rho$, adding it to $D_1L_d(x_{\tilde k},x_{\tilde k+1})\,+\,D_2L_d(x_{\tilde k-1},x_{\tilde k})$ and equating the sum to 0, the claim holds.
\end{proof}
As advanced in Remark \ref{AllVariables}, in spite all variables are present at the same time in \eqref{DELEqs:NS}, which {\it a priori} makes necessary to solve them all simultaneously, we can find particular and meaningful Lagrangian functions \eqref{DiscLag:Dyn} such that the discrete dynamics \eqref{RestDEL:Dyn:a} (we restrict ourselves to the $x$-system since we have just proved that $y$ can be interpreted as $x$ in reversed time) is provided by a discrete flow. That is established in the following algorithm, accounting for the definition of the FVIs:
\begin{algorithm}{\rm Fractional Variational Integrator Scheme}
\label{FractionalAlgorithm}
\begin{algorithmic}[1]
\State {\bf Initial data}: $N,\, h,\,\alpha,\,\rho,\, x_0,\, p_{x_0}.$
\State {\bf solve for} $x_1$ {\bf from} $p_{x_0}=I(x_0,x_1).$
\State {\bf Initial points:} $x_0,\,x_1.$
\For {$k= 1: N-1$}
\hspace{-0.6cm} {\bf solve for} $x_{k+1}$ {\bf from} $D_1L_d(x_k,x_{k+1})+D_2L_d(x_{k-1},x_{k})-h\,\rho\,\Delta_{-}^{\alpha}\Delta_{-}^{\alpha}x_k=0$
\EndFor
\State {\bf Output:} $(x_2,...,x_N).$
\end{algorithmic}
\end{algorithm}
Observe that the initialisation condition in Step 2, i.e. $p_{x_0}=I(x_0,x_1)$, has to be properly determined. This will be object of discussion when defining the fractional discrete Legendre transformation in \S\ref{DLTra}.
\subsection{Discrete mechanical Lagrangian}
Now, let us pick the discrete Lagrangian
\begin{equation}\label{DiscMechLag}
L_d(z_k,z_{k+1}):=\frac{1}{2h}(z_{k+1}-z_k)\,m\,(z_{k+1}-z_k)-h\,U(S^{\kappa}z_k),
\end{equation}
where $S^{\kappa}z_k$ is given in \eqref{KappaRule}, as the usual discretisation of the mechanical Lagrangian \eqref{MechLag}. In this case, \eqref{RestDEL:Dyn} read
\begin{equation}\label{MechDynFracDamp:Disc}
\begin{split}
m\frac{x_{k+1}-2x_k+x_{k-1}}{h^2}&+\kappa\nabla U(S^{\kappa}x_k)+(1-\kappa)\nabla U(S^{\kappa}x_{k-1})+\rho\,\Delta_{-}^{\alpha}\Delta_{-}^{\alpha}x_k=0,\\
m\frac{y_{k+1}-2y_k+y_{k-1}}{h^2}&+\kappa\nabla U(S^{\kappa}y_k)+(1-\kappa)\nabla U(S^{\kappa}y_{k-1})+\rho\,\Delta_{+}^{\alpha}\Delta_{+}^{\alpha}y_k=0,
\end{split}
\end{equation}
where we have divided both sides by $h$. We observe that these equations are a discretisation in finite differences of \eqref{MechDynFracDamp}.
According to what happens in the continous case, i.e. $D^{2\alpha}_{-}= d/dt$ when $\alpha= 1/2$, we expect a particular discretisation of the total time derivative from the term $\Delta_{-}^{\alpha}\Delta_{-}^{\alpha}x_k$. This is proven in the following result.
\begin{lemma}\label{LemmaDisc}
For $\alpha=1/2$ and $k=1,...,N-1$,
\[
h\,\Delta_{-}^{_{1/2}}\Delta_{-}^{_{1/2}}x_k=\sum_{n=0}^k\alpha_n\big|_{_{\alpha=1/2}}\sum_{p=0}^{k-n}\alpha_p\big|_{_{\alpha=1/2}}x_{k-n-p}=x_k-x_{k-1}.
\]
\end{lemma}
\begin{proof} According to \eqref{AlphaDef} we have that $\alpha_0=1$ and $\alpha_1=-1/2$ for $\alpha=1/2$, leading, after expanding the summations, to
\[
\begin{split}
\sum_{n=0}^k\alpha_n\sum_{p=0}^{k-n}\alpha_px_{k-n-p}=(x_k-x_{k-1})+\sum_{n=2}^{k}2\alpha_nx_{k-n}+\sum_{n=1}^k\alpha_n\sum_{p=1}^{k-n}\alpha_px_{k-n-p}.
\end{split}
\]
In this expansion, the value of the sum when $k=0$, explicited in Remark \ref{SumEndpoint}, has been taken into account. The claim automatically holds for $k=1$. For $k\geq 2$, arranging the sum indices we see that the previous expression can be rewritten as
\begin{equation}\label{SumExp}
\begin{split}
\sum_{n=0}^k\alpha_n\sum_{p=0}^{k-n}\alpha_px_{k-n-p}=&(x_k-x_{k-1})\\
+&\sum_{s=2}^{r}\beta_0^sx_{k-s}+\sum_{l=0}^{k-(r+1)}\beta_l^{r+1}x_{k-(r+1)-n}+\sum_{n=r}^k\alpha_n\sum_{p=1}^{k-n}\alpha_px_{k-n-p},
\end{split}
\end{equation}
where, for a fixed $k$, we set $r=k-1$ and
\begin{equation}\label{Betas}
\beta_l^j=2\alpha_{l+j}+\sum_{i=1}^{j-1}\alpha_i\alpha_{l+j-i},
\end{equation}
(it is apparent that $j$ is not a power but a superindex). For a fixed $k=\tilde k$, \eqref{SumExp} acquires the form
\[
\begin{split}
&\sum_{n=0}^{\tilde k}\alpha_n\sum_{p=0}^{\tilde k-n}\alpha_px_{\tilde k-n-p}=(x_{\tilde k}-x_{\tilde k-1}) +\beta_0^2x_{\tilde k-2}+\beta_0^3x_{\tilde k-3}+\cdot\cdot\cdot+\beta_0^{\tilde k-2}x_2+\beta_0^{\tilde k-1}x_1+\beta_0^{\tilde k}x_0.
\end{split}
\]
According to this, it is enough to prove that $\beta_0^j=0$ for any $j$, for which we proceed by induction. From \eqref{Betas} and \eqref{AlphaDef}, it follows that $\beta^2_0=2\alpha_2+\alpha_1\alpha_1$, which vanishes for $\alpha=1/2$. Taking this as the first induction step, it is enough to prove that $\beta^{j+1}_0=0$ assuming that $\beta^j_0=0$. This is shown next:
\[
\begin{split}
&\beta^{j+1}_0=2\alpha_{j+1}+\sum_{i=1}^{j}\alpha_i\alpha_{j+1-i}=2\alpha_{j+1}+\sum_{i=1}^{r-1}\alpha_i\alpha_{r-i}\\
&\hspace{3cm} \quad\quad=2\alpha_{j+1}-2\alpha_r+2\alpha_r+\sum_{i=1}^{r-1}\alpha_i\alpha_{r-i}=2\alpha_{j+1}-2\alpha_r=0,
\end{split}
\]
where we have set $r=j+1.$ Hence the claim follows.
\end{proof}
Using similar arguments, one can prove that
\[
h\,\Delta_{+}^{_{1/2}}\Delta_{+}^{_{1/2}}y_k=\sum_{n=0}^{N-k}\alpha_n\big|_{_{\alpha=1/2}}\sum_{p=0}^{N-k-n}\alpha_p\big|_{_{\alpha=1/2}}y_{k+n+p}=-(y_{k+1}-y_k).
\]
It follows straightforwardly that
\begin{equation}\label{Ffrac:OneHalf}
\Delta_{-}^{_{1/2}}\Delta_{-}^{_{1/2}}x_k=\frac{x_k-x_{k-1}}{h},\,\quad \Delta_{+}^{_{1/2}}\Delta_{+}^{_{1/2}}y_k=-\frac{y_{k+1}-y_{k}}{h},
\end{equation}
showing that $\Delta_{-}^{_{1/2}}\Delta_{-}^{_{1/2}}x_k$, $\Delta_{+}^{_{1/2}}\Delta_{+}^{_{1/2}}y_k$ are the backward and forward (up to a minus sign) difference operators, respectively; thus order one approximations of the velocity. This leads to the following corollary of Theorem \ref{Theo:DiscEqs}.
\begin{corollary}\label{Corolario}
{\it Given $\alpha=1/2$, then \eqref{RestDEL:Dyn:a} is equivalent to the forced discrete Euler-Lagrange equations \eqref{ForcedDEL} when $f^{-}_{L_d}(x_k,x_{k+1})=0$, $f^+_{L_d}(x_k,x_{k+1})=-\rho\,(x_{k+1}-x_k).$ }
\end{corollary}
\begin{proof}
The result follows straightforwardly by replacing \eqref{Ffrac:OneHalf} (first relationship) in \eqref{RestDEL:Dyn:a} and comparing with \eqref{ForcedDEL} for $f^{-}_{L_d}(x_k,x_{k+1})=0$, $f^+_{L_d}(x_k,x_{k+1})=-\rho\,(x_{k+1}-x_k).$
\end{proof}
An equivalent result can be obtained for the $y$-mirror system setting $f_{L_d}^-(y_k,y_{k+1})=\rho(y_{k+1}-y_k)$ and $f_{L_d}^{+}(y_k,y_{k+1})=0$. This discussion makes explicit the relationship between the discretisation of the Lagrange-d'Alembert principle (\S\ref{DiscLdAlem}, \eqref{ForcedDEL}) and the restricted Hamilton's principle developed in this work, that we display in Figure \ref{Diagram1} (where we omit the $y$-system for sake of simplicity).
\begin{figure}[!htb]
\includegraphics[scale=0.42]{DiscLagDiagram2.PNG}
\caption{In the diagram it is reflected that the Lagrange-d'Alembert principle for a general discrete Lagrangian and a particular set of discrete forces, produce the same discrete forced equations as enlarging the discrete phase space, then applying the discrete restricted Hamilton's principle for $\mathcal{L}_d^k$ \eqref{DiscLag:Dyn}, and then setting $\alpha=1/2$.}
\label{Diagram1}
\end{figure}
\subsection{Discrete Legendre transformation}\label{DLTra}
The main guidelines to construct the discrete Legendre transformation in the fractional case are the following:
\begin{itemize}
\item[1.] As in the usual discrete mechanics \S\ref{DiscreteFramework}, we want to reproduce the discrete Lagrangian dynamics \eqref{RestDEL:Dyn} through {\it momentum matching}.
\item[2.] We seek for a fair discretisation of the fractional Hamilton equations in the case of mechanical Hamiltonians \eqref{FracPartiHam}.
\item[3.] We intend to obtain initialisation condition (Step 2) for Algorithm \ref{FractionalAlgorithm}.
\end{itemize}
According to this, we provide the following definition of the discrete Legendre transformation. Previously, we introduce the intermediate variables:
\[
x_k^{\alpha}:=\Delta^{\alpha}_{-}x_k,\quad\quad y_k^{\alpha}:=\Delta^{\alpha}_{+}y_k,
\]
and replace them into the Lagrangian \eqref{DiscLag:Dyn} in order to establish the following definition
\begin{equation}\label{DiscLagInterm}
\begin{split}
\mathcal{L}_d^k(x_{(0,k+1)},y_{(k,N)})&:=\tilde{\mathcal{L}}_d^k(x_k,x_{k+1},y_k,y_{k+1},x_k^{\alpha},y_k^{\alpha},x_{k+1}^{\alpha},y_{k+1}^{\alpha}).
\end{split}
\end{equation}
For the sake of generality, we allow the presence of $x_{k+1}^{\alpha}=\Delta_{-}^{\alpha}x_{k+1}$ and $y_{k+1}^{\alpha}=\Delta_{+}^{\alpha}y_{k+1}$, which are admissible as discussed in Remark \ref{Alphak+1}.
\begin{definition}\label{Def:DLT}
{\it Given the discrete Lagrangian \eqref{DiscLagInterm}, we define the} discrete Legendre transformation {\it by:}
\begin{subequations}\label{LegTrans}
\begin{align}
\begin{bmatrix}
p_{x_k}^{-}\\
p_{y_k}^{-}
\end{bmatrix}&=
-\begin{bmatrix}
\,\,D_{x_k}\tilde{\mathcal{L}}_d^k\,\,\\
\,\,D_{y_k}\tilde{\mathcal{L}}_d^k\,\,
\end{bmatrix}
\,\,\,\,-\,\begin{bmatrix}
\Delta_{-}^{\alpha} & \Delta_{+}^{\alpha}
\end{bmatrix}
\begin{bmatrix}
0&1\\
1&0
\end{bmatrix}
\begin{bmatrix}
\,\,D_{x_k^{\alpha}}\tilde{\mathcal{L}}_d^k\,\,\\
\,\,D_{y_k^{\alpha}}\tilde{\mathcal{L}}_d^k\,\,
\end{bmatrix}, \label{LegTrans:a}\\
\begin{bmatrix}
p_{x_{k+1}}^{+}\\
p_{y_{k+1}}^{+}
\end{bmatrix}&=\,\,\,\,\,
\begin{bmatrix}
\,\,D_{x_{k+1}}\tilde{\mathcal{L}}_d^k\,\,\\
\,\,D_{y_{k+1}}\tilde{\mathcal{L}}_d^k\,\,
\end{bmatrix}+\begin{bmatrix}
\Delta_{-}^{\alpha} & \Delta_{+}^{\alpha}
\end{bmatrix}
\begin{bmatrix}
0&1\\
1&0
\end{bmatrix}
\begin{bmatrix}
\,\,D_{x_{k+1}^{\alpha}}\tilde{\mathcal{L}}_d^k\,\,\\
\,\,D_{y_{k+1}^{\alpha}}\tilde{\mathcal{L}}_d^k\,\,
\end{bmatrix}
,\label{LegTrans:b}\\
\begin{bmatrix}
p_{x_{k}}^{\alpha}\\
p_{y_{k}}^{\alpha}
\end{bmatrix}&=\,\,\,\,\,
\begin{bmatrix}
\,\,D_{x_{k}^{\alpha}}\tilde{\mathcal{L}}_d^k\,\,\\
\,\,D_{y_{k}^{\alpha}}\tilde{\mathcal{L}}_d^k\,\,
\end{bmatrix} \label{LegTrans:c}
\end{align}
\end{subequations}
{\it where we consider the row matrix $[\Delta_{-}^{\alpha}\,\, \Delta_{+}^{\alpha}]$ in the sense of operators.}
\end{definition}
\begin{proposition}\label{MMPropo}
Given the discrete Legendre transformation in Definition \eqref{Def:DLT}, the particular discrete Lagrangian $\tilde{\mathcal{L}}_d^k=L_d(x_k,x_{k+1})+L_d(y_k,y_{k+1})-h\,x_{k+1}^{\alpha}\,\,\rho\,\,y_{k+1}^{\alpha}$ and the intermediate variables $x_k^{\alpha}:=\Delta^{\alpha}_{-}x_k,\,\, y_k^{\alpha}:=\Delta^{\alpha}_{+}y_k,$ the following statements are true:
\begin{itemize}
\item[1.] The momentum matching condition $p_{x_k}^{-}=p_{x_k}^{+}$, $p_{y_k}^{-}=p_{y_k}^{+}$, is equivalent to the discrete Lagrangian dynamics \eqref{RestDEL:Dyn}.
\item[2.] Given the mechanical Hamiltonian \eqref{MechHamilto} and $L_d(z_k,z_{k+1})=\frac{1}{2h}(z_{k+1}-z_k)\,m\,(z_{k+1}-z_k)-h\,U(z_{k+1})$, i.e. we pick $\kappa=0$ in \eqref{KappaRule},\eqref{DiscMechLag} , then \eqref{LegTrans} provides a discretisation of \eqref{FracPartiHam}.
\item[3.] Under the hypotheses of Statement 2, when $\alpha= 1/2$ \eqref{LegTrans} provides a discretisation of \eqref{HamLinearDamp}, i.e. the Hamiltonian dynamics with linear damping.
\end{itemize}
\end{proposition}
\begin{proof}
Statement 1: from \eqref{LegTrans:a}, \eqref{LegTrans:b} and $\tilde{\mathcal{L}}_d^k=L_d(x_k,x_{k+1})+L_d(y_k,y_{k+1})-h\,x_{k+1}^{\alpha}\rho y_{k+1}^{\alpha}$ it follows:
\[
\begin{split}
\begin{bmatrix}
p_{x_k}^{-}\\
p_{y_k}^{-}
\end{bmatrix}&=
-\begin{bmatrix}
D_{1}L_d(x_k,x_{k+1})\\
D_{1}L_d(y_k,y_{k+1})
\end{bmatrix},\\
\begin{bmatrix}
p_{x_{k+1}}^{+}\\
p_{y_{k+1}}^{+}
\end{bmatrix}&=\,\,\,\,\,
\begin{bmatrix}
D_{2}L_d(x_k,x_{k+1})\\
D_{2}L_d(y_k,y_{k+1})
\end{bmatrix}-h\,\rho\,\begin{bmatrix}
\Delta^{\alpha}_{-}\,x_{k+1}^{\alpha}\\
\Delta^{\alpha}_{+}\,y_{k+1}^{\alpha}
\end{bmatrix}=\begin{bmatrix}
D_{2}L_d(x_k,x_{k+1})\\
D_{2}L_d(y_k,y_{k+1})
\end{bmatrix}
-h\,\rho\,\begin{bmatrix}
\Delta^{\alpha}_{-}\Delta^{\alpha}_{-}\,x_{k+1}\\
\Delta^{\alpha}_{+}\Delta^{\alpha}_{+}\,y_{k+1}
\end{bmatrix}.
\end{split}
\]
Now setting the momentum matching condition $p_{x_k}^{-}=p_{x_k}^{+}$, $p_{y_k}^{-}=p_{y_k}^{+}$ it is straightforward to obtain \eqref{RestDEL:Dyn}.
Statement 2: under the hypotheses, we have that \eqref{FracPartiHam} read
\begin{equation}\label{HamFinal}
\begin{split}
\dot x&=m^{-1}p_x,\quad\quad D^{\alpha}_{-}x=-\rho^{-1}\,p_y^{\alpha}, \quad\quad \dot p_x=-\nabla U(x)+D^{\alpha}_{-}\,p^{\alpha}_y,\\
\dot y&=m^{-1}p_y, \quad\quad
D^{\alpha}_{+}y=-\rho^{-1}\,p_x^{\alpha}, \quad\quad \dot p_y=-\nabla U(y)+D^{\alpha}_{+}\,p^{\alpha}_x.
\end{split}
\end{equation}
On the other hand, from \eqref{LegTrans} we get
\[
\begin{split}
\begin{bmatrix}
p_{x_k}\\
p_{y_k}
\end{bmatrix}&=
\,\,\,\,\,\begin{bmatrix}
\,\,m\frac{x_{k+1}-x_k}{h}\,\,\\
\,\,m\frac{y_{k+1}-y_k}{h}\,\,
\end{bmatrix}, \\
\begin{bmatrix}
p_{x_{k+1}}\\
p_{y_{k+1}}
\end{bmatrix}&=\,\,\,\,\,\,
\begin{bmatrix}
\,\,m\frac{x_{k+1}-x_k}{h}-h\nabla U(x_{k+1})\,\,\\
\,\,m\frac{y_{k+1}-y_k}{h}-h\nabla U(y_{k+1})\,\,
\end{bmatrix}-\begin{bmatrix}
h\,\rho\,\Delta^{\alpha}_{-}\,x_{k+1}^{\alpha}\\
h\,\rho\,\Delta^{\alpha}_{+}\,y_{k+1}^{\alpha}
\end{bmatrix},\\
\begin{bmatrix}
p_{x_{k+1}}^{\alpha}\\
p_{y_{k+1}}^{\alpha}
\end{bmatrix}&=
-\begin{bmatrix}
\,\,h\,\rho\,y^{\alpha}_{k+1}\,\,\\
\,\,h\,\rho\,x^{\alpha}_{k+1}\,\,
\end{bmatrix} = -\begin{bmatrix}
\,\,h\,\rho\,\Delta^{\alpha}_{+}\,y_{k+1}\,\,\\
\,\,h\,\rho\,\Delta^{\alpha}_{-}\,x_{k+1}\,\,
\end{bmatrix}.
\end{split}
\]
(Observe that, in the last equation, we have taken a $k$-step forward in \eqref{LegTrans:c}). From this, splitting $x$ and $y$ sides and rearranging terms, we obtain
\begin{equation}\label{DiscHamFinal}
\begin{split}
x_{k+1}&=x_k+h\,m^{-1}p_{x_{k}},\,\,\,\,\Delta^{\alpha}_{-}\,x_{k+1}=-h^{-1}\rho^{-1}p_{y_{k+1}}^{\alpha},\,\,\,\,p_{x_{k+1}}=p_{x_k}-h\nabla U(x_{k+1})+\Delta^{\alpha}_{-}p^{\alpha}_{y_{k+1}},\\
y_{k+1}&=y_k+h\,m^{-1}p_{y_{k}},\,\,\,\, \Delta^{\alpha}_{+}\,y_{k+1}=-h^{-1}\rho^{-1}p_{x_{k+1}}^{\alpha},\,\,\,\,\,p_{y_{k+1}}=p_{y_k}-h\nabla U(y_{k+1})+\Delta^{\alpha}_{+}p^{\alpha}_{x_{k+1}},
\end{split}
\end{equation}
which are a natural discretisation of \eqref{HamFinal}.
Statement 3: we focus on the $x$-system in \eqref{DiscHamFinal}. Replacing the fractional equation into the third one, we obtain
\[
\begin{split}
x_{k+1}=x_k+h\,m^{-1}p_{x_{k}},\quad\quad p_{x_{k+1}}&=\,\,\,p_{x_k}-h\nabla U(x_{k+1})-h\,\rho\,\Delta^{_{1/2}}_{-}\Delta^{_{1/2}}_{-}x_{k+1}\\
&=^{_1}p_{x_k}-h\nabla U(x_{k+1})-\rho\,(x_{k+1}-x_{k})\\
&=^{_2}p_{x_k}-h\nabla U(x_{k+1})-h\,\rho\,m^{-1}\,p_{x_k},
\end{split}
\]
where, in $=^{_1}$ we have employed Lemma \ref{LemmaDisc} and in $=^{_2}$ we have employed the discrete $x$-dynamics, i.e. $x_{k+1}=x_{k}+h\,m^{-1}p_{x_{k}}$. Naturally, the previous equations are a discretisation of \eqref{HamLinearDamp}.
\end{proof}
Furthermore, the discrete Legendre transformation in Definition \ref{Def:DLT} provides a initialisation step $p_{x_0}=I(x_0,x_1)$ (Step 2) for Algorithm \ref{FractionalAlgorithm}. Namely, the $x$-part of \eqref{LegTrans:a} reads
\begin{equation}\label{InitialCond}
p_{x_k}=-D_{x_k}\tilde{\mathcal{L}}_d^k-\Delta_{-}^{\alpha} D_{x_k}^{\alpha}\tilde{\mathcal{L}}_d^k,
\end{equation}
which, for $k=0$, only involves $p_{x_0},x_0$ and $x_1$. For the particular $\tilde{\mathcal{L}}_d^k$ in the theorem above, the initial condition reads $p_{x_0}=-D_{1}L_d(x_0,x_{1})$.
\begin{remark}
The matrix $\begin{bmatrix}0&1\\1&0\end{bmatrix}$ in second term of the right hand side of \eqref{LegTrans:a} and \eqref{LegTrans:b} obeys to the necessity of decoupling $x$ and $y$ dynamics at the discrete level, which we achieved by restricting the variations and setting the critical conditions as (only) sufficient in the Lagrangian side, as shown in Proposition \ref{FullLagProposition} and Remark \ref{RemarkLagrangianMixing}. In other words, it can be considered as a discrete Hamiltonian consequence of the restricted Hamilton's principle. \hfill $\diamond$
\end{remark}
\begin{remark}
It is interesting to note that the result remains the same for any $\tilde{\mathcal{L}}_d^k=L_d(x_k,x_{k+1})+L_d(y_k,y_{k+1})-\tilde\kappa\, h\,x_{k}^{\alpha}\,\,\rho\,\,y_{k}^{\alpha}-(1-\tilde\kappa)\, h\,x_{k+1}^{\alpha}\,\,\rho\,\,y_{k+1}^{\alpha}$,
with $\tilde\kappa\in[0,1]$, which is a way of rephrasing Remark \ref{Alphak+1}. However, the presence of $x^{\alpha}_k$ turns the initial condition \eqref{InitialCond} meaningless from a physical point of view, which makes convenient setting $\tilde\kappa=0$. In that case, the pick of $L_d(x_k,x_{k+1})$, which implies a particular choice of $\kappa$ in \eqref{KappaRule} and \eqref{DiscMechLag}, leads to different discretisations of \eqref{HamFinal} and \eqref{HamLinearDamp}. We remark that the chosen one ($\kappa=0$) preserves the semi-implicitness of variables $x,p_x$ of the symplectic-Euler methods \cite{SS} for \eqref{HamLinearDamp}; say: the final integrator is explicit in the variable $p_x$ and implicit in the variable $x$.
\hfill $\diamond$
\end{remark}
\section{Numerical simulations}\label{Simu}
As a first test example, we employ the linearly damped harmonic oscillator with potential function $U(x)=\mathfrak{c}x^2/2$ and dynamical equation (exactly solvable):
\begin{equation}\label{LinearDamping}
m\ddot x+\mathfrak{c}x+\rho\dot x=0,
\end{equation}
with $m=1$, $\mathfrak{c}=1$, $\rho=0.2$, $x(0)=1$ and $p_x(0)=\dot x(0)=0.5$ in the simulations.
\begin{figure}[!htb]
\includegraphics[scale=0.7]{FracPlot1.jpg}
\caption{Fractional Variational Integrators (FVIs), determined by Algorithm \ref{FractionalAlgorithm}, for several values of $\alpha$, with $h=0.5$ and $N=30$. The Exact (red) line corresponds to the exact solution of \eqref{LinearDamping} for the given set of parameters and initial conditions.}
\label{PlotAlpha}
\end{figure}
\vspace{2cm}
In Figure \ref{PlotAlpha}, we show the outcome of Algorithm \ref{FractionalAlgorithm} with initial condition $p_{x_0}=-D_{1}L_d(x_0,x_{1})$ according to \eqref{InitialCond}, for several $\alpha$'s, where we choose $\kappa=1/2$ in \eqref{DiscMechLag} since it provides the midpoint rule for the potential and it is where the maximum local truncation order (namely 2) is achieved in usual low-order variational integrators \cite{MaWe}. We observe that the FVI approximates properly the solution of \eqref{LinearDamping} when $\alpha = 1/2$, which is natural since that is the case when \eqref{MechDynFracDamp:a} $\rightarrow$ \eqref{LinearDamping} (in other words, $D^{_{1/2}}_{-}D^{_{1/2}}_{-}= d/dt$). Moreover, according to Corollary \ref{Corolario}, that is also the case when the FVI is equivalent to the Forced Variational Integrator (coming out of the discrete Lagrange-d'Alembert principle), Algorithm \ref{ForcedAlgorithm}, when $f_{L_d}^+(x_k,x_{k+1})=-\rho\,(x_{k+1}-x_k).$ This theoretical agreement is numerically tested (and shown) up to machine rounding error in Figure \ref{PositionFigure} (Lower-Left plot), for the different implementations of both Algorithms \ref{FractionalAlgorithm} and \ref{ForcedAlgorithm}. We also show the comparison of the FVI to implicit and explicit Euler integrators for \eqref{HamLinearDamp}, choosing a smaller $h$ for the latter, which is necessary to obtain stable simulations for the explicit Euler scheme. In particular, we compare the $x$-trajectories in Figure~\ref{PositionFigure} (Upper plots) and the energy in Figure~\ref{EnergyFigure} (Left and Middle plots), where naturally, we define the continuous and discrete energies by $E(t)=p(t)^2/2m+\mathfrak{c}\,x(t)^2/2$ and $E_k=p_k^2/2m+\mathfrak{c}\,x_k^2/2$ for $k=0,...,N$; respectively. While implicit and explicit Euler artificially gains respectively looses energy, the FVI respects the energy decay due to the dissipation much better and is very close to the exact solution.
We finally do a numerical convergence study by investigating the global error in both $x$ and $p_x$ variables (Lower-Right plot in Figure~\ref{PositionFigure}) as well as for the energy (Right plot in Figure~\ref{EnergyFigure}). Here, the global error is defined as
\[
\mbox{max}\,|x(t_k)-x_k|,\,\,\,\forall\,\, k,
\]
and equivalently for any other quantity.
For all quantities the convergence is of order $O(h^{0.94})$, i.e.~we obtain a convergence rate of approximately $1$.
\begin{figure}[!htb]
\centering
\begin{tabular}{cc}
\includegraphics[width=7cm]{FVI-Implicit.jpg}&\includegraphics[width=7cm]{FVI-Explicit.jpg}\\
\includegraphics[width=7cm]{Difference.jpg} &\includegraphics[width=7cm]{Error-XP.jpg}
\end{tabular}
\caption{Upper-Left: FVI vs. implicit Euler for $h=0.2$. Upper-Right: FVI vs. explicit Euler for $h=0.1$. Lower-Left: Difference FVI-Lagrange-d'Alembert integrators: $x_k^{_{FVI}}-x_k^{_{LdA}}$ for $h=0.2$. Lower-Right: Log-Log plot of the global error of $x$ and $p_x$ vs. the time step $h$ for FVI.}
\label{PositionFigure}
\end{figure}
\vspace{10cm}
\begin{figure}[!htb]
\centering
\begin{tabular}{ccc}
\includegraphics[width=5.5cm]{Energy-Implicit.jpg}&\includegraphics[width=5.5cm]{Energy-Explicit.jpg}&\includegraphics[width=5.5cm]{Error-En.jpg}
\end{tabular}
\caption{Left: FVI vs. implicit Euler for $h=0.2$. Middle: FVI vs. explicit Euler for $h=0.1$. Right: Log-Log plot of the global error of the energy vs. the time step $h$ for FVI.}
\label{EnergyFigure}
\end{figure}
As for $\alpha\neq1/2$, we employ the following test example:
\begin{equation}\label{BenchExample}
\ddot x+x+D^{_{3/2}}_{-}x=F(t),\quad F(t)=8\,\,\mbox{for}\,\,0\leq t\leq 1,\,\,F(t)=0\,\,\mbox{for}\,\,t>1,
\end{equation}
and initial conditions $x(0)=\dot x(0)=0$. This equation corresponds to \eqref{MechDynFracDamp:a} when $U(x)=x^2/2$, $m=\rho=1$, $\alpha=3/4$ and we add the inhomogeneous external force $F(t)$ in the right hand side (which can be easily carried out by equating the virtual work of such a force and the variation of the actions \eqref{ContAction} and \eqref{DiscAct}, in continuous and discrete scenarios, respectively, under restricted calculus of variations). Equation \eqref{BenchExample} has an exact solution\footnote{Particularly, it is a so-called {\it inhomogeneous sequential fractional differential equation}, i.e.
\[
\left( D^{n/q}_{-}+a_1D^{(n-1)/q}_{-}+\cdots+a_nD^0_{-}\right)\,x(t)=F(t),
\]
with $(n,q)=(4,2)$, $a_1=a_4=1$, $a_2=a_3=0$ \cite{MiRo}.}, but it is of difficult implementation. For that reason, we employ the benchmark numerical solution designed for MatLab in \cite{Vinagre}. We display the performance of the FVI versus the benchmark solution (with a much smaller time step) (Left plot in Figure~\ref{BenchFigure}) and observe a global convergence of order $O(h^{0.97})$ (Right plot in Figure~\ref{BenchFigure}), i.e.~a convergence rate of approximately $1$ as for the $\alpha=1/2$ case.
\begin{figure}[!htb]
\centering
\begin{tabular}{cc}
\includegraphics[width=7.5cm]{Bench-FVI.jpg}&\includegraphics[width=7.5cm]{Error-Bench.jpg}
\end{tabular}
\caption{Left: Benchmark solution ($h=5\times 10^{-3}$) vs. FVI ($h=10^{-1}$). Right: Log-Log plot of the global error of $x$ vs. the time step $h$ (with the benchmark solution as reference).}
\label{BenchFigure}
\end{figure}
\section{Conclusions}
We have developed a restricted Hamilton's principle providing the dynamics of Lagrangian systems subject to fractional damping \eqref{IntEqCont} (continuous setting), \eqref{IntEqDisc} (discrete setting), as sufficient conditions for the extremals of the action. The discrete dynamics \eqref{IntEqDisc} is the result of the discretisation of the mentioned restricted Hamilton's principle (instead of the discretisation of the equations \eqref{IntEqCont} themselves), following the spirit of discrete mechanics and variational integrators \cite{MaWe}. As it is well-know, the variational principles and preservation
properties (symplecticity, Noether's theorem \cite{AbMa}) of the generated dynamics are closely related; we find a particular example in our approach, say the preservation of the presymplectic form $\Omega$ \eqref{TwoForm}, as proven in Proposition \ref{GeomPreservation}. Nevertheless, the dynamical importance of $\Omega$ has to be clarified in future work. This two-form is defined on the fractional state space, Definition \ref{FracPhSp}, which is designed to accomodate the fractional derivatives. It is a vector bundle over the real space, particularly $\R^d\times\R^d$, which is necessary due to the unclear unique definition of fractional derivatives \eqref{FracDer} on a general smooth manifold $Q$. From the geometrical perspective, an interesting challenge for future work is to carry out this generalisation.
The discretisation of the restricted fractional principle leads to the discrete equations \eqref{IntEqDisc}, and to what we denote Fractional Variational Integrators FVIs, via the Algorithm \ref{FractionalAlgorithm}. When $\alpha=1/2$, the theoretical local truncation order of FVIs is 1 \cite{JiObNum}, which is consistent with the observed global convergence order, i.e. $O(h^{0.94})$, in Figure \ref{PositionFigure}, Lower-Right plot. However, this global convergence and the local truncation order 1 represents an improvement from what is expected from the order theorems in \cite{MaWe,PaCu}, since, as proven in \cite{FracOrder}, $\Delta^{\alpha}_{-}x_k$ \eqref{DiscFracDerDef:1} is only a consistent approximation of $D^{\alpha}_{-}x(t_k)$ \eqref{FracDer:RL}, and thus one would expect a slower global convergence. This is an interesting phenomenom to explore in future works. Moreover, the performance of FVI is proven to be superior to other order 1 methods, such as both Euler's. This is particularly evident with respect to the energy, as shown in Figure \ref{EnergyFigure}, accounting for another example of the superior performance of integrators with variational origin in this aspect. In the context of this work, this is furthermore explained thanks to the relationship between the FVI and the Forced Variational Integrators, Algorithm \ref{ForcedAlgorithm}, obtained from the discrete Lagrange-d'Alembert principle, which is established in Corollary \ref{Corolario}. Naturally, we also test the FVI for $\alpha\neq 1/2$.
We notice that the chosen discretisations limit the local truncation order of the FVIs and their global convergence. Thus, as a natural extension of this work, we intend to carry out the application of higher-order techniques, as introduced in \cite{O14,SinaSaake}.
\section*{Appendix: Existence and uniqueness of solutions of fractional differential equations \eqref{ResFracELDamped} }
We study the existence and uniqueness of solutions of \eqref{ResFracELDamped}. It has been proven, Proposition \ref{InvTime}, that $x$ and $y$ systems are equivalent in reversed directions of time. Thus, we focus on the $x$-system and set $d=1$ for simplicity. We are considering $L:T\R\rightarrow\R$ a $C^2$ function; furthermore we shall consider $\left(\partial^2L/\partial\dot x\partial\dot x\right)^{-1}$ smooth. All in all, \eqref{ResFracELDamped:a}, can be expressed as
\begin{equation}\label{System}
\begin{split}
\dot x&=v,\\
\dot v&=f(x,v)+\bar\rho(x,v)D^{\beta}_{-}x,
\end{split}
\end{equation}
with initial condition $x(t_0)=x_0,\,v(t_0)=v_0 $, $\beta=2\alpha$ and $f,\bar\rho:\R^2\rightarrow\R$ given by
\[
f(x,v)=\left(\frac{\partial^2L}{\partial v\partial v}\right)^{-1}\left(-\frac{\partial^2L}{\partial x\partial v}v+\frac{\partial L}{\partial x}\right),\quad\quad \bar\rho(x,v)=-\left(\frac{\partial^2L}{\partial v\partial v}\right)^{-1}\rho,
\]
with $\rho\in\R_+$, adding up for the vector field $F:\R^2\rightarrow\R^2$
\begin{equation}\label{VectorField}
F(x,y):=\left( v\,\,,\,\, f(x,v)+\bar\rho(x,v)D^{\beta}_{-}x\right).
\end{equation}
In proving the existence and uniqueness of solutions of \eqref{System}, we shall take a local approach; in particular we consider the set $\mathcal{B}=I_t\times I_x\times I_v\subset \R^3$, with $I_t=[t_0-\delta_t, t_0+\delta_t],\,I_x=[x_0-\delta_x, x_0+\delta_x],\, I_v=[v_0-\delta_v,v_0+\delta_v]$, $I_t\subset[a,b]$ and $\delta_t,\delta_x,\delta_v\in\R_+$. We consider $\R^d$ as a Banach space equipped with the norm
\[
||\mathfrak{v}||=\mbox{max}\left\{ |\mathfrak{v}_1|,|\mathfrak{v}_2|,...,|\mathfrak{v}_d|\right\},\quad \mathfrak{v}\in\R^d.
\]
Given that, we define the cylinder $\mathcal{C}=I_t\times \mathcal{D}$, with $\mathcal{D}=\left\{(x,v)\in\R^2\,|\,||(x,v)-(x_0,v_0)||\leq \mathfrak{b}\right\}$, $\mathfrak{b}=||(\delta_x,\delta_v)||.$ We establish the following hypotheses:
\begin{itemize}
\item[H1.] $||f(x,v)-f(\tilde x,\tilde v)||\leq K||(x,v)-(\tilde x,\tilde v)||$ in $\mathcal{D}$ for $K\in\R_+$.
\item[H2.] $\bar\rho$ is continuous in $\mathcal{D}$.
\end{itemize}
Note that these two hypotheses follow directly from the assumptions over $L$ and $\left(\partial^2L/\partial\dot x\partial\dot x\right)^{-1}$, say they are $C^2$ and smooth, respectively. Given this, our strategy is to prove that the vector field \eqref{VectorField} satisfy the required conditions to apply both Peano and Picard-Lindel\"of theorems \cite{BookODEs}, ensuring the existence and uniqueness of solutions of \eqref{System} in $\tilde{\mathcal{C}}=\tilde I_t\times \mathcal{D}$, with $\tilde I_t=[t_0-\tilde\delta_t,t_0+\tilde\delta_t]$, $\tilde\delta_t=\,$min$\left\{\delta_t\,,\,\mathfrak{b}/M\right\}$, where $M\in\R_+$ is constructed in the proof below. Moreover, this solution $(x(t),v(t))$ can be set smooth by establishing the proper Banach space of functions when applying the aforementioned theorems.
\begin{proposition}
Given the hypotheses {\rm H1, H2}, the following is true: $F:\mathcal{D}\rightarrow\R^2$
\begin{enumerate}
\item is bounded.
\item is Lipschitz continuous.
\end{enumerate}
\end{proposition}
\begin{proof}
(1) Let $(x,v)\in \mathcal{D}$, then $||F(x,v)||=$max $\left\{ |v|, |f(x,v)+\bar\rho(x,v)D^{\beta}_{-}x|\right\}$. On the one hand, $|v|\leq |v_0+\mathfrak{b}|<\infty$. On the other, by H1 $f$ is continuous in $\mathcal{D}$ and therefore $|f(x,v)|\leq M_f<\infty$. By H2, $\bar\rho$ is also continuous; thus $|\bar\rho(x,v)|\leq M_{\bar\rho}<\infty$. Now, we shall use the Caputo definition \eqref{FracDer:Ca} of the fractional derivative in \eqref{VectorField}, just by using the relationship \eqref{Relation} and setting $x_0=0.$ With that, in $I_t\times\mathcal{D}$ we have
\[
\begin{split}
&|f(x,v)+\bar\rho(x,v)D^{\beta}_{-}x|\leq|f(x,v)|+|\bar\rho(x,v)||D^{\beta}_{-}x|\leq M_{f}+M_{\bar\rho}|D^{\beta}_{-}x|\\
=&M_{f}+\frac{M_{\bar\rho}}{\Gamma(1-\beta)}\Big|\int_{t_0}^t(t-\tau)^{-\beta}\dot x(\tau)d\tau\Big|=M_{f}+\frac{M_{\bar\rho}}{\Gamma(1-\beta)}\Big|\int_{t_0}^t(t-\tau)^{-\beta}v(\tau)d\tau\Big|\\
\leq & M_{f}+\frac{M_{\bar\rho}}{\Gamma(1-\beta)}\int_{t_0}^t|(t-\tau)^{-\beta}| |v(\tau)|d\tau\leq M_{f}+\frac{M_{\bar\rho}|v_0+\mathfrak{b}|}{\Gamma(1-\beta)}\int_{t_0}^t|(t-\tau)^{-\beta}|d\tau\\
\leq & M_{f}+\frac{M_{\bar\rho}|v_0+\mathfrak{b}|}{\Gamma(1-\beta)}\delta_t^{1-\beta}<\infty,
\end{split}
\]
where, according to $\beta=2\alpha$, we consider $\alpha\in[0,1/2]$, since $\beta$ must be less of equal to 1\footnote{Observe that this limits the original range of $\alpha$, i.e.~ $\alpha\in[0,1]$. Allowing that, the displayed proof of existence and uniqueness is not valid as long as we admit $\delta_t\rightarrow 0$, which does not mean that such existence and uniqueness of solutions may be proven in a different way.}. From this, it follows that $||F(x,v)||\leq M<\infty,$ where $M=\left\{ |v_0+\mathfrak{b}|\,\,\mbox{or}\,\,M_{f}+\frac{M_{\bar\rho}|v_0+\mathfrak{b}|}{\Gamma(1-\beta)}\delta_t^{1-\beta}\right\}$, depending on whether the max $\left\{ |v|, |f(x,v)+\bar\rho(x,v)D^{\beta}_{-}x|\right\}$ is achieved in the first or second entry.
\\
(2) We have that
\begin{equation}\label{Norm}
||F(x,v)-F(\tilde x,\tilde v)||=||\big( v-\tilde v,\,f(x,v)-f(\tilde x,\tilde v)+\bar\rho(x,v)D^{\beta}_{-}x-\bar\rho(\tilde x,\tilde v)D^{\beta}_{-}\tilde x\big)||,
\end{equation}
which is equal to the maximum of the absolute value of both entries. On the one hand
\begin{equation}\label{ByConstr}
|v-\tilde v|\leq ||(x,y)-(\tilde x,\tilde v)||,
\end{equation}
by construction. On the other
\[
\begin{split}
&|f(x,v)-f(\tilde x,\tilde v)+\bar\rho(x,v)D^{\beta}_{-}x-\bar\rho(\tilde x,\tilde v)D^{\beta}_{-}\tilde x|\leq |f(x,v)-f(\tilde x,\tilde v)|+|\bar\rho(x,v)D^{\beta}_{-}x-\bar\rho(\tilde x,\tilde v)D^{\beta}_{-}\tilde x|\\
\leq& K||(x,y)-(\tilde x,\tilde v)||+M_{\bar\rho}|D^{\beta}_{-}x-D^{\beta}_{-}\tilde x|=K||(x,y)-(\tilde x,\tilde v)||+\frac{M_{\bar\rho}}{\Gamma(1-\beta)}\big|\int_{t_0}^{t}(t-\tau)^{-\beta}(v(\tau)-\tilde v(\tau))d\tau\big|\\
\leq&K||(x,y)-(\tilde x,\tilde v)||+\frac{M_{\bar\rho}}{\Gamma(1-\beta)}||(x,y)-(\tilde x,\tilde v)||\int_{t_0}^t|(t-\tau)^{-\beta}|d\tau\\
\leq& K||(x,y)-(\tilde x,\tilde v)||+\frac{M_{\bar\rho}\delta_t^{1-\beta}}{\Gamma(1-\beta)}||(x,y)-(\tilde x,\tilde v)||=
\left( K+\frac{M_{\bar\rho}\delta_t^{1-\beta}}{\Gamma(1-\beta)}\right) ||(x,y)-(\tilde x,\tilde v)||,
\end{split}
\]
where we have used H1, H2 and \eqref{ByConstr}. Thus, it follows that
\[
||F(x,v)-F(\tilde x,\tilde v)||\leq\tilde K\,||(x,y)-(\tilde x,\tilde v)||,
\]
where $\tilde K=\left\{ 1\,\mbox{or}\,K+\frac{M_{\bar\rho}\delta_t^{1-\beta}}{\Gamma(1-\beta)}\right\}$, depending on whether the maximum on the right hand side of \eqref{Norm} is achieved on the first or second entry. In both cases, $\tilde K\in\R_+.$
\end{proof}
\medskip\medskip
{\bf Acknowledgments}: This work has been funded by the EPSRC project: `'Fractional Variational Integration and Optimal Control''; ref: EP/P020402/1. FJ thanks Farhang Haddad Farshi for his assistance in the design of Figure \ref{Curves}.
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 4,193
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CapoVelo.com - - UCI Track Cycling World Championships Returns to London Arena
UCI Track Cycling World Championships Returns to London Arena
After a four-year hiatus, cycling fans are set for another thrilling seers of track racing at the Lee Valley VeloPark in London.
During March 2-6, the UCI track World Championships will return to the London velodrome, not only as a prestigious events of its own, but it will also serve as the final qualification opportunity for riders to qualify for the 2016 Olympic Games in Rio.
Great Britain dominated on this track at the 2012 Olympics, winning seven of ten gold medals. However, Event Director Jonny Clay admits that the 2016 UCI Track Cycling World Championships are unlikely to see such supremacy by one nation. "I think the [Great Britain] team is reasonably confident, but not expecting that level of dominance. In terms of the UCI Track Cycling World Championships, this may be the one leading into the Olympics, but all the teams treat it as just as important as any World Championships in terms of competition."
The racing starts on 2 March with qualifying for the team pursuit for Men, the team sprints for Men and Women, and the Women's individual pursuit. Four titles will be awarded on the first day – both the team sprints, the Women's individual pursuit and the Men's scratch race.
On day two, four more titles will be determined over two sessions, with the finals for the Men's team pursuit and kilometre time trial, plus the Women's scratch race and keirin.
Friday, 4 March, will be the first of two days offering three sessions of competition, and four more titles will be decided. Rainbow jerseys will be awarded in the Women's 500m time trial and team pursuit as well as in the Men's individual pursuit and points race. Friday will also see the start of the Men's sprint and Omnium competitions.
Saturday features the busiest schedule of the Championships, with three titles awarded – in the Men's Omnium and sprint, plus the Women's points race. In addition, the Women's sprint and Omnium get under way.
The World Championships conclude on Sunday with four titles contested: the Women's sprint and Omnium plus the Men's keirin and the traditional closer to the Championships, the Men's Madison.
|
{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
| 4,738
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«Things Have Changed» es una canción de la banda sonora de la película Wonder Boys compuesta e interpretada por el músico estadounidense Bob Dylan y publicada como sencillo el 1 de mayo de 2000.
"Things Have Changed" ganó el Premio de la Academia a la Mejor Canción Original, así como el Globo de Oro en la misma categoría. El director de Jóvenes Prodigiosos, Curtis Hanson, creó un vídeo musical para la canción, filmando a Bob Dylan en diferentes localidades y editando la cinta con escenas de la película, tal y como si Dylan estuviera inmiscuido en ella.
Lista de canciones
"Things Have Changed" CD (COL 669333 2)
«Things Have Changed» (Radio Edit) - 3:37
«To Make You Feel My Love» (Live) - 4:10
«Hurricane» - 8:33
*Extraído del álbum Desire
«Song to Woody» (Live) - 4:26
''Grabado en directo en el Santa Cruz Civic Auditorium de Santa Cruz, California, el 16 de marzo de 2000
"Things Have Changed": Dylan Alive! Vol. 3 Japanese EP
«Things Have Changed» - 5:09
«Highlands» (Live) - 11:19
«Blowin' in the Wind» (Live) - 7:10
«To Make You Feel My Love» (Live) - 4:11
Referencias
Enlaces externos
Videoclip de la canción YouTube.
BobDylan.com Letra de "Things Have Changed" en BobDylan.com
Canciones de Bob Dylan
Sencillos de 2000
Canciones ganadoras del premio Óscar a la mejor canción original
Canciones ganadoras del premio Globo de Oro a la mejor canción original
Canciones en inglés
|
{
"redpajama_set_name": "RedPajamaWikipedia"
}
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|
{"url":"https:\/\/www.allfankind.com\/bx9si81w\/8a0c38-pitch-and-helix-of-mini-screw","text":"The full range includes lead screw nuts from 5 self-lubricating iglide\u00ae high-performance polymers, in trapezoidal and high helix threads, cylindrical design or flange options. NOTE 1A: Machine Screw Major Diameter: [machine screw# ( \u00ec. \u00ed \u00ef\u201d)] + \u00ec. \u00f2 \u00ec\u201d. In this case, I don\u2019t want to use Variable Pitch, I want to use Constant Pitch. 66 0 obj Helix or lead angles are given for Unified and other thread series in handbooks such as Machinery\u2019s Handbook. w pitch circle 1 main rotor 2 gate rotor . The screw piles are highly suitable as foundations for all yard and garden buildings. \ufffd \ufffd\ufffd\ufffd\ufffdf\ufffd5w~\ufffd'\ufffd>8\ufffdby\ufffd6O \ufffd%B\ufffd]\ufffd%\ufffdi. The movement speed of the non-rotary part relative to the rotary part was equal to the mini-screw pitch (0.9 mm\/360\u00b0 rotation). p = Pitch of the screw, d = Mean diameter of the screw, \u03b1 = Angle of the screw or helix angle, \u03a6 = Angle of friction, \u00b5 = Co-efficient of friction between screw and nut = tan \u03a6 . The pitch diameter of a crossed-helical gear is given by Equation (6-7), and the center distance becomes: Again, it is possible to adjust the center distance by manipulating the helix angle. <> (1) Friction on the helix plate and friction along the central steel shaft. I wanted my LEDs to be evenly distributed on my cylinder, so there were two options to build this grid: Make LEDs align vertically as they wind around the cylinder, or turn that grid 45 degrees for an alternating-winds alignment. 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The clay nut is constantly being destroyed by p\u2026 Major Diameter of a #10 Screw = [ \u00ec ( \u00ec. \u00ed \u00ef\u201d)] + \u00ec. \u00ec \u00f2 \u00ec\u201d or \u00ec. \u00ed \u00f5 \u00ec \u00ec\u201d This method is extensively used in checking the accuracy of threaded plug gages and other precision screw threads. 0000000023 00000 n Entry Level Mechanical Engineer (Equipment Design) \u2013 (MAN00002R), Entry Level Industrial Designer for General Consideration, Oxford English for Electrical and Mechanical Engineering, Mechanisms and Mechanical Devices Sourcebook, Fourth Edition, CNC lesson 6: Determining Program zero Assignment Values, difference between a helix and screw thread, calculate helix angle with pitch diameter. The Z form of DNA is left-handed.. Some Helix sizes have 40\u00b0 included angle. 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The General Purpose and Centralizing thread forms have a nominal depth of thread of 0.50 \u00d7 pitch and have a 29\u00b0 included thread angle. startxref \u201cAs a modification of this most used standard version, the screw conveyor can also be designed with double helix, conical helix or helix with variable screw pitch. When a cylinder is added to a coil to form the main shaft of a screw, the cylinder must not be tangent to the coil profile. On a taper thread the pitch diameter\u00a0at a given position on the thread axis is the diameter of the\u00a0pitch cone at that position. A \u2026 A protractor may be used to check this angle. Other options New from $35.99. CKD draw 3D part with SolidWork CAD\/CAM soft Helix Screw with varialble pitch Subscribe to see the latest videos from Ph\u1ea1m Duy Anh. 0000001379 00000 n The lead screw pitch, 3.175 mm, divided by the ratio of those two gears, 1.27, gives us 2.5 mm, a much easier number to further divide or multiply into standard metric pitches. 55,00 \u20ac 53,00 \u20ac incl. However, the \u03b1\u2010helix (with the possible addition of the \u03c0\u2010helix, with 4.4 residues per turn) seems to be an exception among biological helices: the protein 3 10 helix (the subscript in protein helix notation counts the number of atoms in a H\u2010bonded cycle, not the screw's translation component!) x\ufffdm\ufffd\ufffdN1@\ufffd|\ufffdG\ufffd\u064e\ufffd\\\ufffd endobj Conveyors are available in configurations from 10 to 20 ft. lengths with a choice of a 3\" diameter helix screw rated to convey up to 150 cu. The helix angle can be expressed as: Helix angle = arctan \u2061 ( 2 \u03c0 r m L ) {\\displaystyle {\\mbox {Helix angle}}=\\arctan \\left ( {\\frac {2\\pi r_ {m}} {L}}\\right)} where. Region parameters (Variable pitch helix only). Miniature Metric Precision Ball Screws are an efficient and cost-effective solution in a small envelope, ideal for use in small spaces. The Paalupiste Helix series is a low cost but high-quality range of screw piles for consumer use. 67 0 obj No Reviews. Most people chose this as the best definition of helical: Of or having the shape of... See the dictionary meaning, pronunciation, and sentence examples. HELIX 1,20M SCREW PILE. Friction resistance increases with helix size because the surface area of the helix in contact with the soil increases with the square of the diameter (see Figure 6-5). In most screws, called \"single start\" screws, which have a single helical thread along their length, the lead and pitch are equal. 68 0 obj If it\u2019s not going in the correct direction, you can select Reverse Direction and insert the pitch amount (in my case 1\/20 because I\u2019m making \u00bc 20 hex cap screw). Then the load is lifted by a height p (pitch of screw). They only differ in \"multiple start\" screws, which have several intertwined threads. Sale . \ufffd ... Standard (free running inserts), a tolerance class 5H or 2B is recommended. During fabrication, the helix flange must start perpendicular (at 90 degrees) to the pile shaft and remain so around the entire surface of the helix. 0000000857 00000 n Also note that pitch and lead are the same for single lead screws. Channel depth, helix angle, pitch and flight clearance are the most important variables in screw geometry. 0000000433 00000 n P is thread pitch (assuming a single start thread) \u03c0 = 3.141 592 654 d 2 is the pitch diameter prex shows the same equation, with \u03c8 = \u03b1 The helix angle is the compliment of the lead angle, so kapitan's equation is correct assuming the terms helix angle and pitch angle are identical. ft.\/hr. %%EOF Arrives before Christmas Only 14 left in stock - \u2026 Fig. L = screw lead (mm) Helix angle. A taper thread is made on the internal or external surface of a cone. 2 1 INTRODUCTION Screw compressors are reliable and compact machines and, as pointed out by Fleming, Tang and Cook, 1998, represent a success story of the 20th century. AN Aluminum-Bronze Screw Gears, Number of Teeth 13, Helix Hand L, Pitch Diameter 18.38: Part Number: AN1-13L: Material: Aluminum Bronze: CAD-Modelle. Teilen Produktauswahl. 71 0 obj The former is known as depth compression ratio and the latter is called volumetric compression ratio. Rolled thread screws are cost effective and are stocked for quick delivery.\" Pitch is defined as the axial distance between adjacent threads on a helix or screw. For special scre\u2026 Required fields are marked *, PITCH DIAMETER, LEAD OR HELIX ANGLE, AND PERCENT OF THREADS. Typically both the hollow centre pipe and the hollow spiral flight are used as a heat exchanger to cool, heat or thaw the material being conveyed. Crest: The surface of the thread corresponding to the major diameter of the screw and the minor diameter of the nut.$36.99 \\$ 36. A clay auger works in a similar manner. endobj endstream The ultimate capacity of multi-helix Screw-Piles in compression and Helical Anchors in tension with a helix spacing\/diameter ration > 3 is often taken as the summation of the capacities of the individual plates: Q M = \u03a3Q H [2.1] where: Q M = Total Capacity of a Multi-Helix Screw-Pile\/Helical Anchor Q H = Capacity of an Individual Helix The standard helix angle for most injection and extrusion screws is 17.6568\u00ba and is calculated as follows (a square pitch screw): Helix Angle = arctangent of Lead\/\u220f x \u2026 Affordable, material S355\/420, helix thickness 4 mm, Made in Finland, guarantee 50 years. We provide in-house fabrication out of a dedicated CWB-certified manufacturing shop located in Coaldale, Alta. 65 0 obj From the down menu, select \u2018 Helix and Spiral Feature Command \u2019. Screw piles \u00d8 150 mm helix HELIX Series. result of the load (thrust force) pushing axially on the nut to create rotary motion The belt needs to be at certain tightness in order for the motor to work properly. Helix Screw piles. Starting immediately, there are DryLin\u00ae lead screw nuts from the full range available for every technical requirement. Screw piles \u00d8 150 mm helix HELIX Series. Now you can go to the command Features > Curves > Helix and Spiral. No Reviews. Metalfab Convey-All\/FSC Flexible Screw Conveyors feature a rugged modular design and provide a highly effective yet economical means to convey powder or granular material without product separation. <> 1 <<>> 2 <<>> 3 <<>> 4 <<>> 5 <<>> 6 <<>> 7 <<>> 8 <<>> 9 <<>> 10 <<>> 11 <<>>]>>\/Type\/Catalog\/ViewerPreferences<>\/PageLayout\/TwoPageRight\/Pages 56 0 R\/OutputIntents[59 0 R]\/Metadata 61 0 R>> Bortolamasi and Fottner [89] reported that the minimum pitch in single-augers must be no less than one-half the screw diameter while the maximum pitch must be the same screw diameter. Lead: The distance a screw thread advances axially in one turn. This multi-start lead screw product provides high linear nut speeds from low rotational screw speeds. Ball screw assemblies range from 4 to 14mm in diameter with leads from 1 to 20mm - all with standard lead accuracies of \u00b152\u00b5m\/300mm. Miniature Rolled Ball Screws are ideal for laboratory, semiconductor, and medical applications. Go to \u2018 features Command manager \u2019 and click on the \u2018 Curves \u2019. 0000040074 00000 n No Reviews. or a 4\" diameter helix screw rated to convey \u2026 Many translated example sentences containing \"screw helix\" \u2013 German-English dictionary and search engine for German translations. The distance from maximum major diameter to maximum major diameter is the nominal diameter. Description: dryspin\u00ae high helix lead screw made from stainless steel, rolled Resistant and made of 1.4301 material (AISI 304), pitch from 2 to 100 mm dryspin\u00ae high helix lead screw made of aluminium Rolled and tribo-optimised coating of EN AW 6082 material, lightweight and . This provides high linear speeds at a low rotational speeds or efficient conversation of linear to rotary movement. 0000001054 00000 n Pitch Size C Pipe OD Std. Peak von Mises stresses of the screw- Pitch variant from 0.6 to 1 mm . 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Spying correct
1 / 1 Jessica Chastain
Jessica Chastain believes new movie 'The 355' reframes the idea of female spies; also stars Lupita Nyong'o, Penelope Cruz
The 44-year-old stars in spy flick The 355 with Lupita Nyong'o and Penelope Cruz and feels it does more than present female spies as "honeypots". Jessica told Empire: "I feel like the film industry has really got female spies wrong. They've portrayed them as honeypots, and that's not the reality. Women weren't being used for their bodies, they were being used for their minds, which is a more interesting concept."
The Molly's Game actor explained she has become more determined to take on more challenging roles. "In the past seven years, I've really looked at the projects I've joined and the parts that I've played in terms of, 'What difference am I making? How can I shape a conversation?' I never really had the dream of being an action hero at all, but the reality is I'm excited to have 13-year-old girls and boys see women in these roles. It's very important for society. We've moved against the status quo, and we're creating our own narrative for it. The film is, in some sense, a political act."
Will Smith: Played a joke on my wife by showing my grandma one of her sex scenes
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Q: U-groups and internal direct product I want to show that if $\gcd (n_i, n_j)=1$ for $i\not= j$ and $m=n_1 \cdots n_k$,then
$$
U(m) = U_{\frac{m}{n_1}}(m) \times \cdots \times U_{\frac{m}{n_k}}(m),
$$
where $\times$ is the symbol for internal direct product, $U(m) = \{x\in \mathrm{Z}^+ | \gcd (x,m)=1\}$, and $U_{s}(m)=\{x\in U(m) \mid x \pmod{s}= 1\}$.
My attempt:
Since $U(m) \cong U(n_1) \oplus \cdots \oplus U(n_k)$ and
$U_{\frac{m}{n_j}}(m) \cong U(n_j)$ (where $\oplus$ is external direct product), it follows that
$$
U(m) \cong U_{\frac{m}{n_1}}(m) \oplus \cdots \oplus U_{\frac{m}{n_k}}(m).
$$
I'm also aware that because $U(m)$ is Abelian, all its subgroups — including the $U_{\frac{m}{n_j}}(m)$ — are normal. I also know that the $U_{\frac{m}{n_j}}(m)$ only share $1$ in common. I've been trying to use these pieces of information.
I think this is very close to finishing, but I'm not sure how to go from here. Any help would be appreciated.
Edit
I tried to show one of the conditions for $U(m)$ being the internal direct product of the previously mentioned subgroups. Namely, that $U(m)=U_{\frac{m}{n_1}}(m)\cdots U_{\frac{m}{n_k}}(m)$.
My attempt:
Since $U(m) \cong U_{\frac{m}{n_1}}(m) \oplus \cdots \oplus U_{\frac{m}{n_k}}(m)$, it follows that all the elements in $U(m)$ are of the form $\mathrm{lcm}(|u_1|,\cdots ,|u_k|)$ where $u_i\in U_{\frac{m}{n_i}}(m)$. Also, $U(m)$ is certainly Abelian, which means for all $x_j\in U(m)$, $|x_1\cdots x_k|$ divides $|x_1|\cdots |x_k|$. When considering these two facts, LaGrange's Theorem, and that the $n_i$'s are pairwise coprime, any $x\in U(m)$ must be a product of $u_i$'s.
I'd like to know if this is wrong in any way.
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# Blanchot
Maurice Blanchot (born 1907) is the author of a significant body of narrative fiction and countless critical essays on both literary and philosophical texts, as well as a substantial amount of political journalism. Straddling the divide between the literary and the philosophical, Blanchot is arguably one of the most challenging and influential figures in twentieth-century writing, whose work has exerted a decisive impact on thinkers as varied as Foucault, Derrida, Deleuze, Bataille, Klossowski, Levinas, Lacoue-Labarthe, Lyotard, Nancy, Barthes, and Kristeva, as well as on some of the most important contemporary writers in Europe and the United States.
Though his main works are now widely available in translation, Blanchot remains relatively unknown in the English-speaking world. _Blanchot: Extreme Contemporary_ is the first book to address the entirety of Blanchot's work in one volume. It examines in detail the literary and philosophical dialogue pursued by Blanchot with Bataille, Heidegger, Levinas, Derrida, Lacoue-Labarthe, and Nancy, as well as Höderlin, Duras, Kafka, Rilke, and Beckett, and shows how for Blanchot literature is inseparable from the radical ethical and ontological questions it asks of philosophy in general. It puts forward a persuasive case for Blanchot's achievement as one of the most compelling writers of fiction of the postmodern age. It also throws new light on Blanchot's controversial political activities before and after the Second World War. This book also contains the most comprehensive bibliography of Blanchot's extensive writings to appear in any language for the last twenty years.
_Blanchot: Extreme Contemporary_ will be essential reading for students of continental philosophy and poststructuralism, literature and French studies.
**Leslie Hill** is Reader in French Studies at the University of Warwick, and the author of _Beckett's Fiction: In Different Words_ and _Marguerite Duras: Apocalyptic Desires._
# Warwick Studies in European Philosophy
Edited by Andrew Benjamin _
Professor of Philosophy, University of Warwick_
This series presents the best and most original work being done within the European philosophical tradition. The books included in the series seek not merely to reflect what is taking place within European philosophy, rather they will contribute to the growth and development of that plural tradition. Work written in the English language as well as translations into English are to be included, engaging the tradition at all levels – whether by introductions that show the contemporary philosophical force of certain works, or in collections that explore an important thinker or topic, as well as in significant contributions that call for their own critical evaluation.
# Blanchot
# Extreme Contemporary
## Leslie Hill
First published 1997
by Routledge
11 New Fetter Lane, London EC4P 4EE
Simultaneously published in the USA and Canada
by Routledge
29 West 35th Street, New York, NY 10001
This edition published in the Taylor & Francis e-Library, 2001.
©1997 Leslie Hill
All rights reserved. No part of this book may be reprinted or reproduced or utilized 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.
_British Library Cataloguing in Publication Data_
A catalogue record for this book is available from the British Library
_Library of Congress Cataloguing in Publication Data_
A catalogue record for this book has been requested
ISBN 0-415-09173-X (hbk)
ISBN 0-415-09174-8 (pbk)
ISBN 0-203-00475-2 Master e-book ISBN
ISBN 0-203-17425-9 (Glassbook Format)
The aim here is simply to test out to what extent it is possible to follow a text and at the same time to lose track of it, to be simultaneously the person it understands and the person who understands it, the person who, within a world, speaks of that world as though he or she were outside it; all in all, to take advantage of the strangeness of a dual work and an author split into two – into absolute lucidity and impenetrable darkness, into a consciousness that knows all and yet knows not where it is going – in order to feign the illusion of a commentary solely preoccupied with accounting for all and yet entirely aware of being able to explain nothing.
Maurice Blanchot, 'L'Expérience de Lautréamont' ( _LS_ , 59)
# Acknowledgements
This book is the third, and final, volume in a series devoted to some of the most challenging areas in contemporary writing in French and essentially concerned with the relationship between the singularity of the proper name and the possibility of literature. Like its two predecessors, on Samuel Beckett and Marguerite Duras, this book has taken many years to finish, and there are many people, friends, colleagues, and others, to whom it owes a debt of some kind or other which it is not possible for me to repay except in words. For their assistance and support at different stages in the writing of this book, I should like therefore to record here my particular thanks to Andrew Benjamin, Geoffrey Bennington, Malcolm Bowie, Saskia Brown, Didier Cahen, Howard Caygill, David Constantine, Simon Critchley, Jonathan Derbyshire, Carolyn Gill, Darren Green, Michael Holland, Alan Jackson, Ian James, Ann Jefferson, Mig Kerr, Philippe Lacoue-Labarthe, Roger Laporte, Peter Larkin, John Lechte, Dionys Mascolo, Philippe Mesnard, Jean-Luc Nancy, Tony Phelan, Douglas Smith, Simon Sparks, Richard Stamp, Chris Turner, and David Wood.
I am grateful to the University of Warwick for the granting of study leave and for the provision of funds that enabled me to carry out the research on which this book is based. I wish in addition to acknowledge further financial assistance from the British Academy during the spring of 1996. I must also thank the University of Nebraska Press, the University of Minnesota Press, the State University of New York Press, and Stanford University Press for permission to quote from copyright material. Finally, I would like to dedicate this book to Juliet, whose very existence is just about as much a surprise to me as the completion of this book.
Extracts from _The Step Not Beyond,_ by Maurice Blanchot, translated by Lycette Nelson, reprinted by permission of the State University of New York Press.
Excerpts from _The Work of Fire,_ by Maurice Blanchot, translated by Charlotte Mandell, reprinted with the permission of the publishers, Stanford University Press. Translation ©1995 by the Board of Trustees of the Leland Stanford Junior University. Original edition ©1949 by Éditions Gallimard.
Extracts from _The Space of Literature_ by Maurice Blanchot, translated, with an introduction, by Ann Smock, reprinted by permission of the University of Nebraska Press, ©Éditions Gallimard, 1955. Introduction and English-language translation ©1982 by the University of Nebraska Press.
Extracts from _The Writing of the Disaster_ by Maurice Blanchot, translated by Ann Smock, reprinted by permission of the University of Nebraska Press, ©Éditions Gallimard, 1980. Copyright 1986, 1995 by the University of Nebraska Press.
Extracts from _The Infinite Conversation_ by Maurice Blanchot, translated by Susan Hanson, reprinted by permission of the University of Minnesota Press. English translation ©1993 by the Regents of the University of Minnesota. Originally published as _L'Entretien infini,_ copyright 1969 by Éditions Gallimard.
# Abbreviations
All references to Blanchot's publisheds writings will be given in the text, using the abbreviations listed below. Titles of English translations, where they exist, are provided in parentheses. In general, page numbers supplied in my text refer to the standard French editions of Blanchot's work; where two figures or sets of figures are cited, the second, wherever possible, refers to the available English translation. At times, for reasons of consistency or accuracy, it has been necessary to modify or revise these versions. Unless otherwise indicated, all additional translations are my own. Full details concerning both texts and translations used may be found in the bibliography of texts by Blanchot at the end of the book.
See Table
# 1: An intellectual itinerary
## Why Blanchot?
To write _on_ is, in any case, without propriety. But to write on the event that is precisely designed (among other things) to make it no longer possible ever to write _on_ **–** by way of epitaph, commentary, analysis, eulogy, or condemnation – is to distort it in advance and to have always already missed it.
'Tracts, affiches, bulletin' ( _BR,_ 204)
There is little doubt today that the name Maurice Blanchot signs some of the most challenging and influential literary and philosophical texts of the last fifty years. Dispersed across a variety of different genres, Blanchot's writings are extensive and numerous. They include countless critical essays on literary and philosophical topics, three full-length novels, ten or more shorter narratives or _récits,_ as well as a significant amount of political journalism, not to mention a host of other fragmentary texts that resist all attempt at stable classification. Despite a reputation for sparseness and impenetrability, they display a matchless lucidity and relentless commitment to explanation, self-commentary, and gloss; taken as a whole, they constitute perhaps one of the most remarkable and enduring of monuments in the whole of recent intellectual history to the perseverance and assertiveness of thought itself.
Blanchot is a writer whose work eludes easy categorisation. His writing straddles literature and philosophy, which is to say that, while it belongs to both, it falls subject to neither. This is not to claim for Blanchot's writing a position of extraterritorial neutrality or metalinguistic immunity; nor is it to suggest that Blanchot's own texts are somehow able to arbitrate authoritatively and decisively on the question of the relation between literature and philosophy as such. Indeed, the case is almost exactly the reverse. While asking fundamental questions of philosophical and literary texts alike, Blanchot's own writing is itself characterised by an awareness of its own irremediable and necessary incompletion, and by the knowledge that, just as all legislation is a response to lawlessness, so limits too are always but a tribute to the limitlessness that inhabits them as an ineluctable and indelible condition of their possibility. Blanchot here pushes philosophy and literature towards the unspoken margins that constitute them as what they are. In so doing he deconstructs and transforms them, radically altering the terms in which today it is possible, indeed necessary, to think not only the question of literature, but also the deeper question of which Blanchot writes in _L'Entretien infini_ ( _The Infinite Conversation_ ) that it is the questioning that eludes and outstrips, precedes and exceeds the question of the whole ( _EI,_ 16–21; 14–17).
Blanchot's intervention, then, is fundamental. What changes, with Blanchot, is not just a localised sector of the literary critical or philosophical landscape, but the manner in which the relationship between literature and philosophy, writing and thought is articulated at all. Blanchot, one might say, is the thinker who has most consistently shown the relevance of philosophy for the practice of literature as such; but he is also the writer who has demonstrated most powerfully the radical extent of the questions addressed in its turn by literature to philosophy as a whole. As a result, Blanchot in his writing has renewed the critical debate concerning the ontological – or non-ontological – status of literature and art in general, and profoundly transformed the manner in which it is necessary to think the question of the ethical demand to which writing is a response. Philosophy, Blanchot contends, cannot escape the words that make it possible, but neither can literature do without philosophy; indeed, as Blanchot puts it in _L'Écriture du désastre_ ( _The Writing of the Disaster_ ),
to write in ignorance and without regard for the philosophical horizon, a horizon punctuated, gathered together or dispersed by the words that delimit it, is necessarily to write with facile complacency (the literature of elegance and good taste). Hölderlin, Mallarmé, so many others, do not allow us this.
( _ED,_ 160; 103)
Blanchot is probably best known as a critic of literature. As such, he is the author of some of the most perceptive and probing essays of the last fifty years. His texts on Sade, Kafka, Hölderlin, Mallarmé, Rilke, Bataille, Char, Beckett, Duras and many others, make him one of the most distinctive and cogent analysts of modern and contemporary literary culture. But Blanchot's contribution does not stop there. He is also an incisive reader of many of the central philosophical texts of the last two centuries, an astringent commentator of Hegel, Nietzsche, Heidegger, Levinas, Foucault, Derrida, and others, while as a novelist he is the author of some of the most trenchant – and intractable – of modern literary texts, texts which, in their limpid clarity and irredeemable obscurity, give voice, as Georges Bataille once wrote, to the vertiginous extremity of that which is itself without limits and which, as Derrida has shown, remains radically uncontainable within the bounds of discourse, including the genre of narrative itself.
For all his exemplary centrality within twentieth-century thought, Blanchot remains a figure of irreducible and striking singularity. Like others of his generation, Blanchot was never an academic, preferring instead to derive his livelihood from his activities as a journalist and from his publications as a literary critic and novelist. As a thinker, Blanchot is in any case a writer deeply alert to that which all systems of thought necessarily repress or exclude and which, more often than not, turns out to be the dissimulated basis of whatever it is they in fact seek to propound. Consequently, Blanchot's critical or philosophical writings are most often couched in the form of fragmentary commentary, and frequently present their arguments by way of a complex movement of citation, paraphrase, and gloss. As a philosopher or critic, Blanchot writes, so to speak, within the margins of other texts, within the interstices of the writings that, by chance or necessity, he encounters as a reader. One might claim as a result, and with some justice, that Blanchot in his own right is therefore not an original thinker at all. He produces – and this is one of the radical departures characteristic of Blanchot's writing in general – few, if any, authentically philosophical or theoretical concepts that may be called his own. Indeed, the terms with which his name is associated, words such as worklessness ( _désœuvrement_ ), the outside ( _le dehors_ ), the neuter ( _le neutre_ ), or disaster ( _désastre_ ), are precisely not terms, not stable points of anchorage, but, as it were, fragments of a singular, still unspoken or long forgotten other language, or language of otherness, made up of words or traces constantly being effaced from within by the strange displacements to which they silently bear witness, always already undoing the discursive oppositions within which they function.
Blanchot's literary or philosophical project, then, like that of Bataille in _L'Expérience intérieure_ ( _Inner Experience_ ), is precisely _not_ a project. This is in part what gives it its singular force and compelling originality. For Blanchot attempts something more essential than simply the task of devising yet another theoretical discourse or conceptual frame within which to arraign art or literature and domesticate the questions it poses to both philosophy and thought. Instead, by persistently occupying the texts of others – from Hegel to Heidegger, Mallarmé to Paulhan, Nietzsche to Rilke – Blanchot tires out these discourses, pushes them to the limit of their endurance. Exhausting them in this way, he is then able to begin to address the fundamental otherness lying dormant within them or having been excluded from them. The consequences here are dramatic. For what comes to be inscribed in Blanchot's writing, both within and beyond philosophy, both within and beyond criticism and literature, is the radically ineliminable character of that – without origin or identity, beyond memory and meaning – which paradoxically both enables and disables the totality of thought as its simultaneous condition of both possibility and impossibility.
One may begin here to measure the scale of Blanchot's importance and the reasons why, in France and beyond, his work has come to exert a decisive influence over literary critics, novelists, and philosophers alike. For Blanchot's remarkable achievement is not only to have written some of the most demanding fiction of the age and to have challenged and displaced many of the basic assumptions that continue to inhabit modern philosophy and literary theory; it is also, in the words of Jacques Derrida, in a double gesture, to have both maintained the necessity of philosophy and yet to have questioned philosophy itself from a site – the site of poetical thinking and writing – which has always resisted the endeavours of philosophy to assign to it its particular – always already philosophical – truth.
'Waiting for us, still to come, still to be read and re-read by the very ones who have been doing so ever since they first learned to read and _thanks_ to him': with these words, in 1976, Derrida once paid homage to Blanchot. What they serve to indicate today is both the urgency and the disjoined temporality with which we contemporary readers are each addressed by Blanchot's writing, enjoined in our turn to transform reading – and reading Blanchot – into what Pierre Madaule once hoped, in a homage of his own, would be a – deadly – serious task.
## An ethics of discretion
Discretion – reserve – is the place of literature.
'Le Rire des dieux' ( _A_ , 194)
Despite his undoubted stature as a key intellectual figure in France from the 1950s onwards, Blanchot is a writer who, for many years, has consistently and single-mindedly resisted the gradual appropriation of literature (especially in France) by the mass media, maintaining as a result an anonymity and a reserve of quite exceptional integrity. There are few photographs of the author, no published accounts of his life, little in his work that would seem to warrant biographical extrapolation. For all that, his intellectual itinerary has often had a fiercely public dimension, and indeed, as though to acknowledge as much, albeit in fragmentary ways, in response to readers' inquiries, in occasional published letters, and in one or two rare autobiographical asides and unprompted vestigial narratives of his own, Blanchot in recent decades has been concerned to disclose, deliberately but allusively, some of the historical circumstances surrounding the turning points in his writing life.
Yet while knowledge of these events does make it now possible to begin to describe some of the pivotal stages in Blanchot's complex and lengthy intellectual career, there remain many areas of the author's life about which little, if any, information is available. Blanchot himself has rarely departed from an overriding commitment to what one might call an ethics of discretion: an ethics that shuns the risk of indiscriminate self-exposure in order to affirm the value of distance and silence and by so doing preserve what in 1984 Blanchot termed 'the right to the unexpected word' ('le droit à la parole inattendue'). As a result, Blanchot's autobiographical moments are never confessional or introspective, nor do they seek to accredit the authority of a single, all-embracing narrative. Memory, Blanchot has often suggested, is itself a function of forgetting; and it is as though, through the fragmentary ways in which he has divulged some of the facts of his autobiography, Blanchot is concerned to remind his readers that a necessary condition of all acts of recollection is the effacement of much else that is essential, and that indeed the essential may in fact best be respected by patient attention to the infinite reserve of forgetting itself.
In Blanchot's fragmentary curriculum vitae, scenes of friendship loom large. So, too, do moments of political hiatus. Both have in common, on Blanchot's reading, their fundamental resistance to the teleology of narrative presentation. Political upheaval and fidelity to friendship belong to history but, more importantly, also interrupt it; they allow the opening of a space in which a relation with otherness intervenes to defer historical closure. Blanchot's discretion, then, both translates a desire for privacy and implies a belief in historical discontinuity; and it is as though, when writing his name into history, Blanchot is more concerned to display, in the margins of that history, the anonymity that precedes the inscription of any name. So when for instance in 1973, in _Le Pas au-delà_ ( _The Step Not Beyond, PA ,_ 9; 2), in one of his earliest covertly autobiographical digressions, he may be found reflecting elliptically on the radical discrepancy between his duties as a professional journalist and his experiences as an apprentice novelist in the years leading up to the publication of _Thomas l'Obscur_ in 1941, Blanchot's purpose is not to provide the reader with a reassuringly linear narrative frame. It is rather to underline that fundamental dissymmetry between workaday activity and artistic non-activity which marked his career at its inception. The effect is to compel the reader to reconsider the relationship between history and writing in such a way that what now becomes primary is not so-called historical truth in its apparent (and misleading) self-evidence, but rather the irreducibility and alterity of that which is exterior to historical narrative because it is what has always already preceded the construction of history _as_ narrative.
In this regard, it is no accident that elsewhere in the same book, in pages first published the previous year in homage to Edmond Jabès, Blanchot also raises the question of how to address the historical – yet more than historical – event of the Holocaust ( _PA ,_ 56–7, 156; 38–9, 114). For here, as in Blanchot's reflections on his own beginnings as a writer, what is at stake is a requirement that one attend, not to the meaning of history, but to the limits of historical understanding, not to the teleology and so-called objectivity of history, but to the burden of responsibility that history imposes, willingly or not, on those who are its actors or protagonists. For Blanchot in _Le Pas au-delà,_ rather than explaining writing by recourse to narrative history, it is more a case of mobilising the resources of writing itself in order to bear witness – impossibly – to that in history which escapes historical representation. It is to uncover, within memory, a question not of history, but of writing, and within that question of writing, an ethical question that is not a question of history as such, but of political responsibility and of experience.
This issue of the relationship between writing and history is at the heart of much of Blanchot's thinking during the 1970s and 1980s. In 1983, Blanchot returned to the topic in another discreet autobiographical self-commentary, where, uncharacteristically, arguably by way of a response to recent controversy surrounding his early involvement with the nationalist right, he acceded to a demand – one he describes as also originating somewhere within himself ( _AC,_ 91; 63) – to supply, under the title _Après coup_ ( _After the Fact_ ), a retrospective commentary to a reissue of _Le Ressassement éternel_ ( _Vicious Circles_ ), a collection of two early short stories from the mid-1930s that first appeared in 1951. Rereading these texts, Blanchot used the occasion to reconsider the connection between those stories and the historical events – more precisely the event of the Holocaust – that at least one of the narratives, 'L'Idylle', in its evocation of a penitentiary workhouse, may be seen to parallel, if not indeed eerily to predict ( _AC,_ 94–6; 65–7). Blanchot, however, remains resistant to this apparent convergence between text and history. 'History', he concludes, 'does not control meaning [ne détient pas le sens], any more than meaning, which is always ambiguous – plural – may be reduced to its historical realisation, were this the most tragic and weightiest imaginable' ( _AC,_ 96; 67). History, then, is not established truth, and provides no authorised translations; indeed, in some cases, Blanchot implies, history itself may turn out simply to have borrowed, in an act of plagiarism, the totalising impetus sometimes to be found within literary narrative itself. But equally, if history is at times just a displaced version of an oppressive, all-inclusive fiction, so, too, in the encounter with historical experience, fictional narrative sometimes also meets its limits. A story such as 'L'Idylle', writes Blanchot – indeed, any story whatever – will only ever fall short of the irreducibly singular historical event of those years, which in his text Blanchot names simply as: Auschwitz. 'No matter when it may be written,' we read, 'every story will henceforth be before Auschwitz' ( _AC,_ 99; 69).
These reservations with regard to narrative representation do not amount to a withdrawal from political exigencies, nor an abandonment of the obligation to bear historical witness. Rather the reverse. To read history, as does Blanchot, as a discontinuous space, irreducible to chronological narrative, is, by rejecting temporal closure, to affirm the necessary futurity of politics and political responsibility. Indeed, it was clearly to reinforce this very point that, some months after _Après coup,_ at the age of 76, Blanchot followed up that book with another short text, _La Communauté inavouable_ ( _The Unavowable Community_ ), based on a discussion of recent work by Jean-Luc Nancy and Marguerite Duras, in which he undertook to draw the philosophical and political lessons of three significant moments in his own political itinerary: the pre-war commitment to action in small volatile groups existing outside of mainstream political parties, the struggle against the Algerian War, and the Paris _événements_ of May 1968. In retracing that history, Blanchot's aim was not primarily to engage in an exercise in retrospective justification or self-criticism, but, more importantly in his view, to affirm the vital actuality of what he describes as 'the demand of communism' ('l'exigence communiste', _CI,_ 9; 1). Indeed, by invoking the promise of communism in these terms, at a time when the word itself had become, in the eyes of almost all other commentators, arguably the most threadbare term imaginable in the whole political lexicon of the West, Blanchot's intention was not just to acknowledge the sorry legacy of a discredited word, but to reaffirm his hope and fears for the future. It is in this sense that Blanchot's discretion – his refusal of historical, autobiographical narrative – demands not to be viewed as a retreat from the political. It is rather that Blanchot chooses to suspend a certain avowable narrative history in order to address his text to the future, with an eye to the risk that in fact makes hope possible at all, to what he describes as _'the always uncertain end_ inscribed within the destiny of community' ( _CI,_ 92; 56, Blanchot's italics).
This same insistence on the determining effects of the future and on both the risks and compelling decisions that are a necessary and indispensable element of any politics whatever also lay behind Blanchot's contribution, the following year, under the title 'Les Intellectuels en question' ('Intellectuals Under Scrutiny'), to the continuing debate in France about the tasks of the intellectual under the socialist presidency of François Mitterrand. In this long retrospective piece, motivated in part by a desire to defend Bataille against the charge of complicity with fascism and sympathy with Vichy France – criticisms which, in their turn, have both been levelled at Blanchot as well – the author gives a personal account of the political challenges that faced intellectuals in France, and elsewhere in Europe, from the Dreyfus Affair to the time of the Algerian War and May 1968. He lingers here and there on the cases of Heidegger and Valéry, and it is by way of conclusion to a discussion of the latter's support for the anti-Dreyfus campaign that Blanchot remarks that: 'In this cursory examination . . . I can find nothing that justifies him [i.e. Valéry] for adding his own name to the list of those who, in the worst possible terms, were calling for the death of the Jews and the destruction of those who sought to defend them.' And Blanchot adds, with obvious disbelief: 'There would thus seem to be, in every life, a moment when the unjustifiable prevails and the incomprehensible is given its due.' These are solemn words for more reasons than one; what they show in 1984, as he reviews the historical circumstances of his own political commitments, is a Blanchot simultaneously aware of the infinite demand of justice and the precarious fragility of all ethical injunctions as such.
In the article, Blanchot also takes the opportunity to recapitulate, obliquely, many of the key turning points traversed or witnessed by him in the course of his own political evolution from the late 1920s to the present and which, taken as a whole, read like a résumé of the principal dilemmas and decisions faced not only by Blanchot in his own political life but by France as a whole from the early decades of the century onwards: the general loss of faith in parliamentary democracy in France that followed the First World War and which, paradoxically, the very methods of military rule that assured victory in 1918 served only to exacerbate; the subsequent discredit into which parliamentary democracy in France continued to fall as a consequence of its instability and weakness; the dangerous impatience with which, in the imperious name of justice, some called for an immediate war against Hitler in the mid-1930s; the dismal failure of the French government to act on 7 March 1936, when, in flagrant contravention of the Locarno Pact, Hitler's army remilitarised the Rhineland and set in train the events that were to culminate in the outbreak of the Second World War; the spineless abjection displayed by the Western democracies when they acceded to Hitler's demands at Munich in September 1938; the decision – in spite of all – on the part of some dissident intellectuals (including Blanchot himself) to defend the republic in the ensuing years; the painful period of the Resistance and the difficult choices that needed then to be made; and, after a lengthy hiatus, the campaign against the Algerian War in 1960, followed by the _événements_ of May 1968; and finally, to close, the fateful year 1943 of which Blanchot writes, justifying the outline sketched in the preceding pages, that 'its return is always possible'. Lessons must be learnt from the past, Blanchot suggests, but it is the future – a future without present or presence – that is the only properly political dimension. This explains Blanchot's persistent watchword, reiterated during the 1980s in relation to the memory of Auschwitz, and inevitably addressed to himself as well as to each of his readers: 'N'oubliez pas!', 'Do not forget!', 'sachez ce qui s'est passé, n'oubliez pas, et en même temps jamais vous ne saurez', 'know what happened, do not forget, and at the same time never will you know' ( _ED,_ 131; 82).
## A share of biography
For me, what mattered was the encounter with others, where chance becomes necessity. The encounter both with people and with places. That is my share of biography.
(Pour moi, ce qui a compté, ce sont des rencontres, là où le hasard se fait nécessité. Rencontre des hommes, rencontre des lieux. C'est ma part de biographie.)
'Les Rencontres (1984)
Retracing these events, Blanchot offers little in the way of private testimony; his concern is rather to show how the flow of history is repeatedly interrupted by so many moments of necessary decision, so many dilemmas, so many failings or slippages, and that it is these that prompt the many sideways turnings that go to make up a single life's itinerary. These moments are not just public ones, for private experience is shaped by them too, with the result that it is the separation between the public and the private that is the most fragile division of all. So it would seem at any rate from the vestigial outline of Blanchot's own personal life given in a rare autobiographical sketch, published in _Le Nouvel Observateur_ in November 1984. In the piece, more aide-mémoire than intimate record, Blanchot dramatises his life's progress in terms of a sequence of dated meetings with a small number of unique and singular others whom he names as: Emmanuel Levinas, Georges Bataille, René Char, and Robert Antelme. As well as a discreet mark of a personal friendship, each of these names, in turn, functions as shorthand for a larger philosophical or political or literary encounter.
As the two men have often recalled, each in his own way, on numerous occasions, Blanchot and Levinas first met as fellow students of philosophy in Strasbourg in 1925. The two had very different backgrounds: Levinas, slightly the older, and a liberal, was recently arrived from Lithuania, while Blanchot, by all accounts still a monarchist, was from a well-off family from Quain in the Saône-et-Loire; yet, almost immediately, the pair swore a pact of friendship to each other whose longevity, terminated only by the death of Levinas in 1995, is at all events remarkable: 'it happened,' wrote Blanchot two years before Levinas's death, 'not because we were both young, but by a deliberate decision, a pact I hope never to break'. The two quickly shared their respective intellectual enthusiasms. Blanchot introduced Levinas to contemporary French literature in the shape of Proust and Valéry, while Levinas for his part helped Blanchot with Husserl and Heidegger, particularly the recently published _Sein und Zeit._ This last encounter in 1927 and 1928 alongside Levinas was one Blanchot still recalled sixty years later as a veritable intellectual shock. Both left Strasbourg at about the same time; and though relations were more distant during the years that immediately followed, it has recently been suggested that it was of Levinas's youngest brother, Aminadab Levinas, who died in Lithuania at the hands of the Nazis shortly before, that Blanchot was thinking when in 1942 he gave the unusual title of _Aminadab_ to his second published novel. It was at any event to Blanchot, at around the same time, while himself still a prisoner of war in Germany, that Levinas entrusted his immediate family, and Blanchot who was instrumental in finding a safe haven for Levinas's wife and daughter in a monastery near Orléans, where they remained until the Liberation.
The second major encounter in his life, Blanchot writes, was his meeting with Georges Bataille. This took place in Paris late in December 1940; again, the friendship between the two was immediate and absolute. Against the background of the Occupation, despite the hardships of the time, the two men maintained a semblance of intellectual activity. Bataille, for his part, was still committed to pursuing, if with diminishing resolve, the experience of community begun before the war by means of such singular, heterodox groups as the Collège de sociologie, or Acéphale; and it was within a similar collective context that much of the discussion took place that was finally to lead to the publication of _L'Expérience intérieure_ in 1943. As Bataille openly acknowledges throughout that text, Blanchot was a major participant in these exchanges, and it is evident that the meeting with Blanchot had a galvanising effect on Bataille. The reverse is no doubt also true, and it may be argued that many of the changes that took place in Blanchot's thinking and writing during the early or mid-1940s may be attributed to his own encounter with Bataille. Their friendship was evidently a two-way relationship; and just as Bataille, in writing _L'Expérience intérieure,_ drew openly on conversations with Blanchot and on the latter's two recent novels, _Thomas l'Obscur_ and _Aminadab,_ so Blanchot, in his first essay collection in 1943, _Faux Pas,_ paid ample tribute to Bataille in return, recalling, apropos of _L'Expérience intérieure,_ Nietzsche's own description of _Also sprach Zarathustra,_ which Blanchot was to remember again, some twenty years later, in _L'Entretien infini_ ( _EI,_ 313; 211): 'this work is entirely apart' ('cette œuvre est complètement à part') (FP, 52).
Coupling Bataille's name, in 'Les Rencontres', with that of René Char, a notable and early active member of the Resistance, Blanchot summarises the war years under the following, highly elliptical headings: 'The call to irregularity. The limit-experience. Opposition to the occupation and the Vichy regime. Underground activity' ('Appel à l'irrégularité. L'expérience-limite. Opposition à l'occupant et au régime de Vichy. Clandestinité'). But today, despite these unambiguous remarks, much obscurity still surrounds Blanchot's wartime activities, and Blanchot's silence about them has rarely passed the bounds of absolute discretion; indeed, it was only in 1994, in a short third-person autobiographical narrative entitled _L'Instant de ma mort,_ that Blanchot finally yielded to his readers, after years of rumour, the sketch – little more than a suspended fragment – of how, fifty years previously, in July 1944, a young man, the 36-year-old Blanchot, faced – for his involvement in the Resistance? – a German military firing squad, and yet unpredictably, unaccountably, and in the end (says Blanchot) unjustifiably, was allowed to escape, allowed to survive the very moment of his own dying. Thus Blanchot's vestigial testimony of the war, the silent murmur of which resounds through much of the fiction and many of the essays published by Blanchot after the war and gives to their exploration of the impossibility of the experience of dying much of its singularity and insistent urgency.
Within a few years of the war's end, Blanchot quickly established himself as one of the most distinctive voices in French post-war literary criticism. He published widely, at times regularly, at times on a more occasional basis, in a host of literary journals and periodicals of diverse political hues. These included _L'Arche,_ which, launched from Algiers early in 1944, was one of the first independent literary monthlies to emerge after the war, and on whose editorial committee, alongside Gide and Camus, Blanchot served for a brief time, as well as more enduring publications such as Sartre's _Les Temps modernes,_ and Paulhan's _Cahiers de la Pléiade._ Probably most important of all among the journals to which Blanchot was a regular contributor in the period following the war was _Critique,_ the monthly review founded by Bataille in 1946 and edited by him, with substantial support from Blanchot in its early years, and in which Blanchot published many of his most influential post-war essays, including a number of celebrated texts on Sade, Char, Hölderlin, Rimbaud, Hegel, Kafka, and Rilke.
In 1947 Blanchot left Paris for Èze-Village, a small town on the Mediterranean coast between Nice and Monte-Carlo, which, as he recorded in 1989 ( _UV,_ 13), faces out to Corsica to the south-east and over Cap Ferrat to the south-west. The decade that followed was a period of astonishing productivity. By May 1947, Blanchot had already completed his third, and last novel, _Le Très-Haut_ ( _The Most High_ ) _,_ on which he had been working since at least 1945. When it finally appeared, in July 1948, it did so simultaneously with _L'Arrêt de mort_ ( _Death Sentence_ ) _,_ Blanchot's first major short narrative or _récit;_ the previous year had seen the first publication of 'Le Dernier Mot' ('The Last Word') and 'L'Idylle' ('The Idyll'), the two stories later collected in _Le Ressassement éternel_ and _Après coup._ Some months before, in mid-January, Blanchot had also finished work on a second, dramatically shortened version of _Thomas l'Obscur_ which, when it was published in 1950, turned out to be no more than a third of the length of the original 1941 text. In 1949, too, Blanchot brought out two volumes of critical essays, _La Part du feu_ ( _The Work of Fire_ ) and _Lautréamont et Sade,_ which together reprinted the bulk of his critical output over the previous four years.
The period between 1948 and 1950 saw the appearance of five major books; but Blanchot in the years that followed was scarcely any less prolific. He continued to publish fiction as well as literary criticism. In 1949 the journal _Empédocle_ had published a short _récit_ entitled simply: 'Un récit'; and three more short narratives quickly followed: _Au moment voulu_ ( _When the Time Comes_ ) (1951), _Celui qui ne m'accompagnait pas (The One Who Was Standing Apart From Me_ ) (1953) and _Le Dernier Homme_ ( _The Last Man_ ) (1957). These, in their turn, were followed, in 1962, by _L'Attente L'Oubli,_ a book that is sometimes cited as Blanchot's last _récit,_ though the text itself contests any simple attribution of genre; essentially unclassifiable, it is a work in which fictional representation finally seems to have become indistinguishable from meditative, philosophical prose, and vice versa.
In January 1953, the _Nouvelle Revue française,_ France's most prestigious pre-war literary journal, was adjudged to have served out the post-war quarantine imposed upon it for continuing publication between 1940 and 1943 under the collaborationist banner of Pierre Drieu La Rochelle, and the journal began reappearing under the pre-war editorship of Jean Paulhan. The first issue carried an essay by Blanchot entitled 'La Solitude essentielle' ('The Essential Solitude'), which was later to become the opening chapter of _L'Espace littéraire_ ( _The Space of Literature_ ); from this point on Blanchot settled into what was to become, until Paulhan's death in 1968, a pattern of regular monthly contributions to the journal. The effect on Blanchot's work as a literary critic was impressive. The freedom he was allowed by Paulhan to write in whatever way he chose about literature in general released him from the constraints of conventional book-reviewing, which had gradually become a source of some frustration to Blanchot. Indeed, it was largely the writer's impatience with the requirement that he discuss topical work by other writers, the format adopted by _Critique,_ that lay behind Blanchot's reluctant withdrawal from active involvement in that journal after 1953.
This left Blanchot able to pursue more radically what he had already begun to do in the latter years of the 1940s, and develop a different kind of critical language – more an oscillating syntax of irreducible paradox than an exhaustive system of totalising concepts – with which to address the question of literature, that fundamental question that Mallarmé, in 1894, had first formulated by inquiring: 'Does something like Literature exist?' ('Quelque chose comme les Lettres existe-t-il') to which he had supplied the answer: 'Yes, Literature does exist and, if you will, it alone, except for everything else' ('Oui, que la Littérature existe et, si l'on veut, seule, à l'exception de tout'), and the consequences of which Blanchot, in 1952, sought to clarify by asking, in more explicitly Heideggerian terms: 'What are the implications for being of the statement that "something like Literature exists"?' ('Qu'en est-il de l'être, sil'on dit que "quelque chose comme les Lettres existe"?', _EL,_ 35; 43). So when in 1955 _L'Espace littéraire_ came out, based on work published in _Critique_ and the _Nouvelle Revue française_ over the previous four years, it was apparent that Blanchot had at his disposal a critical idiom that, alongside its redoubtable philosophical sophistication, manifested, as the publisher's blurb put it, an experiential or experimental dimension entirely its own. The comment is one that holds true for all the essays Blanchot published in the course of the next fifteen years, and it is those texts, published for the most part in the _Nouvelle Revue française,_ that provided Blanchot with the material for the three further ground-breaking volumes that appeared over the next twenty years and on which his reputation as a critic and thinker now largely rests: _Le Livre à venir_ (1959), _L'Entretien infini_ (1969), and _L'Amitié_ (1971).
In 1957, after ten years of solitary writing, Blanchot returned to Paris. One of his first points of intellectual contact in the capital was Dionys Mascolo, whom Blanchot had first met in the early 1940s and who, in July 1958, together with Jean Schuster, was responsible for launching the paper _Le 14 Juillet,_ the main purpose of which was to coordinate intellectual resistance to the return to power of General de Gaulle that had occurred two months earlier. The deep fear at the time, as expressed by Mascolo and Schuster in a flier they produced on 17 May 1958, was that de Gaulle's unconstitutional _coup,_ carried out with the support of the right-wing French military, was the first step towards the setting up of a – fascist – military regime not very different from that of Franco in Spain. Blanchot, on receipt of the first issue of the paper, replied to Mascolo immediately. 'I want to tell you I am in complete agreement,' he wrote. 'I refuse the whole of the past and accept nothing of the present.'
In two subsequent pieces for the paper, 'Le Refus' and 'La Perversion essentielle' ( _BR,_ 167–73), Blanchot developed the substance of his own opposition to the regime. This had essentially to do with de Gaulle's claim – in Blanchot's eyes disturbingly reminiscent of Pétain's self-proclaimed personification of France in 1940 – to embody in his own person the national destiny of France; for by so doing the so-called providential leader of 1958 was seeking to lend his name and prestige to a dangerous, quasi-religious politics, a politics of nationalist mystification that sought mainly to fill the vacancy of power – the necessary void at the centre of politics – with a spurious, authoritarian ideology of national salvation. And in the face of such an 'essential perversion' of the political process the only response that Blanchot deemed to be adequate was one of absolute and categorical refusal.
Blanchot's contribution to the campaign against de Gaulle ushered in an intense period of renewed political activity on his part. Events came to a head when, in September 1960, the philosopher and journalist Francis Jeanson was put on trial, together with twenty-three others, French and Algerians alike, for aiding and abetting the FLN (the Front de Libération Nationale) in the struggle against French colonial involvement in Algeria. To coincide with the trial, Mascolo, Schuster, and Blanchot, with a small group of other intellectuals, issued a solemn declaration supporting the actions of those who had made common cause with the struggle for Algerian independence, and endorsing the right of French conscripts to refuse the draft; the statement concluded with the assertion that 'the cause of the Algerian people, which is making a decisive contribution to the ruin of the colonial system, is the cause of all free men'. After the initial number of signatories, the document soon became known as the 'Manifeste des 121'. It was allowed to circulate, however, only in clandestine form; and indeed the two mainstream journals that sought to print it, _Les Temps modernes_ and _Les Lettres nouvelles,_ were promptly confiscated by the authorities, while those who had signed, under new measures adopted by the government, risked being sentenced to between one and five years in prison. But by then, enough had been done by those involved, activists and intellectuals alike, to discredit France's continued pursuit of the war, which was eventually brought to a close by the Évian accords of March 1962.
Blanchot's hope, already by the Autumn of 1960, was that the 'Manifeste des 121' would be not just an isolated endeavour, but the beginning of a new broadly based intellectual initiative. The vehicle for this longer-term project, he proposed, should be a new journal, a 'journal of _total criticism'_ ('revue de _critique totale'_ ) _,_ as he explained in a letter to Sartre, one which, by combining literary output and criticism with current political and scientific discussion, would demonstrate the necessary interrelation of each of these different areas of thought. The intention was also that the journal should have, as Blanchot put it, an 'essentially collective' existence; to this end it was to be an international enterprise, based on the activities of at least three different national editorial committees, which necessitated lengthy discussions involving potential collaborators from Germany and Italy. But more important still for Blanchot, if the initiative was to succeed, was the need to elaborate a new form of writing, mingling the literary with the political in a coherent but critically open-ended manner, that would allow the journal's commitment to plurality, multiplicity, and collective exchange to find a specific rhythm of its own. The form Blanchot identified as the one best able to achieve this was the fragment; and it is no exaggeration to say that this turn to the fragment was to leave a deep imprint on everything Blanchot was subsequently to write. But by early 1964, despite the efforts of Blanchot, Mascolo, Louis-René Des Forêts and many others, the project for an international journal had failed. But if it did so, Blanchot noted at the time, it did so utopianly; and the very impossibility of success was proof, if any was needed, of the radical necessity of the enterprise.
Within Mascolo's circle in 1958, and as a constant presence in all the different initiatives of the years that followed, Blanchot had encountered another important friendship, that of Robert Antelme. Antelme was by all accounts a remarkable man, and best known at the time as the author of _L'Espèce humaine,_ a powerfully moving account of his experiences during his arrest and deportation to Buchenwald and Dachau for Resistance activity in 1944. First published in 1947, the book made a strong impact on Blanchot, who writes about it at length in an important essay of April 1962 collected in revised form in _L'Entretien infini_ ( _EI,_ 99–103, 191–200; 70–2, 130–5). And when in 1968, as though simply in resumption of the struggle against de Gaulle begun ten years earlier, the whole of France stood paralysed by students' and workers' demonstrations, it was with Antelme, together with Mascolo, Duras, and countless others, indeed – as Blanchot phrases it in 'Les Rencontres' – 'with everyone' ('avec tous'), that Blanchot played what was no doubt a significant if discreetly anonymous part in the May _événements._ Blanchot's main activity, it seems, was as a member of the 'Comité d'action étudiants-écrivains', that fervently anti-authoritarian group of revolutionary students and writers that met from 20 May 1968 till the following spring and which produced in October 1968 a semi-clandestine magazine, entitled _Comité, in_ which over half the texts, although unsigned, were, according to Mascolo, all written by Blanchot.
Admittedly, the committee had eventually foundered by the following March amid fierce internal dissension; around the same time, too, Blanchot himself also abruptly withdrew from the movement, explaining in a letter to Levinas that the reason for his action was the position taken by the extreme left in favour of Palestine and against Israel. Though he acknowledged that the move was not motivated by anti-semitism, he nonetheless took the view that the end result was little different: 'It would seem', he wrote, 'that anti-semitism from now on can count among its allies those who are, as it were, free of anti-semitism'. And he went on to ask: 'Isn't this a strange reversal, one that proves that the absence of anti-semitism is simply not enough?' As though in response to that very question, from 1971 Blanchot began in his writing explicitly to address the issue of the Holocaust.
## Pas de récit?
A story? No. No stories, never again.
(Un récit? Non, pas de récit, plus jamais.)
_La Folie du jour_ ( _FJ,_ 38; 18)
The story that Blanchot tells in the course of these various autobiographical and semiautobiographical texts is incomplete, fragmentary, discontinuous. Everything that is divulged is communicated obliquely. As a result, while these partial narratives are significant for what they affirm, they are also, as Blanchot himself would no doubt concede, equally revealing for what it is they prefer to leave unsaid. It is perhaps unsurprising, therefore, that on the part of some commentators Blanchot's very obliqueness has been viewed with the very greatest of suspicion; and on more than one occasion the writer has been pressed by his critics to begin recounting in more detail what the autobiographical fragments to which I have referred so far leave largely unmentioned: the writer's pre-war involvement with the extremist, nationalist right.
Knowledge of this unfamiliar past was first brought to the attention of the vast majority of Blanchot's readers in 1976 when the journal _Gramma_ published an extensive bibliography of the writer's early published texts. Since then, numbers of readers, at times polemically, at times more dispassionately, have raised the question of the relationship between these pre-war political writings and Blanchot's post-war criticism and fiction. The author's own response to inquiries addressed to him personally has been twofold. On the one hand, he has refused to justify, or even comment in much detail on his pre-war political choices, while on the other hand, particularly regarding specific points of controversy, he has clearly been concerned to preserve the factual accuracy of the historical record. A number of Blanchot's various letters of clarification have found their way into print. Even so, they have failed to satisfy the anxious – inquisitorial – demand for narrative on the part of some of Blanchot's critics who have implied that the writer's objections to historical narrative are a screen behind which to conceal a deeply rooted – and culpable – evasiveness with regard to the past.
It is indeed the case, as we have seen, that Blanchot at times goes to surprising lengths to avoid narrating explicitly the detail of his own particular – albeit far from uncommon – evolution from extreme nationalism to dissident communism. It must be acknowledged, however, that obliqueness or indirection is not in itself proof of historical evasiveness; and in any case, it is simply not true that in his later texts Blanchot eschews all autobiographical self-scrutiny. In _Le Pas au-delà,_ for instance, as mentioned above, Blanchot may indeed be found recalling, indirectly, aspects of his experience as a journalist and writer during the period leading up to the completion of _Thomas l'Obscur_ in May 1940. But in such passages Blanchot's concerns are not anecdotal ones; for what the author is doing is to elaborate a model for a very different understanding of the relationship between writing and history from that enshrined in traditional, retrospective autobiographical narrative, a model that explains not only Blanchot's personal reluctance to satisfy the requirement for autobiographical narrative addressed to him by his detractors, but also the very impossibility of meeting any such demand. This is why it is worth examining in some detail the passage from _Le Pas au-delà_ I have mentioned. Blanchot writes as follows:
I will attempt in vain to represent to myself the person I was not and who, without willing it, began writing, writing in such a way (and thus realising) that the pure product of doing nothing was thereby penetrating into the world and into his world. That took place 'at night'. In the day, there were the acts of the day, the daily words, the daily writing, declarations, values, habits, nothing of importance and yet something that dimly had to be called living. The certainty that by writing he was putting precisely this certainty into parenthesis, including the certainty of himself as a subject of writing, led him slowly, yet immediately, into an empty space whose emptiness (the heraldic, barred zero) in no way prevented the twists and turns of a lengthy itinerary.
( _PA_ , 9; 2)
Writing of his own past in this way, Blanchot resists the temptation to present his experience as a process of teleological maturation. The author's signature endorses neither the fulfilment of a career nor the accomplishment of an artistic project. The emphasis falls instead on difference and incommensurability. Memory becomes split between at least two competing, dissymmetrical pronouns, each instantiating a different textual figure, including, on the one hand, the anonymous narrative voice of _Le Pas audelà_ itself, which speaks in the first person, while referring to itself (as only a few lines earlier) in the second person, and, on the other, the writer of fiction of the early years, to whom the fragment refers in the third person. Crucially, however, this temporal or historical dichotomy does not engender narrative; it serves rather to repeat or re-enact the distance that already, prior to _Thomas l'Obscur,_ separated the author's everyday duties as a journalist from his night-time activities as a writer of fiction. A difference in time introduces a difference in temporality; and there opens up between day and night, between living and writing, a radical – infinite – separation, an interruption which itself does not belong to narrative and which no narrative form can claim as its object nor indeed successfully mitigate or repair, since to do so would be to substitute for the gaping void or futurity that presides over all historical experience a teleological fiction that would serve merely to occlude the very opening of history to itself as an experience of the future. During the 1930s, Blanchot tells the reader, the day was given over to politics, and writing fiction was what – literally – took place at night. But there is no story that can tell the truth of this interval, no narrative that can embrace it, with the result that what counts most, in retrospect, is not the apparent complementarity between the acts of the day and the experience of the night, but the irreducibility of the – futural – void dividing them from each other.
What writing introduces into the world, according to _Le Pas au-delà,_ is a nothing, a gap in intelligibility and an abeyance of all certainty, a vacancy in meaning which alone gives writing its chance. Writing, here, for Blanchot submits to a more exacting requirement than that of telling linear stories. It becomes a response to the disjunction between languages and within language without which writing itself would not in fact be possible at all. In this way, even as it lends itself to a guarded autobiographical digression, Blanchot's text is impelled to question the possibility of any autobiographical contract dependent on the stability and self-identity of a signature. Instead, as a result of its fragmentary structure, a text such as _Le Pas au-delà_ is constrained, by dint of its own conditions of possibility, to bear witness to the very interruption of sense that alone gives it its chance as a text. This is why, in Blanchot's texts, writing and narrative are always, so to speak, in inverse proportion, and why what is asked of writing is that it always contest the power of narrative to capture the meaning of history. This in turn is why Blanchot's diffidence regarding autobiographical narrative asks to be read not as a repression of the past, but rather as a response to the demand that writing itself respect the void or disjunction in language without which, of course, writing itself would not occur at all. In turning discreetly from historical narrative, then, Blanchot not only holds open the promise of the future, he also enjoins the reader perpetually to reconfront the past, not as established story but as futural writing, not as teleology but as eschatology, not as embodied truth, that is, but as pressing injunction.
What matters, then, for Blanchot, is not autobiographical confession or avowal, but the responsibility to writing. This, in turn, is but another name for responsibility to the Other and to justice for the Other. In this context, Blanchot's own principle is clear: it is to respect in language the fundamental disjunction or plurality without which there would be no language, while at the same time rigorously taking responsibility for his own writing, a responsibility that, as Blanchot acknowledges, is incumbent on him alone. As he wrote in 1984 to Roger Laporte, by way of introducing the fullest account yet of his own pre-war political itinerary, 'You know my principle. Let each express himself according to his own responsibility' ('Vous connaissez mon principe. Laissez chacun s'exprimer selon sa responsabilité'). If these words mean anything, it is that the question raised by Blanchot's own particular political evolution is not a question requiring autobiographical disclosure, but, more abruptly and more trenchantly, a question that is inseparable from the issue of ethical and political responsibility.
## The acts of the day (1)
There is no such thing as good nationalism. Nationalism tends always to integrate everything, all values, that is how it ends up being integral, i.e., the sole value.
(Il n'y a pas de bon nationalisme. Le nationalisme tend toujours à tout intégrer, toutes les valeurs, c'est en cela qu'il finit par être intégral, c'est-à-dire l'unique valeur.)
'Sur le nationalisme' (1991)
It is now well known that, during the major part of the 1930s, Blanchot was principally active as a political journalist. As such, he contributed to a wide range of nationalist newspapers and periodicals. By far the most influential and important of these was the _Journal des débats,_ a traditionalist, staunchly conservative evening daily, whose principal share-holder at the time was the iron and steel magnate François de Wendel, and which had as its _directeur,_ whose responsibility it was to determine editorial policy, a long-standing associate of de Wendel, Étienne de Nalèche. By the 1930s, by all accounts, the _Journal des débats_ had known better days. Despite a prestigious history reaching back to the time of the French Revolution, it had entered into a phase of irreversible decline, though in the eyes of the right-wing intelligentsia who made up the paper's main readership it still enjoyed considerable respect for the literary distinction of its articles and editorials. Blanchot's earliest signed contributions to the newspaper date from 1931 and 1932. Relatively soon, however, he had progressed to the position of _rédacteur en chef_ on the paper; in that capacity, working in close consultation with de Nalèche, Blanchot would have been primarily responsible for writing front-page leader articles and generally making sure the paper went to press on time and in good order.
The 1930s in France were of course turbulent and difficult years. Looking back from the perspective of September 1939, Georges Bataille commented: 'The inter-war years is a time when lying was no less necessary to living than drink. The absence of solution defies expression.' At home, growing economic crisis, government instability, corruption in high places, and the increasing militancy of both the left and the extreme right, all left parliamentary democracy in a notoriously weakened state, while internationally the political stage was still largely dominated by the legacy of the First World War, and the fundamental issue for France was still felt to be the question of what policy to adopt in the face of the German threat. As for France's mainstream political parties, these were divided between two different and incompatible approaches to many issues, including foreign policy. On the one hand were those radicals and socialists who, like Aristide Briand, France's ever-present Foreign Minister during the period from 1925 to 1931, had become, after the end of the First World War, the enthusiastic proponents of a foreign policy founded on the League of Nations and on what Prime Minister Édouard Herriot described in 1924 as the threefold principle of arbitration, security, and disarmament. Implemented by successive governments during the latter part of the 1920s and the early 1930s, the policy took shape as a series of gradual revisions or renegotiations of the Treaty of Versailles of 1919, probably the most successful of which, as it seemed at the time, was the conclusion of the Locarno Pact of 1925, by which, in return for admission to the League of Nations the following year, Germany agreed to recognise the Western borders imposed by the Allies at Versailles and accept the demilitarisation of the Rhineland.
Inevitably, this policy of détente was not without its critics. Indeed at the other end of the political spectrum in France were those conservative nationalists who, following the example of Clemenceau, Prime Minister between 1917 and 1920, remained bitterly hostile to any retreat from the provisions of the Treaty of Versailles – irrespective of how unworkable or destabilising they might in reality turn out to be – and who viewed with deep mistrust the attempt to substitute the juridical, contractual framework of the League of Nations for a more traditional diplomacy founded on military strength and a system of alliances between states rooted in mutual self-interest. And it was here that the _Journal des débats_ was to be found, acting for the most part as a daily mouthpiece of the 'Comité des forges', that powerful association of the owners of all the major French steel-producing factories, presided over by de Wendel, which the propagandists of the Popular Front in 1936 were to attack most fiercely, charging it with representing – as it did – the oligarchic might of France's 200 wealthiest and most influential families.
From the outset, in some of his earliest contributions to the _Journal des débats,_ Blanchot displayed an abiding interest in the question of the relationship between force and law. The concern is one that was fundamental to all Blanchot's political thinking of the period, and he went on to rehearse the position time and time again. Politics and violence, for Blanchot, were not irreconcilable opposites, but the two sides of a single coin. All political legitimacy, he argued, rested necessarily on a founding act of force; force, however, was not an abstract entity to be seen in isolation from the interests it sought to promote. Force in politics – for Blanchot the two terms were largely synonymous could be either beneficial or prejudicial, according to whether – put crudely – a particular state of force was in the service of peace or of war; and it was in any case incumbent on national government to secure the peaceful survival of its own national community by recourse, where necessary, to acts of appropriate force. The practical consequences of this position are not hard to see. It followed for Blanchot that, in both the national and international sphere, the recourse to force was both necessary and legitimate if the purpose was to maintain peace; and it further followed that legal or juridical principles that were not adequately defended by appropriate force could only give the illusion of security and were to that extent worse than nothing. In the context of France's struggle first against Germany and then, from 1933, against Hitler, the debate, Blanchot contended, centred not on the speculative question of how war might be avoided between nations, but on the more fundamental issue of how best to defend France's national interest, which began of course with the need to secure a stable long-term peace. The choice facing France was a brutal one: did it want peace, or did it want war? If the answer was peace, as it was for Blanchot, it followed necessarily that the government had to translate that choice into decisive and effective action by recourse to all appropriate means; and though this would begin with tried and tested diplomatic methods, it might end, in the last resort, with military deployment. To precipitate the threat of war by failing resolutely to defend the peace was in this perspective the height of irresponsibility. But to shrink from such action in the belief that war might thereby be avoided was absolutely disastrous, and would serve in fact only to make the prospect of war itself more certain.
This basic political philosophy finds expression in Blanchot's early journalism in a number of ways. In 1932, for instance, he contributed an unsigned obituary to the _Journal des débats_ on the death of Aristide Briand, the main architect of France's postwar foreign policy, writing of Briand that 'his speeches were always triumphs and his actions failures'. The point here was that in politics a fundamental dissymmetry separated words and acts, and that noble sentiments on their own were unable to effect political changes that might be achieved only by decisive commitment to forceful action. Blanchot, however, was far from a cynical apologist for political violence. Indeed, as he contended elsewhere – this was his conclusion to a discussion of Malaparte's celebrated treatise of 1931, _Technique du coup d'État_ – acts of force had to be judged within a specific political context; and acts of revolutionary violence, for instance, would succeed politically – as they had in Russia in 1917 – only when sanctioned by a prior political project enjoying at least some degree of popular support or when the politics of a regime had lost all credibility or coherence. So if the question of politics for Blanchot was inseparable from the question of force, it was also inseparable from the question of the legitimacy of such force. What was fundamental to successful political action, for Blanchot, was an understanding of the necessary reciprocity between force and law, the fact that laws to be respected relied on the recourse to force, but that force itself, if it were to result in stable government, had to be legitimised by reference to the proper interests of the nation; and it was its endemic inability to confront this issue that gave rise to Blanchot's virulent opposition to the woolly internationalism of the League of Nations, which he saw as an organisation motivated more by wishful thinking than by proper understanding of political realities, and blinded to unpalatable truths by its obstinate belief in abstract and fanciful internationalist conventions which it had neither the means nor the will to defend.
Once Blanchot had begun to serve as _rédacteur en chef_ on the _Journal des débats,_ reflecting the fact that the views he was expressing were those of the paper rather than necessarily his own, his contributions to the paper mostly went unsigned. Nevertheless, his statements of opinion on behalf of the journal were no less incisive than those published earlier under his own name; and in the course of the years that followed, though political circumstances varied significantly, the main policies advocated by the paper changed little and the newspaper remained resolutely committed throughout to the nationalist, conservative cause. Being in favour of strong national government, economic stability, the free market, and law and order, the _Journal des débats_ was implacably opposed, as a result, not only to the French Communist Party (the PCF) and Socialist Party (the SFIO), both of which it claimed to be in the pay of the Soviet Union, but also to those sections of the Radical Party who, alongside the socialists and communists, formed between May 1936 and June 1937 the Popular Front coalition that was to carry the hopes of many working people for social reform and peace, and which was to prove the target of some of the paper's most incisive polemic.
In the realm of foreign affairs the single most important political question for the _Journal des débats,_ as for the French right in general, was the threat to security posed by German expansionism. As a result, while the paper displayed, for instance, much affection for Britain, it maintained a consistent and unyielding hostility to Germany, opposing each of the concessions made to Germany by successive governments in the effort to promote better Franco–German relations. The reasons for this were given in the form of a withering critique, on both theoretical and practical grounds, of the internationalism of the League of Nations. An editorial of 17 January 1935, for instance, noted with some irony that, in demanding a plebiscite in the Saar, which eventually resulted in the return of the region to Germany, Hitler (who had taken Germany out of the League of Nations in the autumn of 1933) was merely making profitable use of the Wilsonian principle of national self-determination, and that by endorsing such an abstract juridical convention the League of Nations had done little more than hand Hitler on a plate the means to extend German territorial and economic power and thereby fatally weaken the League of Nations itself. The paper went on to accuse the League of Nations of naivety. Whoever enacts a legal principle, the _Journal des débats_ argued, ought first to consider the question of the force that will ultimately determine the use – or abuse – of that law; as the editorial put it: 'In the beginning was the Word, claim the Geneva parliamentarians. In the beginning was the act, says Adolf Hitler, repeating Goethe's Faust.' ('Au commencement était le Verbe, disent les parlementaires genevois. Au commencement était l'action, répète après le docteur Faust Adolf Hitler.')
Matters came to a head, when, on 7 March 1936, Hitler's troops, in clear breach of the Locarno Pact, reoccupied the Rhineland, and the French government, with its partners in the League of Nations, did nothing. (It is of course now known that Hitler's military commanders had orders to pull back if they encountered French troops crossing the border, adding considerable force to the view, reiterated by Blanchot in 1984, that the real turning point in the West's capitulation to Hitler occurred in March 1936 rather than in September 1938.) Two years later, on 18 March 1938, in the face of another defeat for the cause of peace – Hitler's annexation of Austria – the _Journal des débats_ described the events of 7 March as marking the beginning of a new era, the era of France's squandering of its victory of 1918. In fact, already in 1936, three days after Hitler's army had moved into the Rhineland, the message was much the same: while, as far as the rest of Europe was concerned, it had broken a solemn contract, Germany in its own eyes was only availing itself of an opportunity those other countries had given it in the first place. What the event proved, for the _Journal des débats,_ was the bankruptcy of the internationalism embodied in the League of Nations. Peace, the paper argued, simply could not be guaranteed by contractual or juridical means; as an editorial of 13 March 1936 damningly observed:
If there is one truth that is slowly dawning, ten years too late, it is that the problem that exists between Europe and Germany is not a juridical problem, but a problem of force. . . . Europe and America . . . have believed in a juridical organisation of the world. This novelty implied equal good will on the part of all nations, prepared to accept the map of the world and work together for a better international future. This was to suppose the problem already solved.
( _Journal des débats,_ 13 March 1936)
The logic of contractual responsibility embedded in French foreign policy after 1924 presupposed in Germany, as in all other states, a reliable subject fully committed to honouring its promises; it assumed that, if a country breached international rules, such a rogue state could be arraigned before the tribunal of other nations and persuaded to act otherwise. However, if that state refused to acknowledge the jurisdiction of the court, there was little the tribunal could do about this, except resort to force. But the recourse to force was what the contractual system was specifically set up to eliminate; and while the juridical framework of the League of Nations did allow for the use of sanctions, what this implied in reality was that there was an increased risk of local conflicts escalating to the larger international community and that, accordingly, individual states would be more than reluctant to become involved at all. Indecision and inaction were the only outcome. So the result of reliance on the League of Nations, according to the _Journal des débats,_ was not to lift the threat of war, but in fact to make war more likely by showing the government to be unwilling and unprepared for a properly robust defence of its national interest. This was why, to the paper's ears, the history of Franco–German relations since 1918 sounded like a litany of defeat, with each reversal leading inexorably down the road to war: thus France's early military withdrawal from the Rhineland in 1930, thus the failure of the London conference of 1933 to settle the question of reparations, thus the plebiscite restoring the Saar to Germany in 1935, thus Hitler's remilitarisation of the Rhineland in 1936, thus the Anschluss of March 1938, and thus, finally, the Munich accords of 1938. The lesson seemed clear. The nation had to look to its own moral and material strength to defend itself; in the last resort, enioying the prospect of a stable long-term peace meant preparing for war; as one leader, dated 16 March 1936, put it, using a tried and tested nationalist formula: 'the only guarantee of peace and security lies in moral and material strength [la force morale et matérielle]'. 'Peace', it added, 'is a question of force.'
Blanchot remained with the _Journal des débats_ until the defeat of France, in June 1940, in the war with Germany against which the newspaper had so often warned. Long before then, however, it must be remembered that, in addition to his work at the _Journal des débats,_ Blanchot was also the author of another, parallel, political discourse, one he pursued in a variety of short-lived extremist publications – titles such as _La Revue française, Réaction, La Revue universelle_ or _La Revue du siècle_ – that existed on the further fringes of mainstream right-wing politics. As early as 1931 and 1932, while starting out with the _Journal des débats,_ Blanchot was writing political articles and book reviews in various magazines and journals belonging to what soon became known at the time as the Jeune Droite. This was a loosely defined, highly volatile, disparate ideological movement that was largely the preserve of disaffected younger members of the French monarchist movement, _Action française,_ who, tired of the circumspection and inertia of that party's traditionalist leadership, had turned to more extreme measures in the desire to overcome what they saw as France's decline into mediocrity; indeed, alongside its virulent nationalism, hatred of Marxism, and contempt for parliamentary democracy, what mainly distinguished the Jeune Droite was its diagnosis that France was in the grip of a profound spiritual and ideological crisis that might be remedied only by recourse to radical and violent means that demanded immediate mobilisation.
In some respects, the distance between the Jeune Droite and the more traditionalist French right was less one of substance than tone and emphasis, and, on occasion, the ideas Blanchot could be found expressing in 1932 and 1933 in, say, _La Revue française_ were not far removed from what he had written, in a personal capacity, in some of his early pieces for the _Journal des débats;_ indeed the article 'Les Années tournantes', based on an influential book of the same title by Daniel-Rops, with its evocation of the moral bankruptcy of the current political system in particular and the spiritual emptiness of the modern world in general, turned out, when it appeared in March 1933, to be no more than a digest of a much longer article from _La Revue française_ published some months earlier. And there are strong parallels, too, between the arguments put forward anonymously in the _Journal des débats_ during the middle years of the decade and those contained, for instance, in such extended pieces as Blanchot's two articles on French foreign policy for _La Revue du XX e siècle_ in 1935, in which Blanchot took the opportunity of developing simply at greater length, and in his own name, the same swingeing polemic against the ineffective, juridical, moralistic thinking of the League of Nations as was already in evidence in editorials for the _Journal des débats._
At the same time, as his thinking moves between these two types of publication, from the mainstream, nationalist press on the one hand to the journals of the Jeune Droite on the other, Blanchot's prose acquires a new, more radical inflection. When writing for _La Revue française_ or _La Revue du siècle,_ what Blanchot does is to articulate in a much more far-reaching way the ultimate implications of the critique of government policy presented in the _Journal des débats._ Writing under his own name, Blanchot accelerates and intensifies, so to speak, the discourse available to him within the columns of the traditionalist press. In that process, the political stakes are raised very rapidly, and Blanchot begins to call into question, in the strongest possible terms, not only the wisdom or effectiveness of a whole range of policies carried out by successive radical and centre-left French governments, but also – at least in the form known to him in the early 1930s – the fundamental legitimacy of French parliamentary democracy as such. (As he did so, it has to be remembered that Blanchot was doing no more than to echo the sentiments of many of his contemporaries, on the left as well as the right; indeed, for many witnesses at the time, the most emblematic event of all was the night of antiparliamentary rioting of 6 February 1934, sparked off by a series of notorious financial scandals penetrating to the very heart of government, when activists of the far right and France's war veterans' organisations marched on Parliament, and the authority of the National Assembly hung in the balance.)
From Blanchot's own perspective, the reason for his radicalisation of the constitutional stance defended by the _Journal des débats_ was a logical one. It was that the system of parliamentary democracy was based on similar abstract, juridical assumptions as the League of Nations itself; in both cases, Blanchot suggested, traditional pragmatism, tried and tested over the years, had given way to a politics based on the idolatry of abstract principles. Just as enthusiasts of the League of Nations were prepared to abandon the healthy scepticism of traditional diplomacy in preference for a dream of contractual harmony between states, so the proponents of parliamentary democracy seemed prepared to sacrifice the overriding need to defend the national interest on the altar of the increasingly vacuous principle of political representation. By clothing politics in a fog of moralistic unreality, the argument ran, legalistic principles served not to promote greater security, but merely to exacerbate the inherent weakness of parliamentary democracy by compromising the need for decisive action. It followed therefore that if one rejected the juridical internationalism on which French foreign policy relied, one also had to refuse parliamentary democracy as well; in either case, the key point lay in recognising the existence of a prior imperative, more compelling than the attachment to fetishised conventions, which was the necessity to defend the future of the national community as such. And in order to meet that objective and to restore proper strength and authority to that community, it was essential not only to transform France's political relations with its neighbours, but also to rid the nation, by violence if necessary, of the debilitating effects of a system of parliamentary democracy which reduced the national interest to a sorry charade of factional, party rivalries. What was needed, unambiguously, was a radical, national, possibly violent revolution. This, at any rate, was the conclusion to which Blanchot came in a series of pieces published in the extremist press from 1932 to 1937.
It is important to realise here that Blanchot's relationship to political discourse throughout the 1930s was always, at least, a dual one. On the one hand, he maintained a regular – though anonymous – presence within the _Journal des débats,_ defending with vigour a constitutional politics that, however dismissive it might be of government policy, always stopped short of recommending the violent overthrow of the régime. On the other hand, in a variety of marginal publications, Blanchot attached his name to a rather different discourse whose purpose was to challenge the very legitimacy of parliamentary democracy and to promote the cause of radical, violent revolution. Between these two discourses the relationship was a complex one; though plainly irreducible to one another, they were evidently not mutually exclusive, and it would be misleading to identify Blanchot himself, so to speak, simply with one rather than the other of these positions. In this respect, Blanchot was neither simply a revolutionary nor a constitutionalist, but in some sense always already both. To some extent, the preference for the one rather than the other at any one moment was probably a tactical one. It also has to be remembered that Blanchot was subject to the usual economic constraints of having to earn a living by his writing. But crucial to Blanchot's political evolution throughout the 1930s is the fact that, by holding simultaneously two divergent, dissymmetrical political discourses, he was able to appeal in his writing at one and the same time to two differentiated, incommensurable concepts of the political, which meant that neither discourse ever came to have hierarchical superiority over the other. Never, one may conclude, did Blanchot fall victim to the temptation of only ever subscribing to one single – totalising – political discourse; and it was this that arguably left him relatively immune to the lure of that vision of totalising immanence that some of his nationalist contemporaries found so compelling when they saw it embodied in the total (and totalitarian) states of fascist Italy or Nazi Germany.
Yet the continuity between these two political discourses professed by Blanchot was hardly seamless. Indeed, rather the reverse. Of the two, the first, which was massively present, though not exclusively so, in the editorials for the _Journal des débats,_ corresponded to a concept of politics defined and understood as the day-to-day administration and management of the life of the community; while the second, mainly to be found in some, though not all, of the articles written for extremist journals, embodied a very different concept of politics, of which perhaps the most that can be said is that it functioned as an interruption or suspension of politics in that previous sense. What might be construed there, say, as a politics of the possible, founded on national memory, tradition, and continuity with the past, was interrupted here by a politics of the impossible, turned towards forgetting, refusal, and violent disgust with the present, and investing its enthusiasm in the promise of the future. There were inevitably many areas of antagonism between these two concepts of politics; but the most revealing difference had probably to do with the question of political force or violence. To justify the recourse to force, for instance, Blanchot, according to the first of these two concepts, needed only to refer the question to the criterion of the national interest: force was legitimate to the extent that it was exercised in the resolute defence of peace and for the benefit of the continuing survival of the national community. But the Blanchot of the second of his political discourses was not bound by such prior principles as these. Revolutionary violence is necessarily in excess of any existing legitimising, constitutional framework; its very function from Blanchot's perspective was to introduce into politics a hiatus within which everything might be founded anew; it had therefore to repudiate all existing constitutions in order to perform or invent the future and thereby (re)constitute, so to speak, a new politics, one that, before being established, had to pass through the stage of being a violent refusal of politics, an anti-politics and a non-politics, therefore, since the goal it fixed for itself was not that of safeguarding the present, but summoning up the future by invoking the infinite potential of revolution.
At this point, Blanchot's political project, like that of others among his contemporaries, becomes as daring as it was full of risks. While still preserving – elsewhere – his resolute commitment to a politics of the present, he undertook to imagine the founding of a new politics. The responsibility for this new, revolutionary departure, based on revulsion at the debased character of the present, he entrusted in his writing, as did many others at the time, to youth, for only young people – no doubt like the 26-year-old Blanchot – were deemed to have the enthusiasm, appetite, grandeur, and potential for violence that might release them from the shackles attaching them to the present and the past. This revolution, in Blanchot's eyes, was a revolution whose spiritual and cultural potential far outstripped what, for instance, the Soviet Union had to offer, if only because the USSR by now had become, he argued, an enemy of radical revolution, a bureaucratic and totalitarian power dedicated to state tyranny. For Blanchot the whole purpose of revolution was to introduce into politics and into history a violent hiatus; it was to suspend the claims of possibility by affirming the impossible, and it was to restore to the nation the effervescence it had lost by its acceptance of a decaying regime of parliamentary democracy. That such an act of political refoundation should be presented as necessarily violent was no more than logic or honesty demanded; and the risks this involved were ones that, at any rate in his political writing, Blanchot at the time thought it essential to confront.
In 1933, Blanchot became _rédacteur en chef_ for a short-lived, polemical, independent nationalist daily called _Le Rempart,_ which ran from mid-April till late August 1933. Owned and edited by Paul Lévy, _Le Rempart_ was closely associated with the anti-government and fiercely anti-German policies of the maverick nationalist député Georges Mandel, who, it is said, supplied the paper with much of its inside information. Throughout its brief duration, Blanchot contributed to _Le Rempart_ a series of editorials and news items ranging widely over matters of internal and external policy. But two main motifs are typical of almost everything Blanchot published in the paper. First was a vehement insistence on the ever increasing threat to peace posed by the rise to power of Hitler. Blanchot was unyielding in his total and unconditional hostility to Hitler – whom he was to describe in 1935 as the 'representative of an unacceptable political doctrine' ('le représentant d'une doctrine politique inadmissible') – not only on the grounds that Hitler was pursuing the traditional objectives of German imperialism since Bismarck, but also because National-Socialism itself was no more than a 'perverted nationalism' ('un nationalisme perverti'), based on a 'mystical apotheosis of the nation' ('l'apothéose mystique de la nation'), owing more to religion than to politics and finding expression in many different manipulative and demagogic initiatives, including, wrote Blanchot as early as May 1933, 'the barbarous persecution of the Jews' ('les persécutions barbares contre les juifs'). The second key topic present in Blanchot's contributions to _Le Rempart_ was a deep-seated anger and disgust at the inadequacy of the response to the German threat made by France's centre-left government. And it was this in part that impelled Blanchot to call upon his readers to overthrow not only that government but the whole French parliamentary system of which it was an inevitable and natural product. Revolution here was no longer just desirable; it had become a necessity.
It was under that title, 'La Revolution nécessaire', that Blanchot published, in June 1933, one of the more programmatic and better-known pieces from _Le Rempart's_ brief existence. In it, in typically vitriolic vein, Blanchot lambasts the current parliamentary regime for its weakness, its sectarian divisions, and its irresponsibility in conspiring to squander the peace won at such cost in 1918. In these circumstances, he argued, France had only one chance of salvation; no hope could be placed in a corrupt parliamentary democracy, nor in materialistic capitalism, nor in dictatorial socialism, nor in communism, but only in spiritual revolution, or national revolution. Blanchot concluded:
The spiritual revolution, the national revolution is no longer an image nor a symbol. With every day that passes, events bring it nearer, each day makes it ever more necessary. And, bit by bit, they tell us what it will be: hard, bloody, unjust, our last chance of salvation [dure, sanglante, injuste, notre dernière chance de salut].
These are startling and, for some, shocking words. On one level, no doubt, their vehemence simply measures the degree of Blanchot's anger and impatience in 1933. But it is also important to understand the function of the violence Blanchot invokes here with such unproblematic zeal; for there are at least two reasons why Blanchot's article culminates in the way it does. First, it has to be recognised that Blanchot's words belong less to an established political discourse as such, than to a revolutionary, anti-political, even extra-political one. In writing words such as these, Blanchot's aim is not to describe but to prophesy, not to state but to perform; it is, in anticipation of real events, to effect already, by linguistic means, the political hiatus, the abeyance or vacancy of power that he identifies here as revolution. While that abeyance lasts, however, it can be subject to no constitutional principle or legal system; and this is why the revolution will necessarily be unjust, but also the reason why the purpose of radical revolution is to embody both a hope and a promise: as Blanchot had put it only a few lines before: 'our greatest hope today is that for a free nation, for the defence of man, for culture, will rise up the magnificent promise of revolution' ('notre plus grande espérance aujourd'hui, c'est que pour une nation libre, pour la défense de l'homme, pour les biens de l'esprit, se lève la promesse magnifique de la révolution').
There is also a second reason for this rhapsodic invocation of revolutionary violence; this is directly related to the question: what revolution? In _Le Rempart_ and elsewhere, Blanchot is quick to state what this revolution is _not:_ it is not, for instance, a Marxist transfer of power from one class to another based on 'the myth of collectivism' ('le mythe du collectivisme') and culminating, Soviet-style, in 'the monstrous dictatorship of the State' ('la monstrueuse dictature de l'État'); but neither is it a fascist or Hitlerite irrationalism imposing on its subjects the 'idol of all-powerful community' ('l'idole de la communauté toutepuissante') and whipping up their frenzied enthusiasm in the name of the 'mystical apotheosis of the nation' mentioned earlier. All that can be said about Blanchot's revolution of 1933 is that it will be animated by a violent, visceral refusal of all forms of servitude and of representation, and carried out in the name of national freedom, man, and culture. These words are of course empty ones; what they signal is that the ends of Blanchot's revolution, unlike the sham pseudo-affairs of Hitler or Stalin, cannot be prescribed in advance by political discourse. The revolution is nothing other than the violence of its own unmediated effervescence and the ferment of its own irreducible self-presence. The promise of revolution is precisely what it claims to be: a promise, and as such it cannot be converted into ready currency; for its only mode of existence lies in its futurity and its absence from any system of political representation. And this is why the appeal to violence looms so large in Blanchot's political journalism of the 1930s as a whole, for violence – or, more precisely, one particular kind of violence – is the only manifestation of futurity that political discourse will allow.
In February 1936, Blanchot's name appeared once more on the cover of an extremist nationalist monthly, edited by Thierry Maulnier and Jean de Fabrègues, both influential protagonists of the Jeune Droite movement some years earlier. This new journal was entitled _Combat,_ and it owed its launch in January 1936 to the tense political climate in France surrounding the forthcoming elections in April and May. With its extreme nationalism, its anti-materialism, its anti-communism, its anti-capitalism, its hostility to parliamentary democracy, its endorsement of social and spiritual revolution, and its intermittent anti-semitism, the journal, which by December 1936 was proudly claiming a thousand subscribers, has been described by some recent commentators as being largely fascist in intent, though _Combat_ itself did not explicitly identify itself as such. Inevitably, like many of the journals of the extreme right at the time, _Combat_ was based on an unstable coalition between contending personalities and policies; indeed, Robert Brasillach, who contributed regularly to the journal during the first fifteen months of its operation, and who was already by 1936 a committed fascist and anti-semite, was to complain how the involvement of various liberal intellectuals in _Combat,_ in his words, 'spoiled matters' and caused him to withdraw from the journal after March 1937.
Blanchot's own contribution to _Combat_ comprises a total of eight articles; published from February 1936 to December 1937, all belong to the first two of the journal's three-and-a-half years of existence. In many respects, Blanchot's articles were in much the same vein as those for _Le Rempart._ Violently polemical, full of paradox and rhetorical reversal, they pursue with undiminished vehemence many of the topics dealt with by Blanchot in his other journalism of the period, including the necessity of decisive action abroad to stem German expansion, and the call for radical, violent revolution at home. In 'La Guerre pour rien', for instance, of March 1936, Blanchot castigated the centre-left government for its illogicality in having concluded the previous May what he saw as a vacuous treaty of mutual assistance with the Soviet Union – a state with which France had few, if any, interests in common – with the result, in Blanchot's view, that France was being exposed, in defence of the Soviet Union, to the eventuality of a needless war in which France had no part.
Pursuing the same logic, the following month he lambasted the regime for its reliance – even after the crisis of 7 March – on a foreign policy based on continued support for the League of Nations. Such failings, Blanchot contended, were the inevitable consequence of a system of parliamentary government that lent greater weight to party intrigue than the national interest. By July, after the election of the Popular Front, in a notorious article entitled 'Le Terrorisme, méthode de salut public', Blanchot was arguing once more not only in favour of nationalist rebellion but violent revolution as well. This was the only course of action, he maintained, that was likely to reawaken the French nation from its slumber, and he defended the initiative by pointing to the collapse of all other forms of opposition to the government and by affirming that the interest of the many justified, on the part of the few, the resolute recourse to terrorist acts aimed at bringing down the regime. Was Blanchot's endorsement of violence more rhetorical than real? Was the purpose simply to shock, or was it to lend political support to the growing, menacing street violence of the period? This is difficult to say. Blanchot's invective, at any rate, fell on deaf ears, and by November 1937 he had concluded, in a piece entitled 'La France, nation à venir', that it was the French nation itself that had gone missing and lived on only as a ghostly memory and an ever vaguer hope. The future, it seemed, truly did now belong to the future, a future that was increasingly intangible or remote.
Between these two articles for _Combat,_ from January till October 1937, Blanchot was involved, alongside Maulnier and others, in another journalistic venture. This was _L'Insurgé,_ a weekly broadsheet that was funded by Jacques Lemaigre-Dubreuil, of the vegetable oil manufacturer Lesieur, but which had no titular _directeur_ or _rédacteur en chef,_ which was one reason for its often disparate and at times contradictory editorial stance. Much like _Combat,_ though with a more eager concentration on daily political events, _L'Insurgé_ defended a violently nationalist, anti-parliamentary, anti-capitalist, anti-communist, revolutionary syndicalist agenda; and much like _Combat,_ too, it printed, during its brief existence, its share of intermittent anti-semitic diatribes and cartoons, mostly directed at Léon Blum and his government. Blanchot's involvement in the paper was constant and protracted. Throughout the virtual entirety of its ten-month existence, he wrote a weekly book review, as well as contributing, during the paper's first six months, a regular political column, which suddenly all but stopped after the fall of Blum's government in late June.
In his political texts for _L'Insurgé_ Blanchot returns to many of the foreign policy issues discussed by him elsewhere; perhaps the only notable change, however, is in Blanchot's language, which, in _L'Insurgé,_ becomes even more violently reactive and vitriolic than ever before. The shift in tone is revealing. Gone is the peculiar optimism of _Le Rempart;_ gone, too, is the clear hope that by decisive action war might ultimately be averted. Their place is taken by an even more anguished and desperate polemic against France's parliamentary government. On one level the change is no doubt a direct measure – in Blanchot's estimation – of the ever-increasing probability of war, a war France seemed likely to lose; while on another it is a clear indication that, despite Blanchot's appeals for a violent national revolt to sweep away the government, there was no viable, alternative political force left in France that might fulfil that role. France, for Blanchot, was being precipitated headlong into war by the irresponsibility of **its own** elected government and there seemed little chance the threat might be avoided. It was as though France itself had now fallen into unknown depths of degradation and abjection, France itself that stood dishonoured and divorced from its past, France itself that had fallen victim to a sorry process of separation and decline. All that France now inspired was disgust; and disgust here, for Blanchot, like refusal, was still a revolutionary sentiment; the only hope it held out, however, was that of a violent rejection of what was, and there seemed little prospect of what might be in the future.
## A question of responsibility
I will put it more precisely: that Céline may have been a writer in the grip of a delusion does not make him antipathetic to me, but this delusion expressed itself in anti-semitism; delusion, here, excuses nothing; all anti-semitism is in the last resort a delusion, and anti-semitism, even if it is a delusion, remains _the capital error._
(Je dirai plus précisément: que Céline ait été un écrivain livré au délire ne me le rend pas antipathique, mais ce délire s'exprima par l'antisémitisme; le délire, ici, n'excuse rien; tout antisémitisme est finalement un délire, et l'antisémitisme, serait-il délirant, reste _la faute capitale._ )
Letter to Raymond Bellour (1966)
In recent years, as some readers will be aware, Blanchot has been severely taken to task by a number of commentators for his involvement in _Combat_ and _L'Insurgé;_ and the claim has repeatedly been made that in his political journalism of the period, in the unnecessarily coy words of Tzvetan Todorov, Blanchot 'assumed the role of spokesman for a certain anti-semitism'. The charge is of course a serious one; if proven, it might be felt to damage irreparably the credibility of what Blanchot has to say on the subject of Judaism and about the Holocaust in his texts of the 1960s and 1970s. But what is most startling of all about the accusation that Blanchot himself professed anti-semitic views in the 1930s is that, to date, no evidence of any real substance has ever been produced to support it. At most, the charge rests on a particular interpretation of no more than four articles published by Blanchot in 1936 and 1937. In two of these cases (the articles entitled 'Après le coup de force germanique' and 'Le Terrorisme, méthode de salut public' that appeared in _Combat_ for April and July 1936, respectively) what is at issue are two blunt, even crudely polemical references to the reckless impatience – as Blanchot saw it at the time – of those Jewish émigrés to whom Blanchot refers, with peremptory violence, as 'unbridled Jews' ('des Juifs déchaînés') who, in 1936, wanted to declare immediate war on Hitler, irrespective of the chances of success or the human cost of such a policy. Elsewhere, there are a limited number of passages in _L'Insurgé_ that it is possible to read as drawing in some measure on some of the standard themes of anti-semitic invective; this might be said to be the case, for instance when, in an article entitled, ironically, 'Blum, notre chance de salut . . .', Blanchot refers to Léon Blum – a Jew – as representing 'a backward ideology, a decrepit mentality, a foreign breed' ('une idéologie arriérée, une mentalité de vieillard, une race étrangère'), although, in the context in which the phrase appears, it is more immediately apparent that Blanchot is condemning Blum for his internationalism and support for the Soviet Union rather than attempting to make polemical capital out of his Judaism.
It is true, of course, that there is another, more general dimension to the charge of anti-semitism, which has to do with Blanchot's overall involvement in at least two publications, _Combat_ and _L'Insurgé,_ which, on occasion, if not systematically, did give a platform to anti-semitic views. To some extent this is hardly surprising. Anti-semitism was endemic in the French right and extreme right at the time, and was of course far from unknown in left-wing circles as well; and one might argue in mitigation of Blanchot that having to face the possibility of finding one's own name printed alongside views with which one might, at times, violently disagree was one of the risks the writer had to take if he wanted to be a political journalist at all. In contributing to _Combat_ and _L'Insurgé_ Blanchot was plainly running this risk; and there is every reason to suppose he was both aware of that fact and, up to a certain point, prepared to accept that, in the eyes of his readers, his own name might become associated, albeit indirectly, with some form of anti-semitic discourse. This risk of contagion from anti-semitism was by definition always possible, and though there is no evidence whatsoever that Blanchot was even remotely sympathetic to the anti-semitism of some of the articles that did appear in _L'Insurgé,_ it is clear that these did not discourage him from contributing to the paper in one way or another during the virtual entirety of its existence. Much the same could be said of Blanchot's texts for _Combat;_ and though he eventually withdrew from the magazine after December 1937, Blanchot must nevertheless be reckoned to be fully responsible before that date for allowing his own texts to appear in that magazine, as they frequently did, alongside those of an anti-semite such as Robert Brasillach. Indeed, Blanchot himself concedes as much when he explained in a letter to Diane Rubenstein in 1983 that one of the conditions he laid down for his participation in _Combat_ was the assurance that Brasillach would not be involved in it.
Anti-semitism, then, as Blanchot was well aware, was always present in some form or another in _Combat_ and _L'Insurgé,_ if not over his signature, then at least sometimes alongside it; and this is the reason, as Blanchot implicitly acknowledges in his 1984 letter to Laporte, that it is of course possible to read as indirectly anti-semitic both the two references to the Jews in his pieces for _Combat_ and the ambiguous language used in one or two further articles for _L'Insurgé._ To the extent that the threat of contagion by anti-semitism could never be contained, Blanchot must therefore be held responsible for acquiescing up to a certain point in the anti-semitic language intermittently used in those publications. The possible association of his name with some form of anti-semitic discourse seems therefore to have been a price that, up to a certain point, in 1936 and 1937, Blanchot the journalist was prepared to pay; and if this was the case, it is likely that it was because other political issues at the time seemed urgent enough to warrant the compromise.
But how far was Blanchot prepared to acquiesce in the anti-semitism of some of those around him? There is some evidence that when, in late October 1937, _L'Insurgé_ closed down, it was Blanchot who was chiefly responsible for terminating the venture, and that one of his key motives for doing so was the increasing prevalence of anti-semitism in the paper, as evidenced, in what turned out to be the final issue of _L'Insurgé,_ by one venomous, though hardly uncharacteristic diatribe by Charles Deleuze, the paper's sports correspondent, against Jean Zay, then Jewish-sports and education minister. For his part, Blanchot's own last contribution to the paper, printed in this final issue, was by way of a more radical leave-taking: what it announced was the need for revolutionaries of all political hues to dismantle the binary paradigm of current political action in order to affirm a common future owing nothing to the nationalism of the nation state or the communism of the Soviet Union. Blanchot's involvement in _Combat_ ended in much the same way some weeks later, in December 1937, with a piece which similarly put forward a proposal for a new politics based on a radical redefinition of nationalist as well as revolutionary thinking. Taken together, both pieces amounted to something of an abandonment of any kind of conventional politics, even of the kind espoused by the two publications in which the articles appeared. At any event, as far as the issue of anti-semitism was concerned, Blanchot had by then made his own priorities clear by returning to work alongside Paul Lévy as _rédacteur en chef_ of the satirical, independent nationalist weekly, _Auxécoutes._ 6
The charge of anti-semitism is not the only one to be levelled at Blanchot's pre-war journalism. Some have argued that what is in fact more troubling is the extent to which, in his articles for _Combat_ and _L'Insurgé,_ Blanchot endorses extremist acts of anti-parliamentary, anti-democratic violence; and it is certainly true that he does show little restraint in calling for the overthrow – by whatever means necessary – of the regime, or, for instance, on one occasion, drily envisaging the violent dispatch of those Communist Party leaders pronounced guilty of betraying both the revolution and their country. Perhaps most provocative of all in this respect – as the text of the article implicitly concedes – is the article entitled 'Le Terrorisme, méthode de salut public', published in _Combat_ in July 1936. In the article, written at a time when – Blanchot argued – parliamentary opposition to the Popular Front was ineffectual, and real political debate had become stifled beneath a hollow and oppressive consensus, Blanchot felt justified in appealing, parsimoniously, to a limited campaign of violence which would put an end to the politics of parliamentary complacency and demonstrate the urgent necessity for a more robust defence of France's national interest.
It is important to note that the purpose of such violence was self-evidently not to install fascist dictatorship in France nor to impose a small nationalist clique in government, but rather to interrupt a specific type of oppressive parliamentary politics and thus to allow the possibility of a different political future. So if Blanchot endorsed acts of terror in the piece, it was solely in order to deliver, as he put it, to the very people who might be ready to condemn them – the French electorate at large – the benefits of such an interruption. The purpose was to perform remedial surgery on the body politic. This is why the benefits of such violent action, according to Blanchot, would be primarily therapeutic: a dead or decaying political system would be destroyed, root and branch, and the life of the national community as such would be restored to proper vitality. To this extent, the violence Blanchot was invoking was not directed towards any seizure of power as such; its goal was more symbolic than real, and consisted more in the need to provide the nation with a vivid token or figure of the fundamental illegitimacy of the elected democratic government and the regime it represented. What was clearly at stake here, in Blanchot's view, as it had been in _Le Rempart,_ was the future of the nation as illuminated by the futurity of revolution; and it was no doubt a calculated move on the writer's part that the acts of terror he advocated in the article were justified – at least in Blanchot's presentation – by an unmistakable reference back to the memory of Danton, Robespierre, and St-Just, those earlier revolutionaries of 1793, who, in similar times of national crisis, set up a more famous Comité du Salut Public, or Committee of Public Safety, to defend the nation against the challenge from within and aggression from without, and eventually embarked on a campaign of terror whose purpose was the saving of the Jacobin revolution.
If Blanchot's article espouses terrorism, it was in order to effect a political or historical hiatus; indeed, such a hiatus was indispensable if politics was to be refashioned anew. But what kind of refashioning did Blanchot have in mind? Part of the answer comes in his farewell article to _Combat_ in December 1937, which appeared under the title 'On demande des dissidents'. The conclusion is as follows:
In reality what counts is not being above parties, but against them. It is not to take up the vulgar slogan: neither right nor left, but to be really against the right and against the left. It is evident in these _circumstances_ that the true form of dissidence is that which abandons one position without ceasing to observe the same hostility towards the opposite position or rather which abandons it in order to accentuate this hostility. A true communist dissident is the one who leaves communism, not in order to move closer to capitalist beliefs, but to define the true conditions of struggle against capitalism. In the same way, the true nationalist dissident is the one who neglects the traditional formulas of nationalism, not in order to move closer to internationalism but to combat internationalism in all its forms, including the economy and the nation itself. These two specimens of dissidence seem to us to be equally useful. But they also seem qually rare. Dissidents wanted.
Citing this passage, one influential commentator, Zeev Sternhell, describes it as 'a perfect definition of fascist thinking'; and Jeffrey Mehlman, quoting Sternhell, has taken it as conclusive proof of Blanchot's pre-war commitment to fascism. Summarising the passage, however, Sternhell significantly misconstrues Blanchot's position in the article by portraying it as proposing a political synthesis between left and right, an interpretation he glosses by adducing a quotation from Drieu La Rochelle's _Socialisme fasciste_ of 1934 to the effect that fascism is a politics not of equilibrium or oscillation between left and right, but of fusion.
This notion of fusion is supplied here, however, entirely by Sternhell, and is not to be found at all in Blanchot's actual text. Indeed, the very point of Blanchot's article lies in its rejection of all political parties – of whatever political colouration – as vehicles for radical political change. What it proposes, with some irony, is not fascism, therefore, but, to the extent that they rely on a principle of political representation, the destruction of all political parties in general. By describing Blanchot's article as fascist, it would seem that Sternhell is in some danger of entirely misrepresenting the politics of radical dissidence Blanchot attempts to put forward in the article. For what Blanchot envisages, and what marks, so to speak, his own entry into dissidence – and which is also the reason he specifically rejects the 'vulgar slogan' that gives Sternhell's book its title and main thesis – is not a transcending of right and left or a merging of right and left within some third force akin to the mystified nationalism of Nazi Germany. It is something more complex, that involves a radicalisation of the differences between left and right as well as a fundamental redefinition of all inherited forms of nationalism and communism.
One may doubt of course how and to what extent a programme such as this may be realisable at all in practical political terms; at any event it is clear that Blanchot's programme was designed not to consolidate the political synthesis between left and right that, according to Sternhell, is the hallmark of fascism or National-Socialism, but rather to shatter the totality which such an organic synthesis would aim to embody. The aim is not to fuse right with left, but rather to call into question the whole principle of political representation that would make such fusion possible. This, of course, is why rather than with the prospect of a new, unified political movement, Blanchot's article ends up not with one, but two disparate and irreconcilable forms of dissidence, and with an endorsement of the principle of critical rarefaction rather than of mass mobilisation. What Blanchot's parting manifesto sets out in this way, then, is not a recipe for fascism, as Sternhell maintains, but something more clearly resembling a destruction of representational politics in all of its received forms.
Any challenge on this scale to the stability of inherited political discourse, as Blanchot acknowledges, is necessarily founded on an act of radical, violent refusal. But violence, according to Blanchot, is inherent in politics; indeed, as I have argued, the two are for Blanchot like the two sides of the same coin. Admittedly, there are many different kinds of violence. But what is it, one may ask, about the therapeutically destructive violence of the kind recommended by Blanchot that distinguishes it from the violence being done to the nation by parliamentary democracy or from the violence of war itself? To these questions Blanchot provides in fact only a partial answer; for what justifies the revolutionary recourse to violence in Blanchot's texts is not so much the national interest, which necessarily has to be suspended by the hiatus in politics that the revolution effects, but rather the purity of the act of refusal itself (and this is why, in some of the later texts in _Combat_ and _L'Insurgé,_ Blanchot often seems to write not as a nationalist, but a violent anti-nationalist). 'Refusal tolerates no conditions,' Blanchot wrote in 1933, 'except that of never recanting.'
However, with regard to some of the texts it is supposed to govern, the purity of this refusal is arguably far less radical than Blanchot's project required. For as Jacques Derrida has convincingly argued, any act of foundational or refoundational violence is necessarily always already iterative. To the extent that the violence of refusal, by interrupting politics, offers itself up as the foundation and beginning for a new politics, it inevitably always already forms part of that new politics; the act of rebellion turns retrospectively into an act of foundation and the act of refusal becomes an act of assertion or endorsement. The implications of this reversal are serious ones for Blanchot's political project, for they show with what ease the violence of refusal may be inverted and take on the mantle of a very different type of violence, a violence committed not on behalf of the nation or its people but against it. What at one stage may be construed as a pure interruption of politics necessarily runs the risk of turning out to have been only an impure episode _in_ politics. So, however much the article on terrorism, for instance, may be read as a defence of the promise of the future against the oppressive, dictatorial violence of a sham parliamentary consensus, it is always possible – and necessary – for the piece to be read or misread as endorsing a mode of violence that, far from interrupting a violent politics, turns out to have already been contaminated by that which it sought to oppose. At the very point he was evidently seeking to break out of the cycle of the repressive politics of representation, there was every danger that Blanchot might end up, in his extremist journalism, appearing merely to propose a mirror image of that politics. There is – and was – always the risk that the endeavour to put a violent end to a politics of violence might itself be metamorphosed into just another sequence in the violence of that politics. And rather than being the opponent of oppressive violence, it is therefore always possible – and necessary – for the Blanchot of _Combat and L'Insurgé_ to be read in fact as simply another exponent of the self-same political violence.
A deep and worrying instability, then, affects the whole of Blanchot's revolutionary project; and this is surely one reason why that particular project was pursued by him only intermittently, in moments of great national complacency or crisis. The risks involved, however, were not contingent ones; for, as Blanchot recognised, there could be no politics at all without the necessary recourse to force or violence. So, while it is simplistic to describe Blanchot as an apologist for terrorism, it is evident that the unstable violence of _Combat_ and _L'Insurgé_ was far from an aberration on Blanchot's part. At any event, the uncontrollable political equivocation inherent in his texts for _Combat_ and _L'Insurgé_ marks a clear limit to Blanchot's revolutionary nationalist project of the 1930s. One symptom of that limit is the concept of the nation itself; and it is striking that throughout the 1930s, as he rejects all forms of political or parliamentary representation in the name of the promise of future revolution, Blanchot conceives of that future almost exclusively in terms of the self-presence of the nation, that is, beyond representation, the nation's self-identity and proximity to itself as political subject and origin. Throughout, it is the nation in arms that operates in Blanchot's discourse as the only acceptable or credible agent of political action and the ultimate source of legitimacy for any politics at all.
Though understandable in the context of continental Europe after the First World War, this deep attachment to the nation as the subject of history nonetheless severely dates and limits the whole of Blanchot's thinking in that period, and inscribes it within a certain political, ideological, and philosophical closure, a closure in relation to which the other – the future – is thought of only in terms of the concept of nation as source of identity. What this suggests is that the crucial blind spot of Blanchot's politics throughout the 1930s was the failure to think of the question of community except in terms of homogeneous nationhood, and to conceive of futurity not in relation to the alterity of writing, as he would do later, but by recourse to a concept that, despite Blanchot's strenuous efforts to make it appear otherwise, it is difficult to understand in any other way than 'primarily as tradition, order, certainty, truth, and every form of rootedness' (PA, 9; 2), in short, in terms of those very concepts that, gathered under the rubric of the relation to being, writing itself will later put into question.
Which is why, on the one hand, Blanchot is right to disown the texts from _Combat_ and _L'Insurgé_ that carry his name; and why, on the other, it is nothing less than a measure of Blanchot's philosophical rigour and fundamental probity that the question that comes to dominate the whole of his post-war thinking, with regard to literature, philosophy, and politics, beyond the concept of subject or nation, is the question of the Other and the question of community, and the question of that responsibility to the Other and for the Other which founds, but thereby also precedes, the law.
## The acts of the day (2)
[Drieu La Rochelle] renewed his previous offer to me [i.e. that Blanchot take over editorial responsibility for the _Nouvelle Revue française_ ]. I refused absolutely, saying: I could not ask writers to collaborate on a journal for which I myself would not wish to write. We went our separate ways, and I did not see him again, except once, in late '43 or early '44, when I came across him on the Champs Élysées. He told me: 'So it was you that was right. They really are too stupid' (I imagine he meant the occupying forces or the collaborators). I replied: it's not a matter of stupidity, but horror.
Letter to Jeffrey Mehlman (26 November 1979)
By early 1938, to all intents and purposes, Blanchot had abandoned the discourse of nationalist revolution he had been pursuing, at regular intervals, since the beginning of the decade; but he still continued to work as _rédacteur en chef_ for both the _Journal des débats and Aux écoutes._ At any event, 1938 was a watershed for other reasons; for late in September that year Chamberlain and Daladier, though they pretended differently to their electorates, set the final seal on the policy of appeasement of Hitler and, as Blanchot and others had predicted, launched Europe on the inevitable path to war. Admittedly, this view of Munich was not universally shared at the time; the agreement which forced Czechoslovakia to cede to Hitler the German-speaking territories along its western border with Germany was hailed by many on the French right as a resounding success. Maulnier, in _Combat,_ for instance, though still deeply critical of the parliamentary system as such, applauded the government for averting the immediate threat of war. The _Journal des débats,_ however, was much less sanguine. Though the paper was strongly in favour of closer Franco–British collaboration, and though it took the view that the issue of the Sudeten Germans could be resolved peacefully, it also recognised that Hitler remained a threat, and concluded that the most urgent priority was for the Western democracies to arm in expectation of war.
The response from _Aux écoutes,_ which had formerly been sympathetic to _Combat_ and had greeted its launch in January 1936 with some enthusiasm, was more outspoken still. One unsigned editorial, for 10 September 1938, complained that 'There have been too many gentle words when what was needed was robust language.' A fortnight later, Paul Lévy delivered the paper's considered verdict: in preparing to appease Hitler, he wrote, Britain and France had 'suffered the most serious defeat in their history'; elsewhere in the same issue, with evident approval, the paper cited Churchill's famous remark, pronounced before a group of Czech journalists a few days before, to the effect that 'France and Britain had to choose between war and shame. They chose shame, and they will get war, too.' It is true that this hostility to Munich on the part of Lévy and Blanchot was hardly unexpected, for it was entirely consistent with everything the two men had written throughout the whole of the preceding decade; no-one was surprised, therefore, least of all _Je suis partout,_ which could only think of accusing Lévy of being a war-mongerer.
Two-and-a-half years later, in the months leading up to France's defeat by the German army, Blanchot was still writing for the _Journal des débats_ and _Aux écoutes._ But when defeat came, the _Journal des débats,_ like many other titles, withdrew to Clermont-Ferrand. For his part, François de Wendel, writing to de Nalèche in a letter of 24 July 1940, made plain his wish in the circumstances to see the paper die a natural, honourable death, and withdrew his financial backing; the _directeur,_ however, for private reasons, decided to keep the paper going and, after a brief interval, it began appearing again from offices in Clermont-Ferrand, from where, with financial support from the Vichy government, which it received together with all the other official Vichy newspapers, and in spite of the disapproval of de Wendel, it continued to appear regularly till mid-August 1944. There is evidence that Blanchot too made strenuous representations to persuade de Nalèche to suspend publication; and though he persisted up until the fall of France in writing for the _Journal des débats,_ he was increasingly to find his editorials censored or refused, leaving Blanchot little option than to withdraw at that point from all political involvement in the paper.
With _Aux écoutes,_ the situation was initially much the same. It too retreated to Clermont-Ferrand and, by mid-July, began appearing again; however, instead of the (Jewish) name of Lévy, it now carried that of Blanchot (as _directeur_ ) on its masthead, and Blanchot continued in this role as Lévy's proxy till the end of the month, bringing out two further editions. Only three more issues appeared, however, in August, without Blanchot's name; and shortly afterwards the paper folded entirely, subject to a censorship order from Laval. During the brief interregnum of late July, Blanchot, while paying lip service to the new Vichy ideology of national revolution, seems to have attempted to gain for the paper enough latitude to allow it to explain to its readers, for instance, how, a week before the armistice, high-level discussions had taken place between Churchill and the French Prime Minister Paul Reynaud concerning a plan for the creation of a single Franco–British Union under which France would be declared an integral part of Britain. Had it been accepted, the plan would have guaranteed what the paper referred to, approvingly, as the 'relentless pursuit of the struggle' ('la continuation acharnée de la lutte') against the occupier. However, despite the support of Reynaud, the initiative failed, and France, instead, alone amongst the occupied countries of Europe, embarked upon a policy of overt collaboration with the enemy.
There is little public record of Blanchot's actions during the Occupation; but what is known suggests that throughout that period Blanchot followed a dual strategy of endeavouring on the one hand to maintain some kind of public role for himself within the literary and cultural sphere, while on the other safeguarding, as much as was possible, his intellectual independence. The result was a series of short-lived initiatives on Blanchot's part, such as his brief involvement early in 1941 with the Jeune France cultural association, and also, in 1942, his readiness, temporarily, to support Paulhan in his scheme to regain editorial control over the _Nouvelle Revue française,_ which since the defeat had been under the collaborationist editorship of Drieu La Rochelle, a ploy which at the very least had the effect of destabilising the journal and hastening its demise by July 1943. In both cases, it seems, though determined in the first instance to explore the possibilities for cultural intervention, Blanchot quickly withdrew his co-operation as soon as those limits became too constricting. And when, to a large degree for personal financial reasons, it seems, he agreed to resume writing for the _Journal des débats,_ which he did for three-and-a-half years, from 16 April 1941 until 17 August 1944, eight days before the Liberation, it was to contribute only a regular weekly book review to the paper and not to endorse its generally enthusiastic support for the politics of Vichy. In December 1943, some fifty of these pieces, many extensively edited, together with three other articles and an unpublished introduction, and at least one article which had previously fallen foul of Vichy censorship, were brought together, at the suggestion of Gaston Gallimard, in what was to be Blanchot's first volume of critical essays, _FauxPas,_ a title chosen, Blanchot later wrote, to reflect the fact that _FauxPas_ really was a false move.
## A temptation?
Literature contemplates itself in revolution, it finds its justification in it, and if it has been called Terror, this is because its ideal is indeed that moment in history when 'life endures death and maintains itself in it' in order to secure from it the possibility and truth of speech. This is the 'question' that seeks to realise itself in literature, and which is its being.
La Littérature et le droit à la mort' ( _PF,_ 311; 321–2)
Writing in _Critique,_ in November 1947, under the title 'Le Règne animal de l'esprit' ('The Spiritual Animal Kingdom'), borrowed for the occasion from Hegel's _Phenomenology of Spirit,_ Blanchot speaks of a temptation ( _PF,_ 308; 318). The temptation arises, Blanchot puts it, when literature and history appear to coincide in one absolute, seemingly apocalyptic interval, when the suspended event of writing finds itself mirrored and justified in the pure advent of revolutionary Terror. At such a moment, writes Blanchot, freedom and death are as one, the possibility of death becomes inseparable from the possibility of life, and absolute negation speaks with the same authority as absolute affirmation. Between Blanchot the revolutionary journalist during the 1930s and Blanchot the literary critic of the post-war years, the parallels, as Blanchot himself seems to imply, are compelling; and it may seem as though, obliquely and in retrospect, Blanchot is seeking to legitimise some of his pre-war political decisions by associating with them a belief in the radical, contestatory force of literature itself.
But there is here a crucial difference. Though writing may find its justification in revolutionary violence, its goal is in fact not justification, but justice; and though it may find itself mirrored in the spectacle of the Terror, its aim is not self-reflection, but disappearance. Literature is irreducible to the sovereignty that comes from absolute power; the only injunction to which it falls subject is that of its own effacement, absence, and radical exteriority to power and possibility. If literature, for Blanchot, belongs to destruction, it is because its role is not to assert authority, but to contest all authority, including its own. Unlike the dangerously ambiguous texts of 1936 and 1937, always liable to find themselves re-inscribed within a discourse reliant on the political self-presence of the nation, literature here retains a reserve and distance that renders it irreducible to the political simplification that threatens and ultimately disqualifies those early political texts. For if it is true that literature, like death, seizes itself paroxystically within the immediacy of a paramount moment of absolute freedom, as the rhetoric of terror would suggest, it is also the case that literature, like death, is an encounter with the limitless impossibility of that moment, and with the lack of power that leaves writing forever suspended as an absent event that can never properly come to be, since the only domain it occupies is the domain of worklessness, impotence, and disaster.
To compare Blanchot's extremist texts of the 1930s with the texts on literature written after the war is to be struck irresistibly by apparent similarity and irreducible difference. Here as there, a principle of representation or mediation is being challenged in the name of the effervescent immediacy of a terrible and sublime moment. But where the pre-war political writings fall victim to the contagious effects of an unmediated nationalism that, in its very appeal to the future self-presence of the nation, is in fact, though it may not itself be aware of this, just another discourse of political representation, literature turns aside from the temptation of such a moment to resist the lure of immediacy. Literature, so to speak, maintains a discretion that Blanchot's political texts, in their impatience to realise themselves, were never able or willing to display, which was why they were always vulnerable to compromise, liable to be contaminated by the very violence they claimed to challenge. For its part, literature refuses that contamination; and what Blanchot's account of literature and terror in 1947 shows is the extent to which, if Blanchot's polemical texts of 1936 and 1937 failed, it was not because they were extreme, but because they were in a sense never extreme enough; it was not because they refused a certain legitimate politics, but because their refusal of that politics was in fact never pure enough, and because the interval they sought to effect in politics was never separate enough from the politics of representation for Blanchot to envisage a politics no longer founded on the concept of nationhood and on the future self-presence of the nation as the model par excellence of community as such.
Between 1937 and 1947, much that defies words had somehow taken place: France's defeat and Occupation, the Second World War, the resistance, the discovery of the death camps, the beginnings of the cold war. During that period, Blanchot seems to have spent the bulk of his time writing: writing the weekly essays that were to be partially collected in _FauxPas_ and the numerous articles that established his reputation as a critic shortly after the war; writing too the novels _Thomas l'Obscur, Aminadab,_ and _Le Très-Haut,_ not to forget the thick manuscript confiscated in 1944 that Blanchot mentions in _L'Instant de ma mort_ ( _IM,_ 15). This discrepancy between public script and private scribbling may appear immense. But it is also somewhat deceptive. For what Blanchot arguably came to understand in the course of this protracted exposure to writing, from the late 1930s onwards, was not only the extent to which the political discourse to which he, like many other thinkers and writers of the non-conformist right, had appealed in his pre-war revolutionary texts had been covertly, indeed primarily, an aesthetic one; but also, more importantly, that this grounding of the political in aesthetics had relied on an ultimately untenable conception of the work of art as a form of deferred self-presence.
For such had largely been the view of literature put forward by Blanchot in the many book reviews he contributed throughout 1937 to _L'Insurgé,_ the main purpose of which, he wrote at the time, was to deal with such works as were distinguished by 'a certain indifference to futile things, of which there are many, a certain pride which forces them to refuse common resources, a certain ruthlessness towards themselves and towards ourselves'. As these lines suggest, the literary critical standpoint adopted by Blanchot at the time was primarily an aesthetics of refusal, an aesthetics that privileged in the work of art its ability to contest all forms of social and political representation by the immediacy of its own self-presence. The position was of course not merely an aesthetic one; much the same view informed Blanchot's political texts for _Combat_ and _L'Insurgé,_ which were also devoted to the attempt to discredit all current forms of political and social representation in order to affirm, beyond representation, the possibilities and potential of France's future self-presence as a nation. In both the aesthetic and the more properly political sphere, the message then was largely the same; and there is an evident logic in the fact that Blanchot, in his literary critical essays at the time, may be found affirming in the work of art the voluntaristic purity of a future act of politico-aesthetic refoundation.
But in Blanchot's case, aesthetic ideology and the practice of literature soon parted company. As Blanchot was quickly to discover in writing his own early novels, to rely on a view of literature that privileges foundation over the responsiveness to alterity and asserts the identity of a subject over the contestation that comes from the Other is to cling to an impoverished, and in the end untenable conception of art. It is to prefer an illusory idea of art's self-sufficiency and autonomy to the radical questions raised by the possibility – or impossibility – of the work of art as such; and it is to repress the realisation, as Blanchot himself was subsequently to demonstrate, that art and literature are necessarily deeply inimical to all notions of cultural foundation or beginning, all conceptions of artistic truth or essence. The years between 1937 and 1947 were plainly eventful ones. But perhaps the most decisive factor of all that makes the Blanchot of 1947 no longer the Blanchot of 1937 is simply that, in the interval, the writer had begun at last to take seriously his own experience as a novelist, and to abandon the lingering commitment to the project of (re)founding, in literature, the self-presence of the French nation. The shift was no doubt not a chance event, but the result of much self-criticism and scrutiny; it was also not a sudden one, and the complex temporality of the change in direction that ensued may be seen to have left many marks on Blanchot's work, not least on the differing speed with which Blanchot the literary critic and Blanchot the political thinker caught up with Blanchot the writer of fiction.
Some ten years after 'La Littérature et le droit à la mort', in 1958, as he prepared to join the struggle against de Gaulle and the pro-colonialist military who had brought him to power, Blanchot found himself embarking on what was to be a new phase in his political itinerary. There seems little doubt that the writer was then brought to reflect upon the lessons to be learned from his own political past. As always, Blanchot did so obliquely, in an essay first published in August that year, by considering the fate of two other thinkers whose writings raise complex political questions of their own: Nietzsche and Heidegger. Blanchot begins by reminding the reader how the texts of the first, however implausibly, re-interpreted, re-edited, and compiled anew by his family and heirs, were made to serve as philosophical or ideological justification for some of the most repellent policies of the Third Reich. But how, asks Blanchot, was such falsification possible? Blanchot's judgement is as scrupulous as it is unforgiving. Clearly, Nietzsche and the Third Reich had nothing in common; but Nietzsche, having at one point announced, in spite of himself, a forthcoming systematic work of philosophy, which he did not write, has to bear some responsibility for the fact that others were able, on the basis of that statement, to concoct such a work as 'The Will to Power' and, in so doing, give to Nietzsche, posthumously and no doubt against his will, the role he was made to play in the 1930s and 1940s with such dire political consequences.
Reading Nietzsche's fragments, then, Blanchot is aware that meaning is always contextual; and that by cynically manipulating the context of his writings, Nietzsche's heirs were able to ascribe to him views he surely would have found offensive. But while this shows that statements cannot be taken in isolation and are decisively bound to their context, it also confirms that contexts can and do change, and that texts whose meaning becomes contaminated in that process have perhaps some responsibility to bear for the possibility of such changes. Assuredly, Blanchot concedes, the principle may be reversed; and if the thought of Nietzsche is dangerous, he writes in _L'Écriture du désastre,_ this may simply prove that danger is an essential element in the movement of thought itself and that, without such risks, thinking itself is no more than bland conformism ( _ED,_ 189; 123). However, Blanchot's analysis, if it applies to Nietzsche, must also be seen to apply to Blanchot; and it is at any event worth noting that in the principle of contamination which Blanchot invokes to describe Nietzsche's responsibility for the misreading of his own texts, there is at work precisely the same logic of re-inscription as that which affects Blanchot's own earlier political writings, a logic that dictates that statements made in one context can, unpredictably and uncontrollably, always be reread and misread otherwise than how they were intended, and even turn into something more nearly resembling their opposite. The responsibility for thought, Blanchot concludes, is infinite; as in the case of Nietzsche, it is from the very outset always at least twofold, since the responsibility of thought is responsibility not only to the contestatory violence of thought but also for the incalculable implications of that violence.
In a long footnote to the article, which was extensively reworked prior to republication in 1969, Blanchot turns to Heidegger's reading of Nietzsche, first pursued in Freiburg in a series of lectures and seminars during the period from 1936 to 1939. On one level, those lectures, as Blanchot freely admits, represent an important moment in Heidegger's endeavour to distance himself from the official philosophers of National-Socialism and from the regime he had first supported with such enthusiasm. Even so, there remains the question of how and why Heidegger had been willing to put his own philosophical writing to work in the service of a racist, totalitarian political system. Blanchot is attentive to the nuances of Heidegger's reading of Nietzsche; but as with Nietzsche, he is also rigorously unforgiving. Indeed, he concludes by observing how, in his public endorsement of Hitler shortly before the fateful plebiscite of November 1933, which took Germany out of the League of Nations and began the long road to war, Heidegger was guilty of committing his own philosophical language – the language of _Sein und Zeit_ – to the cause of the Nazi state and of thus compromising his own writing by the politics it was used to defend. The contamination of the one by the other remains irredeemable; and for having thus put his own philosophical probity into the balance, writes Blanchot, Heidegger's own discourse will henceforth always remain under suspicion ( _EI,_ 210; 451).
At this point, some readers of Blanchot have remembered Blanchot's own troubled political past. Are there parallels to be drawn between Heidegger and Blanchot? Should one apply to Blanchot the strictures whose principle he clearly and convincingly articulates with regard to Heidegger? The question is without doubt a legitimate one. But to confuse Blanchot with Heidegger is also to neglect a number of essential, absolutely irreducible differences: Blanchot never endorsed Hitler politically, never held high office in a state institution with the explicit support of the Nazi Party, never threw the weight of his philosophy behind the inner truth of the Nazi movement, and consistently refused the temptation of silence when it came to responding to the question of the Holocaust. And from 1938 onwards he began the lengthy process of dismantling his own uncritical reliance on what Philippe Lacoue-Labarthe has termed national-aestheticism, and in articulating that turn to the ethics of alterity that was to be the main focus of all Blanchot's work in the years to come.
This is not to say Blanchot's early political writings do not run substantial risks; nor is it to suggest that Blanchot's readers should not, like Blanchot, be rigorous and unforgiving in questioning the implications of those early political texts. Respect for Blanchot's later writing demands nothing less.
# 2: The (im)possibility of literature
## Founding fictions
Literature is perhaps essentially (but neither solely nor overtly) power of contestation: contestation of established authority, contestation of that which is (and the fact of being), contestation of language and the forms of literary language, lastly contestation of itself as power.
'Les Grands Réducteurs' (A, 80)
Blanchot's two early novels – _Thomas l'Obscur_ (1941) and _Aminadab_ (1942) – are arguably among some of the most compelling yet obscure of all modern literary texts. More than fifty years after publication, their resistance to interpretation remains undiminished. This inscrutability, however, is not merely the product of unfamiliar contextual factors, nor does it derive simply from the systematic withholding of information by the author or narrator. Indeed, contrary to repute, these early novels of Blanchot are neither full of esoteric allusions nor obsessed with silence; they are, rather, outrageously garrulous and self-explanatory, and given to endless digressions designed to elucidate the import and purpose of the stories they tell. In that process, however, each of these texts persistently reveals itself to be more than the sum of its parts, with the result that what is most enigmatic of all is the extent to which, as one reads, the local and specific mysteries posed by Blanchot's writing give way to the larger, more intractable puzzle of the nature of the discourse that has somehow engendered them. The only secret of such texts, one might say, is the secret of their own possibility; and the main question they raise, therefore, is the question of their foundation as works of literature.
Admittedly, one of the most obvious reasons for the obscurity of Blanchot's fiction is the sceptical disdain the texts themselves adopt towards conventional norms of novelistic verisimilitude or narrative coherence. These are texts that, unlike others, make little attempt to conceal the violence with which they interrupt what the reader is enjoined to consider, at least implicitly, as a previously stable state of affairs. The technique is one already exploited to challenging effect in Blanchot's two short stories 'L'Idylle' and 'Le Dernier Mot', written in 1936 and 1935 respectively. In the first of these, the tale begins with the sudden arrival of a foreigner within an (unnamed) city, without it ever being explained from where he has come or why, and what purpose his visit might have; yet this does not prevent the narrative from recounting his brief life in the city, oppressed by the terrible – and terribly ambiguous – spectacle of a sham life of happiness, until he is eventually punished for attempting to escape from the city and dies as a consequence, only then to enjoy the hollow privilege of a grand funeral, after which everything returns to the dubious normality of domestic routine. 'Le Dernier Mot', in turn, commences in a similarly abrupt way, with an unexplained exchange of conversation between anonymous interlocutors about some presumably metatextual password, or watchword, of indeterminate character ('le mot d'ordre', _AC,_ 57; 39) that is somehow no longer available; pursuing this half-hearted theme into something more akin to allegory or myth, the text keeps going up to the point where, like the Tower of Babel to which Blanchot's text refers in conclusion, everything in the story collapses into ruins, bringing to an end, among other things, this particular story.
But the uncompromising anti-realism even of stories such as these appears tame and unexceptionable when set alongside the outlandishness of the first version of _Thomas l'Obscur._ Part philosophical inquiry, part _Bildungsroman,_ part inner experience, part self-reflexive _mise-en-abyme,_ part Pentecostal fable, part apocalyptic rhapsody, part ironic romance, part stylistic _tour de force,_ this is a work that displays few, if any, of the standard features of conventional novels of the time. True, it retains a residual narrative structure that follows a vaguely circular pattern; and while at the outset the protagonist, Thomas, is found slipping into the sea, embarking, in the mist, on what is described, no doubt self-referentially, as a new and unfamiliar itinerary on his part, and which leaves him bereft of all previous points of orientation, so he is discovered at the end of the novel by the side of the sea once again, hurling himself, at least metaphorically, into what the book describes in closing as 'a flood of crude images' ( _TO1,_ 232; cf _TO2,_ 137; 117). Between these two rhyming moments, each detailing the protagonist's self-possessed immersion into the watery deep, little of consequence seems to have taken place, except for the dispatching of Thomas, this 'keeper of the impossible' ( _TO1,_ 231; cf _TO2,_ 136; 116), along with the reader, on a bizarre sequence of fantastical textual experiences and encounters, involving love, death, destruction, and recreation, and which, while they may remind the reader of much that belongs to the literary past, endeavour nonetheless to affirm their absolute autonomy from conventional expectations and hold within themselves the irreducible mystery of their unfolding.
Published one year after _Thomas l'Obscur, Aminadab_ presents what seems to be the next stage in Thomas's story; it opens in the broad light of day with a scene in which the protagonist, after passing by an unassuming shop and resisting the invitation to go in, then, for no clear reason, responds more positively to an enigmatic gesture – invitation? sign of friendship? farewell? or warning? – from a woman waving from one of the windows of the building opposite, somewhat resembling a run-down boarding house. In a decision that is lightly taken but heavy with consequences, Thomas enters this second building. But by launching him – like the reader – on an endless journey through the various rooms and floors of this allegorical establishment, about which nothing seems certain, and on an ultimately fruitless quest for the woman who appeared to signal to him, Thomas's initial resolve proves to be the source of all his subsequent misfortunes, misfortunes which are brought to an end – though not necessarily a conclusion – in the closing scene when Thomas at last submits to what is described as a necessary, if still enigmatic fate. All the while, in an explicit echo of the book's opening, the light in the room is shown to be slowly fading, as though to suggest that Thomas's life and the narrative have become synonymous and that the falling silent of the one serves to announce the extinction of the other.
Such textual moments of beginning and ending are evidently performative as well as descriptive; by invoking the fictional possibility of a particular finite world, they ground Blanchot's own narratives within the space of that possibility. By asserting their own freedom, so to speak, Blanchot's novels legislate for their own autonomy as literary artefacts. But in order to do so, and however strenuously they endeavour to carry out this project, they are necessarily faced with a fundamental paradox. Beginnings and endings can only be thought in their own absence. Any act of foundation, be it an act of aesthetic framing, of ontological grounding, or ethical injunction, has to confront the necessary circumstance of its own prior absence; indeed, for an act of foundation to be possible at all, it must first be preceded by an absence of foundation. Such prior absence is thereby a necessary condition of any possibility of foundation; however, to the extent that it insists, as it necessarily must, on the belatedness and fragility of any such moment of foundation, any such condition of possibility is also a condition of impossibility. The laying of foundations, as it were, is an activity that may take place only within a bottomless abyss; and it is the bottomless abyss that constitutes the only reliable foundation, albeit a foundation that is always already an absence of foundation. Such an absence of foundation is not just an ironic flourish: it is an essential condition that, to the extent that it enables the act of foundation, by that very token also disables it. The price of the possibility of foundation, one may say, is the impossibility of that foundation.
In this perspective, it is clear that none of Blanchot's narrative beginnings or endings functions unproblematically. Without exception, all these moments of narrative inception or closure prove in fact to be remarkably indeterminate; and, throughout, the self-certainty of the initial founding moment is tempered – if not indeed radically challenged – by the awareness that the world founded in this way fails to embody itself as an inaugural, fully transparent presence. Instead, the act of foundation remains irreducibly opaque; and it is as though the fictional world Blanchot's writing opens up already has its distant origin elsewhere, in an unspoken and largely irretrievable anteriority that resists narrative exposition and veils the clarity of the text in impenetrable but indeterminate obscurity. At times, as in the case of the perplexing sign from the third floor to which Thomas rashly responds in _Aminadab,_ not knowing what it signifies, nor even if it signifies at all, the possibility of beginning, in the form of the presumed meaning of that gesture, becomes synonymous with the paradoxical impossibility of beginning that the gesture's failure to communicate seems to imply. And as a result, like Blanchot's other incipits, the opening of _Aminadab_ remains suspended on the threshold of its own necessary violence, which the writer makes little attempt to mitigate or transform. The violence of the origin is never effaced; to the dismay, puzzlement, frustration, and ultimate fascination of the reader, the beginning of the text is displayed as pure interruption, as a hiatus in whatever is held to precede the beginning.
Much the same is true of Blanchot's endings. In _Thomas l'Obscur,_ by way of bringing to a close an extraordinary scene of burgeoning vernal creation, albeit one that by the end undergoes a catastrophic fall into solitude, darkness, and tawdry imagery, Blanchot has his own protagonist cast himself into the waves of these images 'as if', writes Blanchot, 'shame had begun for him' ( _TO1,_ 232; cf _TO2,_ 137; 117). Put in these terms, the ending deceives as well as disappoints; Blanchot's prose, like Thomas himself, seemingly lingers on, surveying everything that has just taken place, delaying the closure the narrative already seems to have reached, and thus outliving itself, so to speak, echoing what the 1950 version terms the 'empty word of Thomas' ( _TO2,_ 135; 116). The reference to shame in this context, though, is revealing; what it recalls is Levinas's early contention that 'what manifests itself in shame is . . . precisely the fact of being chained to oneself, the radical impossibility of fleeing from oneself in order to hide from oneself, the irredeemable presence of the self to itself'. The ending of _Thomas l'Obscur,_ then, is not the ending of Thomas; rather it is like an ultimate condemnation of both Thomas and the literary discourse that carries his name, the final proof, so to speak, that the only possible end to writing is in fact the interminable impossibility of such ending.
Such paradoxes are also much in evidence in _Aminadab._ There, at the very moment Thomas is busy receiving from Dom, his erstwhile and future companion, the news that it is time for them to leave, he hesitates, prolonging once again the questions that have never ceased to be asked in pages gone by, yet never answered; even at this late stage, as Thomas hovers between life and death, silence and speech, the threat of darkness and the yearning for light, he seems still to be hoping for a response to the question that will allow him to throw some illumination upon all that has occurred ( _AB,_ 227). Unabashed by the information (of indeterminable purport and unverifiable accuracy) given by the young woman Lucie, to the effect that he was not the one to whom the inaugural sign was intended in the first place ( _AB,_ 220), and told that at any event, as far as the boarding house was concerned, according to Dom ( _AB,_ 212), he would have been better advised going down to the basement rather than up to the floors, Thomas brings the novel to an end with an unanswered, still almost preliminary question, which he addresses to Lucie, unless it is to the light itself (assuming the two to be different): 'Who are you?' ( _AB,_ 227). The question is one Anne, in _Thomas l'Obscur,_ had already asked of Thomas, only to regret having done so a moment later; and as the Thomas of _Aminadab_ echoes it once again, it becomes evident that this novel too, like its predecessor, can only end on a sentence that is more a repetition that a resolution, more a rehearsal than a moment of closure. The failure to reach conclusion, in both texts, is a token of the extent to which both novels, as they seek to convert the violence of beginning or ending into the self-sufficient law of their own necessity, cannot escape the realisation of their own irredeemable excess; they remain traversed to the very end by the enigma of everything that, by necessity, cannot in fact be spoken and which yet seemingly constitutes the only reason for their existence.
Blanchot's two novels, then, begin without beginning, and end without ending. Their possibility as autonomous, self-bound artefacts is traversed by the impossibility of such self-legislation; and what is at one moment a condition of possibility is at another a condition of impossibility. Beginnings are a function of the absence of beginnings, and endings an interval imposed on the endlessness of language; each and every point that claims self-presence is shown here to be already inhabited by the other from which it derives, on which it depends, and which therefore has the potential to disable its claim to be what it is. For all that, Blanchot's novels do not lapse into anti-mimetic playfulness. Rather, a new kind of monstrous logic – an ante-logic as well as an anti-logic – takes over within Blanchot's writing, one whose effects are signalled from the outset, in _Thomas l'Obscur,_ by a series of strange and mind-numbing paradoxes and inversions by which clarity turns to obscurity, plenitude to void, presence to absence, decision to paralysis, and freedom to constraint. Standing for instance on the shore again, Thomas looks on intently, with increased acuity, at the distant sight of another swimmer – perhaps his own doppelgänger – appearing and disappearing off the horizon, and is filled with a feeling of boundless freedom, somewhat akin to the freedom of beginning itself. But already the euphoria of his telepathic intimacy with the far-off swimmer becomes mingled with discomfort and distress; and it is as though boundless freedom has suddenly turned into something more nearly resembling opaque compulsion, in much the same way that, in the pages that follow, it is by closing his eyes that Thomas sees more deeply into the darkness, and while refusing to walk that he finds himself imperiously being propelled along.
From this point on in Blanchot's text, writing becomes inseparable from what might be described as an experience of nothingness. Necessarily, however, _Thomas l'Obscur_ challenges the adequacy of such terms. Like the inner experience of Georges Bataille, Blanchot's project can be described only in antithetical or paradoxical terms; and just as, like that of Bataille, Thomas's experience is an experience that is not an experience, so nothing in Blanchot's novel is not nothing. Or so it would seem from what is one of the novel's most famous passages, Thomas's encounter with the night. Blanchot writes:
The night soon appeared murkier and more terrible to him than any other night, as though it had really come forth from a wound in thought that could no longer be thought, thought treated ironically as an object by something other than thought. It was night itself. Images that made up its darkness flooded over him, and his body, changed into a demonic mind, sought to picture them to itself. He could see nothing, and, far from being overwhelmed by this, he made this absence of vision the culminating point of his gaze. Useless for seeing, his eye took on extraordinary proportions, grew beyond measure, and, stretching to the horizon, let the night penetrate to its centre in order to create for itself an iris. So it was, by virtue of this emptiness, that his gaze and the object of his gaze mingled together. Not only did the eye which could see nothing apprehend something, it apprehended the cause of its vision. It saw as an object that which prevented it from seeing.
( _TO1,_ 14–15; cf _TO2,_ 17–18; 14–15)
This night – night 'as such' – is not the luminous night of the concept nor of dialectics, but the night before the concept and before dialectics; it is the night that precedes the night, the night that is itself impenetrable to the thought of night but which is the necessary prior condition of that thought. As such, its status is uncontrollably paradoxical. In so far as it is irreducible to dialectical reason and logical negation, Thomas's night before night cannot be thought as night, nor can it be addressed as a prior cause of night; it remains foreign to all forms of identity as well as all narrative chronology. This night is not a night, therefore, nor does it occur before night; it is the night thinkable only as that which is external to the concept of night, and, to that extent, may be described only as a radical impossibility of night. And yet because this is the night which does precede the night, it is necessarily also that which enables the night – both word and thing – to appear as such; and to that degree it must logically already hold within itself the possibility of the appearance of night.
The night 'itself', then, in Blanchot's description, features here as both possibility and impossibility; indeed possibility at one level is entirely dependent on impossibility at another. This explains the peculiar experience undergone by Thomas as he peers into the depths of night. For as he gazes on, in the darkness, with his unseeing eye, at his own inability to see, all he can make out, in the lack of any visible object, are the circumstances that might make vision possible. However, those circumstances do not themselves constitute an object of vision; what they do is rather to present Thomas with the spectacle of an absence of vision, with the result that Thomas, blinded by this absence of sight, is faced, so to speak, with the vision of the circumstances that prevent him from seeing anything at all. So as Thomas stares on, unseeing, into the night, what he finds himself looking at is not just the absence of a particular visible object, nor simply the absence of visibility as such, but, more radically, the presence of that darkest of other nights (as Blanchot terms it elsewhere) which is the only truly nocturnal night of all, the night that is impenetrable to all visibility and synonymous with an originary impossibility of vision which is nothing other than the prior condition of visibility as such.
Throughout _Thomas l'Obscur,_ possibility in general is retraced, as here, to the originary impossibility that must be excluded or negated in order for possibility to be constituted as such. Blanchot dramatises this here with regard to vision; elsewhere he does so in relation to language. The logic is much the same. For the night before night that confronts Thomas in the darkness is both irreducible to the word night and yet its necessary condition of possibility. The night 'itself' is both prior to the word night and the word's most original inspiration; yet while the other, darker night, to which the word night pays silent homage, cannot itself be incorporated within language, the – impossible – attempt to name that night before night is nevertheless, in Blanchot's account, what constitutes the purpose of literature as such. According to this Orphic logic, by which possibility is only ever a function of prior impossibility, and an object only ever grasped at the very moment of its ineradicable loss, the unnamable night before night constitutes the only (im)proper object (or absence of object) that literature may claim as its very own, even though for it to do so is for literature to be reminded all the while that what counts as its own is also that which is irrepressibly alien to it. The paradox is one that has radical implications for Blanchot's fiction. On the one hand, the night 'itself', by tracing the limit that passes between language and that which is irreducible to it, is evidently – as mystery or enigma – what enables narrative to take place at all; on the other hand, however, in so far as it cannot be enclosed within the bounds of narrative, such a night necessarily also transgresses those bounds, disables narrative, and throws the very possibility of narrative into question. Narrative is no longer in control of itself, and is left instead, as in the case of _Thomas l'Obscur,_ to grapple with the impossibility of naming that constitutes it as narrative. Such self-reflexive turns are crucial to Blanchot's fiction, as the example of _Thomas l'Obscur_ shows; and just as the impossibility of vision reveals what is arguably most extreme and fundamental about sight itself, so the most radical and far-reaching of novels for Blanchot are those that pursue narrative to their originating point, which is that of their own dissolution.
As all readers of Blanchot's fiction know, the impact of this bizarre yet rigorous logic on Blanchot's work is persistent and dramatic. What it does to _Thomas l'Obscur,_ for instance, is to turn the novel into a monstrous and heterogeneous infusion of oxymoron, aporia, and paradox; and it is from this potent verbal brew – more central than one might think to mainstream literary thinking in the twentieth century – that flow the many baffling figures and tropes that litter Blanchot's text. Witness, for instance, the eerie spectacle of Thomas, this very first of human beings ( _TO1,_ 133), who, in the depths of night, finds himself
grappling, in dreadful solitude, with something absolutely inaccessible, absolutely foreign to all being, absolutely inconceivable, something which he could say did not exist, yet which filled him with terror and which he could sense lurking within the space of his own solitude.
( _TO1,_ 24; cf _TO2,_ 30; 27)
Thomas, we read, is a character who, were he to be seen with his own eyes, 'would have appeared in the form of an entity that existed only in so far as it did not exist entirely' ( _TO1,_ 41); who is thus 'really dead and yet at the same time excluded from the reality of death' ( _TO1,_ 48; cf _TO2,_ 40; 36), and is 'in every circumstance anonymous and entirely without history' ( _TO1,_ 64; cf _TO2,_ 56; 55).
Thomas it is, the novel adds, who is 'in each human act the dead person that simultaneously renders it possible and impossible' ( _TO1,_ 212; cf _TO2,_ 106; 93); who is at bottom a figure 'whose true essence was not to be, on whom the fact of not being conferred neither a diminution nor an aspiration to being, who could not be considered as lacking but as superabundant' ( _TO1,_ 216); and who, to close, avers: 'I am indeed the origin of that which has no origin' ( _TO1,_ 226; cf _TO2,_ 129; 108). And all these attributes are not limited to Thomas; many of them are found, too, in the figure of Anne, the novel's Eurydice, who, at the moment of her own death, is one of the few to realise to what extent 'death, destroying everything, could also destroy the possibility of annihilation' ( _TO1,_ 204; cf _TO2,_ 96; 85). As she does so, she demonstrates clearly how the preoccupation with dying, throughout Blanchot's work, is less in the service of negativity or nihilism than of radical passion, irreducible extremity, and boundless affirmation.
In the language of Heidegger, the nothing ('das Nichts') that Blanchot stages in this way in _Thomas l'Obscur_ is of course not nothing, because the question it raises, as Heidegger maintains, is the question of Being itself. One is reminded here of Blanchot's early review of _La Nausée,_ in which he described Sartre's novel as aspiring – with only partial success – to the status of 'a kind of novel of Being' ('une sorte de roman de l'être') and went on to recall the importance of Heidegger for a proper assessment of the challenges facing modern art. Texts such as the recently translated 'What is Metaphysics?' of 1929, claimed Blanchot, 'allow one to gauge the force and creative will of [Heidegger's] thinking which, in the infinite contest between laws, understanding and chance, offers art a new point of view from which to contemplate its necessity'. This endorsement of Heidegger in the review explains why Blanchot was somewhat critical of what he describes as Sartre's undue concessions to traditional psychological analysis and conventional plotting. Blanchot was evidently both too much and too little of a philosopher to be tempted by the project for an existential anthropology implicit in _La Nausée,_ and when _Thomas l'Obscur_ appeared, three years after _La Nausée,_ it accordingly made only minimal concessions to the demand for either character or plot. But despite the profound differences between them, _La Nausée and Thomas l'Obscur_ share a similar objective, which is that of pursuing within fiction a project that has all the signs of a foundational philosophical enterprise. But where Sartre was ultimately to rely on the self-transparency and self-reflexive freedom of the narrating subject, Blanchot was to pursue some of the implications of the Heideggerian 'nothing' in a more radical way, one that, as we have seen, begins gravely to compromise the foundational possibilities of language and literature as such.
Blanchot's first move in _Thomas l'Obscur_ is in fact not to adhere to the economy of Heideggerian Being at all, but rather to transpose or translate the Heideggerian principle of the 'nothing' into the rather different language of Emmanuel Levinas, more particularly into the Levinasian topos of the _il y a,_ that phrase signifying 'there is', which, for Levinas, replaces the originary generosity of the Heideggerian gift of Being (as expressed in the German phrase: _es gibt_ ) with the horror and anonymity of being. This use of the phrase _il y a_ was not exclusive to Levinas, however, for it was cited in much the same way by Blanchot in his story 'Le Dernier Mot', where it is identified – by the story's first-person narrator – as constituting 'no doubt' the last word mentioned in the title ( _AC,_ 66; 45). But if for Blanchot's story _il y a_ is that last word, the narrator explains, it is because 'the last word cannot be a word, nor the absence of a word, nor anything other than a word' ( _AC,_ 77; 53). In this respect, the _il y a_ is a strangely ambiguous moment of ontological foundation; on the one hand, since the _il y a_ is logically prior to all propositions, including negative ones, and cannot itself be negated, it necessarily serves as a moment of foundation for being; but, on the other hand, as Levinas insists, if the _il y_ _a_ necessarily precedes the constitution of any world whatever, be it a fictional or non-fictional one, it follows that the _il y a_ poses an implicit, ineliminable challenge to the autonomy and stability of that world. Considered from the point of view of the _il y a,_ the world derives its possibility from prior impossibility. So if the _il y a_ is a founding moment, what it founds is in fact an absence of world; what it inscribes is a series of always anterior repetitive traces whose origin is always lost. Accordingly, when the _il y a_ does reveal itself, Levinas argues, it does so in the form of eternal wakefulness; and what it dramatises is not self-presence nor originary giving, but the necessary intrication of being and otherness and the irreducible dependence of all giving on what is a prior and thus infinitely irredeemable debt.
To the extent that it becomes synonymous with the necessary failure of all beginnings and endings, the _il y a_ demonstrates what Levinas describes in _De l'existence à l'existant_ as 'the impossibility of death, the universality of existence even in its annihilation'. But if absence is always already a form of presence, presence itself must be at some risk of forfeiting the very proximity and nearness at hand that make it what it is. Presence is always already marked by its own alteration and by the inherent possibility of non-presence. Ineluctable alterity threatens from the outset; and being becomes inseparable here from its own boundless absence and impossibility of foundation. The implications of _Thomas l'Obscur_ are much the same. What Blanchot's novel suggests is that, if the attempt is made in the language of fiction to grasp being for itself, so to speak, then being turns out necessarily to be always already inhabited – and thereby dissolved – by alterity, loss of origin, infinite repetition, absence, nothingness, the impossibility both of beginning and of ending. By a dizzying and outrageous paradox, being comes to function like an instantiation of its own groundless absence. As far as Blanchot's writing is concerned, the result is both poverty and plenitude, nothingness and infinity, a radical emptiness that, by interrupting the all, finds itself facing the fact of its own boundless exteriority. Literature, here, oscillates uncontrollably between that which is and that which refuses to be; what writing founds as possibility at one moment, it dismantles as impossibility at another.
If _Thomas l'Obscur_ is a novel of the impenetrability of night 'itself', _Aminadab,_ the second novel involving Blanchot's Thomas, is a novel of unremitting daylight. This is what is stressed in the novel's opening sentence, which reads: 'It was broad daylight' ('Il faisait grand jour', _AB,_ 7). Compared to _Thomas l'Obscur, Aminadab_ is a novel of mimetic representations and artefacts. But from the outset, this mimetic impetus in the novel is sharply and ironically curtailed. This becomes clear very soon after the beginning of the story, when Thomas, having boldly entered the boarding house, is invited to select a room for himself. Shortly after, in typically uncanny fashion, Thomas finds himself in a place containing a strange painting contraption, comprising, among other things, a complicated assemblage of pullies and ropes, several stools, an easel, a mirror, a spotlight, a sundial, and a collection of palettes dripping paint on to the floor. Thomas sits down on one of the stools, and, looking at the unfinished painting, realises that much of the detail of the room he is in has already been faithfully copied on to the canvas, including the very stool on which he is sitting. Eventually the painter completes Thomas's portrait; and though the painter seems most satisfied with his work, Thomas for his part can find no resemblance at all between his own face and the portrait; and it is as though Thomas, his features aged and blurred by the painting, has been incorporated into the picture under false pretences, as an effaced simulacrum of himself.
In the course of this description, the narrator remarks: 'It was difficult to tell what was there to do the painting and what was there to be painted' ('Il était difficile de savoir ce qui dans cet ensemble devait servir à être peint ou à peindre') ( _AB,_ 19). A similar reversibility or indecision applies to Blanchot's description as well; for over and above the story of Thomas, it is apparent that what is being depicted in this scene of mimetic reproduction is the process of mimetic reproduction as such. On Blanchot's part, the description of the studio is plainly a novelistic _mise-en-abyme,_ whose purpose is to dramatise and test out whatever claim might be made by this novel – and all other novels in general – to be adhering to some form of mimetic realism. But Blanchot's text rapidly shows that the theory of mimesis, like much else in the novel, rests on a fundamental paradox. The only perfect imitation of an object would be in fact that object itself. Such redoubling would produce, however, not resemblance, but repetition. Before any object may be perceived as an exact, mimetic copy of another, a margin of alterity must first have differentiated object from copy in order that the relation of resemblance between the two may be instituted at all. But in so far as resemblance is founded in differentiation, mimesis in the true sense proves to be an impossibility. Mimesis, it transpires, only functions at all in so far as the imitation is an imperfect one. Indeed, it may be said that it is only because of the impossibility of mimetic identity in the true sense that mimesis is possible at all. But, by that token, the relationship between object and copy falls subject to a logic that necessarily escapes the concept of mimesis, with the consequence that any theory of mimesis is left with the insurmountable problem of accounting for the existence of the mimetic process itself, since that process, though it may produce objects that are judged to be mimetically accurate, is not itself an imitation. An ineliminable residue remains, which necessarily exceeds mimesis; like the question of the inaugural light that illuminates – without clarifying – the world depicted in _Aminadab_ itself, something radically other than the fictional world itself continually survives, intelligible only as an obscure enigma defying translation into anything other than itself.
Throughout _Aminadab,_ the reader is faced with ineluctable, aporetical impasses of this sort. The result, as far as the text is concerned, is radical uninterpretability. Moreover, Blanchot's text is itself largely made up of conjecture, commentary, interpretation and gloss, attributed to a variety of characters, all claiming to make sense of the internal workings of the enigmatic world of the boarding house in which the novel is set. Paradoxically, the sheer number of these often contradictory internal commentaries makes interpretation not easier, but harder. Indeed, one of the ways in which the novel resists interpretation is by already incorporating within its textual fabric numerous other, already available mythological, philosophical, religious, literary, or rhetorical discourses. On one level, these serve to exacerbate the apparent interpretability of Blanchot's narrative; at the same time, by overlaying the narrative with a palimpsest of rival interpretations or commentary, they also obliterate the narrative and leave the reader, as it were, with little hermeneutic business of his or her own. Yet these internal discourses, though they may exhaust the text, are themselves never exhausted by it, if only because, to a large degree, they are what constitute the text as such. The bizarre result is a writing in which everything already seems to possess somewhere in the novel its own implicit or explicit interpretation, except for the process of commentary itself, which remains uninterpreted and, one might add, boundlessly uninterpretable, with the result that what seems lucid at one stage becomes quite opaque at another. The inhabitants of this boarding house, Thomas reflects at one point, in free indirect speech, are many of them liars and rogues ( _AB,_ 33). What could be more lucid? We are reminded, however, elsewhere in the text, that this is only Thomas's – negligent and entirely short-sighted? – interpretation of what is going on, in which case that thought too is just as likely to be completely false. So as far as the reader is concerned, all Thomas's opinion might be held to show is how far Thomas, too, is nothing but a liar and a rogue.
But it would be wrong to conclude from this kind of hermeneutic paralysis that all in the novel is equally mendacious or arbitrary. To do so would be to ape Thomas's own fondness for unreliable generalisation and fall, with the protagonist, into the trap of negligence and misrepresentation (assuming for a moment – _concesso non dato_! – that the house is not indeed full of liars and rogues). In place of this kind of lazy relativism, what Blanchot's text suggests instead – and this is the particular inflection that _Aminadab_ gives to the question of the _il y a_ – is that when it comes to interpretation it is always already too late. Reading, writing, speaking are weighty obligations, from which it is impossible to withdraw, that remain incumbent on all of us, whether we wish it or not. Like Thomas during most of the early part of the novel, to go anywhere at all is always to be accompanied by the irksome, inevitable presence of that other who, in the form of the character Dom – whose name evokes the Lord to whom Thomas is forced continually to play the part of reluctant bondsman – is literally chained to him as his very own special, alien companion. Just as Thomas himself is a sometimes unwilling reader of the discourses he encounters on his journey through the house, which he cannot elude, so the reader, too, so long as he or she is a reader, cannot avoid becoming entangled in this text, enchained to a narrative and a controversy that may be without resolution but are not without making their own urgent, ineluctable appeal for the reader's attention.
But within or beneath the text, disturbing its foundations, just as the basement – according to Dom – runs under the house, offering a possible space of freedom far from the light of day, there is always the lure of something else, an outside, which seems to offer hope of escape. To suggest this finally is perhaps the purpose of the title of _Aminadab._ The name, unexplained until some fourteen pages before the end, arrives suddenly, referring only to some perhaps legendary gatekeeper whose role it is to guard the great gate leading below the house to the basement. But the existence of such a figure, the reader is told, is probably a myth, and the great gate only a wooden fence; in any case, if Thomas were to leave by the basement, all he might find would be the impossibility of dying and the cruel torment of rebeginning. Better, says Dom, to stay inside the house. But to do so would of course be not to find rest, but to remain, like Aminadab the mythical gatekeeper, hovering between house and basement, inside and out, fiction and truth, legend and reality, always embarked, like at least one of his earlier, Biblical namesakes, on the endless journey out of Egypt, together with the other children of Israel, towards an ever distant promised land. This novel, like history, fails to conclude, and the light the novel throws on such events remains obstinately opaque.
Throughout the whole of both _Aminadab_ and _Thomas l'Obscur,_ the question arises of how these texts are to be read. Neither book allows one to entertain for long the possibility of reading them as conventional realist texts; and the reader is obviously tempted instead to attempt to decode, if not perhaps _Thomas l'Obscur,_ then at least _Aminadab,_ as Sartre was the first to propose, as an allegory. But if _Aminadab_ is an allegory, it is not because – as Sartre contends – the novel is simply a coded, fantastical representation of the metaphysical dereliction of modern man. If _Aminadab_ is readable at all as a second-degree, allegorical narrative, one that incorporates within itself, so to speak, the perpetual possibility of another, figurative layer of interpretation or legibility, it is because, by its very resistance to interpretation, the novel implicitly appeals at every turning to the possibility of there being another text, another interpretation, another commentary able to frame the text and somehow efface its startling indeterminacies.
However, nothing in Blanchot's novel allows itself to be deciphered in this way. That other, parallel discourse into which the novel might be translated is nowhere accessible, and the logic of allegory, ironically conceding its own failure in the form of the novel's refusal to be paraphrased or translated except into the text it always already is, serves here only to reaffirm the impenetrable clarity and radiant obscurity of Blanchot's writing. In Paul de Man's phrase, allegory supplies here only the story of an 'impossibility of reading'; it operates not as a trope of self-reflexive interiorisation or self-representation but as a figure of exposure to what Foucault, glossing one of Blanchot's later essays on Kafka, terms the outside, that alterity that can never be included within conceptual thought because, like the night before night of _Thomas l'Obscur,_ it is what constitutes the unspeakable condition of thought itself. If it is an allegory, _Aminadab_ is at best an ironic or impossible one, and the phantom, parallel, other reading that the novel may be thought to propose is in fact nothing other than the novel itself perpetually repeated and continually reread, with the result that the relationship between the text and its own reading is a relationship of radical strangeness, both infinite proximity and infinite distance.
As the story of Aminadab the gatekeeper serves to illustrate, there is in Blanchot's novels no exit from the labyrinth of language and fiction. Reading here is possession of the reader by the text, not interiorisation of the text by the reader. Early in _Thomas l'Obscur,_ Blanchot dramatises this strange reversal. As Thomas sits at his desk, reading, he is devoured by the text before him as though by a praying mantis; and in this singular sexual aggression, it is not just a case of Thomas reading the text, but rather of the text reading him. The Thomas of _Aminadab,_ as his decisions too become the object of conjecture and interpretation by the other inhabitants of the boarding house, undergoes a similar mutation; and the effect, in both fictions, is not just to portray Thomas as an obstinate reader, but to transform the reader into a double of Thomas, unable to withdraw from the strangely circular, empty obligation of always being a reader of words continually read by words. Reading here is a process of endless submission to language. Indeed, it transpires that even to interrupt reading is still to be caught in the act of reading words as well as being read by them. The only option that remains is to repeat, recite, reinvent words with a view not to reasserting the apparent presence of words to themselves, but by responding to their fundamental absence from themselves. Which is what Blanchot did in 1950 by signing a second version of _Thomas l'Obscur,_ one that, by virtue of the many changes it makes to the 1941 text, itself has the status of a detailed reading or interpretation of the earlier work, but which, for all that, is also a restatement of the previous version in its fundamental difference from itself. In this rereading of _Thomas l'Obscur_ by Blanchot there is thus no progress towards clarification, no lifting of ambiguity, no added profundity. All the new version of the novel does is to add further to the burden of reading. By so doing, what it shows is that works such as _Thomas l'Obscur_ are always other than what they seem, and that if they seem to belong to impenetrable night, to the impossibility of beginning and the impossibility of ending, they respond to that condition with the unrelenting persistence of an act of limitless affirmation.
In this way, as they tirelessly endeavour to found their own possibility as fictions, Blanchot's novels are impelled by the logic of that ambition to contest without end the possibility of all such acts of temporary foundation. Literature is not an entity coincident with itself and possessed of the purity of self-presence, as Blanchot's prewar or wartime criticism might lead one to believe; it is more like a constant movement of disappearance and effacement that is perpetually put into crisis by its own lack of stability, identity, or definition. In the end, what Blanchot's novels demonstrate is the fundamental aporia and impossible possibility of literature 'as such'. For literature founds itself only upon the abyss; and whatever literature founds, therefore, including literature itself, is necessarily without foundation. The implication, too, is that the proper name Thomas, that doubter and twin to his own other self, is only ever a mark of the writer's anonymity, a lingering trace of the original namelessness that for Blanchot the novelist represents both the appeal and necessity of literature as such.
## How is literature possible?
On one of its sides, poetry makes sense, but on another it unmakes it. It distances speech, and if it restores it to us, it is from afar. It binds dangerously the possibility of speaking to an impossibility that becomes, as it were, its very condition. It allows us to write 'I am unhappy', but this _initial_ expression of unhappiness, by depriving us of already-formed, familiar, and reliable thoughts, exposes us to an experience full of risks, and more than that, to a silent drone, a stammering, whose perfection does not prevent us from acknowledging it as a lack.
'Le Paradoxe d'Aytré' ( _PF,_ 75–6; 71)
Blanchot's first published texts as a literary critic date from his earliest years as a political journalist. The two functions, in Blanchot's case, often overlapped; they did so most extensively in 1937, when, for almost the entirety of its ten-month existence, Blanchot contributed to _L'Insurgé,_ alongside his better-known political pieces, a regular literary column in which he reviewed, among others, a host of recent books by such authors as Julien Benda, Georges Bernanos, Drieu La Rochelle, Aldous Huxley, Marcel Jouhandeau, Thomas Mann, François Mauriac, Charles Maurras, Victor Serge, and Virginia Woolf. But while Blanchot's work as a critic began in a space determined by political activity, it would be wrong to conclude that his criticism was in a narrow sense political. The truth is rather the reverse. It was Blanchot's consistent belief during that early period that in the contest between the literary and the political it was literature that enjoyed foundational status, and not politics. For Blanchot, only literature – not politics – was in a position to tell the truth about the political as such, while politics, in its dealings with literature, was destined always to encounter the limits of its power and authority. Which is to say that, alongside many of his nationalist contemporaries, Blanchot's own critical writing of the late 1930s and early 1940s – and this was arguably to remain the case until the end of the war – was profoundly indebted to what Philippe Lacoue-Labarthe, commenting on Heidegger's withdrawal from active politics after his disastrous endorsement of Hitler and the National-Socialist state, has termed national-aestheticism, the idea, that is, that art, to the extent that it is what founds all community, is what founds the political too.
Blanchot's own pre-war version of national-aestheticism, however, unlike that of other, more conservative writers, was bound to a project that was explicitly (re)foundational and to that extent profoundly and resolutely contestatory. This is clearly demonstrated not only in Blanchot's literary journalism, but by his early fiction as well. In 'L'Idylle' of 1936, for instance, the city – the _polis_ – seems able at best to provide its citizens with the terrible (and suspect) spectacle of happiness; literature alone, in the form of Blanchot's own narrative, might be seen to embody Alexandre Akim's irreducible yearning for freedom and his desire to quit the city for good. First articulated in a work of fiction, this principle of the supremacy of art as both political and metapolitical foundation soon became a central theme in Blanchot's literary journalism. So when in January 1937 he entitled his programmatic opening article for _L'Insurgé:_ 'De la révolution à la littérature' ('From Revolution to Literature'), this was plainly not a call for writers to put their work in the service of (national) revolution, but more an appeal to the boundless revolutionary potential of literature as such, which he proposed to measure, significantly enough, by the capacity of works of literature not only to found a higher order of reality, but to give rise to other art works too. Far more important, at any event, than the oppositional sympathies of the individual writer, Blanchot asserted, was
the oppositional force [la force d'opposition] expressed in the work of art itself and measured by its capacity to do away with other works or abolish a section of ordinary reality, as well as the capacity to summon up new works, as forceful or more forceful than itself, or to determine a higher reality.
Blanchot's activities as a literary critic for _L'Insurgé,_ like that paper itself, were to be short-lived, and it was not until April 1941, in a very different political context, that Blanchot embarked on what was to be his second sustained period of activity as a literary journalist: the three-and-a-half years spent at the _Journal des débats_ as one of the paper's regular reviewers of current novels and essays. In that role, Blanchot was clearly intent on pursuing the literary critical project he had begun before the war. Indeed, despite the changed circumstances brought about by the Occupation and the setting up of the Vichy government, Blanchot's writing about literature, though it had to be voiced more discreetly than ever before, had lost none of its fundamentally oppositional impetus. From the outset, Blanchot's concerns in the texts for the _Journal des débats_ were at least threefold. First, in the abeyance of all normal public and political life, Blanchot was clearly committed to the notion that, in so far as it represented a veiled sign of national (and nationalist) resistance, cultural activity under the Occupation should carry on in spite of all; and it was in these terms that he justified in his own column, albeit discreetly, his own preoccupation with contemporary cultural matters. There were other reasons, too, which had to do with Blanchot's own work as a novelist; and it is more than likely that, in such pieces as 'Mallarmé et l'art du roman' ( _FP,_ 189–96; _BR,_ 43–8), and in the essay on Lautréamont published in Maulnier's _Revue française des idées et des œuvres_ in April 1940 (FP, 197–202), Blanchot was seeking to provide his readers with something like an implicit commentary on his own novelistic practice. More generally, too, Blanchot in his articles was clearly aiming to articulate in detail, and for the first time, a coherent aesthetic discourse of his own, one that combined a vigorous opposition to literary realism with a defence of what Blanchot thematises initially as the foundational autonomy and purity of the literary work.
Throughout the 1940s the name Mallarmé was one of Blanchot's most important points of literary reference for such a foundational view of literature, and in the articles for the _Journal des débats_ the legacy of the poet looms large. But at the time, this choice of Mallarmé as the founding poet of French modernity was scarcely a neutral one. For the monarchist right, in the words of Maurras, Mallarmé was no more than a belated, backward romantic ('un romantique attardé'), a symbol of politico-cultural as well as poetical anarchy, and one of the sources of the sickness afflicting the modern French nation. As far as anti-semitic, collaborationist circles were concerned, the diagnosis was much the same; for Robert Brasillach, writing in June 1940, shortly before France's capitulation to Germany, Mallarmé was an alien presence at the heart of the nation ('un étranger dans sa nation', as Brasillach put it), while Drieu La Rochelle, the following year, for his part declared Mallarmé's poetry a masterpiece of onanism and inversion, and attacked the poet for sinking to the lowest depths of physical, i.e. sexual, degradation ('le dernier degré de la déchéance du corps, la déchéance sexuelle'). Admittedly, hostility of this kind was not universal. Some years before, the influential _Nouvelle Revue française_ critic, Albert Thibaudet, though a staunch supporter of Maurras, had written more warmly, if still somewhat faintly, of Mallarmé's heroically self-denying decadent preciosity, while Thierry Maulnier, in his _Introduction à la poésie française_ of 1939, similarly paid tribute to Mallarmé's gifts as a verbal alchemist.
For Blanchot, however, Mallarmé was far from the poet of the coruscating – or barbaric – word. Initially, Blanchot's own position was more closely aligned With that of Paul Valéry, for whom the lesson of Mallarmé lay in the realisation that poetry, though it might use the same words as everyday discourse, functioned according to entirely different demands and constraints, and that if prose-writing resembled walking, as Valéry was wont to explain, poetry itself was more like dancing. In Mallarmé's case, the poet's exemplary lucidity and total mastery over the resources of language gave his texts an unequalled purity of purpose and effect. Blanchot's account, though in many respects similar, radicalises this argument. Mallarmé in his poetry, Blanchot contends, raises the status of the work of art to that of an absolute, pure creation that obeys its own necessity and embodies its own law. All true literature, he argues, aspires to the status of myth and founds itself in the effort to overcome chance; it eschews all facile concessions to the expectations of the audience, including the demand for conventional realism or verisimilitude; it offers a radical challenge to the degradation, mediocrity, and inauthenticity of everyday life, taking instead as its essential theme the communal destiny of a national people in particular and of humanity in general. At this point, as we shall see, Blanchot's account of Mallarmé takes on a markedly Heideggerian tone; and there are a number of evident similarities between the role ascribed to Mallarmé in these texts of the early 1940s and that played by Hölderlin in Heidegger's writings on poetry (or 'Dichtung') during the previous decade.
But of all Blanchot's wartime articles perhaps the most significant were those devoted to Jean Paulhan's critical essay _Les Fleurs de Tarbes_ in the autumn of 1941. In that book Paulhan had set about diagnosing what he described as a characteristically modern sickness or neurosis: the claim, originating with nineteenth-century romanticism, that rhetorical commonplaces – clichés, readymade expressions, banal statements and received platitudes as well as established literary rules and conventions – were an impediment to original thought and should be purged from all genuine literary work. Because of its belief in the essential purity of thought and its profound mistrust, even hatred of words, Paulhan described this attitude as a form of Terror; and in _Les Fleurs de Tarbes,_ by demonstrating its internal contradictions, what he sought to do was to dispel the effects of Terror and rehabilitate what he referred to as Rhetoric, that now discredited, classical view of the relation between language and thought that held that linguistic commonplaces were a necessary and integral part of all communication and expression, including literature itself. While Paulhan's rhetorician was willing to affirm that literature was a matter of adhering to proven, well-established norms, his terroristic opposite number refused all rhetoric in the name of the purity of thought and the authenticity of original poetic expression.
'In short,' as Michael Syrotinski explains, 'Terror is the precedence of thought over language, and Rhetoric the priority of language over thought.' Paulhan in his book has few difficulties in illustrating the inconsistencies in the terrorist case. Not only did Terror display an impoverished understanding of the poetic potential of commonplaces, he argued, but it was self-contradictory even in its own terms; indeed the terrorist who, by waging war on words, sought by his words to establish the priority of thought over language was at best only likely to win a Pyrrhic victory. In the debate between thought and language, Paulhan concluded, all was a question of perspective; and what may be proof of the priority of thought over language for the writer of a text, may be for the reader of the self-same text exactly the reverse, and vice versa. But as Paulhan pursued this argument, he slowly became aware that the opposition between Terror and Rhetoric was less clear-cut than it first appeared; and in the end it is even as though the terrorist protesting against his dependence on a given state of language has more in common with the rhetorician than the rhetorician himself. Despite their seeming incompatibility, then, it transpires that Terror and Rhetoric have far more in common than seemed to be the case; and Paulhan, even as he promotes the hope of a (rhetorical) reconciliation between Rhetoric and Terror, is also left in conclusion quizzically undermining the foundations of his own argument. What had begun as a crusade against Terror ends up following a vicious circle of its own making; for Rhetoric rediscovered, as Paulhan concedes, is not far removed from Terrorist purism reaffirmed.
The main source of the peculiar self-defeating or self-questioning paradoxes running through _Les Fleurs de Tarbes_ is not hard to find. For the crux of Paulhan's book turns on the peculiar logic of linguistic commonplaces themselves; and it is this that lies behind not only the inconsistencies of Terror that Paulhan exposes, but also the aporias and self-inverting paradoxes of his own demonstration. For all commonplaces, as Paulhan points out, are by nature uncontrollably duplicitous, which is why they provide ammunition for both rhetorician and terrorist alike. To the extent that they can be repeated in any and every circumstance, they are proof of the precedence of language over thought; but in so far as they may be used in each case as though for the very first time, they are evidence of the inventiveness of human thought in its encounter with language. Being common, their purpose is to enable communication; but being unoriginal, they just as easily defeat it; they are a prime instance of the poverty and insufficiency of everyday speech; but they are sometimes full of the most surprisingly florid turns of phrase or figures. Commonplaces, it would seem, can function simultaneously in each of these contradictory registers. As such, they follow a peculiar logic of hyperbolic paralysis reminiscent of that described by Lacoue-Labarthe in his reading of Hölderlin's fragmentary texts on tragedy. Indeed in Paulhan, as in Hölderlin, it is as though sense relations themselves have begun to behave according to a strangely aporetic pseudo-dialectic of hyperbolic intensification that, far from accrediting the mediating power and all-embracing unity of sense itself, instead has the effect of ruining the possibility of mediation at all. Statements are inverted into their opposites without transition, and paradox becomes the only viable figure of thought at all. In the case of Paulhan's commonplace, the effects are impressive: the more transparent a commonplace seems, the more impenetrable to thought it becomes; the more communicative it is, the more uncommunicative it proves; the more unambiguous it is, so the more ambiguous it turns out. And vice versa.
Reading Paulhan, however, Blanchot is not primarily concerned to assess the validity of the critic's case. As with the work of other writers, Blanchot's strategy is twofold: it is first to generalise the essential proposition of the text to its fullest possible extent; and second to radicalise that argument to the point where it becomes consumed by its own impossibility. It is immediately evident in this context that Paulhan's Terror is not a simple romantic aberration. Rather, it follows from Blanchot's own post-symbolist view of literature as an act of pure, absolute creation that what Paulhan is addressing as Terror is in fact the essence of literature as such. The so-called sickness of words, Blanchot rejoins, is inseparable from the health of words; and any authentic literature, as it seeks to found itself within the terms of its own freedom, affirms itself necessarily, according to the logic of Paulhan's Terror, as a radical refusal of literature, a hatred of language, and a repudiation of all established literary conventions.
But as literature embarks on this contestatory project, it has to negate its own conditions of possibility, for it is clear that without language and without the existence of previous texts, and without the conventions regulating and enabling them, there can be no recognisable thing such as literature at all. If literature chooses to exist only on the basis of the purity and originality of its endeavours, it is soon faced with the realisation that on those terms it cannot exist at all, except as endless banality or mute silence. In such circumstances, Blanchot asks, how is literature possible at all? Only, he replies, by virtue of a double illusion: if literature claims to be pure or original, it is because language has already proven it wrong: the purity of literature is already an impurity, and its alleged originality already a lack of originality. The essence of literature, Blanchot concludes, lies in its radical non-essentiality; the purest and most original texts, one might say, are those that expose themselves with utmost purity to their absence of purity. Literature here becomes affected with the same vertiginous logic as Paulhan's commonplace, and it becomes increasingly clear for Blanchot, in the texts written for the _Journal des débats_ during the Occupation, that the essence of art lies not so much in the foundational purity of the work, but rather in the aporia that turns that act of foundation into no more than the impossibility of a possibility. As a result, writing in Blanchot comes increasingly to be described in terms of the contestatory extremity of the inner experience – experience without experience – of writing as such, and it is here that one may sense the crucial importance for Blanchot not only of his own work as a writer of fiction, but of his friendship with Georges Bataille. At any event, the Kantian project adumbrated by Paulhan in _Les Fleurs de Tarbes,_ namely that of placing literature on a more secure, rhetorical footing, as Blanchot shows, and as Paulhan was himself no doubt aware, inevitably founders. As it does so, the attempt to produce a coherent aesthetic based on the purity of the work of art gives way to a teeming array of uncontrollable paradoxes that deprive Paulhan's would-be Copernican revolution of any intelligible foundation, and announce to Blanchot, in respect of literature, the radical inadequacy of all transcendental arguments as such.
The exchange between Paulhan and Blanchot turns principally on a question of literature. But there is more here at stake also. It would be to trivialise matters unduly simply to assume, as some have done, that Blanchot in his account of Paulhan's Terror was primarily concerned with settling an account with his own guilty, pre-war activist past. But it is nevertheless evident that the debate between Blanchot and Paulhan about the possibility of literature also touches crucially on the political question of the future of the (national) community in its time of distress. Both for Paulhan and for Blanchot, in their differing ways, it is clear that the question of writing was inseparable from the question of its relation to the wider political and cultural community at large. It is this that gives Blanchot's account of Paulhan its particular urgency in the context of the France of 1941. Indeed, the paradoxical status of literature in Blanchot's essay depends largely on its simultaneous appeal to the future of the community and its resistance to all established, current and prior forms of that community. Literature itself, in these texts, so to speak, is the real, yet always contestatory commonplace of the community; it grasps itself, in Blanchot's account, as an impossible interval, one that belongs necessarily to the language to which it is addressed, but which also projects itself beyond the present state of that community to voice what is nothing less than a promise of radical futurity. Blanchot says as much explicitly, by way of conclusion, when he reminds his readers that in order to imagine the future of literature 'all that is needed is to conceive that _true_ commonplaces are words torn apart by lightning and that the rigours of the laws found the absolute world of expression, beyond which chance is nothing but sleep' ( _FP,_ 101; _BR,_ 60).
In their capacity to contest all that is external to them, such _true_ commonplaces, as Blanchot calls it, are the works of literature themselves. The role imparted to them here is impressive; but what is perhaps most striking of all is the extent to which, by committing literature to a politics of opposition to the status quo, Blanchot endeavoured still to entrust to literature the capacity to found anew the national community it sought to address. In the France of 1941 this was of course an unambiguous message, and one that provided little comfort for supporters of the Vichy regime or the Occupation. But it meant that Blanchot also found himself in the rather contradictory position of claiming for literature a capacity to refound the current political and cultural order, while also having to acknowledge elsewhere that the only foundation of which literature was capable was an impossibility of foundation. The realisation that literature's boundless power of contestation derived from its necessary and irredeemable antagonism to all acts of foundation was one that was already implicit in _Thomas l'Obscur and Aminadab;_ but it was arguably to take Blanchot the literary critic several years more before he was able fully to draw the necessary philosophical conclusions from this. But this was no doubt why, once the Liberation did come, one of the writer's first tasks was to embark on a detailed philosophical explication with one of the most influential philosophical sources of his thinking at the time: the writings of Heidegger.
## 'Naming the gods': from Heidegger to Hölderlin
The just relation to the world is in turning aside, and this turning aside is just only if it is maintained, in separation and distance, as the pure movement of turning aside.
(Le rapport juste au monde est le détour, et ce détour n'est juste que s'il se maintient, dans l'écart et la distance, comme mouvement pur de se détourner.)
'Le Rire des dieux' ( _A,_ 194)
The reference to Heidegger is rarely explicit in Blanchot's wartime texts for the _Journal des débats,_ but its importance is hard to underestimate. In 'Mallarmé et l'art du roman', for instance, first published on 27 October 1943, and chronologically the last of the articles to be collected in _Faux Pas,_ Blanchot writes as follows:
Language is what founds human existence and the universe. Man who reveals himself in dialogue and finds in dialogue the event of his foundation, and the world that is put into words by an act that is its fundamental origin, both express the nature and dignity of language. The mistake is to believe language is an implement man has at his disposal in order to act or manifest himself in the world; language in reality has man at its disposal to the extent that it guarantees him the existence of the world and his existence in the world. Naming the gods, transforming the universe into discourse, that alone is what founds the authentic dialogue that is human existence, and that too provides the fabric of such discourse, its shimmering and mysterious figure, its form and constellation, far from the everyday expressions and rules that apply in ordinary life. (Le langage est ce qui fonde la réalité humaine et l'univers. L'homme qui se révèle dans un dialogue où il trouve son événement fondamental, le monde qui se met en paroles par un acte qui est sa profonde origine, expriment la nature et la dignité du langage. L'erreur est de croire que le langage soit un instrument dont l'homme dispose pour agir ou pour se manifester dans le monde; le langage en réalité dispose de l'homme en ce qu'il lui garantit l'existence du monde et son existence dans le monde. Nommer les dieux, faire que l'univers devienne discours, cela seul fonde le dialogue authentique qu'est la réalité humaine et cela aussi fournit la trame de ce discours, sa brillante et mystérieuse figure, sa forme et sa constellation, loin des vocables et des règles en usage dans la vie pratique.)
( _FP,_ 191; _BR,_ 45)
From his use of the expression 'la réalité humaine', introduced in the 1930s by Henry Corbin as a way of rendering Heideggerian 'Dasein' into French, it is evident that, in this passage, as elsewhere in _Faux Pas,_ Blanchot is quoting. He does so, however, as is often the case, both approximately and anonymously. Happily in this instance, the source of the quotation is not difficult to identify: it is Heidegger's famous lecture, 'Hölderlin und das Wesen der Dichtung' ('Hölderlin and the Essence of Poetry'), delivered in Rome in April 1936 and first published in French in 1938, though it is clear that Blanchot subsequently read the text in the German original. In the lecture, Heidegger quotes at length from Hölderlin, notably from a fragment usually held to belong to a preliminary draft for the unfinished poem 'Friedensfeier' ('Celebration of Peace') and which runs:
Viel hat erfahren der Mensch. Der Himmlischen viele genannt,
Seit ein Gespräch wir sind
Und hören können voneinander.
Michael Hamburger gives the following English version:
Much men have learnt. Have called by their names many of those in Heaven
Since we have been a discourse
And able to hear from each other.
Glossing these words in the lecture, Heidegger explicates them in the following terms, providing Blanchot with much of his own emphasis on the foundational role of language with respect both to human Dasein and the opening of a world:
From the moment language happens authentically as discourse, so the gods come to speak and a world appears. It is important, on the other hand, to see that the presence of the gods and the appearing of the world are not just a consequence of the happening of language, but are simultaneous with it. So much so that it is in the naming of the gods and the becoming word of the world that consists the authentic discourse which we are ourselves.
(Seitdem die Sprache eigentlich als Gespräch geschieht, kommen die Götter zu Wort und erscheint eine Welt. Abet wiederum gilt es zu sehen: die Gegenwart der Götter und das Erscheinen der Welt sind nicht erst eine Folge des Geschehnisses der Sprache, sondern sind damit gleichzeitig. Und das so sehr, daß im Nennen der Götter und im Wort-Werden der Welt gerade das eigentliche Gespräch besteht, das wir selbst sind.)
( _E,_ 40)
Late in 1943, then, in a deliberate and sustained manner, Blanchot can be found paraphrasing Heidegger who is himself paraphrasing Hölderlin. To understand what is at stake here it is perhaps worth recalling the particular weight that, in 1943, for Blanchot, attached to the two names of Heidegger and Hölderlin. Heidegger, of course, thanks to Levinas, had been familiar to Blanchot since the late 1920s; and by the early 1940s, as the passage cited above attests, he had evidently become for Blanchot the thinker most deeply and purposefully engaged in articulating philosophically the question of the foundational nature of language in general and poetic language in particular, and in challenging the inherited presuppositions of aesthetic theory as such, including received notions of literary mimesis or traditional beliefs regarding the duality of form and content. In raising these issues, Heidegger, for Blanchot, was clearly setting the agenda for modern literature and art in general; and this was why, in formulating his own view of the foundational dimension of language, Blanchot needed in fact to do little more than simply transpose into French Heidegger's central contention, contained in the 1936 lecture, that 'language is not an implement to be had at one's disposal, but that very event that has at its disposal the highest possibility of being human' ('die Sprache ist nicht ein verfügbares Werkzeug, sondern dasjenige Ereignis, das über die höchste Möglichkeit des Menschseins verfügt') ( _E,_ 38).
In the lecture, Heidegger went on to reverse the traditional hierarchy between ordinary language and poetic language, arguing that poetic language – which Heidegger reformulates, in a more originary turn, as 'Dichtung', i.e. as that which necessarily precedes, as their common origin, all such derived notions as poetry or literature – was what opened, not only the very possibility of everyday language as such, but also the possibility of the coming of a people – notably the German Volk – into the knowledge of its historical destiny and its own particular task within the history of Being. 'Dichtung', according to Heidegger, is thus, as he puts it, the originary, foundational tongue of a historical people:
Poetry is the founding naming of Being and of the essence of all things – not just any saying whatever, but the saying by which everything which we subsequently discuss and deal with in everyday language first comes into the open. That is why poetry never treats language as material that is present-at-hand, but poetry itself first makes language possible. Poetry is the originary language of a historical people.
(Dichtung ist das stiftende Nennen des Seins und des Wesens aller Dinge – kein beliebiges Sagen, sondern jenes, wodurch erst all das ins Offene tritt, was wir dann in der Alltagssprache bereden und verhandeln. Daher nimmt die Dichtung niemals die Sprache als einen vorhandenen Werkstoff auf, sondern die Dichtung selbst ermöglicht erst die Sprache. Dichtung ist die Ursprache eines geschichtlichen Volkes.)
( _E,_ 43)
Developing this account, Heidegger argues, with specific reference to Hölderlin, that the proper task of the work of art, by virtue of the foundational dimension of poetic language, is to articulate for the historical community what he calls the between ('das Zwischen'), that median space or time that separates, while thereby connecting them, both gods and humans; the vocation of poetry is thus to integrate as one, as Heidegger puts it, the signs that come from the heavens and the voice of the people ( _E,_ 46–7).
It was Hölderlin, for Heidegger, who most radically dedicated his poetic word to this task, by describing in his poetry the task of mediation ascribed to 'Dichtung' at the very moment he endeavoured to perform that task. Hölderlin, for Heidegger, was the poet who most intently aimed to understand the necessity of poetry and carry out its demand in the time of distress that is the time of modernity. Hölderlin in his work, Heidegger affirms, thereby 'puts into poetry the essence of poetry' ('dichtet das Wesen der Dichtung') ( _E,_ 47). This is what gave Hölderlin, in Heidegger's eyes, his crucial philosophical importance; the work of Hölderlin constituted an essential turning point in the history of the West, a turning point that, as Heidegger was wont to remark, had been all but overlooked by the West itself. But no matter, even in his solitude, indeed no doubt because of that very solitude, Hölderlin remained for Heidegger quite simply the essential poet (the poet's poet, 'der Dichter des Dichters', _E,_ 47); and the phrase is one Blanchot in his turn cites approvingly, noting for instance of René Char, Heidegger's friend, in October 1946, that 'his poetry is the revelation of poetry, the poetry of poetry, and, as Heidegger says more or less of Hölderlin, the poem of the essence of the poem [comme le dit à peu près Heidegger de Hölderlin, poème de l'essence du poème]' ( _PF,_ 105; 101).
But the importance of Hölderlin for Heidegger was not just philosophical or poetological; it was political as well. Heidegger began lecturing on Hölderlin in the winter of 1934, soon after his resignation, in April, from the post of Rector of Freiburg University, a position that, as is now well known, he took up in April 1933 as part of a co-ordinated campaign to win over the German university system to National-Socialism. The context of Heidegger's turn to Hölderlin was therefore crucial, and there is little doubt that on one level it was undertaken by Heidegger as a means of redefining and defending – philosophically as well as politically – his own disastrous political venture of 1933 and his own particular commitment to Nazism (of whose 'inner truth', it is now clear, Heidegger was to remain convinced for many years to come). Some of the actions of this political Heidegger were no doubt familiar to Blanchot by the early 1940s. In any case, the figure of Hölderlin was itself no stranger to political controversy. Already for some years, since the end of the First World War, the poet had been viewed by some – including the Nazi ideologue Alfred Rosenberg – as Germany's own national poet, an identification that Heidegger arguably sought to cultivate in his own way by dedicating his 1936 lecture to the memory of Norbert von Hellingrath, Hölderlin's early editor, killed in the trenches in 1916, and which culminated in June 1943 in celebrations throughout the Reich, including Paris, on the occasion of the centenary of the poet's death, under the patronage, amongst others, of Josef Goebbels.
In the dialogue between Heidegger and Blanchot about the poems of Hölderlin, then, there are many different issues at stake; these bear not only on the question of the relationship between literature and philosophy, between metaphysics and its other, between poetry and ethics, but also on the debate about language, community, nationhood, and great art. At the same time, however, it is also noticeable how little explicit engagement with these broader issues is visible in Blanchot's recourse to Heidegger in the essays of the early 1940s from which I have quoted so far. In 1942 and 1943, it is as though Blanchot was content simply to incorporate Heidegger's reading of Hölderlin into his own thinking without much reservation. It was not until three years later, in 1946, that Blanchot began explicitly to articulate in his published critical work the outlines of a more thorough, and more demanding engagement with Heidegger. This he did in an essay entitled 'La Parole "sacrée" de Hölderlin' ('The "Sacred" Word of Hölderlin'), first published in _Critique_ in December 1946, and collected, in revised form, in _La Part du feu_ three years later ( _PF,_ 115–32; 111–31). The essay is in effect a detailed commentary on Heidegger's essay of 1941 devoted to Hölderlin's famous poem about the role of the poet, 'Wie wenn am Feiertage . . .' ('As on a holiday . . .', in Hamburger's translation). It was written at impressive speed. As late as mid-October Blanchot was confessing in a letter to Bataille, who had evidently suggested he review Heidegger's article for _Critique,_ that though he was tempted by the proposition, he had still not yet seen the recent issue of the magazine _Fontaine_ containing the French translation of Heidegger's text; by December, though, Blanchot's article had been delivered to the publisher and was already on sale in bookshops. All of which suggests that, even before the essay was written, Blanchot had spent considerable time reconsidering his relationship to Heidegger and rereading Hölderlin.
As elsewhere in his work, in the essay 'La Parole "sacrée" de Hölderlin', Blanchot proceeds indirectly. He writes largely alongside Heidegger, that is to say, with Heidegger, but also, simultaneously, against him. This is a common strategy in Blanchot, who in all his writing, rather than endorsing or opposing a philosophical argument, tends instead to accompany it through its various twists and turns in order then to prise it apart by confronting it with the limits of its own possibility. In reading Blanchot's article on Heidegger, therefore, it is less important to attend, in themselves, to the overt conceptual statements that occur in the essay, many of which can be traced back to Heidegger, than to examine the particular, often paradoxical treatment to which Blanchot subjects some of those formulations. So while the essay does open with a series of circumspect criticisms regarding the legitimacy of Heidegger's style of commentary, including his prosaic neglect of Hölderlin's poetic rhythms and his reliance on etymology as a mode of argument, most readers of the article have declared Blanchot's account of Heidegger's reading of Hölderlin to be extremely faithful to Heidegger, so much so, for instance, that Herman Rapaport, in a discussion of the relationship between Heidegger and Derrida, credits Blanchot – somewhat ironically in the circumstances – with being little more than a particularly astute, foresightful 'mediator of Heidegger.'
If one reads Blanchot's text with greater attention to detail, however, there are signs of a more deep-seated disaccord with Heidegger. This is borne out by the minor modifications that Blanchot incorporated into the essay when revising it for inclusion in _La Part du feu,_ as, too, do many subsequent remarks made by Blanchot about Heidegger's account of Hölderlin. Admittedly, as far as the 1946 article is concerned, the evidence for Blanchot's departure from Heidegger's reading is at times very faint. This does not of course make it any the less real. Indeed, as though to illustrate this very point, Blanchot at one moment offers a sly reflection on the elliptical nature of his own intervention. Heidegger, in his 1941 piece, is glossing Hölderlin's use of the word _Natur_ (Nature) and quotes the following lines from the poem 'Am Quell der Donau' ('At the Source of the Danube'):
Wir nennen Dich, heiliggenöthiget, nennen
Natur! dich wir, und neu, wie dem Bad entsteigt
Dir alles Göttlichgeborne.
(We name you, under a holy compulsion we
Now name you Nature, and new, as from a bath
From you emerges all that's divinely born.)
Heidegger points out that, in the manuscript, these lines are in fact crossed out; this is taken to support the contention that the Latinate term Nature _(natura)_ is no longer sufficient to Hölderlin's purpose, but has silently begun to bear witness to Greek _(φυσιζ_ in its capacity as original, founding word or _Grundwort_ ('This overcoming', says Heidegger, 'is the consequence and the sign of a more original saying' ['Diese Überwindung ist die Folge und das Zeichen eines anfänglicher anhebenden Sagens'] _E,_ 58). This is, of course, in Heidegger's thinking, a recurrent strategy, one that consists in the re-translation of so-called founding words back into their original Greek forms. Blanchot's rejoinder to dais move is to repeat it, but at the same time also invert it: 'That nature', he writes, '. . . owes to the Sacred its most essential qualities ... is something that is affirmed by these lines from "At the Source of the Danube", lines that, although subsequently crossed out, were nevertheless still formulated' ('Que la nature . . . doive au Sacré ce qu'elle a de plus essentiel.. . c'est ce qu'affirment les vers d' "Aux Sources du Danube" qui, quoique biffés par la suite, n'en ont pas moins été formulas') _(PF,_ 123; 119).
The point is more than a philological nicety. For Blanchot follows up his remark almost immediately with the claim, which is at the centre of his disagreement with Heidegger, that by interpreting the Sacred as chaos, even chaos suitably redefined as an originary gaping and the opening or awakening of nature itself to the light, Heidegger falls victim to a misleadingly traditional view of the poet. The darkness of original night, Blanchot objects, mourned by the poet as he is illuminated by nature's awakening, has no place in Hölderlin's experience. (Throughout the essay, this emphasis on Hölderlin's experience is a significant feature of Blanchot's reading, and one that places his essay within the context of an exchange not only with Heidegger, but also Bataille.) For Blanchot, paradoxically, it is at the very moment when Heidegger begins reading Hölderlin as though the text were written in a language closer to the pre-metaphysical origins of Western thought, that he succumbs to a misreading that fails to do justice to Hölderlin's originality. The point may seem trivial, but more is at issue than a question of poetical interpretation; for while both Heidegger and Blanchot agree that the Sacred precedes both humans and gods as their common space of possibility, they differ sharply as to the relationship between the language of poetry and this necessarily unapproachable and incommunicable origin.
For Heidegger, the Sacred, seen either as originary chaos or as the very essence of nature itself, marks the very opening of being to itself. For the poet, it is like a moment of originary quickening that, like the nature – _φυσιζ_ – that is his all-inclusive and prescient companion, ushers him into the light of both world and work. Hölderlin, in this way, appears in Heidegger as the poet who, out of darkness, receives the vision of light clarified, transformed, shaped and mediated through poetry; he is the poet who, as Heidegger puts it in his 1943 postface to 'Was ist Metaphysik?', in naming the Sacred, fulfils the same task as the thinker who speaks Being. The figure is that of the poet as mediator; yet though Blanchot acknowledges the importance of that role in at least some of Hölderlin's poems, he is quick to see its limitations _(PF,_ 118; 114). Indeed, his own account of the poet could not in fact be more different; for Blanchot's Hölderlin is not the poet of the burgeoning origin, but of the origin dispersed, not the poet of the night transfigured, but of the day affirmed, not the poet of the lighting of Being, but of the light which precedes the light. In Hölderlin, Blanchot writes, 'night and chaos always end up vouching for law, form, and light' _(PF,_ 123; 120); and he continues as follows:
The Sacred is the light of day: not the day as contrasted with the night, nor the light as it shines from above, nor the flame that Empedocles searches for below. It is the light of day, but prior to the day, and always prior to itself, the day before the day, a light before light to which we are closest when we grasp the dawn, the infinitely far-off distance of daybreak which is also what is most intimate to us, more inward than inwardness itself.
_(PF,_ 124; 121)
In their account of the Sacred, Heidegger and (after him) Blanchot both cite a prose commentary by Hölderlin on a translation of a fragment from Pindar. The text, which Hölderlin titles 'Das Höchste' (and which an exactly contemporary Blanchot work suggests one might venture to translate into French as 'Le Très-Haut', 'The Most High'), seeks to formulate the dichotomy between what Hölderlin terms the immediate or the incommunicable ('das Unmittelbare', that which cannot be mediated) and the law, as he puts it, of rigorous communicability ('die strenge Mittelbarkeit'). While the first of these, Hölderlin maintains, is impossible for both gods and mortals, since both are subject to the need to make distinctions (i.e. mediate contrasts or oppositions) in order to judge or to know, the second imposes on both mortals and gods, by the most elevated of laws, the stringent obligation to communicate or to mediate between the two realms of heaven and earth. Hölderlin's words are elliptical and enigmatic, and pose formidable problems of interpretation. For his part, Heidegger resolves these by aligning Hölderlin's distinction with the concept of ontological difference, i.e. the difference between Being ('Sein') and beings ('das Seiende') that is at the centre of all Heidegger's own work. On this reading, Hölderlin's paradox of communicability and incommunicability may be reformulated in such a way as to refer to Heideggerian Being, to the extent that being, for Heidegger, necessarily resists all mediation or communication because it is precisely what enables all communication or mediation to take place at all. And it is this logically prior realm, which is not in itself an entity, but the condition of existence of all possible entities that, in Heidegger's view, constitutes the essence of Hölderlinian _φυσιζ_ ; and it is this that Hölderlin's Sacred word is held to name. As Heidegger explains:
Immediate all-presence is the intermediary for all that is communicated, i.e. that which is communicable. The immediate itself is never something communicable, but on the other hand the immediate, in the strict sense, is communication, i.e. it is the communicability of the communicable, because this makes it possible in its essence. 'Nature' is all communicating communicability; it is 'the law'. (Die unmittelbare Allgegenwart ist die Mittlerin für alles Vermittelte und d.h. für das Mittelbare. Das Unmittelbare ist selbst nie ein Mittelbares, wohl dagegen ist das Unmittelbare, streng genommen, die Vermittelung, d.h. die Mittelbarkeit des Mittelbaren, weil sie dieses in seinem Wesen ermöglicht. Die 'Natur' ist die alles vermittelnde Mittelbarkeit, ist 'das Gesetz' .)
_(E,_ 62)
In his own paper, Blanchot rallies initially to much the same view, and cites Heidegger accordingly: 'What is the Sacred?', he asks. 'It is the immediate, says Heidegger, taking his lead from a prose fragment by Hölderlin, the immediate that is never communicated, but is the principle of all possibility of communicating [l'immédiat qui n'est jamais communiqué,mais est le principe de toute possibilité de communiquer]' _(PF,_ 123; 120).
At this point further difficulties arise. For if the Sacred is that which by essence cannot become an object of communication or mediation, and if it is nonetheless the task of Dichtung to name the Sacred, how is the poem possible? Does it not, in an act of fatal hubris, simply turn the Sacred into an object, the very reverse of itself? This danger exists, Heidegger concedes, and the law of the Sacred is thereby necessarily threatened. But in the last resort, Heidegger rejoins, that which is derived from the origin is powerless against that origin ('So vermag das dem Ursprung Entsprungene nichts gegen den Ursprung') ( _E,_ 74). The Sacred –φυσιζ or Being – is that which makes the poetic act possible; poetry, which owes its existence to the Sacred, cannot change the Sacred into what it is not, if only because the Sacred already includes its own opposite. Though the poet may suffer at the hands of the Sacred, he belongs to its opening and is given, by that opening, both the duty and the means to speak it in its constancy or permanence, and submit to its law. The poet's part, for Heidegger, corresponds to a kind of originary fervour, and his stance is one of silent, prescient absorption in the all-enclosing presence of Being. Whatever the anguish of the originary chaos represented by the incommunicability of the sacred, the poet, on Heidegger's submission, is nevertheless able to domesticate it by his silence and transmute it into the gentleness that characterises all poetic disclosure. As Heidegger writes, in a famous passage:
The violence of chaos, that offers no pause, the terror of the immediate, that thwarts all access, the Sacred is transformed by the silence of the sheltered poet into the gentleness of the communicable and communicating word.
(Die Erschütterung des Chaos, das keinen Anhalt bietet, die Schrecknis des Unmittelbaren, das jeden Zudrang vereitelt, das Heilige ist durch die Stille des behüteten Dichters hindurch in die Milde des mittelbaren und vermittelnden Wortes gewandelt.)
( _E,_ 71)
For Heidegger, then, the poem is possible, in Hölderlinian terms, on condition that the incommensurability or discontinuity that exists between the demand for poetic communication and the elusive nature of the incommunicable object can somehow be transmuted into a relationship that admits of mediation, a mediation whose purity is vouchsafed both by the suffering of the poet and the intensity of his silence.
Despite appearances to the contrary, Blanchot refuses this moment of reconciliation between Dichtung and Being. Blanchot insists, with more radical commitment than Heidegger, on the irreducible incommensurability of the relation – the relation without relation – that binds the necessity of poetic communication to the incommunicability of the Sacred, and vice versa. Hölderlin's poem, he argues, is the site of what can only be described as a fundamental aporia, an impossible possibility. As Blanchot asks Heidegger:
How can the Sacred, which is 'unexpressed', 'unknown', which is what opens only on condition that it is not unveiled, which reveals because it is itself unrevealed, how can it fall into speech, let itself be alienated into becoming, itself pure interiority, the exteriority of song? In truth, indeed, this is not possible, it is impossibility itself [à la vérité, justement, cela ne se peut pas, c'est l'impossible].
( _PF,_ 128; 126)
While Heidegger spirits away the incommensurability between the communicable and the incommunicable to discover in that relationship a movement of gentle mediation, Blanchot insists instead on the groundless impossibility of such mediation. The poem, for Blanchot, remains the place of an irretrievable tension, one that cannot be unified within any kind of dialectic or within the totalising embrace of Heideggerian Being; instead of being construed, as it is by Heidegger, as an inaugural act of historical or ontological foundation, what Hölderlin's poem affirms, for Blanchot, is the irresolvable double bind that imposes on poetry the obligation of writing while confronting it perpetually with the impossibility of writing. Hölderlin's text, throughout, is read by Blanchot as a dramatisation of this aporia. Thus, for instance, the disabling circularity that dogs Hölderlin's project to name in poetry the wholeness of nature, a wholeness that comes into being only by the act of poetic naming that constitutes it, but which cannot come into being because, in order for there to be a poet to carry it out, the act of naming needs already to have taken place. The poet is the product of the poetic work, which confers on him his existence; before the work is written, therefore, the poet has no proper existence; but if that is so, no poetic work can in fact be produced, with the result that the poet survives only in a ghostly future perfect, divided from himself and without necessity, confronted with the aporia of an act that cannot accomplish the task which it is called upon to fulfil.
Such aporias, Blanchot insists, are the price as well as the condition of poetry as such; for, as Hölderlin's work attests, poetry somehow, in spite of all, remains a possibility. But this possibility rests on its own necessary impossibility; and this is what Blanchot's essay raises in objection to Heidegger: the question of the impossibility of the possibility of poetry. (In later texts, alongside Levinas, Blanchot will ask of Heidegger a very similar question, not in respect of Dichtung, but in relation to what, in a celebrated passage of _Sein und Zeit,_ Heidegger addresses as the question of Dasein's 'ownmost possibility', the possibility of its impossibility, that is to say, the question of death.) To press home the point, Blanchot cites Heidegger once more, offering his own version of the passage from the commentary on 'Wie wenn am Feiertage . . .' given above. Blanchot writes:
Directly, the Sacred cannot be grasped, even less can it become speech, yet through the silence of the poet, it supposedly lets itself be pacified [il se laisserait apaiser], transformed, and finally transported into the speech of song. Possibly so. But what has become of the problem? It has taken another form, but it still remains a problem; or rather, it has now been replaced by a double enigma: how and why can 'the violence of chaos that offers no pause, the terror of the immediate that thwarts all access, the Sacred' ['l'ébranlement du chaos qui n'offre aucun point d'appui et d'arrêt, la terreur de l'Immédiat qui fait échec à toute saisie directe, le Sacré'] let itself be transformed and attained by silence? And, in turn, how and why can silence let itself be attained by speech?
( _PF,_ 128; 126)
Crucially, here, at the very moment he incorporates into his own text this description of the poetic act, Blanchot truncates, interrupts Heidegger's sentence. The move is an emblematic one; for as he does so, Blanchot signals the decisive moment of his own intervention into Heidegger's discourse, that moment where Heidegger's own discourse founders on the impossibility to which it fails to do justice. It is important to note here to what extent Blanchot's interruption of Heidegger is founded not only on a poetological difference, but an ethical one as well. Crucial to Blanchot's divergence from Heidegger is the question of the law, of that which Hölderlin in his Pindar commentary names as 'das Höchste', the Most High. For Heidegger, the question of the law is simple, since what is at stake in Dichtung for Heidegger is the law of Being itself; it is this of course that enables Heidegger to treat poetry as enacting a gesture of political inauguration by telling a historical people the truth about its historical, ontological destination. (And it is worth recalling here Heidegger's notorious claim, made in 'Der Ursprung des Kunstwerks' ['The Origin of the Work of Art'] of 1936, that, in its relation to truth, poetry takes its place alongside a number of other such moments, including the act of founding of the state and of essential sacrifice.) But for Blanchot, however, the question of the law is never simple; it is always double, it always involves having to respond, beyond the possibility of the law, to the impossibility that both founds and exceeds the law: the law beyond the law, so to speak, that, beyond being, formulates a demand that, by definition, can never be satisfied. As Blanchot writes, giving his own version of the Most High: 'Speaking is what is required, that, that alone will do. And yet speaking is impossible' ('Parler, il le faut, c'est cela, cela seul qui convient. Et pourtant parler est impossible') ( _PF,_ 129; 127).
From this ethical difference between Hölderlin and Heidegger flow all the other points of divergence touched on by Blanchot in his article: the emphasis on poetry's resistance – as poetic rhythm – to the totalising designs of philosophy; the distrust of Heidegger's confident reliance on the truth of etymology; the insistence on the specificity of Hölderlin's own poetic experience; the wary reminder, contra Heidegger, that any poetic act is confronted from the outset with the impossibility of beginning, and that poetry as a result cannot serve as a moment of stable ethical or political foundation; and the repeated affirmation, as Blanchot puts it, 'Le Grand Refus' ('The Great Refusal', _EI,_ 69; 48), that while poetry may name the possible, its task is to respond to the impossible, and that, in the guise of the immediate, what Hölderlin's poetry is forever in the process of encountering is not Being, but, as Blanchot formulates it in 1969, in a heavily reworked passage from _L'Entretien infini,_ the 'infinite presence of what remains radically absent, presence always infinitely other in its presence, presence of the other in its alterity: non-presence' ( _EI,_ 54; 38). Hölderlin's poetry, for Blanchot, founds no homogeneous community possessing within itself the truth of its own future destiny. Contrary to what Heidegger explicitly and implicitly uses Hölderlin to assert from 1934 onwards, the community that poetry installs for Blanchot is a community that exists outside of truth, in the interruption of history, in its exposure to the worklessness of the outside.
Blanchot ends 'La Parole "sacrée" de Hölderlin' with a reference to the impossible possibility, and with a final quotation:
In its very impossibility, the reconciliation of the Sacred with speech demanded of the poet's existence that it most nearly approach non-existence. It was then, for a moment, that itself seemed possible, when, just before foundering, it allowed itself to be affirmed in song, in words issuing from an already silent body and uttered by a dead voice, in such a way that the hymn alone, worthy of the essence of the day, rose from the depths of the vanished day, and that spirit was glorified by its own distraction, but not because the most high is darkness, nor because in the end the spirit must be bound to its own destruction, but, as the All made itself language to say: because whoever wishes to encounter darkness must seek it in the light of the day, stare at the light of the day, and become the light of the day for themselves:
Enigma is the pure surging of what surges forth
The deep that shakes all, the coming of the day.
[Énigme est le pur jaillissement de ce qui jaillit
Profondeur qui tout ébranle, la venue du jour.]
( _PF,_ 132; 131)
These closing two lines, cited as though from Hölderlin, with which Blanchot more or less concludes his article, bring together three key motifs from Blanchot's account: poetry's pure affirmation of the impossibility of its own possibility, the absence of foundation that puts everything into question, the always prior arrival of the light. Yet despite appearances to the contrary, these two lines are in fact not from Hölderlin at all; or, more precisely, they are on Blanchot's part both more and less than a quotation. In reality, only the first line is from Hölderlin, namely from the fourth stanza of the poem 'Der Rhein' ('The Rhine'), which reads: 'Ein Rätsel ist Reinentsprungenes'. The second line that Blanchot reproduces is not however the verse which follows, nor is it easy to locate at all in the whole of Hölderlin's published work. It is more readily interpretable as Blanchot's own summary and apocryphal anthologising of two powerfully recurrent Hölderlinian motifs: the plunging abyss, the coming of the light.
What should one make of such slapdash scholarship? Laziness? Negligence? The result of writing too fast? Or just a poor memory? All this, no doubt. But also: the implicit affirmation, on Blanchot's part, that philosophers – such as Heidegger – have for too long endeavoured to interpret Hölderlin's poems; the task now is to rewrite them.
## The limitlessness of the limit
But if we call the day to account, if we manage to cast the day aside in order to discover what lies before or beneath it, we find that the day is already present, and that what is before the day is always already day, in the form of the inability to disappear rather than the power to make things appear, as obscure necessity and not radiant freedom. The nature, then, of what is before the day, of existence prior to the day, is the dark side of the day, and that dark side is not the undisclosed mystery of its beginning but its inevitable presence, a 'There is no day' [un 'Il n'y a pas de jour'] which merges with a 'There is already day' [un 'Il y a déjà du jour'], its appearance coinciding with the moment when it has not yet appeared. The day, in the course of the day, allows us to escape from things, it lets us understand them by making them transparent and, so to speak, null – but the day itself is what one cannot escape: in it we are free, but itself the day is fatality, and the day as fatality is the being of what is prior to the day, the existence from which it is necessary to turn aside in order to speak and understand.
'La Littérature et le droit à la mort' ( _PF,_ 318; 329 – 30)
The relation between Blanchot's critical essays and fiction is complex and multiple. The two series of texts do not belong to a single homogeneous space, but possess a different rhythm and temporality, a divergent external history and an incommensurable set of internal constraints. For this reason alone, it would be short-sighted to impose on the relation between Blanchot's discursive prose and fictional writing a secure hierarchical structure or polarity. Indeed, particularly in Blanchot's case, where the divisions between essay and fiction are less clearly demarcated than in other writers, there is arguably little reason to attribute special qualities – of originality, profundity, or subversive potential – to one group of texts rather than another. Often it is the similarities between the essays and fiction that are more striking than the differences between them; and, frequently, motifs that first occur in a fictional context return as subjects of discussion in Blanchot's criticism, while issues debated in the literary essays often subsequently feature in barely modified form in fictional works.
This apparently easy circulation of meaning is deceptive. It ignores the effects of the peculiarly suspensive violence with which literature, Blanchot argues, interrupts the workaday continuity of meaning and causes a banal sentence such as Kafka's: 'the head clerk rang' ( _PF,_ 79; 74) to take on a very different set of functions according to whether it appears in a work of fiction or is used in the course of daily office routine. The argument is one that has often been put forward by theorists of literature inspired by phenomenology or by the structuralist neo-formalism that in many respects is its heir. Admittedly, it is no doubt essential for literary criticism to do justice to the disruptive force – the difference – that is put in play by the fictional deployment of language in literary texts. All too often, however, on the part of literary critics, the result has been to confer on the work of art the status of a superior, ideal, or idealised object whose principal distinguishing feature lies in the extent to which it is hoisted clear of so-called everyday discourse. The consequence of this, in turn, has been to determine the specificity of literature purely from the perspective of the very norms literature on this account is in fact deemed to transgress; at which point literature finds itself, albeit negatively, entirely reliant after all on the very rules and conventions that it is one of writing's main characteristics most strenuously to challenge.
The originality of Blanchot's own approach to the issue of the specificity or distinctiveness of literature, though it has often been misrepresented, may be measured by the extent to which he resists these pitfalls. Indeed, Blanchot refuses to see literature as being essentially transgressive if by that what is implied is that the fate of literature is entirely bound, dialectically, to societal or discursive norms; and he similarly rejects any idea of the work of art as an autotelic, intransitive, or self-referential entity entirely devoted to the aesthetic closure of its own essential self-presence. The so-called specificity of literature, for Blanchot, if indeed it exists at all, is not to be found in this concept of the self-legislating autonomy of literary language, nor in the idea that literature displays a richer potential for communicativeness than so-called ordinary speech; nor does it lie in literature's greater veracity or truthfulness to experience, nor – as a neo-formalist critic might claim – in its greater awareness of its own inherent falsity as a verbal artefact.
In Blanchot's terms, what is singular about literature – to the extent that the question: what is literature? is susceptible of any determinate answer whatsoever – is rather the infinite paradoxicality of its essential non-essentiality and the boundless peculiarity of its relation to its own limits. Limits, of course, as Jacques Derrida has shown, obey a disconcerting and duplicitous logic that might be said to be entirely their own, were it not for the fact that limits, by their very nature, are endowed with infinite scepticism whenever it comes to determining the limits of that which is proper or improper to them. Limits in this sense are not figures of authority or power, but evanescent traces traversed by an abysmal and aporetic logic. Once they are traced or inscribed, they serve to discriminate between a certain order and a certain disorder; and what falls within a given limit is declared to be subject to a law that does not hold for what is beyond that limit. Limits, however, can be instituted or proclaimed as such only by an authority that exists beyond the limit to be determined and which is itself not subject to the law to be enforced. To institute a limit therefore is not only to circumscribe and contain, but to yield paradoxically to the fragility of any such attempt at circumscription or containment; and it is to make the limit itself a function or effect of the limitlessness on which it is necessarily premised. The limit, then, no longer operates as a term in its own right, but rather only as a response to the interminability that thereby comes to inhabit it. And though its existence is implied by the limit, the limitlessness beyond the limit obeys a different law to that limit, a law that cannot of course be named from the perspective of the limit, except on condition that the limit itself be suspended. The limitlessness beyond the limit limits the limit by turning its law into a secondary, derived entity, and affirms its own necessity as an always prior demand, a demand that is thereby infinitely at odds with the law of the limit.
At this point it appears that the limitlessness beyond the limit itself constitutes something resembling a law, albeit a law that is necessarily irreducible to any law of the limit and infinitely dissymmetrical with it. Yet though the two exigencies – that of the limit and that of the limitlessness of the limit – are disjoined from one another, they do not exist in isolation from each other; and while the limitlessness beyond the limit belongs to the limit, so to speak, as its very condition of possibility and impossibility, the reverse is also true, for without the interval inscribed by the limit the limitlessness that lies beyond it would not be possible either. In this way, the limitlessness beyond the limit may be said both to follow and precede the limit; just as the demand of limitlessness exceeds that of the limit, limitlessness itself is inseparable from its own necessary interruption. As Blanchot insists, the limitlessness is itself inherent in the limit; what it implies is not the lure of an effusive, all-embracing totality, but the rigorous necessity of infinite fragmentation, boundless discontinuity, and endless finitude. This logic is both bizarre and paradoxical, but it is absolutely fundamental to Blanchot's account of literature and of the literary. For Blanchot, one might say, literature is what arises when the relation between limitlessness and the limit is pursued to the point of its limitlessness. In literature, so to speak, limits are simultaneously maintained and suspended, and their effects intensified as well as effaced. What literature names is the strange movement by which, without ever gathering itself into dialectical unity, language poses limits and affirms meaning, but even by doing so silently reaches beyond those limits to respond to the limitlessness that is inseparable from those limits and which those limits in turn necessarily serve to punctuate and interrupt.
Three years after the publication of 'La Parole "sacrée" de Hölderlin' in 1946, there appeared in a literary journal a short story by Blanchot with a puzzling double title. The work was announced on the cover of the periodical as 'Un récit?'. In the table of contents, however, and on the first page of the text, it was simply called 'Un récit', until eventually, in 1973, the story was reissued under the now more familiar name of _La Folie du jour_ ( _The Madness of the Day_ ). As Derrida has convincingly shown, the slippage between these two – subsequently three – titles is more than a case of localised lexical permissiveness. The effects extend to the story or text as a whole, which may thus be read not only as a protracted commentary on the hesitation inherent within its own title(s), but also as a radical questioning of its own unity and wholeness as such. Indeed, in that each proposed title doubles both as a naming of the text and an integral part of it, as an address to the text and a quotation from it, Blanchot's story puts itself in the double and dissymmetrical position of placing itself at one and the same time under the jurisdiction of one law, the law of literary genres and of narratives, while also responding to another law, the law of the text's own original namelessness and irredeemable madness. The result is to shatter the apparent self-identity or self-presence of the story as such.
Indeed, throughout the text of _La Folie du jour,_ Blanchot exploits to radical effect the many paradoxes – governing the relationship between inside and outside, first and second, general and particular – that are produced by the structure of the limit, the frame, and the margin. The result is that complex structure of repeated self-imbrication that Derrida has famously described as 'chiasmic double invagination'. This is not in itself the product of any particular subversive decision on Blanchot's part; as Derrida maintains, it is an uncontrollable consequence of the general structure of framing as such. What is distinctive, however, about Blanchot's text is that it pushes the paradox of the limitlessness of the limit to the extreme point of its necessary impossibility; and it is at that point – that limit – that everything changes, and the reader of Blanchot's story is faced with the task of interpreting – or, better, not interpreting – a textual space which no longer allows one to discriminate with finality between what is inside and what is outside, what is general and what is singular, what is subject to the law and what is beyond it.
Texts published in periodicals, however, as Derrida points out, always carry at least one supplementary title in the shape of the magazine's own, and it is worth noting that the journal in which 'Un récit' (or 'Un récit?') did first appear in 1949, edited by René Char, Albert Béguin, and Albert Camus, was the literary monthly _Empédocle,_ which had taken its own name from the eponymous hero of Hölderlin's unfinished – and probably unfinishable – tragedy of 1797–99, _Der Tod des Empedokles_ ( _The Death of Empedocles_ ). Admittedly, this encounter between Hölderlin's Empedocles and Blanchot's _récit_ may be entirely coincidental; but, as Blanchot argues elsewhere, apropos of Mallarmé's 'Un coup de dés', chance is not without its own – unpredictable – necessities. (Indeed, one might ask, was the hesitation between 'Un récit' and 'Un récit?' not itself the result of an inspired misprint?) And if the law is one of chance, chance will of course make its own irresistible demands. Which is why there is perhaps some justification for reading _La Folie du jour_ from the edge or margin of its Hölderlinian resonances, and as an attempt at fictional exploration of the issues at stake in 'La Parole "sacrée" de Hölderlin'. By doing so it will be possible not only to examine more closely the relationship between Blanchot's essays and fiction, but to understand better, also, the impact on Blanchot's fictional writing of the engagement with Heidegger examined above.
Hölderlin's Empedocles is a figure who, combining intense hubris with intense devotion to the gods, aims to embrace the All in an act of willed self-destruction. In writing the play, which went through three different versions, each more fragmentary than the last, it is now generally accepted that Hölderlin was exposing himself to enormous artistic as well as personal difficulties. But as Blanchot argues in a later essay on Hölderlin in _L'Espace littéraire_ ( _EL,_ 283–92; 269–76), the project for the play also ran another risk, namely that of falling victim to an inauthentic illusion, what Blanchot describes there, recalling much of his earlier analysis, as the lure of the immediate, the impossible. In all his subsequent work, however, the temptation, Blanchot writes, was one Hölderlin strenuously resisted. 'The more Hölderlin is subject to the ordeal of "the fire of the heavens",' Blanchot writes, 'the more he expresses the necessity of not surrendering to it immeasurably [sans mesure]' ( _EL,_ 288; 273). There is here, in Blanchot's reading, a further instance of what Philippe Lacoue-Labarthe has described as Hölderlin's 'hyperbologic', that strange pseudo-logic of indefinite exchange between contraries in the movement of their limitless intensification. Fidelity to gods and to men for Hölderlin is purchased only by infidelity to both; and poetic purity is at the price of turning aside from both the heavens and the earth to maintain intact the interval that separates them and challenges their immanence or self-identity; indeed, writes Blanchot, it is as that pure, rendingly empty distance of separation between the gods and humankind that the Sacred may most properly – improperly – be formulated ( _EL,_ 289; 274).
Blanchot's reading of Hölderlin turns crucially on an awareness of the oscillating tension between two dissymmetrical forces: between that which is without limits and always threatens to overwhelm meaning, and the artistic – prosodic – measure needed to contain the limitlessness while also being interrupted by it. The affirmation of this dual logic is a crucial priority in Blanchot's account of Hölderlin; but it is also an essential element in all Blanchot's critical writing on literature in general. Indeed, it may also be said to be at the centre of a fictional text like _La Folie du jour;_ and for this reason it is not difficult to perceive many striking similarities or convergences between Hölderlin's poems and Blanchot's own _récit._ Both depend, for instance, on the double contention, first, that the only proper goal of literature as such – precisely because it is not an object but the prior condition of any subject or object at all – is the limitlessness of the impossible; but, second, that the boundlessness of what is unlimited can only be manifested in literature by rigorous insistence on separation and difference. These are the terms within which Blanchot's reading of Hölderlin moves; but they characterise the thematic and narrative structure of _La Folie du jour_ as well.
Indeed, _La Folie du jour_ begins, as though in homage to Hölderlin's Empedocles, with an extraordinary testimonial to the limitless extremity of both living and dying:
I am neither clever nor stupid. I have known joys. That is saying too little: I am alive, and this life gives me the greatest of pleasure. Death, then? When I die (perhaps shortly), I will feel immense pleasure. I am not talking about the foretaste of death, which is insipid and often disagreeable. Suffering numbs the mind. But such is the remarkable truth of which I am sure: I experience boundless pleasure in living and will find boundless satisfaction in dying [j'éprouve à vivre un plaisir sans limites et j'aurai à mourir une satisfaction sans limites].
( _FJ,_ 9; 5)
These are impressively powerful and affirmative words. But they are less transparent than might at first appear. Despite the text's suggestions to the contrary, living and dying, like pleasure and satisfaction, are not absolute events or experiences; for they may be grasped or determined only within the limits that circumscribe them and constitute them as what they are. Necessarily, then, Blanchot's references to living and dying, pleasure and pain, inscribe those limits, but they do so seemingly in the absence of those limits. No sooner are the limits of experience implied than they are overwhelmed by the boundlessness of the affirmation that spills beyond them. The opening sentences, with their cautious clauses of denial, their moment of self-confessed understatement, and the allusion to the impending (impossible?) possibility of death, all serve to mark the limits within which the narrator is speaking; but paradoxically, by the very fact of marking these limits, Blanchot's sentences place themselves beyond them. To draw attention to such limits is to efface them, and the principle of limitation turns unaccountably into one of excess. Writing itself, of course, neither lives nor dies; and it is that very impossibility that enables it to locate itself, as here, within the space between the possibility of living and the possibility of dying as they present themselves to the narrator. But if writing can occupy the ground between life and death, it is only because of its essential limitations; for it is the essential irrelevance and impossibility of the difference between life and death as far as writing is concerned that allow Blanchot's text to test to the extreme the possibilities embodied in those two equally impossible states. Writing is an experience of the limit; and precisely because of this it is also an act of limitless affirmation. In other words, it is the limitedness of writing that is the key to its limitlessness. And the experience of dying, when it too comes, serves as an instantiation of much the same logic, as both a realisation of inescapable finitude and an occasion for boundless gaiety ( _FJ,_ 12; 7).
Blanchot's text experiments throughout with this strange, mutual implication of the experience of the limit and the limitlessness of experience. Telling stories, for instance, is a task that, as Blanchot's narrator reminds his readers at the end, falls only to those capable of reasoning with distinction ( _FJ,_ 37–8; 18); and it is indeed clear that, for a narrative to be articulated at all in a text, that text must endow itself with recognisable limits. The condition is one to which _La Folie du jour_ subscribes without difficulty; indeed it does so to excess, with the result that the narrative structure of the _récit,_ shot through with interruptions of one kind or another, remains painfully discontinuous and, as the 1973 title implies, seems never far from seeming quite deranged. At many points the _récit_ is like a long sequence of fragments, of disconnected scenes, elliptical beginnings, and moments of charged apostrophe, which the impressively staccato, incisive rhythm of Blanchot's prose manages to carry along with a violence all its own. At one stage, the narrator has a brief vision, but what promises to be an all-embracing moment of plenitude turns out to be no more than a peculiarly truncated – limited – glimpse of a man half entering, half leaving the entrance to a courtyard, allowing a woman with a pram to pass between him and the gate. 'This brief scene', comments the narrator with beatific enthusiasm, 'whipped me into a frenzy [me souleva jusqu'au délire]' ( _FJ,_ 19; 10).
Some pages later, with more dramatic effect, at the exact mid-point of the text, a telling fissure interrupts even this precarious account: the protagonist has glass crushed into his eyes, and he is briefly blinded by the excessive exposure to the light that ensues. With his eyes wrapped in bandages for the apocalyptic duration of seven days, he is forced to contend, not with the illumination afforded by the light (which has to be limited if normal vision is to take place), but with the blindness caused by the limitlessness of the light. Light here is what enables the narrator to see but also prevents him from seeing; and much the same double bind confronts the narrator when he attempts to narrate the event of the destruction of his sight: such an event, standing as it does at the centre of _the récit,_ is what allows the text to aspire to story-telling at all, except that such an event, no longer belonging to the realm of the visible, refuses to be narrated, and leaves the text of Blanchot's _récit,_ like the light itself, oscillating madly between being a condition of possibility and one of impossibility, simultaneously opening and closing the narrative itself ( _FJ,_ 22; 11).
What follows, however, serves only to intensify the narrator's desire to re-experience the boundless extremity of this impossible encounter, this encounter with the impossible:
Even when I was cured, I doubted this to be the case. I could neither read nor write. I was surrounded by Northern mists. But what was strangest of all was this: despite the memory of my awful encounter [le contact atroce], behind curtains and dark glasses I was slowly wasting away. I wanted to see something in broad daylight; I had had my fill of the ease and comfort of the shadows; I had the same desire for the daylight as for water and fresh air. And if seeing was fire, I demanded the plenitude of fire, and if seeing was an encounter with madness, I madly desired that madness.
( _FJ,_ 23–4; 12)
From this point on, _La Folie du jour_ begins to supply to its readers, almost for the first time, something resembling a continuous narrative thread; but, in order to do so, it is as though the text has to accomplish, after Hölderlin, a categorical return of its own, and turn aside from the fire of the heavens to speak with the sobriety and restraint of a Northern twilight. Indeed, the protagonist's brush with the madness of the light can only be narrated thanks to these belated, if piecemeal concessions to story-telling. And at this stage, the story falls victim to a double bind reminiscent of that experienced by Hölderlin in his struggles with Empedocles: on the one hand, to the extent that the story to be told hangs essentially on the limitlessness of the event that is, so to speak, its focal centre, the narrative must endeavour to speak of that which is immediate and impossible; but, to the extent that the limitlessness of that event is what makes the story impossible to narrate, the narrative must also turn aside from what lies at its centre to linger on its problematic margins, fissures, and borders. Such, for Blanchot, is literature's inescapable demand and its infinitely challenging resource: its affirmation of the limit and its responsiveness to the limitlessness beyond the limit.
But at no point do these two competing demands ever become reconciled by Blanchot within a unified dialectical structure. The light in Blanchot never turns into an originating moment able to incorporate within itself both its limitlessness and its own limits; it behaves instead according to an aporetical logic of originary repetition. The light, like the _il y a,_ always precedes itself both as what it is and as its own absence. The light, therefore, is always already something other than light; in itself, so to speak, it neither illuminates nor makes dark, since it is what enables these founding ocular metaphors to appear at all, and thereby relieves them of any other foundation than that of their own prior absence. In its very madness, Blanchot's _récit_ refuses the law of narrative in its everyday demands for clarity, luminosity, and illumination. While it repudiates this familiar figure of the law, which is 'severe and unpleasant' ('rigoureuse et peu agréable') ( _FJ,_ 29; 15), it does not transgress the law to encounter lawlessness, for it responds to the demand of another law, one that is glimpsed lurking behind the doctors' backs. This other law is very different from the repressive authority represented later in the _récit_ by the ophthalmologist and the psychiatrist who come to interrogate the narrator and demand from him a story. Personified as a singular, sexually differentiated, explicitly female character, this other law is a seductive and impulsive figure, whose persuasiveness rests, paradoxically, not on a set of legitimate and authoritative principles, but is strangely contemporaneous with the language of the narrator and his text; as the law herself declares: '"The truth is, we can no longer go our separate ways. I will follow you everywhere, I will share the same roof, we will enjoy the same sleep"' ( _FJ,_ 31; 15).
This encounter with the law constitutes what might be described as the _récit's_ transgressive moment. Transgression here is not a simple movement beyond the jurisdiction of the law of narrative accounting (or recounting), identified with the ophthalmologist and psychiatrist, towards the institution of a better, more authoritative statute. For one thing, as Blanchot's narrator discovers, there already is beyond that law of accounting another, more compelling figure, who calls herself the law and has little in common with the repressive authority of the state. The law, so to speak, is itself not single, but always at least double, which is one reason why this other figure of the law encountered by Blanchot's narrator is not described in terms of a universalising abstraction, but as a sexually differentiated, singularised female character. (As we have seen, it is this knowledge of the irreducible division at the heart of the law that constitutes for Blanchot one of the lessons of the Hölderlinian topos of 'Das Höchste', the Most High.) Whatever her frivolity and impenetrability, the demands of this other law are both inescapable and unanswerable. But rather than showing the remorseless uniformity and self-identity of the law, these twin traits more nearly draw attention to the essential insecurity and fragility of the law, its insatiability and persistence. This is arguably why, in _La Folie du jour,_ though the figure of the law is the one constantly harassing the narrator and inciting him to action – though without prescribing or imposing upon him what that action might be – it is the figure of the law who feels terrified and under threat, not the narrator. If the law is the Most High, she is also that which is most vulnerable; and if the law is both of these, this is because her demand for justice is always absolute, always singular, and as such can never be satisfied.
As a result, in _La Folie du jour,_ as in the poems of Hölderlin, the law always expresses itself as an irreducible double bind. The requirement, in Blanchot's case, is for the text to testify to the madness of the day by simultaneously supplying a narrative and doing justice to what escapes narrative; and while it is between these irreconcilable demands that Blanchot's _récit_ comes to be written, it is the result of the dissymmetry between them that it proves impossible for the text adequately to satisfy either one. This affirmation of impossibility is the measure of Blanchot's paradoxical enterprise; and it is by respecting the essential incompatibility between the two requirements that _La Folie du jour_ demonstrates the necessity of responding simultaneously and dissymmetrically to the need for narrative and the demand for justice. If literature is an act of transgression, one might argue, it is only to the extent that transgression is understood not as a breach of the law in the spuriously anarchic name of lawlessness, which results only in a reinforcement of the law, but rather as an act of contestation of the repressive authority of psychiatrists, ophthalmologists, and police superintendents, in the name of a more exacting demand, one which traverses literature without being identical with it, and which requires – impossibly – that one respect simultaneously both the limit and the limitlessness of the limit.
One can perhaps say now why _La Folie du jour_ is not simply an illustration of the arguments Blanchot presents in his critical essays on Hölderlin, and also why Blanchot's fiction is radically irreducible to the idea that the work of art is founded on anything other than its own impossibility. To have succeeded in either of these aims, _La Folie du jour_ would have to have been deducible from a single authoritative law, embodying a homogeneous structure of representation, based on the fundamental equivalence between the two moments or instances of the law portrayed within the story. However, the relation in Blanchot's text between the limitlessness of experience and the experience of the limit makes such a structure untenable. The paradox here is not solely decorative, nor does it serve simply to express scepticism; it is a response to the sheer impossibility of subordinating difference, discord, or contradiction to any single totalising concept. Paradox in Blanchot, one might say, is not the sign of a dialectical imagination, it is rather the proof that dialectical synthesis or unity is no longer even imaginable. The madness of the day, in this way, does not belong to the order of representation but effects a pure interruption of all representation. Like a Hölderlinian caesura, it intervenes into narrative as a vacant interval, a syncope that suspends representation in order to respond to the law beyond the law, the law that enables but also disables representation as such. If literature here proves transgressive of philosophical discourse, it is not because it can more, but rather because it can less; not because it confronts the law of representation from a position of greater authority, but because it turns aside from representation in order to affirm the other law – the law of the other beyond all law – that interrupts all representation.
# 3: Writing the neuter
## From work to worklessness
The time of the absence of time is not dialectical. In that time, what appears is the fact that nothing appears, the being that grounds the absence of being, which is when there is nothing, and which already is no longer as soon as there is something: as if there were beings only by virtue of the ruination of being, when being is lacking. The reversal which, in time's absence, constantly refers us back to the presence of absence, to this presence as absence, to absence as the affirmation of absence, an affirmation in which nothing is affirmed, in which nothing does not cease to be affirmed, with the monotonous insistence of the indefinite, this is not a dialectical movement. Contradictions do not cancel each other out, nor do they become reconciled; only in time, for which negation is power, is the 'unity of contraries' possible. In time's absence what is new renews nothing; what is present is no longer of the moment; what is present presents nothing, represents itself, and belongs henceforth and for all time to the movement of return.
'La Solitude essentielle' ( _EL,_ 21; 30)
In November 1947 and January 1948 Blanchot published, in two parts, what still remains his most programmatic philosophical account of literature in general. The first of these two papers was called 'Le Règne animal de l'esprit' ('The Spiritual Animal Kingdom'), the second 'La Littérature et le droit à la mort' ('Literature and the Right to Death'); and it was this latter title that Blanchot was subsequently to re-use for the essay as a whole when it appeared largely unchanged as one continuous text in 1949 in _La Part du feu._ The two original titles were not without significance. For while the phrase 'La Littérature et le droit à la mort' was loosely based on a quotation from Hölderlin's Empedocles, the title 'Le Règne animal de l'esprit', for its part, was explicitly borrowed, via Bataille and Kojève, from Hegel, namely from the first section of the third part of chapter five of the _Phenomenology of Spirit,_ the section entitled: 'Das geistige Tierreich und der Betrug oder die Sache selbst' ('The Spiritual Animal Kingdom and Deceit, or the "Matter in Hand" Itself').
As commentators have acknowledged, Hegel is the initial source of much of Blanchot's thinking in 'La Littérature et le droit à la mort'. Blanchot's own reading of Hegel is closely based on that of Kojève, whose _Introduction à la lecture de Hegel_ also first appeared in 1947 and to whose general thesis – that Hegel's philosophy is a philosophy of death – Blanchot subscribes with little, if any qualification. Moreover, Blanchot's specific interest in the section on the 'spiritual animal kingdom' in the _Phenomenology_ was itself prompted by Kojève's remarks on the passage in his _Introduction,_ and reinforced by Georges Bataille's own earlier response to Kojève which, in a famous draft letter of 1937, later reprinted in _Le Coupable_ ( _Guilty_ ), he himself describes as 'that of an animal caught with its leg in a trap'. The reason why Blanchot chose to focus his attention on the implications of this particular section from the _Phenomenology_ is not hard to see. For on Kojève's submission, within the general framework of the teleological movement of Hegelian spirit through history, what Hegel's account of the 'spiritual animal kingdom' sought to address was the moment of the inactive and uncreative literary intellectual.
'Man,' declared Kojève, 'the real presence of nothingness in being (time), is action, i.e. struggle and work: – this and nothing else.' But in Kojève's scheme, the literary intellectual fell far short of human destiny as defined in these terms. The artist, for Kojève, fails to engage in work in the everyday, real world; as a result, he is incapable of negating and transcending himself and remains only ever able to express his naturally given talent in a solitary manner; and this failure was what condenmed him, in Kojève's eyes, to a purely individual, unmediated, animal-like existence. The literary intellectual, for Kojève, though he might claim to sacrifice his own private interests to absolute, timeless values such as truth and beauty, was thus destined to miss the appointment with history and misunderstand the nature of the matter at hand. What he did instead, according to Kojève, was to perpetrate on himself and the world what amounted to a deception; he was thus guilty of what Hegel termed 'Betrug', and what Kojève – followed by Blanchot – rendered into French as: 'imposture', or fraud (the term recurs at a number of crucial moments in Blanchot's text). And Kojève concludes:
the Intellectual negates nothing, therefore creates nothing; he displays only his 'nature': he is a 'spiritual' animal (the spiritual animal kingdom). . . . The ideal universe he sets in opposition to the world is merely a fiction. What the Intellectual offers to others has no real value; he is therefore deceiving them.
In his letter to Kojève reproduced in _Le Coupable,_ Bataille on one level largely accepts this diagnosis; but, as befits a writer who famously describes himself elsewhere as having been 'broken, crushed, killed a thousand times over' ('rompu, broyé, tué mille fois') by Kojève's lectures, he dramatises the implications in a very different manner. One has to realise, Bataille observes, that Hegel himself also belonged, to a certain degree, to the spiritual animal kingdom. By accentuating the discrepancy between Hegel's authorship of the _Phenomenology_ as a relative and particular individual and the claim of his thought to enact the movement towards absolute knowledge, the remark was the occasion for a more far-reaching challenge. For if history was at its end, or at the very least finite, as Kojève maintained on the basis of his reading of Hegel, and if the essential driving force in history was, as Kojève asserted, what Hegel had conceptualised as negativity, then the problem remained of the ultimate fate of such negativity once history was over. On logical grounds, Bataille objected, history itself could not be relied upon to eliminate the negativity that, having provoked it into existence at the outset, necessarily exceeded its closure at the end. Negativity, he continued, was thus in fact irreducible to the dialectic of history it was meant to propel; instead of history's obedient servant, it more closely resembled a form of inescapable, originary violence.
From Bataille's perspective, the problem of the end of history was not an abstract or fanciful one; nor was it to be taken literally. Rather, by delimiting the restricted economy of the historical dialectic, it raised a fundamental question as to the status and significance of whatever fell short of that economy. More specifically, as Bataille suggested in his letter to Kojève, it threw into sharp new focus the issue of the meaning and necessity (or lack of meaning and necessity) of inner experience in particular, and art, sacrifice, and religion in general. These were the things, according to Bataille, that, having appeared at the dawn of history, were more than likely to survive its ending. From Bataille's perspective, Kojève's imperious dismissal of the literary intellectual served, then, only to dramatise more acutely still the puzzle as to the relationship between philosophy and art, history and experience, a puzzle Bataille formulated in response to Kojève by coining the – for Kojève – entirely self-contradictory notion of unemployed negativity, 'negativité sans emploi'. Such negativity, Bataille explained, is without historical purpose or justification; if it remains thinkable as negativity at all, it is a mode of negativity emptied of content and without possibility of absorption, negativity that has become synonymous with the impasse of its own violence, ineliminable excess, uselessness, and absence of truth.
Approaching this debate in his turn, Blanchot's own concerns in 'La Littérature et le droit à la mort'were at least twofold. First, there was an evident polemical purpose. This involved the need to formulate a critical response to one other important or influential account of the relationship between the literary intellectual and history put forward in the course of 1947: Sartre's _Qu'est-ce que la littérature?._ 9 In that essay, Sartre had sought to portray the writer of prose (though, symptomatically, not the poet) as an agent of history, subject to the necessity of choice and thus responsible for all his actions, including literary ones. 'Speech is action' ('Parler c'est agir'), wrote Sartre; and it is no exaggeration to say that the whole doctrine of 'engagement' or commitment was designed to assert this identification of prose writing with action in the world, and thereby assimilate literary prose to a moral project founded on intersubjective freedom and existential authenticity. In response to this, Blanchot for his part aimed to enlist the support of Kojève and Hegel in challenging the philosophical basis of Sartre's position; and Blanchot's essay may in some ways most easily be read as a sustained attack on Sartre's endeavour to equate literature with action in the world, and as a refusal of the normative implications that flowed from this subordination of literature to moral propaganda.
Like Sartre, Blanchot took from Hegel the basic principle that all action is negation; but, unlike Sartre, he went on to distinguish between the negativity characteristic of literature as such (and not just, in an impoverished version, of lyric poetry, as Sartre had allowed) and the negativity governing work in the world. If literature was action, Blanchot rejoined, it was action of a very different kind to the work performed by the ordinary citizen in the course of his or her daily labours. The main difference lay in the immediacy or absoluteness with which literature as a whole negated the totality of the world as such. Literature, Blanchot argued, was not subject to the same dialectic of progressive and necessary mediation as work that takes place in the world; it obeys instead a dialectic – or, more exactly, a pseudo-dialectic – that knows not an infinite number of mediated and graduated positions, but only two: All or Nothing. For Blanchot, literature is in fact not dedicated to producing meaning in the world; its goal is rather to abolish the world absolutely in order to put in its place an absolute absence of world and thereby to substitute for real, functional objects a series of imaginary, absent objects. By definition, literature could not be made subject to worldly, moral criteria such as authenticity or intersubjective freedom; to be a writer at all, for Blanchot, was to be committed to what, for Sartre, made sense only as verbalism and bad faith.
The only truly committed literature, Blanchot maintained, was the literature of Kafka or Sade, for theirs was a literature committed to the absoluteness of literature's negation of the world and, by that token, to the absoluteness of its self-affirmation in the face of the world. The fatal contradiction of Sartrean 'engagement' was that, by conceiving of writing as mediated action, it was unable to take seriously the peculiar specificity of writing as absolute negation (and affirmation). By imposing on literature a moral goal outside of literature, all Sartre's theory demonstrated, in spite of itself, by its failure to control reading and convince the audience of the truth of its position, was the inherent unreliability and ambiguity of literature, its propensity to turn into objects of fundamental irony the most fervently defended of moral or political beliefs. As Sartre was to discover in the ensuing decades, 'engagement' was thus at a cost: not of ruining literature, as Sartre's traditionalist critics had charged, but, more disturbingly, as Blanchot had argued, of fatally weakening those very values Sartre wished literature to embody, and which literature, by virtue of its ineliminable bad faith, was never able unambiguously to endorse. (Thus began, according to Blanchot, the irresistible process of disengagement recounted in _Les Mots,_ by which Sartre was finally to abandon literature altogether.) Literature and worldly action, for Blanchot, if they were to converge at all, might meet not within the pages of Sartrean 'littérature engagée', but, as we have seen, only on those rare, extreme occasions when the absoluteness of the negation and affirmation embodied in literature coincided with a similarly radical moment in the political sphere, as it did for instance at times of convulsive, revolutionary upheaval. This was the logic, as far as Blanchot was concerned, behind Sade's support for the Jacobin revolution and his contempt for all reformist half-measures; and it was the ever future prospect of further instances of this unmediated convergence between All and Nothing that testified to the solidarity that existed, for Blanchot, not between literature and day-to-day political struggle, but between literature and revolution, between that writing that could no longer recognise itself in the literary institutions of the past and the political transformation that is premised on radical refusal of what had hitherto been accepted as politics as such.
But if Blanchot was ready, in 'La Littérature et le droit à la mort', to exploit Kojève's account of Hegel's 'spiritual animal kingdom' for his own purposes in the polemic with Sartre, he was willing to accompany the Hegelian dialectic only up to a certain point. Ultimately, like Bataille in _Le Coupable,_ Blanchot turns Kojève's Hegel against himself. The work of the literary intellectual, rather than marking a failed moment within the progressive unfolding of the dialectic, comes to figure instead a recalcitrant impasse testifying to the failure of the dialectic; what for Kojève had turned into a tribute to the teleological possibility of the dialectic, is for Blanchot more like the affirmation of its necessary interruption. This strategy of inversion and displacement is one Blanchot uses throughout 'La Littérature et le droit à la mort'. At the outset, Blanchot reiterates the claim, made by Hegel towards the beginning of the chapter on the 'spiritual animal kingdom', that individuality is initially caught in a vicious circle, in which, Hegel argues, in order to know what the purpose of action is, individuality must first know what it is itself, but where in order to know what it itself is, individuality must first have acted. Blanchot in his text applies the paradox to the figure of the author, who is placed by way of a similarly vicious circle in the predicament of being unable to be a writer until he has written a work, but unable to write a work until he becomes a writer. In the end, of course, in order to begin at all, both Hegel's individuality and Blanchot's author are forced simply to act; yet what for Hegel is construed as a paradigmatic case of dialectical mediation and passage remains for Blanchot an encounter with the impossibility of beginnings and the interminability of writing; as Blanchot puts it elsewhere: 'In order to write, it is necessary already to write' ('Pour écrire, il faut déjà écrire') ( _EL,_ 184; 176). Hegelian Aufhebung turns here into a – literally – aporetic impasse, one Blanchot resolves in 'La Littérature et le droit à la mort' only by abrupt rhetorical fiat: 'Let us suppose', he writes, 'the work to be written: with it the writer is born' ( _PF,_ 297; 305).
Elsewhere, much the same pattern is repeated, with the result that Hegel's dialectic of spirit is rewritten by Blanchot as a series of discontinuous and incompatible demands rather than a progressive ascent towards absolute knowledge. As Blanchot observes of the writer:
The difficulty is not only that the writer is several figures in one, but that each of these moments negates all the others, demands everything solely for itself and accepts neither conciliation nor compromise. The writer must respond simultaneously to several absolute and absolutely different commands, and his moral code is made up of the encounter and opposition between implacably hostile rules.
( _PF,_ 303; 312)
In this way, what may appear at first sight to be a respectful homage to the logic of Hegel's text proves at a second look to more closely resemble a parody of Hegel, with the result that the Hegelian dialectic comes to be doubled in Blanchot's text with a pseudo-dialectic that is more like a logic of paralysis than of progression. What the word literature comes to name in Blanchot, therefore, is not a moment in the unfolding of the dialectic but rather an interruption or suspension of the dialectic. Indeed, as Rodolphe Gasché has convincingly argued, the more one reads Blanchot's essay, the more one is aware of how little in fact remains, despite appearances to the contrary, of Hegel's original project: no essence, no spirit, no Aufhebung, no resolution of contradictions, indeed no contradictions at all.
Blanchot, for his part, concedes in a footnote that his reading of Hegel does not aim to give a faithful presentation of the author of the _Phenomenology of Spirit_ ( _PF,_ 295; 302). But it would be wrong to conclude that Blanchot is pleading guilty to the charge of wilful misreading. With Hegel, Blanchot's strategy is much as it was with Paulhan or Heidegger: it is to accompany the text along a certain trajectory in order to propel it into an aporia of its own making. With regard to Hegel, the principle is one Blanchot formulates only much later, in _L'Ecriture du désastre._ There, he points out the danger of opposing Hegel directly; for to do so is inevitably already to endorse Hegel. This means of course that the only way to read Hegel's text, as it were, is to read it without reading it and, as Bataille had suggested almost forty years previously, to take as the guiding principle of that reading the necessary complicity between Hegel and the fraudulence of the 'spiritual animal kingdom':
It is not possible to 'read' Hegel, except by not reading him. Whether to read him or not read him, to understand or misunderstand, or reject him, all this is Hegel's doing or else does not happen. Only the intensity of this non-event [ce non-lieu], in the impossibility of there being one, readies us for the death – of reading, of writing – that leaves Hegel alive, in the fraudulence of finished Meaning [dans l'imposture du Sens achevé]. (Hegel as the fraudster [l'imposteur]: this is what makes him invincible, mad with his own seriousness, a counterfeiter of Truth [faussaire de Vérité]: 'putting one over on us' to the point of becoming, without realising it, a master of irony – Sylviane Agacinski.)
( _ED,_ 79; 46)
In 1947, then, Blanchot's purpose in reading Hegel (and Kojève) was not to use Hegel as a foundation for his own thinking, as has been too quickly assumed by some; his project was rather to explore the margins of Hegel's text and put the dialectic of history into relation – albeit necessarily a relation of non-relation – with what it forcibly excluded. This is why 'La Littérature et le droit à la mort' is written simultaneously as a commentary on Hegel and as an interruption of Hegel, and why Blanchot positions himself obliquely both inside and outside Hegel's system, which is to say neither inside nor outside, but along its margins and borders. It is no accident in this respect that Blanchot comments on the strategic infidelity of his essay to Hegel only from the edge of his own text, that is to say, from the window of a footnote. Indeed it is indicative of Blanchot's trajectory in the essay as a whole that of the text's five footnotes the first is devoted to Hegel (and Jean Hyppolite), the next two to Hegel and Kojève, and the last two to Levinas, whose name appears here for the first time in Blanchot's published work, as though to bear witness, from below the page itself, to at least one of Hegel's exclusions. The appeal to Levinas is a key move in the argument; for it allows Blanchot to deploy, with regard to the relationship between philosophy and literature, the deconstructive force of what, after Levinas, Blanchot calls the _il y a,_ and which he summarises in 'La Littérature et le droit à la mort', citing _De l'existence à l'existant,_ as
the anonymous and impersonal flow of being that precedes all being, the being that at the heart of disappearance is already present, that in the depths of annihilation still returns to being, being as the inescapability of being, nothingness as existence: when there is nothing, _there is_ being [quand il n'y a rien, _il y a_ de l'être].
( _PF,_ 320; 332)
The _il y a_ functions as an important crux for the whole of Blanchot's argument in 'La Littérature et le droit à la mort'. Its implications are many. First, as far as the reading of Hegel is concerned, the effect is to delimit the Hegelian dialectic and open it up to what in _L'Espace littéraire,_ speaking of Kafka, Blanchot calls the outside, and which has to be understood as that which precedes or exceeds the limits of the dialectic, yet which does not belong to the dialectic as a moment of necessary transgression. According to Levinas, as Blanchot reminds the reader, the _il y a_ affirms simultaneously both the presence of being and the presence of the absence of being. The _il y a_ always presents being, Levinas argues, and cannot itself be negated; indeed, since the necessity of affirmation always precedes the possibility of negation, any attempted negation of the _il y a_ , however it may begin, will always end, by an inescapable necessity, as a perpetual recapitulation – a 'ressassement éternel' – of the _il y a_ itself. Similarly, since it cannot itself be negated, the _il y a_ cannot therefore be assimilated within the mediating logic and ascensional dynamic of Hegelian Aufhebung. It remains irreducible to Hegelian conceptuality, and has to be seen as constituting a radical challenge to the totalising pretensions of dialectical thought itself. Like Bataille's 'négativité sans emploi', the _il y a_ refuses incorporation within the totality of history or time; and to the extent that it may be described as occurring in time at all, it is an event or absence of event that may be situated, in the vocabulary of _L'Espace littéraire,_ only in the time which is the time of time's absence, the time that is not the substance of spirit's culmination but of futurity, of deferral and return, recurrence and difference.
There is a second reason why the _il y a_ threatens the totalising closure of the Hegelian dialectic. For not only is the _il y a_ irreducible to Hegelian conceptuality, it is itself indispensable for the very possibility of dialectical thought as such. Indeed, precisely because the _il y a_ presents to thought the original possibility of both being and non-being, it is – like Heidegger's 'Nichts' in 'What is Metaphysics?' – what enables logical negation to be formulated at all. Without the _il y a,_ therefore, it is hardly conceivable how the dialectic might itself be possible. In 'La Littérature et le droit à la mort', then, as in the early texts of Levinas, the _il y a_ occupies an important double function; it is both unassimilable within the dialectic, yet necessary for the dialectic to begin; it both allows logical negation to occur yet refuses to comply with it; it is at one and the same time both what enables and what disables the possibility of the dialectic. It is the rock on which the dialectic is founded but also founders; it is in that respect not so much a concept as a pre-concept, or pseudo-concept, a trace or inscription that is itself, like that to which it refers, irreducible to the distinction between spirit and body, idea and matter, concept and thing. The _il y a_ was derived by Levinas, in the first instance, from Heidegger's celebrated distinction between being (Sein) and beings (das Seiende); and like Heideggerian ontological difference, the _il y a_ also belongs to both philosophy and something other than philosophy. This is clear from the self-contradictory, impossible status of the phrase; for what the _il y a_ seeks to name, in its total generality, is the pre-conceptual singularity of being. The condition is a peculiar one, but crucially, for Blanchot, it is a condition that the _il y a_ shares with literature. Indeed, Blanchot argues, what literature, in its turn, seeks as its object or goal is not the reality of the world as limply described by language, nor is it the pure concept brought into being by the annihilation of things by words, nor is it the sonorous echo of words released from the obligation to mean, but something more radical and originary still, since it is the very condition of all these other operations: the pre-conceptual singularity of things as they were before their destruction by words. As Blanchot writes:
[S]omething was there, which has now gone. Something has disappeared. How can I retrieve it and look over my shoulder towards what comes _before,_ if all my power consists in turning it into what comes _after_? The language of literature is a quest for this moment that precedes it. Generally, it gives this the name of existence; it wants the cat as it is, the pebble seen _from the side of things,_ not man in general, but this man and, in this man, what man rejects in order to speak of him, which is the founding of speech and which speech excludes in order to speak, the abyss, Lazarus in the tomb and not Lazarus returned to the light, the one already beginning to smell, who is Evil, Lazarus lost and not Lazarus saved and raised from the dead.
( _PF,_ 316; 327)
Literature here, for Blanchot, like the _il y a,_ dedicates itself, not to the resurrection embodied in conceptual thought, but to the unthinkable singularity that precedes the concept as its simultaneous condition of both possibility and impossibility.
Blanchot's position in 'La Littérature et le droit à la mort' is never simple, always double. Throughout, the purpose is to demonstrate that Hegel's famous thesis – that language, as Blanchot paraphrases it, is 'the life that bears death and maintains itself in death' ('la vie qui porte la mort et se maintient en elle') ( _PF,_ 324, 336) – is susceptible of two radically divergent readings, depending on whether death is seen as triumph or defeat, as the apotheosis of the labour of negation, or the rending apart of being by its own unmasterable limits. Such ambiguity, Blanchot insists, cannot be resolved, since ambiguity is always already its own only answer. Death here is the ultimate example – the example that cannot by definition be an example – of such radical ambiguity. Throughout Blanchot death insists, at one and the same time, as a figure both of extreme possibility and radical impossibility. Indeed, if the philosophy of Hegel, according to Kojève, is a philosophy of death, this is because death in Hegel is what supplies the key to all possibility as such; and Blanchot documents at length in his essay the implications of this argument, which are that language, and thus humankind in general, find their truth in the finitude embodied in death, because death is the source of the negativity that separates sign from object and by making language possible makes both humanity and literature possible, too: 'It is therefore entirely accurate to say that when I speak,' as Blanchot puts it, 'death speaks in me [la mort parle en moi]' ( _PF,_ 313; 323).
But if death in this way for Blanchot, as it was for Hegel and for Heidegger, is on one level the most proper of human possibilities, it is also something irreducibly other: it is an experience that is never accessible as such for any human self or subject, who is effaced in dying and unable to address dying as in any sense an individual, personal experience belonging to any self-present subject, self, or agency at all. Moreover, if death is the origin of all human possibility, it is an event that has as its own paradoxical outcome the very withdrawal of death from the realm of possibility. If death is possible, then, it is only with the effect of making death impossible. Death as possibility is abruptly inverted here, without mediation or transition, into death as impossibility; and the possibility of death becomes a limitless non-experience of the impossibility of dying. The encounter with finitude becomes an encounter with infinity, and the relation to the limit a relation with immeasurable alterity, a relation only conceivable, as Levinas will later argue, as a relation of non-relation. Death here quits the realm of the possible; it requires no ability, or power, or authenticity on the part of whoever dies, who forfeits all possibility in the act – which is not an act – of dying itself. Both as experience and performance, then, death is impossible. For Blanchot, as for Levinas, it ceases here to be a symbol or proof of humankind's humanity; it ceases even to be the possibility of impossibility that it was for the Heidegger of _Sein und Zeit_ ; it becomes instead, simultaneously but non-contemporaneously, the impossibility of all possibility.
Pursued to the limit in this way, the entire dialectic of possibility conceptualised by Hegel and by Kojève, accompanied at length by Blanchot in his writing, becomes unhinged. On one level, Blanchot's commentary ironically leaves much of the dialectic in place, while wildly exacerbating its effects in order to turn it into an unwieldy and lopsided embodiment of its own partial logic; but, on another, it serves to announce the arrival of an alterity – the alterity of 'désœuvrement', of worklessness or unworking – which is irreducible to all dialectic, and survives only as a residue that cannot be taken up into the process of negation, mediation, and work. Literature stands here divided between possibility and impossibility, between its ability to name things in their absence and conceptual immateriality, and the impossible obligation to respond in words to the pre-conceptual singularity of existence as it once was before language made this possible. Writing is traversed by fundamental ambiguity, split apart by two (or more) competing requirements, two demands, so to speak, that recur in one form or another throughout the entirety of Blanchot's work, but whose differences cannot ever be equalised, reconciled, or mediated by any dialectic whatsoever, be it of history or of being, but only ever affirmed in their fundamental incompatibility, dissymmetry, and discord. And it is to this that literature in Blanchot is called upon to bear witness.
## The worklessness of being
But what might be the difference (if there is one) between death by suicide and death by other means? The first of these, by entrusting itself to dialectics (entirely founded on the _possibility_ of death, the use of death as power), is the obscure oracle which we cannot decipher, but with whose help we can sense, while constantly forgetting it, that whoever has pursued to the limit the desire for death, invoking the right to death and having power of death over himself – thereby opening what Heidegger has called _the possibility of impossibility –_ or alternatively, believing himself to be master of what is beyond all mastery – allows himself to be caught in a sort of trap and is immobilised for all time – if only for an instant, obviously – at the moment when, ceasing to be a subject, losing his stubborn freedom, and as an other to himself, he comes up against death as that which does not happen or as that which (giving the lie to the dialectic, as in a fit of madness, by letting it reach its goal) reverses itself into _the impossibility of all possibility._
_L'Écriture du désastre_ ( _ED,_ 114–15; 70)
In _L'Espace littéraire,_ his fourth collection of critical essays, published in 1955, Blanchot develops the problematic opened up in 'La Littérature et le droit à la mort'. Like the 1947 essay, _L'Espace littéraire_ as a whole is framed by the question posed to being by the existence of literature as such; and like the earlier text too, it stages a prolonged encounter between a post-Hegelian dialectic of possibility and action in the world and the paradoxical logic of Levinas's _il y a,_ that ineluctable apprehension of the presence of the absence of being that Blanchot thematises not only as an interruption in the dialectic, but also as an inescapable demand and impersonal affirmation synonymous with the language of literature itself.
The question posed to being by literature is in the first instance, according to Blanchot, a question literature poses to itself; and it is for this reason that _L'Espace littéraire_ is principally concerned with the drama of literature's self-questioning in what, after Hölderlin, Blanchot terms a time of need or distress ( _EL,_ 258; 245). Blanchot's approach is driven, however, less by system and concept than by acute attention to the writing experience of a number of exemplary yet irreducibly singular literary figures, such as Mallarmé, Kafka, and Rilke. These are the protagonists in the drama Blanchot unfolds, and which begins, as it were, with a remark like that of Mallarmé, made in 1891 in a famous letter to Vielé Griffin, that the whole mystery of poetry lay in a word: the word _'c'est',_ 'it is' ( _EL,_ 35; 43). But this abrupt insight into the mystery of poetry, Blanchot contends, gives rise in Mallarmé not to a single realisation, but an experience of being that was necessarily divided. Mallarmé's poetic experience commences, then, like that of Hölderlin, not simply with the realisation of being, but with an awareness of the inescapable dissymmetry between the existence of the work of art as an object or entity that is, and the vast generality of being which necessarily withdraws from all particular determination, but alone suffices, precisely because it is not an object, as the properly intangible, mysteriously unavailable object of poetry as such.
The dichotomy at issue here is a familiar one to readers not only of Blanchot but of Heidegger too. For the dichotomy within being that Blanchot articulates in response to Mallarmé's remark is plainly derived from a reading of Heideggerian ontological difference. In _L'Espace littéraire,_ however, Blanchot does not linger for long with Heidegger; instead, much as he had in the late 1940s, he follows Levinas in recasting Heidegger's account of ontological difference as a relation between existence and the existent, between on the one hand the worldlessness, anonymity, and neutrality of the _il y a_ and, on the other, that suspension or interval in the _il y a_ which, in his post-war texts, Levinas refers to as hypostasis, and which marks the event-like emergence of time as an evanescent present and brings in its train the possibility of being in the world, positionality, subjectivity, consciousness, and freedom. Hypostasis introduces a break into the namelessness of existence by turning existence into an attribute able to be embodied by a speaking self; from within the impossibility of negation that is existence, it hosts the arrival, so to speak, of an existent, who comes to be by asserting responsibility for itself as a necessary consequence of its dependence on the inescapability of existence.
As far as _L'Espace littéraire_ is concerned, it is clear that much the same structure of difference and dependence as obtains between existence and the existent also governs the relation between the work of art and the worklessness that is the concealed origin of the work; and, indeed, Blanchot dramatises the genesis of the work of art in a manner that very closely echoes, in content if not necessarily in its vocabulary, Levinas's description of the emergence of the human existent from the anonymity of existence. In order to produce the work, for instance, Blanchot writes, the work that is an inaugural event as well as a possibility of communication, the poet must impose silence on the sterile repetitiveness of the origin and thereby master its interminable chatter; and the writer too must affirm from within the debilitating circularity of existence the possibility of a limit and a measure with which to sustain the work as such. In that movement, Blanchot continues, the work is born; yet, as it emerges into the world, the work has also to recognise its precarious vulnerability and its dependence on the very worklessness to which it owes its existence and which will, in the last resort, always ruin its own demand to exist ( _EL,_ 29; 37).
But though Blanchot adopts as his own much of the conceptual framework of Levinas's early writings, this is not to say he reproduces Levinas's argument without some characteristic shifts in emphasis. Blanchot, for instance, unlike Levinas, tends to assimilate the thought of hypostasis – and of all that is posterior, so to speak, to the moment of hypostasis – to a regionalised or delimited version of the Hegelian dialectic, which obviously, as a result, ceases to be a totalising dialectic at all, but survives nonetheless in Blanchot's text as a thought of being and non-being, of the necessity of death and freedom in the world. In addition, Blanchot underscores with arguably much greater force than Levinas not only the logical circularity of the _il y a_ but also, despite the negative terms in which it is thematised, the inescapably affirmative character of the _il y a_ ; writing for instance, at one point, of the powerlessness of the work of art, Blanchot rejoins that the work, in its very weakness, points to a realm in which impossibility is a mark not of privation, but affirmation ( _EL,_ 232; 223).
Though Blanchot's borrowings from Levinas (and Heidegger) are extensive, not all Blanchot's thinking in _L'Espace littéraire_ can be attributed to their influence. At least as much in the book is the result of a continuing debate between Blanchot and Kojève's Hegel (and, occasionally, Marx), largely mediated through the seemingly marginal but in fact crucial figure of Georges Bataille. As a result, in so far as Blanchot's writing may be presented in systematic fashion at all, it has to be said his position in _L'Espace littéraire_ is manifestly and deliberately heterogeneous. What Blanchot defends in the book is not systematic theory, but a persistently double logic which finds difference, discord, and extreme reversal at every stage in the argument it puts forward. On one level, _L'Espace littéraire_ puts forward a post-Hegelian dialectic of being and non-being, negation and affirmation, death and freedom, which is largely reminiscent of the dialectic left in place, if only partially or parodically, at the end of 'La Littérature et le droit à la mort' in 1947. On another level, however, the book disables and delimits this dialectic by showing it to be necessarily dependent on the – non-dialectical – impossibility of negation embodied in the _il y a,_ and which persists within the work, as the neutral force Blanchot terms worklessness, from where it continues to act both as a prior condition of the work's emergence and as a force of dissolution that cannot be embodied in the work, and therefore undoes the work and ruins its very possibility. As Blanchot writes, for instance, in a passage that appears convoluted only because of the extreme hypotactic intricacy of his thinking:
That the work should necessarily be the sole light of what is extinguished and by which all is extinguished, that it should come to be only when the extremity of affirmation is verified by the extremity of negation, this is a demand we can still understand, though it is contrary to our need for peace, simplicity, and sleep, indeed, it is a demand we understand intimately, as the intimacy of the decision that is ourselves and that gives us being, only when, at risk only to ourselves, we reject, by fire and the sword, by silent refusal, its permanence and favour. Yes, we can understand that in this the work is pure beginning, the first and last moment when being presents itself as the precarious freedom which in our sovereignty has us deny being [l'être], without, however, yet including it in the appearance of beings [dans l'apparence des êtres]. But as for the demand that makes the work into what declares being at the extreme moment of the break – 'the very word: _is_ ' ['ce mot même: _c'est'_ ] –, the point it brings to incandescence, even as the demand is itself consumed by the flame, this too we must grasp, and learn that it makes the work impossible, because it is what never allows the work to be reached, is what precedes the work, is the point at which, of being, nothing is made, at which nothing is fulfilled, and is the deep recess of the worklessness of being [la profondeur du désœuvrement de l'être].
( _EL,_ 39; 46)
Two demands, the demand for the work and the demand of worklessness, vie with each other within the non-dialectical movement of art's self-affirmation; the conflict however is unequal, and the outcome, though unexpected, is also a foregone conclusion. Despite this, the nature of the relationship between these two demands is crucial to Blanchot's thinking; and it is an issue he addresses explicitly in a paper entitled 'Le Regard d'Orphée' ('The Gaze of Orpheus'), which, in a prefatory note, he commends to the reader as the compelling yet elusive centre of _L'Espace littéraire_ as a whole. This duality of status is in turn itself reflected in the paper's figurative, mythological complexion, which places it mid-way, so to speak, between the fictional experience of night – 'l' _autre_ nuit', the _other_ night ( _EL,_ 169–78; 163–70) – as dramatised in _Thomas l'Obscur,_ and the philosophical account of literature as pre-conceptual singularity given in 'La Littérature et le droit a la mort'. By taking a detour through myth, 'Le Regard d'Orphée' aims to satisfy simultaneously the demand both for philosophical exemplarity and for literary singularity. However, by claiming to meet both requirements at once, it fulfils neither of them properly, with the result that, while on one level the story of Orpheus, as Blanchot configures it, indeed reprises and clarifies in theoretical terms many of the underlying arguments and themes common to Blanchot's work as a whole, it also functions transgressively as a _mise-en-abyme_ of the excessive, paradoxical logic it aims to describe – which is none other than the logic of law and transgression itself – and thereby disables the claim of philosophy or literary theory to be able to rescue literature from the otherness of darkness and bring it into the light.
In the story of Orpheus, the poet descends to the underworld to retrieve Eurydice from premature death; this, exceptionally, he is allowed to do on condition that he not look at her again. This condition Orpheus of course fails to respect, and he loses Eurydice again, and is himself subsequently torn apart and destroyed. The tale, in one perspective, Blanchot argues, is an exemplary account of the necessity of obliqueness and indirection, since it is the detour through partial blindness and invisibility that alone allows Orpheus to penetrate to the depths of the other night, that night before night, in order to bring Eurydice back from the dead. This law of indirection that requires Eurydice remain invisible is inescapable; for it is the condition of Orpheus' successful return to the light and of the work as such. That which is immediate, Hölderlin might have said, is impossible, and can only be grasped by way of a detour or turning. Which is to say, for Blanchot, that the work is itself primarily concealment, dissimulation: 'the depths do not give themselves up directly,' he writes, 'but reveal themselves only through dissembling themselves in the work' ('La profondeur ne se livre pas en face, elle ne se révèle qu'en se dissimulant dans l'œuvre') ( _EL,_ 180; 171).
But Orpheus disobeys this law of necessary obliqueness. He gazes upon Eurydice, and destroys the work; and it is as though what Orpheus yearns to bring back to the light, and which is the reason he cannot not look upon her, is not the Eurydice who will be restored to life and visibility, but – like the figure of Lazarus already beginning to smell to which Blanchot refers in 'La Littérature et le droit à la mort' – the other Eurydice who belongs to the other night and thus will always remain invisible, dying of a death that is without possibility or end. And it is for this other, irreducibly invisible Eurydice that Orpheus squanders both the work and his beloved's resurrection. This act of betrayal and sacrifice, which carelessly forfeits the work and seals Eurydice's fate, according to Blanchot, is itself as unavoidable as it is necessary; it is a response to another, more demanding requirement, to the law of the origin and of worklessness itself, which asserts that what is essential is not the work, but the darkness without which there would be no work at all, and in relation to which the work itself, though it may be the most extreme test of Orpheus' power as a poet, is of lesser and almost negligible importance: 'The work is all for Orpheus,' Blanchot writes, 'except for that desired look in which the work is ruined, with the result that only in that look can the work reach beyond itself, be united with its origin, and be consecrated in its impossibility' ( _EL,_ 182–3; 174).
From the point of view of the work, Orpheus proves culpably impatient and should have settled for bringing back Eurydice in song. However, as Blanchot reminds us, even in Orpheus' song Eurydice was always already absent and Orpheus himself already about to be dismembered for his singing. Orpheus' impatient gaze was therefore necessary. Without it, Orpheus would not have been exposed to the patience that is but another name for the care or concern for the origin of the work that is worklessness itself; had he remained patiently faithful to Eurydice, he would not have experienced the torment of limitless patience that is the origin of the work. Patience and impatience begin here, in Blanchot's text, to gather together and disperse according to a peculiar movement of similarity and difference, reciprocal intensification and abolition. At times, while it remains a calculated decision to refrain from precipitate action, patience seems like the very opposite of impatience; at others, when it is a refusal of the tempting lure of intervention, patience seems more like the only valid impatience that there is, the impatience that experiences to the limit the torment of the wait. Impatience also, though it too may take the form of a calculating act accomplished in order to abolish the necessity of waiting, may itself become transformed at the next moment into the most excruciatingly patient wait there is. And if at one moment he can write that 'true patience does not exclude impatience, it is intimacy with it, impatience suffered and endured without end' ( _EL,_ 181; 173), Blanchot too can go on to add, some three pages later, that 'impatience has to be the core of profound patience [le cœur de la profonde patience]' ( _EL,_ 184; 176).
Patience and impatience function here according to a pattern that is not without recalling once more the 'hyperbologic' of chiastic intensification found, as Philippe Lacoue-Labarthe has argued, at the centre of Hölderlin's theory of tragedy. Indeed, in this respect, Blanchot's Orpheus finds himself in an analogous position to Hölderlin's Oedipus: if patience is impatience deferred, it is also impatience intensified, and if impatience is patience deferred, it is also patience intensified. But though patience and impatience may seem here to function like dialectical contraries, all dialectic remains absent: there is no moment of synthesis or of unification, no time in which to reconcile the two, only the time of the absence of time which belongs not to progress but to return, not to work but to worklessness, not to the world, but to worldlessness. And this is why the movement of patience and impatience in Blanchot gives rise not to a dialectic of the work, but to an awareness of the fundamental dissymmetry between work and worklessness, as a result of which, in the rivalry between the demand for the work and the demand of worklessness, it is necessarily always the latter that prevails. Orpheus' sacrifice of Eurydice does not lead therefore to the work, but to the sacrifice of the work, and to the affirmation of the impossibility of the work as the secret of its origin.
## Outside
What occurs [ce qui arrive] through writing is not of the order of that which occurs. But, in that case, what gives you the right to claim that something like writing ever does occur? Or is it that writing might not ever need to take place [advenir]?
_L'Écriture du désastre_ ( _ED,_ 102–3; 62)
In the final section of _L'Espace littéraire,_ under the heading 'La Littérature et l'expérience originelle' ('Literature and the Original Experience'), Blanchot turns to the question of the relation between the work and the origin. This last chapter first appeared, in a substantially longer version, in two instalments in _Les Temps modernes_ in 1952. On one level – and this is probably the reason it was initially published in that journal – the text is a commentary on Hegel that covers many of the same issues as 'La Littérature et le droit à mort' and offers a similar critique of Sartrean 'littérature engagée' as that earlier article. In 'La Littérature et l'expérience originelle', however, Blanchot does shift the terrain of the discussion to consider the implications of Hegel's famous claim, put forward in the first of his _Lectures on Aesthetics,_ that, as far as the modern world is concerned, art is and remains, in Hegel's words, 'from the point of view of its highest determination, as far as we are concerned, a thing of the past' ('nach der Seite ihrer höchsten Bestimmung für uns ein Vergangenes'). Hegel goes on to explain that, having become divorced from its 'authentic truth and vitality', art in the modern age has forfeited its inner necessity and lost its prominent place within reality to become instead simply an object of literary critical, aesthetic inquiry. But while Blanchot explicitly cites Hegel's diagnosis as the starting point for his discussion, it is clear that since 1947 the debate has moved on; and there are numerous clues within Blanchot's text which suggest that, as well as a contribution to the debate with Hegel, Kojève, Sartre, and Bataille, this concluding chapter – particularly the sections entitled 'Les Caractères de l'œuvre d'art' ('Characteristics of the Work of Art') and 'L'Expérience originelle' ('The Original Experience') should also be read as an oblique response to Heidegger's recently published essay 'Der Ursprung des Kunstwerks' ('The Origin of the Work of Art'), which concludes with an Afterword devoted to a brief discussion of the self-same passage from Hegel's _Aesthetics._ 2
Admittedly, Blanchot mentions Heidegger's name, in a footnote, only once in the 1955 version of the essay (and not at all in the 1952 article), and otherwise refers to him, seemingly in passing in the body of the text, only as 'a contemporary philosopher' ('un philosophe contemporain') ( _EL,_ 251; 240). Indirection, however, is a law to which Blanchot subscribes not only as a critic of literature but as an essayist as well; and closer examination of Blanchot's essay clearly demonstrates how far 'La Littérature et l'expérience originelle' shares with 'Der Ursprung des Kunstwerks' a number of key emphases and formulations. Among others, these include the distinction drawn by Blanchot, here as elsewhere, between the work of art and what Heidegger terms equipment, and which is most likely derived, if not directly from _Sein und Zeit,_ from Heidegger's early Hölderlin lecture of 1936. Similarly, the view that art has its origin in art itself and not in the work of art nor the person of the artist, has a clear Heideggerian resonance, as does Blanchot's account of the structure of the work as resting on a violent contest between what Heidegger, in 'Der Ursprung des Kunstwerks', thematises as earth (Erde) and world (Welt), and which Blanchot, in his essay, describes as the antagonistic coming together within the work of art of what he calls elemental depth ( _EL,_ 233; 223) and the presence of being ( _EL,_ 235; 225).
Elsewhere, too, Blanchot is at one with Heidegger in rejecting all aesthetic philosophy – which necessarily means any aesthetic philosophy – founded on the opposition of subject and object, form and content. Finally, when he refers to the reader as one who consecrates or preserves the work, or, even more explicitly, when he argues that 'the work is history, it is an event, the event of history itself, and this is because its most steadfast claim is to give to the word beginning all its force' ( _EL,_ 238; 228), Blanchot is obviously echoing some of Heidegger's most provocative and original formulations. At one point Blanchot even uses the same pseudo-etymological pun when he refers to the work as an original 'leap', or 'Ur-sprung' ('saut', _EL,_ 257; 244); and in other places, even some of the same names recur, notably that of van Gogh, one of Heidegger's few – though problematic – examples, to whom Blanchot, with only the faintest of ironies, attributes the stolidly peasant-like, pseudo-Heideggerian pronouncement that 'My attachment is to the earth' ('Je suis attaché à la terre', _EL,_ 234; 224).
Of all these shared motifs that pass between Heidegger and Blanchot, the most important is without doubt the common reference to Hegel's _Aesthetics._ Indeed, for Heidegger and Blanchot, Hegel's remark that art today is a thing of the past identifies the essential predicament of modern art in time of distress. The two thinkers understand the thesis, however, in sharply differing ways. According to Blanchot, art after Hegel, so to speak, finds itself divorced from the truth of history for the simple reason, as Kojève had testified, that history in its essence has now been accomplished. The present and future belong not to art, but to action in the world; and if there is a book to be written in these modern times, Blanchot quips, that book is most likely not _War and Peace_ but _Das Kapital._ But though history has severed its relationship with art, and though art no longer belongs to the world and to the work and to truth, the question of art still remains; and it remains as an absolute question that can no longer be adequately met either by world, work, or truth. The time of distress may be a time of crisis for art, but it is a time in which, perhaps for the first time, the question of art is posed by art itself without immediately being referred back to some already established theological or philosophical discourse. Modern art's exteriority to world, work, or truth is grasped here not as a condition to be lamented, but rather as a chance for art. For if the time of distress is inseparable from the radical exteriority of art to world, work, truth, it is because, for Blanchot, that time has begun to resemble the time outside time of the origin itself, and because the origin of art is not to be found in the rootedness of art within the classical past, but in the worldlessness, worklessness, and truthlessness of the art of the future.
Heidegger, for his part, begins with what is in many respects a similar diagnosis of art's isolation in this time in which, as he puts it, following Hölderlin, the gods are both no more and not yet. Glossing Hegel's remark in his turn, Heidegger rephrases it as follows: 'Is art still an essential and necessary way in which the truth occurs [geschieht] that is decisive for our historical existence [unser geschichtliches Dasein], or is art no longer this?' The question, according to Heidegger, still hangs in the balance; and the time of distress is measured exactly by the uncertainty that grows for Heidegger, while the world hesitates between the truth embodied in a poem of Hölderlin's, to which, he complains, the moderns have failed to harken for already a century-and-a-half, and the benighted desolation that is the consequence of the world's domination by metaphysics and its monstrous offspring, modern technology. Modernity for Heidegger, here, is synonymous with the world's inability or failure to heed the truth of art; indeed, so dire is this time of distress, contends Heidegger in 1946, that it is not just the gods that have departed, leaving behind only a trace of the sacred, but the traces of that lost trace itself that are now barely legible at all.
Importantly, what for Hegel and Heidegger, for different reasons and in vastly differing ways, is evidence of the beleaguered autonomy or isolation of art, is for Blanchot the very beginning of art. For it is only when art is not reducible to world, work, or truth, Blanchot maintains, that the question of art can be asked at all, and asked not under the much diminished rubric of aesthetic pleasure and learned criticism, as Hegel had predicted in the _Aesthetics,_ but with direct relevance to the question of being in the time of distress itself ( _EL,_ 224; 216). For his part, Heidegger's response to the retreat of art from truth is to seek to ground afresh the relationship between art and world, art and work, art and truth. This he does of course by elaborating a radically new account of truth, truth not as _homoiosis_ or adequation, but _aletheia_ or unconcealment. From Blanchot's perspective, the gain here is also loss; what is achieved by Heidegger's redetermination of the work of art as the 'setting-into-work of truth', the 'Ins-Werk-Setzen der Wahrheit', is obfuscation rather than clarification, in that Heidegger substitutes for the radical demand of art an answer founded in world, work and truth, which are the very things, in Blanchot's eyes, that the existence of the work throws into question. Blanchot insists that the question be left open, that art be affirmed in its extreme refusal of world, work, and truth. And it is this Blanchot no doubt had in mind when beginning his discussion of Heidegger with a sentence that might appear to have come unchanged from 'Der Ursprung des Kunstwerks', but which, from Blanchot's pen, takes on a very different, not to say diametrically opposed meaning, and reads: 'A just answer is rooted in the question' ('Une réponse juste s'enracine dans la question' ( _EL,_ 219; 211).
A number of crucial issues divide Blanchot and Heidegger here. The first has to do with the relationship between art and the work. For Heidegger, though the two are construed as distinct, they are also treated as inseparable, with the result that, as 'Der Ursprung des Kunstwerks' testifies, the questioning of art by Heidegger inevitably culminates in a thinking not of art, but of the work. For Blanchot, however, to identify art first and last with the work, like Heidegger, is to place art exclusively on the side of worldly possibility, power, and truth, and to do violence to the very possibility of art, which lies in its fundamental absence from itself, its essential inessentiality, its paradoxical impossibility. (And it is also, Blanchot implies, to do violence to the refusal of the world enacted by, say, the poems of Hölderlin by enlisting them in the defence of the German Volk in its so-called time of distress.) True, the work for Heidegger functions as a foundational beginning, as an inauguration of itself and of what it founds, and to an extent Blanchot agrees; but if the work begins by beginning in this way, it can only do so because it is itself grounded (without in fact being grounded at all) in a worklessness which is the measure of the sheer impossibility of its beginning: 'The work says the word: beginning,' Blanchot writes, 'and what it claims to give to history is that initial right [l'initiative], the possibility of a point of departure. But itself it does not begin. It is always prior to any beginning, it is always already completed' ( _EL,_ 239; 228).
For Heidegger, however, if the work is a beginning, it is because the work has a privileged relationship to truth. Here Blanchot protests, and does so at his most incisive. Admittedly, truth, in Heidegger's text, is a concept that is new and difficult. It must first be taken as _aletheia,_ unconcealment; second, it has to be read not as established verity, but as the dynamic self-presentation of truth out of what Heidegger calls un-truth, _Un-wahrheit._ Truth, he writes, is in its essence un-truth. This does not mean that truth is falsity, nor that truth is born from its dialectical opposite, but rather that the truth of a work cannot be derived from received knowledge: 'The setting-into-work of truth', Heidegger explains, 'thrusts up the awesome and at the same time thrusts down the ordinary and what we believe to be such. The truth that discloses itself in the work can never be proved or derived from what went before.' ('Das Ins-Werk-Setzen der Wahrheit stößt das Un-geheure auf und stößt zugleich das Geheure und das, was man dafür hält, um. Die im Werk sich eröffnende Wahrheit ist aus dem Bisherigen nie zu belegen und abzuleiten.') And Blanchot comments:
As soon as the truth it is thought to yield is brought into the light of day and becomes the life and labour of the day, the work closes upon itself as something that is alien to this truth and without meaning, for not only in relation to established and reliable truths does the work seem alien, the very scandal of what is monstrous and un-true, but always it refutes the true [car ce n'est pas seulement par rapport aux vérités déjà sues et sûres qu'elle paraît étrangère, le scandale du monstrueux et du non-vrai, mais toujours elle réfute le vrai]: whatever this is, even though it may come from the work itself, the work overturns it, takes it back, buries it and hides it away. And yet the work says: beginning, and matters powerfully to the day. It is the first light that may be said to precede the day. It is an initiation, an inauguration. 'A mystery that inaugurates', says Char, but the work itself remains mysteriously excluded from initiation, exiled from the light of truth.
( _EL,_ 239; 228–9)
Blanchot goes on, some pages later:
In relation to the world in which truth founds and bases itself on decisive affirmation as a place from which it can arise, art is an original representation of the anticipation and scandal of absolute error [l'erreur absolue], of something un-true but where the prefix 'un-' does not have the incisive character of a limit [de quelque chose de non-vrai, mais où le 'non' n'a pas le caractère tranchant d'une limite], for it is rather the full endless indeterminacy on which the true has no purchase, which it has no power to win back, and faced with which it determines itself only by becoming the violence of the negative.
( _EL,_ 255; 243)
Un-truth, in Heidegger's sense, Blanchot argues, is a violence that belongs to truth in the world; for that reason alone it could not be more different from the absolute error – the absolute exteriority – of literature to all possible concepts of truth that Blanchot endeavours to address in the closing sections of _L'Espace littéraire._ Indeed, as Levinas points out, it is at those moments, when Blanchot seems closest to Heidegger, that he is at his most distant.
Art for Blanchot is inseparable from the risk it represents not only to work, world, and truth, but also to the existence of the poet as such. Here, as in his discussions of Hölderlin, Blanchot shows himself sensitive in the extreme to the artistic experience of the writers he examines. In this questioning of experience, the fundamental point on which Blanchot's writing turns, as I have said, is the (non-)experience of death. Heidegger, in famous pages in _Sein und Zeit,_ had written of death as the most extreme and proper possibility of Dasein; and in his account of Rilke in _L'Espace littéraire_ Blanchot reflects at length on that formulation. But as Rilke's work suggests, if death is extreme possibility, it is also extreme impossibility. In dying, no authentic self comes to be; rather, death cannot be experienced as such at all, and only results in the withdrawal of the possibility of dying. If death is impossible, it is because in death I do not die; the one who dies is the impersonal, anonymous 'on'. With death no relation is possible; death is that which is radically other. Indeed, death is not a constant or self-identical experience at all, rather the space of a violent reversal that tips without mediation from possibility into impossibility. In articulating death in this way, Blanchot in effect disposes of one of the key hierarchies in _Sein und Zeit,_ the distinction between Dasein proper ('authentic' Dasein) and the Verfallenheit of Dasein, that fallen anonymity that is life in proximity to third-person Heideggerian 'man', Blanchot's 'on'. Being here loses its relation to itself as possibility, authenticity, or propriety; it dissipates itself to a point of extremity where it becomes synonymous with the affirmation of its own impossibility; and this, most insistently of all, is what sums up the experience of the origin Blanchot dramatises in _L'Espace littéraire._
Art, for Blanchot, in the time of distress, does not lapse into dereliction, for it is only then that it discovers itself as what it is (not): as the worldlessness, worklessness, and truthlessness of the origin. The time of distress is a time of double infidelity to both humankind and the gods, and though Blanchot borrows from Heidegger some of the phases in Heidegger's historical scheme – the presence of the gods, the absence of the gods, the absence of the presence of the absence of the gods – Blanchot's prognosis differs crucially from that of Heidegger. While for Heidegger it was an article of conviction, as he puts it in his posthumous _Spiegel_ interview, that 'only a god can still save us', for Blanchot the time of distress is not a time of salvation or nostalgia for salvation, but the time of art, the time of the absence of time. Far from humankind needing a new god and a new divine, it is for Blanchot only the absence of the presence of the absence of the gods that allows the question of art to be raised at all. Art in time of distress is inseparable from the questioning to which there was and still is no answer; and it is this persistence of the question and the radical absence of the answer that Blanchot's writing affirms in the imminence and alterity of what Blanchot names elsewhere: a turning.
'Will you accept the certainty that we have reached a turning?' asks one of the interlocutors in _L'Entretien infini._ '– If it is a certainty,' replies his partner, 'it is not a turning. The fact of belonging to this moment at which a change of epoch (if any such exists) is taking place also affects the certain knowledge that would wish to determine it, making both certainty and uncertainty inappropriate. Never are we less able to circle around ourselves than at such a moment, and the discrete force of the turning lies first in this' ( _EI,_ 394; 264).
## Neutre
The Neuter, if it may be called neuter, could be said to be that which withdraws while withdrawing and withdrawing even the act of withdrawing [ce qui se dérobe en dérobant et dérobant jusqu'à l'acte de dérober], without anything appearing of what thereby disappears, an effect reduced to an absence of effect: the neuter, at the articulation of the visible–invisible, the inequality still of the equal, the answer to the impatient question (that classifies and determines in advance by incautiously dividing into two, asking: which one?), albeit an answer that immediately and imperceptibly, even as it seems to entertain the question, modifies its structure by its refusal not only to choose, but even to accept the possibility of a choice between two terms: such as the one or the other, yes or no, this or that, day or night, god or man. 'Which of the two?' – 'Neither the one nor the other, the other, the other', as if the Neuter could only speak in the form of an echo, yet thereby perpetuating the other by way of the repetition that difference, always included in the other, albeit even in the form of the bad infinite, ceaselessly calls forth, like the swaying of the head of a man exposed to endless commotion.
_Le Pas au-delà_ ( _PA ,_ 107–8; 77)
In 1963, in an essay on the use of adjectival nouns and substantival infinitives as expressions of the unknown in the poems of René Char, Blanchot wrote as follows:
Research – poetry, thought – relates to the unknown as unknown. This relation discloses the unknown, but in a disclosure that leaves it veiled; in this relation there is presence of the unknown; the unknown, in this presence, is rendered present, but always as unknown. This relation of presence must leave intact – untouched – what it conveys and not unveiled what it discloses. The relation will not consist in an unveiling. The unknown will not be revealed, but indicated. (La recherche – la poésie, la pensée – se rapporte à l'inconnu comme inconnu. Ce rapport découvre l'inconnu, mais d'une découverte qui le laisse à couvert; par ce rapport, il y a présence de l'inconnu; l'inconnu, en cette présence, est rendu présent, mais toujours comme inconnu. Ce rapport de présence doit laisser intact – non touché – ce qu'il porte et non dévoilé ce qu'il découvre. Ce ne sera pas un rapport de dévoilement. L'inconnu ne sera pas révélé, mais indiqué.)
Six years later, in the text of the essay reproduced in _L'Entretien infini,_ the lines given above read as follows:
Research – poetry, thought – relates to the unknown as unknown. This relation discloses the unknown, but in a disclosure that leaves it veiled; in this relation there is 'presence' of the unknown ['présence' de l'inconnu]; the unknown, in this 'presence' [en cette 'présence'], is rendered present, but always as unknown. This relation [ce rapport] must leave intact – untouched – what it conveys and not unveiled what it discloses. The relation will not consist in an unveiling. The unknown will not be revealed, but indicated.
( _EI_ , 442; 300)
From one version to the other, three changes have occurred: twice the word 'présence' has been replaced by '"présence"' , while in the phrase 'ce rapport de présence', the word 'présence' has been deleted entirely. These are not isolated cases. Elsewhere in the essay many similar modifications have been made. In 1963, for instance, using the infinitive form as a kind of hypothetical impersonal injunction, Blanchot had written of the requirement to 'relate to the unknown without unveiling it, by a relation of presence [par une relation de présence] that would not be a disclosure [qui ne serait pas une découverte]'; in 1969, however, while making the very same point, the sentence also came to state something more nearly resembling the opposite of what it initially asserted: it was essential, Blanchot wrote this time, to 'relate to the unknown without unveiling it by a relation of _non-presence_ [par une relation de non-présence] that would not be a disclosure' ( _EI,_ 443; 300; emphasis mine).
Presence, then, in Blanchot's essay, between 1963 and 1969, becomes '"presence"', or non-presence, or else is silently erased. Why these bizarre, multiple shifts affecting the one, apparently self-identical term? One explanation is that Blanchot became aware only belatedly of the inconsistency in his text between the concept of presence and the Heraclitean logic of veiled disclosure he was endeavouring to describe, and amended his text retrospectively to translate more effectively into words his original thought. Equally, the converse might also be true, and it is arguable that in the intervening years Blanchot simply changed his mind and, as Roger Laporte has suggested, withdrew the word presence largely as a result of his encounter with the work of Derrida, particularly Derrida's deconstruction of the privilege conferred on presence by a long-standing and tenacious metaphysical tradition. But historical or circumstantial explanations of this kind provide at best only a partial answer. For not only do they founder on the sheer impossibility of distinguishing between authorial intention and verbal expression, but also, more importantly, they fail to address the more fundamental issue as to what made Blanchot's rewriting of his text possible at all and thus enabled him, whatever the motivation, to rename as '"présence'", or 'non-présence', or by means of an erasure, what he had initially formulated as 'présence'.
On one level, Blanchot's deletion of the term 'présence' in 1969 was evidently not the result of an arbitrary decision; for it must have rested at the very least on the realisation that, from the outset, the word presence was deficient in some respect and already sufficiently lacking in self-presence, so to speak, for its inappropriateness to become manifest. Presence in 1963, one might conclude, was necessarily already sufficiently divided from itself for the precision and adequacy of its meanings to be queried; for only on that basis was it possible for the text to demand and then accommodate the alternative, antagonistic reading introduced by Blanchot six years later. Already in Blanchot's original text, then, the integrity of the word '"présence'" seems to have been fundamentally flawed; and one could go on to argue that 'présence' itself in Blanchot was thus already marked in advance in 1963 with the possibility of its qualification (or disqualification) as '"présence"' or as 'non-présence'. That this was so is already clear from the context of Blanchot's remarks at the time on the necessarily veiled character of the unknown. For it would follow from that account that what the author sought to indicate by the term 'présence' in 1963 was not in fact the fullness and immediacy of the present at all, but rather the withdrawal of presence from itself and the movement of self-differentiation that made such withdrawal possible and in turn allowed Blanchot in 1969 to overwrite the word 'présence' with the series of ghostly alternatives listed above.
However, even this reading remains somewhat unsatisfactory, for it too is ultimately dependent on the self-presence of Blanchot as the author of his own text and the stability of Blanchot's own proper name. For what is at stake here is in fact the status of that name and of all names in general. Underlying the very possibility of the changes Blanchot makes to his text on Char is a more general issue that has to do with the degree of autonomy, self-identity, or self-coincidence enjoyed by any name, in any text, once the conventional link between the name and that which it purports to name is severed or loosened, as it is for instance by the substitution of a term such as 'non-présence' for the 'présence' that preceded it. As that example implies, the act of naming is a good deal less reliable or definitive than it might appear.
To give something a second name, if only obliquely, is always in effect to challenge the adequacy of its first name. What is true of renaming here is necessarily also applicable to naming in general. To supply any name is always tantamount to erasing a name; and any name, it would seem, can only ever exist in response to a prior absence of name, which is also a demand for a name. Any act of naming is dependent therefore on what one might call a condition of original namelessness. But any such condition is the source of an irreducible double bind. On the one hand, it implies the possibility, indeed necessity, of naming in general, without which there would be no names at all, and not only no meaning or sense available to discourse. On the other hand, if what presides over the giving of names is a principle of original namelessness, it follows that the relationship between a name and what it names can only ever be a secondary or derived one, with the result that every name, so to speak, is only ever a name in name only. Indeed, if namelessness is what generates the obligation and possibility of naming, it must also be responsible for all those cases – which might in principle include all cases – where the names in question turn out merely to be provisional, false, or assumed ones. In this circumstance, there would seem to be little difference – other than a purely conventional one – between a proper name and an improper one. A fundamental aporia would come to affect the relationship between the name and its referent, with the result that the whole stability of conceptual discourse as such might be called into question, leaving the reader with no reliable criteria, other than purely linguistic ones, to distinguish between that which is simply 'presence', in Blanchot's text, and that which is '"presence"', even though the difference between the two, irreducible as it is itself to any received concept of presence, is indispensable – as difference – to the very possibility of thinking as such.
The act of naming behaves here according to two counterbalanced, but irreconcilable conditions. At one moment, in responding to the demand for naming, words are required to confer identity on what they name; at the very next, however, they find themselves compelled to yield to the lack of identity that gives rise to the name and inhabits it as a condition of its deployment. To the extent that they are meaningful, one might say, words are necessarily also meaningless. Namelessness is a condition that is exacerbated rather than appeased by the imposition of the name; and the fate of each and every name is thus to be ravaged to the core by the namelessness to which it owes its existence. If they are to function as names at all, one may conclude, names necessarily remain precarious and arbitrary, subject to endless multiplication and revision; rather than providing any stable point of anchorage in reality, what they trace in language is both the possibility and necessity of a process of constant slippage from one word to the next.
Namelessness itself is, in turn, of course only a name, and without the prior existence of names there would be no state of namelessness at all. To this extent, the original namelessness to which I have referred is in fact not original, nor is it indeed nameless. It functions in discourse not as a prior state of mute self-coincidence, but more like a dubious spectre of originality, whose status is secondary rather than primary, and whose purpose is to challenge the adequacy of words rather than confirm their authority. Instead of a self-present origin, it operates, as far as words are concerned, rather as an endlessly excessive possibility of erasure and substitution, deferral and re-inscription. Paradoxically, it is by withdrawing identity from words that language provides for their constant displacement, alteration, and proliferation; indeed, as Blanchot has often suggested, what from one perspective seems like the sickness of words, from another is no more than the very principle of their health. Words in this respect behave according to a homeopathic principle, which is evidence of the strange circumstance that, within words and beyond words, there is an infinite reserve of language (in both senses of the word) that is a source both of plenitude and absence and which results in the fact that while on the one hand language delimits, suspends, effaces, and puts at a distance, so in that very same movement it displaces, supplements, disseminates, and transforms. Language is inseparable from a fundamental paradox, one that traverses both name and namelessness, that implies that language is always able to say more, and less, than it is capable of saying. Language and sense are never commensurate with one another; just as there are always too many words for too few meanings, so there are always too many meanings for too few words.
It is this principle of paucity and excess, suspension and alteration, effacement and proliferation, that in Blanchot comes to be called the neutre. Despite the name, the neutre is not an entity, nor is it, properly speaking, a concept. It is rather a name for the namelessness of the name, a concept whose purpose is to conceptualise that which precedes all concepts. Neither immanent nor transcendent with regard to language, it is both a word and not a word, a modest trace that in its very discretion bears witness to the discretely nameless character of that to which it refers. In this way, like language itself, the neutre is perhaps best understood as a movement of perpetual effacement and re-inscription that is logically prior to all conceptual distinctions. It therefore cannot be subordinated to the opposition between the visible and the invisible, the present and the absent, the intelligible and the sensible, all of which it precedes, suspends and displaces. As such, it is in excess of all positionality, all principle of being (or non-being), subjectivity, or truth. But while being marginal to thought, it is also fundamental to it; for what it names, by a detour, in its own resistance to naming, is the unthinkable and impossible margin of thought itself, the alterity that is at the origin of all thought as such.
As Blanchot's treatment of a term such as presence indicates, what the neutre names in particular is the capacity of words silently to suspend and remark themselves, to arrest and displace their function, so that they cease to mean what they mean, but begin to oscillate uncontrollably between what they still do mean and the always other possibility that they mean something different, something that inhabits them as their own fundamental alterity. Blanchot's responsiveness to this unrepresentable excess – which is never present in writing as itself, but only as a perpetual movement of withdrawal and re-inscription – is one of the most striking aspects about the changes he makes to the text on Char, and the reason why, contrary to appearances, Blanchot in that text is in fact not mainly concerned to revise his text and rectify its sense by imposing upon it, retrospectively, some subsequent, teleologically determined truth, but rather, by means of a detour of (re-)naming that silently invokes the namelessness that inhabits each and every name in his text, to produce a kind of abysmal demonstration of the silent and abysmal workings of what is in fact the neutre itself.
This, then, is what is most clearly at stake in Blanchot's peculiar reinscription of the term presence as that which is always already less than present just as it is always already more than present. For '"presence"', in Blanchot's text, evidently does not quite mean the same as 'presence', but nor is it the same as 'non-presence'. It is, one might say, both presence itself, and something other (or less) than presence; it is presence mentioned rather than used, but thereby used rather than mentioned; as such, it is perhaps readable only as an enigmatic erasure of presence, as presence, so to speak, in its fragility as erasure, presence always already effaced and just about to be effaced, presence deferred and dispersed, transformed into a possibility of otherness, into a spectral '"presence"' or 'non-presence'. This other ' "presence"' does not belong to presence, since it is what suspends presence, postpones it, and separates it from itself. In so doing, what it does is to introduce into the writing of the word (assuming that '"presence"' is still a word), in the form of a set of suspensive quotation marks and an invisible erasure, a supplementary trace that, being itself beyond presence, is by that token also beyond being and non-being alike. What it presents to thought instead is an alterity that is beyond ontology. So if the neutre in Blanchot functions on one level as manifestation and erasure, it also exceeds those concepts, which it suspends and defers. Manifesting itself as erasure, the neutre discreetly erases that manifestation; erasing itself as manifestation, it manifests itself as an absent erasure that has always already disappeared.
The neutre, however, is not just a name; it is a response to that which always exceeds the name, which is why it gives rise in Blanchot to a discursive strategy or syntax founded on the movement of withdrawal and re-inscription to which one can also attribute each of the other rhetorical displacements – substitution, deletion, and interpolation – employed by Blanchot in rewriting 'René Char et la pensée du neutre'. As in the case of naming, these are not the preserve of rewriting as such, but are inseparable from writing in general. Here the neutre in Blanchot, to the extent that it serves to formulate the non-coincidence of language with itself, also operates as a name (without name) for writing itself, and just as the resources of the neutre are infinite, and infinitely oblique, so too are the possibilities – and impossibilities – of writing, and it is these, in response to the infinite alterity of the neutre, that Blanchot in his later texts explores at length, by fragmenting the page, by suspending generic boundaries and pushing them to the limit, by the systematic use of paranomasia to displace the borders between words and reconfigure the relations between them, and by dividing words from themselves in order to problematise such distinctions as that between use and mention, quotation and text, original and copy.
If the neutre is what makes words mean other than what they mean, it is arguably also what allows them to mean at all. And if meaning comes to be through the neutre, it follows that the neutre itself is beyond meaning. While the neutre is essential to the production of meaning, then, it also outstrips it; it refers both to the limitlessness of language and to the limit that gives rise to meaning. The neutre in this respect, Blanchot explains, is not diffidence or complacency, rather the reverse; indeed, he notes, the neutre is 'neutral' only 'in relation to meaning; not indifferent to it, but haunting the possibility of meaning and non-meaning by the invisible gap [écart] of difference' ( _EI,_ 448; 304). The neutre bears witness to the necessary excess of discourse over words, syntax over semantics, gesture over sense, saying over said, experience over subjectivity. By inscribing within writing the possibility of the always other word or non-word, the neutre is what turns any text – and not just Blanchot's own – into a palimpsest constantly divided from itself and dismantled from within by an alterity that it cannot internalise.
Words and names necessarily harbour within themselves the trace of their possible deletion and substitution by another name or word; by that very token, they also display in their inscription itself, and the necessary possibility of their effacement, the risk and chance of subsequent deletion, alteration, and transformation. For that reason, both as logic and as name, despite the singular number of the term, the neutre cannot be subordinated to the unity or totality of being. It therefore cannot be thematised as One, but only addressed as multiple exteriority and immeasurable alterity. This explains why ultimately the neutre cannot be contained in fact within the horizon of the possible, nor can the alterity it presents to thought be thematised in relation to possibility. The neutre necessarily invokes that which exceeds the possibility of names; it thus bears witness, in Blanchot's writing, not to the finite possibility of the other, but rather to the limitless alterity that, while it generates the infinite proliferation of names, is also beyond all possibility of naming. Intelligible neither as immanence nor as transcendence but infinitely other, the neutre is not just a name for the possibility of the other within being; it is a response to the namelessness and infinite alterity of the impossible beyond being.
Many are the words belonging to Blanchot's critical or philosophical lexicon that in 1969 come to be altered in response to the thought of the neutre. Indeed, the changes it prompts in Blanchot's texts, as they are revised and rewritten in detail, are far-reaching and systematic. On occasion, the word used to neutralise an earlier term is the word neutre itself. As such, the terms it displaces are most often words such as: 'impersonnel' (impersonal) or 'anonyme' (nameless or anonymous). Elsewhere, 'immédiat' (immediate) comes to be rewritten by Blanchot as '"immédiat"', 'être' (being) as '"être"', 'logos' (logos) as 'différence' (difference), 'authentique' (authentic) as 'juste' (just, valid, fair), and even 'désœuvrement' – suspected perhaps of being too easy a hostage to dialectical negativity – is deleted and re-inscribed as the phrase: 'absence d'œuvre' ('absence of work') ( _EI,_ 45; 32). But while these changes – together with the numerous other modification there is not space to mention – clearly demonstrate the extent to which, in the mid-to late 1960s, Blanchot embarked on a major reformulation of his critical and philosophical position, they also show, conversely, that what continues to count in Blanchot's text are not individual words or concepts, but, more profoundly, the very texture of his writing in the way it inscribes concepts only to displace and transform them. Blanchot is a writer of the oblique; and it is rarely a question of rejecting one set of concepts in favour of another, more a question of accompanying the available critical or philosophical discourses to the stage where they encounter – as indeed they must – the irreducible alterity that, albeit at different strategic points, inhabits each and every one of them. In the end, the only remarkable fact about Blanchot's rewriting of some of his own earlier articles in _L'Entretien infini and L'Amitié_ is that the available discourse Blanchot seeks indirectly to question in this way is found not in the work of another but in texts signed by the nameless name of Blanchot.
As far as Blanchot's philosophical itinerary is concerned, the emergence of the neutre in the texts of the 1960s marks an important turning point. As we have seen, the thought of the neutre, like Derridean différance, with which it shares many features, marks an attempt to re-articulate in a compelling new way a host of issues relating to the question of being, language, and the history of philosophy which, in one form or another, had been at the centre of much of Blanchot's previous work. By introducing into his work the possibility of a new style of argument and new (non-)conceptual language, the thought of the neutre gave Blanchot's writing an astonishing fresh creative and critical impetus, which, among other things, allowed the writer radically to re-examine some of the longest-standing philosophical underpinnings of his own discourse. Most prominent of all in Blanchot's engagement with contemporary philosophy were, of course, his readings of Heidegger and Levinas; and it is no surprise therefore to find that one of the first topics against which the thought of the neutre comes to be deployed in Blanchot was the question of being itself.
During the 1940s and 1950s, as I have argued, Blanchot's critical writing was largely reliant on the impersonality and self-defeating paradoxicality of the Levinasian _il y a_ to challenge the basis of Heidegger's foundational conception of Being. By the early 1960s, however, subsequent to his return from Èze and his renewed involvement in political activism, Blanchot, both in parallel with Levinas and independently of him, had begun to reformulate for his own purposes some of the implications of the thought of the _il y a._ Blanchot did this in terms that were somewhat different from those in which Levinas's own thinking had begun to develop, and this divergence is what was plainly at issue in an otherwise odd exchange between Blanchot and Levinas on the relationship between the neuter and Heideggerian Being. In a famous remark towards the end of _Totalité et infini_ in 1961, in which he declared his complete break with what he termed the philosophy of the Neuter, using the expression to refer to the thought of Hegel and Heidegger, Levinas credited Blanchot with having drawn attention to the impersonal neutrality ('la neutralité impersonnelle') of Heideggerian Being of beings ('l'être de l'étant'). But already some years before, in an article of October 1958, responding to an earlier paper by Levinas, 'La Philosophie et l'idée de l'infini' ('Philosophy and the Idea of Infinity'), Blanchot had paid Levinas exactly the same compliment; but at the same time, while plainly endorsing Levinas's criticisms of Heidegger, Blanchot had nonetheless baulked at the use of the term neutre, on the grounds, as he put it, that Heidegger's neuter was 'un Neutre un peu honteux' ('a poor man's Neuter', even 'an apology for a Neuter'). And it is as though, despite their common polemic against Heidegger, Heideggerian Being is for Blanchot not nearly neuter enough, while for Levinas it is already too neutral by half.
On one level the difference here is simply one of terminology; and the apparent inconsistency between Blanchot and Levinas on the subject of the neuter is merely the result of two divergent concepts of neutrality. For Levinas, since the neuter names the conceptual totality of Being, it follows that the neuter is inseparable from a reduction of the Other to the Same. For Blanchot, however, as I have suggested, the position is almost exactly the reverse, since the neutre, in Blanchot's perspective, by its refusal of all conceptual self-identity, is indeed what effects an interruption of totality. And as Levinas himself was later to admit, the neutre according to Blanchot is not an effacement of difference, but a mode of radical transcendence, more akin to what Levinas in later texts thematises as the infinite reserve of Saying ('Dire') as opposed to the Said ('Dit').
It is true that there are a number of wider issues at stake here between Blanchot and Levinas which reach beyond these questions of terminology, and which have a significant bearing on Blanchot's response to Levinas's later work of the 1960s and 1970s. I shall return to these in due course. In the meantime, it is perhaps worth noting the impact of the thought of the neutre on Blanchot's writing in general. Indeed, in Blanchot's thinking between the late 1950s and 1960s, there is evidence of a remarkable shift, prompted no doubt in part by Levinas, the nature and extent of which may best be measured by a brief comparison between two versions of a passage from the essay 'Comment découvrir l'obscur?' ('How to Discover the Obscure?'), published in November 1959, and incorporated in revised form in _L'Entretien infini_ ten years later. Reiterating the post-Hegelian notion that negation is possibility and that 'the struggle for possibility is the struggle against being' ( _EI,_ 67; 47), Blanchot then adds:
But should we not also say: impossibility is what in being has always already preceded being, being that always precedes itself and always ruins the decision of the point at which it may be said to reach completion? Certainly, we should! This would seem to imply that in possibility it is still being that lies awake, and that if it negates itself therein, it is in order the better to preserve itself from the _other_ assertion that always precedes it and is always more primal than that which names being, and which is the assertion the Ancients revered as Destiny, that which turns aside from all destination.
(Mais ne devons-nous pas dire aussi : l'impossibilité est ce qui dans l'être a toujours déjà précédé l'être, l'être qui toujours se précède et toujours ruine la décision du terme où il s'accomplirait? Assurément, nous le devons! Cela revient à pressentir que c'est bien l'être encore qui veille dans la possibilité et que s'il se nie en elle, c'est pour mieux se préserver de cette _autre_ affirmation qui toujours le précède et qui est toujours plus initiale que celle qui nomme l'être, affirmation que les anciens révéraient déjà sous le nom de Destin, cela qui détourne de toute destination.)
In _L'Entretien infini_ in 1969 the lines given above were revised as follows:
But should we not also say: impossibility, neither negation nor affirmation [ni négation ni affirmation], indicates what in being has always already _preceded_ being [indique ce qui dans l'être a toujours déjà _précédé_ l'être] and yields to no ontology [ne se rend à aucune ontologie]? Certainly, we should! Which would seem to imply that in possibility it is still being that lies awake, and that if it negates itself therein, it is in order the better to preserve itself from the _other_ experience [cette _autre_ expérience] that always precedes it and is always more primal than the assertion that names being and which is the experience the Ancients doubtless revered as Destiny, that which turns aside from all destination, and which we seek to name more directly in speaking of the _neutre._
( _EI,_ 67; 47)
Between these two versions of the 'same' passage, three important changes in emphasis and substance have been introduced. First, impossibility in Blanchot is no longer thematised as extreme affirmation, but redetermined, in accordance with the thought of the (non-dialectical) neutre, as that which escapes affirmation and negation alike and exceeds all such dialectical oppositions or contraries. Second, what is described in 1959 as that in being which precedes being, and is attributed by Blanchot to the interminable self-preceding or self-anticipation of being itself, has in 1969 lost its self-precedence and been displaced into an anteriority that has already preceded being (the deletion of the present tense and italicisation of the past historic are both symptomatic here) and thus can barely be assimilated to it at all, which is why Blanchot can declare it irreducible to any ontology whatsoever, including, presumably, the residual ontology contained in the _il y a._ The forces that had given rise to the worklessness of being, so to speak, instead of persisting as a recalcitrant part of being, have now seemingly seceded entirely from the domain of being (and non-being) as such. As they do so, in what is the third important change to the text, they become testimony to what Blanchot describes in 1969 as the experience of destiny, an experience to which Blanchot gives here the name of the neutre, and which functions in the argument as a figure of non-ontological alterity.
As he deploys the neutre to redescribe Destiny in this way, Blanchot once more encounters Heidegger. Indeed, it is difficult not to read the reference to the figure of Destiny as a direct response to Heidegger's account of that which, in Parmenides, under the name Moira, is deemed to embody the presence of the present that is the destiny of Being, 'the gathered and thus unfolding destination of presence as the presence of the present' ('die in sich gesammelte und also entfaltende Schickung des Anwesens als Anwesen von Anwesendem'). But while Moira in Parmenides serves Heidegger as a means of developing the concept of ontological difference in terms of the twofold of presence and the present, and thus of thinking the manifestation of Being as such, Blanchot reads it very differently. Indeed, in its oblique resistance to destination as truth, the destiny that is the neutre is a turning aside from presence. Destination is thus not identified with the manifestation of presence and the truth of Being. Instead, as an instantiation of the neutre, the figure of destiny for Blanchot becomes a figure of limitless wandering beyond being or non-being, and a trace of that which has always already turned aside from itself and always already bears the mark or re-mark of an encounter not with being, but with impossibility and the other.
The thought of the neutre accelerates and reconfigures much that had been in preparation in Blanchot's writing of the 1950s, but had also remained implicit within it. In particular, it marks a decisive abandonment of all points of ontological principle, even one as precariously self-defeating as the _il y a._ The neutre in Blanchot is no longer here an apology for being; and as the centre of gravity of Blanchot's writing shifts, so the question of being – as simultaneously in the work of Levinas – becomes secondary, relegated as it were to subsidiary status by the question of the other. This is the displacement in Blanchot's thinking that is most clearly responsible for the radical new emphasis adopted in many of the texts collected in _L'Entretien infini._ The thought of the neutre allows Blanchot not only to pursue the delimitation of Hegelian dialectics that the logic of the _il y a_ had already allowed, but also to suspend the question of Being as formulated by Heidegger both in the fundamental ontology of the 1920s and 1930s and his subsequent critique of ontology, and, in so doing, to begin to dismantle the commitment to unity and totality that, for Blanchot, under the name of the One, remains the common and abiding legacy of the metaphysical tradition as such. Blanchot points this out in a footnote from 1969 which carries much of the philosophical weight of _L'Entretien infini_ as a whole; as Blanchot writes:
dialectics, ontology and the critique of ontology share the same premise: all three rely on the One, either because the One is what accomplishes itself as all, or because it understands being as a gathering, and as the light and unity of being, or because, above and beyond being, it affirms itself as the Absolute. Faced with these assertions, should we not say: 'the most profound question' is the question that escapes reference to the One? It is the other question, the question of the Other, but also the always other question.
( _EI,_ 34; 440)
No longer synonymous – unlike the _il y a_ – with the unity of being, the neutre is necessarily irreducible to the truth of Being in its Heideggerian sense; indeed, from the very outset, as we have seen, the neutre is affirmed by Blanchot as that which is indifferent to the movement of concealment and unconcealment, veiling and unveiling, that is constitutive of Heideggerian _aletheia._ The neutre has nothing to do with being or non-being; it signifies without concealing or unveiling, and it does not belong to the realm of the visible or invisible. As Blanchot reminds his readers, speaking is not seeing; and there is little doubt that, in rejecting that optical metaphor, Blanchot was endeavouring to break not only with Heidegger's account of the manifestation of truth in particular, but also with Western metaphysics and Greek thought in general. And it was to that end that Blanchot in all his writing began to reaffirm the radical primacy and irreducible alterity of that in language that was beyond language and belonged to the outside, by virtue of worklessness, poverty, errancy, fragmentation, or multiplicity.
The impact of the thought of the neutre on Blanchot's work was arguably threefold. First, from the mid-1960s onwards, it allowed him to articulate, within philosophy, what in retrospect has all the hallmarks of a distinctive deconstructive strategy. Second, the neutre gave him the means to reformulate the terms of the challenge put to philosophy by literature and writing as such. This Blanchot did in a sequence of essays written from the early 1960s onwards, principal among which were his two articles from 1964 on Kafka, 'Le Pont de bois' ('The Wooden Bridge') ( _EI,_ 568–82; 388–96) and 'La Voix narrative' ('The Narrative Voice') ( _EI,_ 556–67; 379–87), and a long essay on André Breton and Surrealism published three years later under the title 'Le Demain joueur' ('Tomorrow at Stake') ( _EI,_ 597–619; 407–21). In those essays, and others written during the same period, literature comes to be seen by Blanchot in a surprising new light. No longer is the fate of literature bound to the question of being in time of distress; instead, according to Blanchot, it now appears that literature has already intervened to suspend the alternative between being and non-being, and that, by passing beyond that alternative, writing has begun to respond to an urgency, a demand, that cannot be thematised in relation to being or non-being, but figured only as the infinite relation of non-relation with the Other. As Blanchot suggests, perhaps tentatively still, in 'La Voix narrative':
all language begins by making statements and in making statements affirms. But it may be that to produce a narrative (to write) is to draw language into a possibility of saying that would say without saying being and without denying it either [qui dirait sans dire l'être et sans non plus le dénier] – or again, more clearly, all too clearly, to establish the centre of gravity of speech elsewhere, where speaking would neither affirm being nor need negation to suspend the work of being that is accomplished ordinarily in all forms of expression. In this respect, the narrative voice is the most critical of all, which, unheard, might give to be heard [la plus critique qui puisse, inentendue, donner à entendre].
( _EI,_ 567; 386–7)
This shift beyond being and non-being is an essential one for Blanchot. In its discretion and reserve, what it marks is the turning of a change of epoch. Under the impetus of the neutre, literature puts itself at a distance not only from all immediacy, but from all mediation too, according to the detour that constitutes it as always other to itself. Irreducible either to visibility or invisibility, writing here may be said neither to veil nor to unveil; it calls into doubt the logic of manifestation and all the optical metaphors that have hitherto dominated all philosophical thinking on truth, language, meaning, literature, art. As writing for Blanchot suspends here all relation both to being and to non-being, it dissolves the dependence of thought on the figure of the One. The multiple and the fragmentary in writing are what come to take the place of the unity of being or thought, effacing the appeal to the totality of presence in order to invoke a future without present or presence, a future that is dedicated, in Blanchot, not to identity, meaning, or self, but to the infinite alterity of that which is unknown, to a beyond that corresponds to an idea neither of transcendence nor of immanence.
The implications of this deep-seated displacement in Blanchot's thinking were not limited to his essays on literature; they had a dramatic effect, too, on Blanchot's own practice as a writer of literature. Indeed, just as the key turning point in Blanchot's early career was arguably the experience of writing _Thomas l'Obscur, Aminadab,_ and _Le Très-Haut,_ so perhaps the crucial moment in Blanchot's writing during the late 1950s and early 1960s was his protracted exposure to fragmentary writing forms, forms that, as in the case of _L'Attente L'Oubli_ or _L'Entretien infini_ itself, were no longer subject to generic distinctions such as the division between narrative and discursive prose. Under the attraction of the neutre, writing in Blanchot comes henceforth not only to be thought, but practised as well, according to a (non-)fundamental (non-)principle of radical fragmentation, fragmentation construed neither as dismemberment or failure, presence or absence, but as the only exacting response, beyond being and non-being, to the limitless infinity of the outside. To write the neutre is thus always to interrupt being and non-being alike, and announce in words the promise, the coming of the always irreducible, dissymmetrical other.
## July 1948: writing, dying; dying, writing
Rilke's assertion that there is, so to speak, a double death, or two relations to death, one commonly called authentic and one inauthentic, echoed as it is in philosophy, only expresses the _doubling_ within which an event such as death withdraws as though to preserve the emptiness of its secret. Inevitable, but inaccessible; certain, but ungraspable; that which gives meaning, nothingness as power of negation, the force of the negative, the end on the basis of which man is the decision to be without being, is the risk that rejects being and is history and truth, death as the furthest point of power, as my most proper possibility – but also the death which never happens to me, to which I can never say yes, with which no authentic relation is possible, which I evade at the precise moment I think I can dominate it through resolute acceptance, for then I turn away from what makes it essentially inauthentic and essentially inessential: in this perspective, death disqualifies 'being _for_ death', it does not have the solidity to sustain such a relation, it is indeed that which happens to nobody, the uncertainty and the indecision of what never happens, to which I cannot give serious thought, for it is not serious, but is its own fraudulence, a crumbling away, a self-consuming emptiness – not the term, but the interminable, death not as propriety but as anonymity, not true death but what Kafka calls 'the sneer of its capital error'.
'Rilke et l'exigence de la mort' ( _EL,_ 161; 155)
In July 1948, some months after 'La Littérature et le droit à la mort', two works of fiction by Blanchot appeared simultaneously. The first, _Le Très-Haut_ ( _The Most High_ ), was classified as a _roman,_ the second, _L'Arrêt de mort_ ( _Death Sentence_ ), as a _récit._ This genre distinction, which has numerous precedents in French and in other European literatures, Blanchot glossed in 1954 in an essay entitled 'Le Chant des sirènes' ('The Sirens' Song') that, under the title 'La Rencontre de l'imaginaire', was later used to open the volume _Le Livre à venir._ 2 Crucial to the difference between the _roman_ and the _récit,_ Blanchot suggested, alongside all the more usual criteria concerning conventions of plausibility, complexity of plot, and narrative voice, was the relationship to time: while the _roman,_ having its origins in the rich diversity and familiarity of the world of human experience, belonged to the time of the everyday, the _récit_ on the other hand existed only in relation to another time, a time outside of time and at a distance from time, a time that is the time of the _récit_ itself. As a result, Blanchot argued, the _récit,_ unlike the _roman,_ has the curious property of existing only as an enactment of the singular event that constitutes the _récit_ itself; and this is why, as Blanchot put it, there is paradoxically no such thing as the _récit_ in general, but also why there are so many _récits._ As Blanchot explains:
The _récit_ is not the relation of the event, but that event itself, the approach to the event, the location in which this event is called upon to take place, as an event still to come and by virtue of the attractive power of which the _récit_ can itself hope to become reality.
( _LV,_ 13; _SS,_ 62).
As Blanchot's publisher's note indicates, _Le Très-Haut_ and _L'Arrêt de mort,_ though very different, share some common threads. These range from numerous recurrent thematic motifs and similar narrative cruxes to broader philosophical and political concerns, all of which echo back and forth from one text to the other like reciprocal quotations, though in point of fact the two works intersect explicitly on perhaps no more than one occasion. _Le Très-Haut,_ which is staged in the first instance as an ironic allegory of Kojève's end of history, begins by exploring the relationship – both discrepancy and interdependence – between political modernity and ancient familial myth, between the conceptual realisation of the state and the persistence of distant pre-historical origins which come to be expressed in graphic bodily or physical terms, as 'something very old, criminally old' ('quelque chose d'ancien, de criminellement ancien') ( _TH,_ 58; 54). As the novel proceeds, this complex rivalry between modern reason and primordial myth comes to be figured as a relationship between the law and its outside, between the immanence of the political and the transcendence of death, disease, and love; and indeed, in its final stages, as Blanchot's would-be Kojèvian state is beset with arson, plague, and open rebellion, the novel finds itself dramatising the circular logic by which, in the novel, all opposition to the state ends up becoming, by dialectical reversal, a reassertion of the power of the state; and at the same time the novel endeavours also to transcend this circle by introducing into the fiction a relation of radical otherness, embodied at the end in the love between Jeanne Galgat, his carer, and the protagonist Henri Sorge, the Most High of the title, a love enacted in the mortal alliance which the pair contract at the end, at the very moment when death seems both absolutely inevitable and perpetually impossible.
By comparison, though it shares many of these concerns, _L'Arrêt de mort_ is a far more elliptical and episodic work, and one that dispenses with even the residual paraphernalia of plot and counter-plot, character, and setting, found in _Le Très-Haut._ The result is both enigmatic discontinuity and infinite fascination, and this is arguably one reason why, of the two books, indeed of Blanchot's fiction as a whole, _L'Arrêt de mort_ is the one text that has attracted by far the greatest volume of critical commentary. The story is made up of two semi-autonomous first-person narratives. The first relates the death, then the apparent resurrection, and, finally, two days later, the second, now definitive death, brought about by a final injection administered by the narrator, of a woman friend named in the text simply as J. The second narrative, in which only the narrator and the subsidiary figure of the doctor in fact reappear, is apparently a continuation of the first, and tells of the narrator's subsequent relations with a series of women – named in the text as C(olette), N(athalie), and S(imone) – which culminate in a scene in the subway during the bombing of Paris, in which the narrator, entirely out of character, as he admits, finds himself impelled to propose marriage to N(athalie) in a language not his own, in which he can barely make himself understood; there follows a strange and climactic encounter between the pair that night in the narrator's hotel room, until one week later, to the narrator's horror, Nathalie tells of having had a cast made of her hands and face the very day of the bombing by the same sculptor as J. had done in the earlier part of the book, and whose name Nathalie had found in the narrator's wallet, at which point, save for the brief, metatextual epilogue deleted from the 1971 version of the text, the narrative abruptly ceases.
Apart from the highly charged and deeply allusive character of the events in the book, what is most striking of all in _L'Arrêt de mort_ is the strange logic of suspense and completion – of suspense as completion and completion as suspense – governing the narrative; this is reflected most obviously of course in the bizarre double meaning contained in Blanchot's title, which utilises an otherwise perfectly commonplace French expression – were it not, as Blanchot had put it seven years before, that commonplaces themselves might best be seen as words torn apart by lightning – and which is made here to signify both (idiomatically) a sentence prescribing death and (literally) a suspension in the enactment of death. The strategy is one Blanchot exploits to great effect in other ways too, and _L'Arrêt de mort_ draws throughout, in an understated yet knowing way, on a vast range of commonplace intertextual material of mythical, biblical, historical, political, literary, and philosophical provenance. _L'Arrêt de mort,_ in this respect, is very much a sum of many previous texts, and it is a text that seems to offer itself to the reader in several readily accessible ways: as fictional memoir, as ghost story, as love story, even as a _roman personnel_ in the tradition of Constant's _Adolphe._ 6 But each of these angles of approach is quickly found wanting, and none of them proves persuasive enough to circumscribe Blanchot's text. Commonplaces, in _L'Arrêt de mort,_ are less a source of clarification than of obscurity; read as a personal memoir, for instance, the _récit_ soon turns into a memoir without a person; and the case is much the same with each of the other approaches: if the book is a ghost story, say, it is a ghost story without ghosts; if a love story, a story without lovers, and so on.
The effect is of a text in which literary conventions are simultaneously incorporated yet frustrated; each enigma proves, as the narrator puts it on one occasion, to be no more than the shell of an enigma, but all the more puzzling or fascinating as a result. This process of allusion and disappointment, withdrawal and displacement is everywhere in evidence in _L'Arrêt de mort_ ; but it is particularly striking in the numerous references made throughout, explicitly or implicitly, to specific historical dates. Of all Blanchot's _récits, L'Arrêt de mort_ is the only one seemingly to be located at a particular historical moment, and there are many tempting parallels to be drawn between the events that unfold in the _récit_ and the contemporary historical events mentioned in the book. Most suggestive of all among these historical references are those made, at the beginning of _L'Arrêt de mort,_ to the Munich accords, signed at the end of September 1938, the aftermath and consequences of which seem to form the background to the events surrounding J.'s death and apparent resurrection in the early hours of the night between Monday 11 and Tuesday 12 October, and her second death twenty-four hours later, early on the morning of Wednesday 13 October 1938. Also evoked by Blanchot's text, with a similar degree of historical precision, is the first bombing of Paris on 3 June 1940, during which the meeting in the subway between Nathalie and the narrator takes place, and the day on which, it is realised in retrospect, she had the cast taken of her hands and face. And no doubt some significance too attaches to the fact that Blanchot's narrator, in a state of intense 'désœuvrement', first writes down his story, only then to destroy it immediately afterwards, during the aftermath of the Fall of France, in the first months of the Occupation, i.e. throughout July and August of 1940 ( _AM,_ 8; 1).
There are many other similar moments in _L'Arrêt de mort_ when private or personal events seem in some way to be encoding a reference to events occurring on this broader historical stage. At one point, for instance, it is reported that J. had been sick for at least ten years, i.e. since 1928, much like France itself in the eyes of the early Blanchot ( _AM,_ 12; 4); that J.'s doctor had considered her dead since the time of the Popular Front ( _AM,_ 13; 5); and that, on the same authority, the narrator in 1940, who had been existing on borrowed time since Munich, had only six months to live ( _AM,_ 14; 5). To these possible parallels between the personal and the political may also be added a number of other, covert literary allusions, among which might be, as Michael Holland has argued, a reference to _Nadja,_ André Breton's quasi-autobiographical _récit_ of 1928, in which case Blanchot's _récit_ may be read as a critical reworking of Breton's 'always future book', as Blanchot calls it ( _EI,_ 615; 419), which not only deals with the way in which Breton himself is literally haunted by a series of unexpected chance encounters, but also gives an important role in the plot to the afternoon of 13 October 1926. And it is not impossible too, as Derrida and others have proposed, that Blanchot in _L'Arrêt de mort_ is also drawing on a series of more esoteric, private allusions, perhaps including a reference to the life-story of Bataille's lover, Colette Peignot, better known under the name Laure, who died of tuberculosis at the age of 35 in November 1938, three-and-a-half weeks after the fictional J., and whose dying words – 'La rose!' – which Bataille recorded in a diary entry for 12 October 1939, are strangely reminiscent of J.'s in Blanchot's story ( _AM,_ 44; 25). The rose, as Blanchot notes some years later, was of course the Orphic emblem par excellence, the very symbol of poetical activity and of death ( _EL,_ 163; 157).
There seems little doubt, then, that Blanchot in _L'Arrêt de mort_ was seeking to set the events in the story within a much larger historical and intertextual frame; and the many readers who have endeavoured to decode the _récit_ as an allegory containing a political or historical message are in this respect following an interpretative strategy already in part suggested by Blanchot's text itself. Indeed, the principle of convergence between private and public experience is itself clearly established from the outset by the fact that Blanchot's narrator, like the author himself in earlier years, is a political correspondent involved in the day-to-day writing down of historical events. Here, though, some of the ambiguities of the story begin. For in his capacity as a dutiful journalist, Blanchot's protagonist cuts a strangely detached and irresolute figure. He admits for instance that, as the events of Munich unfolded, instead of urgently hurrying back to Paris, as public and private responsibilities demanded, he for no obvious reason continued to linger in the holiday resort of Arcachon, with the result that, on this occasion at least, the narrator's involvement in history was, paradoxically, in the form of his absence from it.
For all its apparent arbitrariness, the incident is far from an isolated one, for a similar disjunction or dissymmetry between the narrator's private endeavours and the broad canvas of contemporary history is also in evidence when the narrator agrees to act for a friend in a duel arising from some purely private matter of the heart, on (or about) the very day in 1940 when, Paris having been abandoned by the government, the capital was declared an open city and left to the devices of the occupying forces ( _AM,_ 117; 73). Elsewhere, too, the parallels between historical events and the events occurring in Blanchot's _récit_ are more problematic than it might seem. Whereas an allegorical, historico-political interpretation of the text might require, for instance, that J.'s proposed cure be read as a symbol of the drastic remedy accepted by Chamberlain and Daladier at Munich, the fact is that Munich in reality delays J.'s treatment and makes it impossible for it to be administered at all. Munich in this regard is more condemnation than salvation; and while this certainly emphasises to what extent, in Blanchot's own view, Munich, by vainly attempting to defer war, served only to hasten it, it also suggests that between historical events and private experience there are more discrepancies than there are correlations; indeed, in the light of J.'s heroic resistance in the weeks that follow Munich, and given that the narrator's voluntary absence from Paris, which, he claims, made the events surrounding her death possible ( _AM,_ 24; 12), was in spite of Munich and not because of it, one might equally conclude that personal experience serves not to confirm history, but rather to contest it, and that it is only a simplistic reading that attempts, reductively, to deduce the one from the other.
The relationship between public and private events in _L'Arrêt de mort,_ then, needs to be examined with some care. But remarkably enough, in the haste to establish correspondences between the two series of occurrences, critics of Blanchot's _récit,_ it seems, have ignored what is arguably the most peculiar feature of all regarding what, in the opening pages of _L'Arrêt de mort_ ( _AM,_ 11; 4), the narrator claims to be the only date of which he is sure, and which marks the day on which J., having apparently already died once, dies again for the second time: Wednesday, 13 October [1938]. A cursory glance at the calendar quickly reveals that in fact 13 October 1938 fell not on a Wednesday at all, but on a Thursday. The date that is given in the text as the only one of which the narrator is certain is therefore decidedly uncertain. At the very moment when, by way of the date, Blanchot prompts the reader to scour the text in order to forge an encounter between private experience and historical reality, the possibility of that encounter, in the form of the calendar, is in fact withdrawn. History as such and the events in the _récit_ prove irreducible to one another; and it is far from evident that J.'s death, to the extent that it takes place within time at all, occurs within the same temporal frame as that occupied by the historical record. This disjunction is not simply a case of metafictional playfulness on Blanchot's part, for it addresses a dissymmetry that is very much at the heart of J.'s whole existence; indeed, when, at one moment, commenting on the awkwardness with which she endeavours to die, her doctor is heard to exclaim: '"Ah! par exemple"' ('"Typical!"', _AM,_ 46; 26), what is immediately clear is that the life of J., like her forthcoming death, is exemplary or typical only because it is neither exemplary nor typical, and that her death, if it is interpretable at all, belongs not to the teleology of history, but to the unspeakable, the irremediable, and the impossible that exist only within the margins of history.
There were, and are, of course, many years when 13 October, the date mentioned by Blanchot's narrator, did fall on a Wednesday; these include, among others, the years 1937, 1943, and, most curiously of all, the one in which _L'Arrêt de mort_ was itself published, 1948. As far as the reader is concerned, the last of these is perhaps the one that disturbs the most, for at the very moment it appeared to be anchoring itself reliably in the past, Blanchot's _récit,_ when first published, seems to have been looking forward, covertly, to its own ghostly repetition. Inevitably, as the contemporary reader knows, the future of that repetition was destined in its turn to become past; and what _L'Arrêt de mort_ reminds us, by the use it makes of the ghostly date of Wednesday 13 October, is not only that the past is a condition of the future, but that the future is a condition of the past, and that in addressing the past _L'Arrêt de mort_ was also addressing the futurity implicit within that past. Temporality here is repetition and recurrence, a factor that empties time of all presence and denies the narrative that is history any monopoly either on meaning or experience. Indeed, what seems clear is that, if it indeed takes place as claimed by the text, i.e. on Wednesday 13 October 1938, J.'s death cannot be said to take place within history, or in relation to any present moment at all; so if it does take place, then, it does so only within Blanchot's writing, in which it occurs without occurring, so to speak, as an event that is simultaneously made possible by the structure of repetition and difference that is the calendar and yet, equally well, rendered impossible by the calendar: Wednesday 13 October 1938 exists within time, one might say, only to the extent it exists outside of historical time. This is perhaps why, as far as _L'Arrêt de mort_ is concerned, each event is also always already a non-event, and why what transpires in the _récit_ is not only that the time of narration takes precedence over the time of the plot, but also that the time of reading takes precedence over the time of narration, with the result that reading itself, in its very futurity, becomes a re-enactment of the absent events of the _récit._ Whence, perhaps, in the closing pages of the 1948 version of _L'Arrêt de mort,_ the fearsome warning, were the reader to supply an end to the text in exchange for its effacement, that the reader's end would be the narrator's beginning, and that there would cease to be any escape at all from this infinite round of inscription and erasure.
The temporality of writing implied here is a temporality that contests the teleology of history and of historical narrative. Writing instead is in the form of a series of repetitions or recurrences in which to advance is always already to return, and in which an event such as the death of J., though it marks a date and is marked by a date, belongs in fact neither to past nor to present nor future, but rather to another time, beyond presence, between past and future, in relation to which time itself, like the certain date of J.'s death, is continually, so to speak, beside, outside, or beyond itself. Blanchot's _récit_ is thus not a story dedicated to memorialising the past in the belief that narrative is truth, nor is it a work bound confessionally to the future as a moment of deferred revelation. It is a text that belongs to a time outside of time which is the time of return. Return here, however, is not a function of permanence or identity, but of the alterity which is the possibility of time itself. So if _L'Arrêt de mort_ is a text about historical events, it is not about the priority of history over experience, but about the repetitive futurity of events as such, their status as promise or condemnation, tragedy or hope. To adopt the terms of the narrator's encomium to the timeless moment that now concludes _L'Arrêt de mort,_ if Blanchot's _récit_ is about boundless unhappiness, it is also about boundless affirmation; if it is about the necessity of time, it also about the interruption of time. What it thus inscribes within time and finitude, so to speak, is the infinity of a timeless instant.
If the structure of temporal return is what is at stake in the dates everywhere to be found in _L'Arrêt de mort,_ the same is also true of the narrative structure of the _récit._ This too is in the form of a series of potentially endless repetitive enactments which imply both finality and incompletion, time and timelessness. From the outset, the reader is told, Blanchot's narrator has tried several times, in novels and in other books, to give written form to the events of which he is required to speak ( _AM,_ 7; 1). But on each occasion the weakness and unreliability of words have conspired to undermine the writer's confidence in the possibility of ending. Instead of expressing truth, the narrator concedes, all words have ever done is to withdraw from truth. To write at all, for the narrator, is to write in the knowledge that the whole truth will always remain inaccessible; but if this is so, it is not because the truth is an embarrassing or unmentionable secret, but because truth, as far as words are concerned, is what has always already been lost. This implies no nostalgia, only the realisation that the absence of truth is the very secret of the possibility of words. The secret however is one that, by definition, cannot be uttered; it is in fact, to that extent, anything but a secret.
It is here, with the secret that resists words precisely because it is the secret of words, that Blanchot's _récit,_ much like 'La Littérature et le droit à la mort', in fact begins. As it does so, it is necessarily obliged to provide a name, albeit a name without name, for that unspeakable secret. That name, however, is not hard to find, for it is the name without name of death itself. As elsewhere in Blanchot's work, death in _L'Arrêt de mort_ is both extreme possibility and extreme impossibility, finitude and infinity, limit and limitlessness, experience and anonymity, meaning and meaninglessness. It is, as it were, the only secret worth revealing precisely because it is not a secret and cannot be revealed. The resulting paradox has many implications for language as such, and, in their dealings with Blanchot's narrator, words, in _L'Arrêt de mort,_ behave accordingly. They are keenly aware of the limits imposed upon them by the simple fact they are just words; but they realise too that those limits are not only the end, but also the beginning of their very existence, the basis of both their possibility and their inexhaustibility, their impotence and endlessness. And this is why, as the past recedes from the narrator's grasp and refuses to be spoken, the absent past imposes itself upon him nevertheless as the condition of his very speaking; the events surrounding the dying and the impossibility of dying traversed by J. become like a repetition of a past that, lying beyond the limits of language, carmot be spoken but which, to the extent it gives rise to the limitlessness of language, everywhere has to be spoken, albeit therefore in its absence, in silence, and beyond possibility. The logic of Blanchot's text, like the remedy proposed to J. by her doctor, an eccentric versed in Paracelsus ( _AM,_ 19; 9), is not allopathic, but homeopathic. Death, like life, knows no opposite and cannot be cured except by itself. What is true here of dying is also true for writing; and the only antidote to the limitations of language proves to be the limitless exposure to those limits.
Writing here, then, becomes an insistent repetition of that which by definition it cannot say, and what Blanchot's _récit_ does, in the inability to narrate the events of the past, is in fact to repeat them as a process of endless incompletion or limitless finality. Repetition functions accordingly as both failure and compulsion, worklessness and demand. The past, so to speak, is what never happens to the extent that it is what has always already happened, in exactly the same way that death, precisely because it always already belongs to the past, is only ever the non-event of its own impossibility. In the same way, the story Blanchot's narrator in _L'Arrêt de mort_ is impelled to tell is a story that never happens as an event present to itself, and this is not only why none of the events in the story ever really takes place except as a ghostly occurrence or repetition of another event, but also why as a result the narrative is in fact more a narrative of the very limits of narrative, and why the narrator's story continues perpetually to take place as a constant repetition of itself. Whence, in _L'Arrêt de mort,_ the elisions and interruptions in the narrative, the perpetual recurrences, and the process of replication or duplication that turns each person or event into a version of another; each event, it seems, is like a memory or a premonition that is full of meaning only to the extent that it is simultaneously empty of all meaning.
Once Blanchot's _récit_ begins, then, the story proves to be both singular and multiple, vestigial and compelling; and what takes place does so according to a movement of constant recurrence and repetition. Even before it is realised, for instance, that the _récit_ as a whole is in two parts that mirror each other, the reader is introduced to a story involving two sisters, not just one. Later, the narrator, on a mysterious impromptu visit to S(imone), finds that she lives in an apartment comprising 'a single room, without a hallway, divided in two by a large curtain, with one side for the day, the other for the night [d'un côté pour le jour, l'autre la nuit]' ( _AM,_ 72; 43), and it is difficult in retrospect not to see the text of _L'Arrêt de mort_ itself similarly split between day and night, _jour_ and _nuit,_ J. and N(athalie). The text as a whole is full of suspended vignettes of this kind; towards the beginning of the book, for instance, in similar fashion, Blanchot provides a _mise-en-abyme_ of the story to come in the form of a brief scene, said to have taken place towards the end of 1940, which involves an unnamed woman and a mysterious object – perhaps the lifeless cast of J.'s once living hands, though this is nowhere confirmed – concealed in a wardrobe, the sight of which, as though by contagion, provokes in the woman a reaction – convulsive shaking and a deathly rattle – which recalls the process leading to J.'s death while, as far as the reader is concerned, also foretelling those earlier events. The effect is uncontrollable recurrence, a movement of return in which, as by the delayed action of some mysterious poison, the future turns into a ghostly premonition of the past and the past into a deathly repetition of the future.
To this temporal inversion is added even greater and more disturbing intensity as a result of the identity of the object glimpsed by the woman. Nowhere is it revealed what this is; and the only indication that the so-called 'living proof' relating (but how?) to the events in the story, and which lies hidden in the wardrobe, is indeed the cast of J.'s hands, is the narrator's admission that he can see those hands before him as he writes and, as he puts it, 'that they are alive [qu'elles vivent]' ( _AM, 21;_ 10). But if the object in question is the cast of J.'s hands, the effect, if taken together with the closing invitation to the reader (in the 1948 version) to imagine the hand responsible for writing _L'Arrêt de mort_ itself, is to frame the narrative as a whole with an enigmatic double reference to hands: to hands that are dead and alive because they have been uncannily re-embodied in statuesque effigy, and to hands which are alive and dead because they are the hands that write.
Writing features here in two guises, as extreme affirmation and extreme dispossession, as that which passes beyond the limit of living and that which exposes itself to the limitlessness of dying. Throughout _L'Arrêt de mort,_ the motif of hands recurs time and again as a bodily figure of this double belonging of writing to living and dying and thus, in a sense, of its subservience to the demands of neither, its responsiveness instead to that which has no name except for the nameless name of the impossible. Thus, for instance, the statement of J.'s last wishes, scribbled down by her by hand, setting out what was to happen in her own final absence ( _AM,_ 13; 5); her handwritten correspondence with the narrator, possible only in his absence, but mysteriously sustaining her in the repeated encounter with death ( _AM,_ 24–5; 12); and the lines in J.'s hands, preserved in the cast as a monument to both life and death, but offering beyond life and beyond death not only a kind of physical inscription of her discontinuous life-story, but also a reminder of her final communication in death – never more alive, nor more lucid, says the narrator – which is to squeeze the narrator's hand 'with all the affection and tenderness she could' ( _AM,_ 51; 30). Thus, too, the scarred hand of Colette, full of mystery and significance ( _AM,_ 60; 35); and, most dramatic of all, in the dark, the exchange of hands, as from Orpheus to Eurydice, between the narrator and the unnamed N., sealing the pact of love that binds them to one another momentarily, eternally, across the foreignness of their languages, for the duration of a story that concludes with the knowledge that N.'s hands, too, like those of J., are alive only to the extent that they are also destined to die, and that, ominously, as though in advance of that end, N., too, has had a cast made of her hands in a procedure that, the narrator points out, is startling and dangerous when performed on living people ( _AM,_ 120; 75), no doubt because it changes them into the survivors of an impossible death, a realisation which is not limited to those who have casts made of their hands, but one that extends, by contagion, to all those who are subject to the temporality of return, be it through speaking, writing, or reading.
Early in the first part of _L'Arrêt de mort,_ reference is made in passing to a photograph of the Turin shroud, hanging in the doctor's surgery, and displaying two overlaid or interwoven images, the one showing the face of Christ, the other that of Veronica, the woman credited traditionally with wiping Christ's face after the crucifixion, thereby imprinting on the cloth, in blood, the image of Christ's death. The description is another suggestive _mise-en-abyme,_ and one that invites the reader of Blanchot's _récit_ everywhere to seek within each seemingly truthful icon (the etymology of the name Veronica) the traces of duplicity and otherness. For in this particular photograph, the features of Christ serve not only to confirm death and announce the possibility of resurrection, but also reveal in that very possibility the assertive force of a woman's face. The point here, however, is not to profess a belief in resurrection in any theological sense, but to indicate how, in death, uncontrollable contagious forces belonging neither to life nor death are inscribed within works of art, overwhelming limits and crossing boundaries, such as to undercut a traditional image of spiritual transcendence with a decidedly adverse image of earthly pride and beauty. The implications of this are not lost on Blanchot's _récit,_ for not only does the story proper, relating the narrator's first encounter with J., begin with her mysterious appearance one night in his hotel bedroom, seemingly in response to a sense of the threat of dying hanging over him, an almost complete stranger ( _AM,_ 16; 7), but it continues with a whole series of similarly mysterious comings and goings in rooms at night, bearing witness to the fragility of spatial distance and the overwhelming force of unconscious desires: at one point, the narrator bursts in on Colette, his neighbour, apparently mistaking her hotel room for his own ( _AM,_ 59; 35); shortly after, a strange presence, that of Nathalie, materialises uninvited in the early hours of the morning in the narrator's own hotel room ( _AM,_ 64; 37); later still, for no good reason, the narrator finds himself in the middle of the night pushing open the door to S(imone)'s apartment in her absence ( _AM,_ 72; 43); until, finally, once again, the narrator's own hotel room finds itself mysteriously penetrated by N. who, having appropriated the key from the narrator, locks him out, only for him, to his manifest astonishment, to force the door open by striking it with his fist ( _AM,_ 107; 66), upon which there follows the climactic, almost immobile scene of the exchange of hands.
Moments of repetition, recurrence, or return of this kind are everywhere to be found in _L'Arrêt de mort._ But when textual motifs are repeated, what is it that repeats itself? 'Think about it,' says Henri Sorge, the protagonist of _Le Très-Haut,_ when faced with a similar question, 'all the events in history are there around us, just like dead bodies [comme des morts]' ( _TH,_ 88; 88). Except, he adds, that 'Now, they are going to exist properly, now is the time, everything is reappearing, all is being revealed in clarity and in truth' ( _TH,_ 89; 88). Repetition here is not reinforcement, but a process of transformation by which events are stripped of identity and substance, and addressed in their singularity, intensity, and irreducible alterity. The finite possibility of death is turned into the infinite impossibility of dying, the possibility of time into the impossibility of time passing, and the reader is faced, as though in a premonition or repetition of an apocalyptic day of judgement, with what in _L'Arrêt de mort,_ referring to the mysterious appearance of N(athalie) in his hotel room, in the guise of the dead J., the narrator, using the adjectival noun in the impersonal or neuter form, describes as: 'l'irrémédiable', the irremediable ( _AM,_ 65; 38). The irremediable, as the narrator explains, operates too according to the homeopathic principle; like death itself, it is that which cannot be remedied and which immobilises and interrupts all thought. There is no remedy for the irremediable; but it can still be thought, albeit as an interruption of thought, as a thinking of that in thought which escapes thought. This is the trait of the irremediable that likens it to the thought of the _il y a_ and the thought of the neutre; and if the irremediable can be thought, implies Blanchot's narrator, it necessarily therefore must be thought, if only because, as a thought, the irremediable not only falls subject to the necessity and the compulsion of that which knows no negation, but is in fact itself the thought which knows no negation. Accordingly, it can only figure in thought as absolute negation, which is why in turn it figures in thought as measureless affirmation, as the timeless invocation of the singular moment that, as Derrida has shown, Blanchot addresses here, as elsewhere, with the apocalyptic summons: 'Viens', 'Come'. It is this thought that _L'Arrêt de mort,_ after all, is written to affirm.
The irremediable, here, interrupts meaning; like the neutre, it introduces into language a distance, neither internal nor external, neither visible nor invisible, that withdraws events or statements from themselves and re-inscribes them as other than what they are. The narrator explains how this occurs (without occurring) by recourse to a figure that is to be found increasingly in Blanchot's later _récits._ The figure concerned is that of the 'vitre', the window pane or glass. Referring at one stage to his intermittent relations with S(imone), the narrator recalls one occasion when, after six years of absence, he suddenly caught sight of her through a shop window ('la vitre d'un magasin'); he comments: 'Whoever has disappeared completely and is suddenly there before you, behind a pane of glass, becomes a sovereign figure' ('Quelqu'un qui a tout à fait disparu et qui, brusquement, est là, devant vous, derrière une glace, devient une figure souveraine') ( _AM,_ 72; 43). The phenomenon – what Blanchot calls here, as in later texts, 'the phenomenon of the window-pane', 'le phénomène de la vitre' ( _AM,_ 79; 48) – functions here as a complex play of presence and absence, incompletion and finality; it inscribes within a scene or an event a difference that suspends or neutralises them, strips them of presence, and returns them to what the narrator next describes as 'a timeless past' ( _AM,_ 80; 48). All that takes place takes place, so to speak, within quotation marks, both as a vestige and as an alteration of itself; and the effect is to bring about a striking transformation in the narrator's relationship to the characters that feature in his narrative, whom he now begins to forget, or at least no longer to remember as beings, but rather as a series of interruptions in being. The first hint of this process occurs in the Paris metro, when the narrator realises that, as her name suggests, his neighbour C(olette) has suddenly become in his memory, as he puts it, 'a huge, impersonal hole, though still alive, a kind of living lacuna, from which she emerged only with difficulty' ( _AM,_ 63; 37).
In the case of C(olette), this particular dissociation seems to have remained largely without consequence; but when similar disturbances begin to characterise his dealings with Nathalie, this other Eurydice, with whom the narrator takes refuge in the underworld of the Paris subway on 3 June 1940, the effects are more startling. For it is here that words come to acquire a life and death – an absence of life and death – all their own, and it is this that is most clearly responsible for the narrator's impetuous proposal of marriage to Nathalie, a proposal he is able only to make in spite of himself, in words he only barely understands. Next, carried away by his violence, the narrator begins addressing N. in his own language, with the result – chance or fate? – that the pair lose contact in the crowd and each goes their separate way – Nathalie, it would seem, to the appointment with the sculptor, the narrator to the aimless wait that awaits him. The pair meet again, of course, that night, for the climactic exchange that has already been mentioned, but, unbeknown to them both, it is clear that something irremediable has happened in the meantime, were it not that what has happened had in any case always already happened in the distant past.
Now, though, the compulsion to repeat that irretrievable past takes on a more violent, less forgiving dimension, and while on one level the narrator forfeits any control over the words he still takes to be his, he is also compelled, with those words, to entertain the thought that those words demand of him; as he does so he is compelled to affirm that thought with all the force he can muster, for both the limitations of those words and the limitlessness to which they give rise are his thought and his thought alone. The legacy of words is violently double: while the finite nature of language offers only misfortune, the limitlessness of language makes it possible, by means of a word that is thereby no longer a word, to address the other, the loved one from whom the narrator, as from Nathalie, is infinitely distant and to whom, as to Nathalie, he is also infinitely close, with a single, solitary word, which, borrowed from the Book of Revelation, puts an end to the world in order to speak to the other beyond all world. Repeated thrice over in _L'Arrêt de mort,_ it is the word: 'Viens', 'Come', that, to the extent it belongs to no language at all, is both the final limit and the infinite boundary of writing as such.
What is at stake then in _L'Arrêt de mort_ is the force of a demand that does not belong to the world, but precedes, traverses, and survives the world, allying itself to a dimension that may only be described as eschatological, and which, outside of history and outside of narrative, can only be thought as the relation of non-relation that thought itself entertains with that in thought that is always other than thought. This thought that suspends ontology, history, and temporality is an obligation that cannot be refused, in the sense that to refuse it is already to affirm it, to endow it with a force that is beyond being or non-being, making it intelligible perhaps only as a kind of singular, pre-conceptual trace, prior to being, prior to thinking, and even prior to language, though equally it is none of these things and may only be embodied by a pact, an alliance with the other, founded not on a statement of common destiny, not on a hierarchical injunction, not on the universal pretensions of a moral code, not on rules or norms or assertions of principle, but on a word that is not a word and which, in its endless repetition, can only ever be pronounced in the singular: 'Viens', 'Come'.
# 4: The absence of the book
## Beyond philosophy?
'I reject these words with which you address me, this discourse into which you seek to entice me in such assuaging tones, and the duration of your words, one after the other, by which you delay me in the presence of an affirmation, and above all this relation you create between us by the simple fact of speaking to me even in my unresponsive silence.' – 'Who are you?' – 'The refusal to engage in discourse, to negotiate with the law of discourse.' – 'Do you prefer tears, laughter, the madness that does not move?' **–'** I speak, bu **t** I do not speak in your discourse: by speaking I prevent you from speaking, by not speaking I oblige you to speak; there is no recourse for you, no instant in which to rest from me, who am there in all your words before all your words.' – 'I invented the great logos of logic to protect me from your incursions and to allow me to say and know by saying, according to the peace of well-developed words.' – 'But, in your logic, I am there also, denouncing the oppression of a coherence that takes on the role of law and I am there with my violence, affirming itself beneath the mask of your legal violence, the sort that subjects thought to the grip of understanding.' – 'I invented poetic irregularity, the errancy of broken words, the interruption of signs, and forbidden images in order to put you into words and, by putting you into words, silence you.' – 'I am silent, and straight away, in the hollowing out of day and night, you can hear me, all you do is hear me, no longer hearing anything, then hearing everywhere the distant murmur that, with every simple word, the cries of torture, the sighs of happy people, the spinning of time, the disorientation of space, has now passed into the world in which I speak.' – 'I know that I am betraying you.' – 'You are not in a position to betray me, nor be faithful to me. I do not know faith, I am not the unspeakable demanding secrecy, the uncommunicable that muteness would make manifest, I am not even the violence without words against which you might defend yourself with the violence that speaks.' – 'Nonetheless, affirming when I deny, denying when I affirm, ravaging in an ever thoughtless act of wrenching separation: I denounce you as the word never uttered or still too many that aims to except me from the _order_ of language and tempt me with another way of speaking. You torment me, it's true, even by leaving me in peace, but I can torment you too: justice, truth, truth, justice, these terms you reject with your pre-emptive sneer pursue you in their turn even to the point of the _other_ into which you reverse them. You do me Good by harassing me with the charge of injustice, and I might even say you are the Kindness that is never caught doing anything kind.' – 'You can say this, I accept everything, confess my part in everything.'
_Le Pas au-delà_ ( _PA ,_ 159–60; 116–17)
Throughout the late 1940s and 1950s Blanchot's critical writing maintains a close and privileged relationship with the philosophical work of Emmanuel Levinas. But only rarely during that period does Blanchot explicitly refer to Levinas; and it was not until 1961, when the publication of _Totalité et infini_ ( _Totality and Infinity_ ) signalled an important new departure in Levinas's thinking, that Blanchot began to address Levinas's work at all directly and in any detail, eventually devoting to Levinas not only a sequence of three chapters in _L'Entretien infini,_ but also a substantial portion of the fragmentary texts published in _L'Écriture du désastre_ in 1980, as well as a lengthy tribute to Levinas, under the title 'Notre compagne clandestine' ('Our Clandestine Companion'), later the same year. That Blanchot was drawn to address Levinas's work increasingly in this way is an indication not only of the urgency for Blanchot of Levinas's thinking, but it is also a sign of the distance between the two thinkers, a distance that was a necessary and integral part of their friendship and which manifests itself throughout in divergences of language and idiom, strategy and context. Difference here, though, does not necessarily constitute opposition, nor should one attribute the undoubted community of questioning that exists between the two authors to the influence of the one upon the other. Instead, to read the texts of Blanchot alongside those of Levinas, and vice versa, is to be aware – beyond influence and beyond opposition – of the singular rigour of a response to infinite debt, a debt that Blanchot, for his part, when speaking of Levinas, frequently acknowledged with unalloyed personal affection.
Infinite debt, however, does not imply slavish repetition, but almost exactly the reverse. Properly to pay tribute to another means paying tribute to another's difference; and this is achieved not by effacing distance but only by accentuating it, not by faithful repetition but by recognising the necessity of infidelity. There is here a peculiar double bind that governs all relations between texts and which requires that fidelity, if it is to be possible at all, must necessarily be couched in the form of infidelity. The paradox is one that is inherent in all commentary, and Blanchot describes its implications in detail when considering the challenge of writing about that other friend who was Georges Bataille. Fidelity, he writes, is not what it seems.'A commentator', remarks Blanchot, 'is not being faithful when he faithfully reproduces; the words or sentences he quotes, by the very fact of being quoted, change in meaning and become immobilised, or else assume excessive prominence' ( _EI,_ 301; 203). Infidelity, it seems, is an unavoidable condition of all commentary. But if this is so, how then is it possible to respond faithfully to a text at all and repay, so to speak, the infinite obligation it thus imposes?
The – provisional – tactic Blanchot adopts in response to Bataille mingles modesty and respect with uncompromising defiance. Proximity, for Blanchot, is only possible from a distance. To write on Bataille, he suggests, it is essential to avoid repeating the key words or concepts – terms such as despair, horror, ecstasy, rapture – that feature so powerfully in Bataille's work, in order to attend instead to the internal demand – the 'exigence' – of Bataille's writing, that which draws it outside or beyond itself and confounds the act of naming. To read Bataille in this way is to begin by suspending all the names the text provides itself and for itself, including that of the author, in order to affirm what escapes all naming and which is rather to be found in what Blanchot calls the 'surprise' of Bataille's language – both Bataille's surprise at language and the surprise of language at Bataille – and in the unique tone of the silent discourse of Bataille's text. The result is a practice of reading that refuses to privilege conceptuality, but instead allows itself to be led in the text by the repetitive movement of withdrawal and re-inscription, retraction and retracing, that Blanchot elsewhere addresses as the neutre.
Language, under the effects of the neutre, testifies here not to the stability of the name according to its status of universal particular, but, beyond all naming, to the nameless alterity inscribed in a singular movement of a writing, such as that of Bataille. Language here becomes synonymous with the movement that carries it beyond its own limits and, as a result, allows it to think more than it is able to think and to evoke an alterity whose only distinguishing trait is that, like dying, it is beyond all possibility. As Blanchot goes on to remark:
what no existent can attain in the primacy of his or her own name, what existence itself, in the seduction of its chance individuality [dans la séduction de sa particularité fortuite], in the movement of its sliding universality [dans le jeu de son universalité glissante], can in no way embrace, what therefore decidedly always escapes, all this language gratefully accepts; and it not only maintains it, but it is indeed by affirming it as such, as always other and always at a distance, as the impossible and the incommunicable, that language speaks and finds its origin, just as, in such language, thought thinks more than it is capable of thinking.
( _EI,_ 312; 210)
Here it seems that, in order to respond faithfully to a text, it is necessary to respond, not to those elements in a text – concepts or arguments, topics or themes – which are easily repeated, and which, once repeated, in fact betray the text, but rather to that in the text – a tone, a slippage, an accident, or an incoherence – which cannot be named as such and thus seemingly resists repetition. The position is a paradoxical one, as Blanchot freely admits, in that what is thereby adjudged to be most singular or recalcitrant in a text is not that which is most original in it, but that which is already most repetitive in it and that by virtue of the displacements wrought by the iterative movement of writing itself marks the text with the trace of its irreducible difference from itself. What must be affirmed, in response to a text, one might say, is neither the letter of the text nor its spirit, but, beyond the binary saw of that alternative, that which constitutes a text's peculiar signature: its chance and its necessity, its displacements and detours, its obsessions and interruptions, its infinite difference from itself and irreducible alterity. In reading Bataille, then – and much the same is true of his reading of Levinas – what is at stake for Blanchot is not conceptuality but writing, not the ideality of meaning but the transformation effected in thought by textuality itself. And this is why, in considering Blanchot's texts written in response to Levinas, it is crucial to read those essays and fragments not simply for their argument, but with an attention to their own density and singularity as texts.
By the time _Totalité et infini_ appeared, the association between Blanchot and Levinas had been a close one for already some thirty-five years. Shortly before 1961, as the debate on the question of the neutrality of Being in Heidegger suggests, there are signs that the philosophical relationship between the two men had entered into a new phase, one that served ultimately to redefine the nature of the accord between them. For in 1956, reviewing _L'Espace littéraire,_ Levinas credited Blanchot, alongside many others, Hegelians, Marxists, and Heideggerians alike, with the contention that philosophy was henceforth at its end: 'The present age', wrote Levinas, 'will have been the end of philosophy for everyone!' ('Ce siècle aura donc été pour tous la fin de la philosophie!'). And though Levinas went on to argue that Blanchot's interpretation of the work of art was fundamentally incompatible with that of Heidegger, he charged Blanchot – no doubt as a friend, but also (and for that very reason) with undisguised impatience – with the task of travelling further, nomadically, across the desert of the 'un-serious' and the 'un-true' that the Blanchot of 1955, in the footnote that concludes the main body of the text of _L'Espace littéraire_ ( _EL,_ 260; 247), had presented as the source of art's so-called 'authenticity'. But if Blanchot was to respond to that exhortation, Levinas implied, it would mean confronting, not simply the question of being and non-being, but, more urgently, in Levinas's view, the question of ethics, a question that for Levinas was embodied in the biblical figure of Amalek, still waging war on the Jews, as he had first done at the time of the early Hebrews' journey out of Egypt, and who, Blanchot agreed in 1989, following ancient tradition, was the personification of Evil itself.
In developing this argument, Levinas's main concern was to pursue a vigorous polemic against Heidegger, and in particular against Heidegger's subordination of justice to the question of Being, and total neglect of the philosophical legacy of Judaism. But by making the charge in a review of _L'Espace littéraire,_ Levinas, if only indirectly, seems to have wanted to take issue with Blanchot too, not because Blanchot was at all close to Heidegger, as Levinas himself demonstrates in that same text, but rather because, in his opposition to Heidegger, according to Levinas, Blanchot nevertheless shared something with Heidegger, which was the refusal, common to both, of ethics, or, at the very least, of the concept of ethics. For Levinas, this reluctance made little sense on Blanchot's part, particularly since ethics, for Levinas, was precisely that which was 'un-true' in the radical sense of not being subject to values of truth or falsity, however defined; and it is this that led him, referring to Blanchot's footnote, to issue the following challenge:
If the authenticity of which Blanchot speaks is to mean anything more than an awareness of the un-seriousness of edification, anything more than a mockery – the authenticity of art must proclaim an order of justice, the slave morality that is absent from the Heideggerian city.
Though Levinas had begun his discussion by suggesting how in Blanchot the impossibility of art radically transcends the realm of possibility that is philosophy, his review ended not by testifying to art's challenge to the philosophical order, but by demanding of philosophy – not of art – that it break with the truth of being and address the question of justice. As far as Levinas was concerned, what was beyond philosophy, so to speak, was not literature, but philosophy still: philosophy understood not as totality, but as a response to infinity.
A quarter of a century later, in 1980, writing in homage to Levinas in 'Notre compagne clandestine', Blanchot begins by repeating Levinas's remark – which Levinas had previously addressed to him – about the relation between literature and the end of philosophy implied in _L'Espace littéraire._ As he does so, however, Blanchot reaffirms the sceptical and ironic punctuation of Levinas's statement, which is enough, he suggests, to suspend and reverse its literal meaning. Blanchot does not demur at the precipitate violence, even injustice of Levinas's earlier challenge regarding his residual affinities with Heidegger and his (implied) neglect of the question of ethics. Indeed, he endorses Levinas's conclusion by assigning to philosophy – and not to literature – the dominant place in the trivium of art, philosophy, and science: in the face of these its rivals, philosophy, claims Blanchot, answering Levinas, is what both has the last word and has no last word.
Philosophy in this context, for Blanchot, is not invoked in the guise of the systematic exposition of conceptual truth. Instead, it is presented under the auspices of the now fifty-five-year-long bond of friendship that first brought Blanchot and Levinas together as students of philosophy in Strasbourg in 1925. The gesture is an affectionate and personal one; but its significance is more than autobiographical. For Blanchot goes on to claim that philosophy itself, as he puts it, is in fact but another name for friendship:
Philosophy [he explains, recalling the hopes shared by Levinas and himself as students] would forever be our companion, by day, by night, were it even to mean her losing her name, turning into literature, knowledge, non-knowledge, or absenting herself, the clandestine friend in whom we respected – loved – that which did not allow us to be bound to her, even as we sensed that there was nothing wakeful in us, vigilant even in sleep, which was not due to her difficult friendship. Philosophy or friendship. But philosophy is precisely not an allegory.
Philosophy here is addressed both as philosophy and as something – someone – more than philosophy, which – or who – can still be named as philosophy but also exceeds that name and stands, as it were, beyond philosophy. As Blanchot explains, reading Levinas for Blanchot means primarily having to respond to this dual status of the word philosophy, as though what was singularly repetitive in the texts of Levinas, his signature, so to speak, were his multiple relationship with the order and tradition of philosophy itself. In order to read Levinas, Blanchot argues, it is necessary to traverse philosophy; but Levinas also demands something more, which silently transcends the name of philosophy. This is why, though he reads Levinas philosophically, Blanchot pays Levinas the tribute of reading him not as a philosopher, but as a friend who is both more and less than a philosopher. Rather than endorsing or carrying out a critique of Levinas's conceptuality, Blanchot responds instead, within his own writing, but without mediating concepts of his own, to the singular inflection, the obsessive demand, the pressing urgency, the assault on philosophy that Levinas's own texts insistently rehearse. As he does so, Blanchot necessarily multiplies the slippages between his own thinking and that of Levinas, which are visible throughout Blanchot's philosophical homage in 'Notre compagne clandestine'. Whence, in that essay, the numerous borrowings from Levinas's own distinctive vocabulary, but at the same time, the admission that Blanchot is quoting from memory, and without authority, as is confirmed by the many only partially accurate quotations in the essay and by the lengthy paraphrases, often based on unidentified citations, which give the impression of a Blanchot haphazardly ventriloquising Levinas's discourse. The effect is deceptive. For while Blanchot speaks in the language of Levinas, he does so with discretion: by which I mean that he does so with both affectionate reserve and an awareness of the complex differences of both emphasis and substance that separate them from each other; with the result that Blanchot's account of Levinas, in its very fidelity to Levinas's language, also reads like a silent testimony to the fundamental dissymmetry between the two. In paying homage to Levinas, Blanchot does not write from a position of neutrality, but by borrowing extensively from Levinas dedicates his text to the impossibility of remaining faithful to Levinas, which is Blanchot's way of remaining faithful to Levinas, and to the divergences between himself and his friend, divergences that are all the more marked, in their discreet invisibility and silent suspension, by the fact that Blanchot writes in a language that is not so much his own as that of Levinas. Addressing the Other in this way, Blanchot remarks, is to speak not of the Other, but to the Other.
By baptising philosophy as friendship, and by singularising the love of wisdom that is philosophy itself, Blanchot turns aside from the idea of philosophy as conceptual totalisation. Friendship here, Blanchot writes in 1993, again echoing Levinas, is no longer thought on the model of Greek _philia,_ but according to a logic of non-reciprocity and dissymmetry, in response to the disclosure of the other as unknown and the pre-eminence of the Other for whom one is responsible. Friendship both implies community and throws the immanence and homogeneity of community into doubt. Never the property of an individual, always at least double, never general but always unique, beyond event or anecdote, the fruit of both chance and necessity, it is what inscribes the different and the multiple within all relation and escapes all possible gestures of totalisation. The friendship between Blanchot and Levinas is more than an external circumstance of thought, it is its persistent manifestation and metamorphosis. Philosophy here stands beyond in-difference and neutrality (in Levinas's sense of the word). It becomes instead the site of its own excess, its own alterity, its own difference with itself; and if philosophy is invincible scepticism, as Blanchot puts it, citing Levinas, it is because, as the thought of the neutre affirms, the borders and limits of thought are necessarily always suspended at the very moment they are determined, and because thought, as it puts itself in doubt, thereby finds itself in the disconcerting predicament of always thinking more than it thinks itself capable of thinking.
Philosophy and literature here are no longer in opposition. Not for Blanchot therefore the postmodernist enterprise of treating philosophy as a branch of literature, nor the temptation to abandon the austerity and rigour of thought in favour of the effusive figurality of the literary. Instead, Blanchot maintains the separation between philosophy and literature; but he does so not in order to preserve the autonomy and self-identity of each, but in answer to the namelessness of the demand of (and for) otherness to which both, in their divergent ways, give voice. For Blanchot, philosophy and literature have become oddly synonymous with the absence of their own name; and to the extent that they refuse that name, they persist as other than what they are and as a response **–** and responsibility – to what is beyond being and non-being. Across the divide that separates them, literature and philosophy answer, then, in common, but without communality, to and for the irrepressible infinity of language. And if Levinas is right to claim that it is 'indiscretion with regard to the unsayable [l'indiscrétion à l'égard de l'indicible] which is probably the very task of philosophy', what this says is that philosophy, by being bound to that in language which is beyond language, is also inseparable, like literature itself, from the thought of the neutre, from what, using Levinas's words, but in a formulation unmistakably his own, Blanchot describes as the
Enigma of a Saying [d'un Dire] as of a God speaking in man, man who relies on no God, for whom there is no such thing as dwelling [pour qui il n'y a pas à habiter], who is exiled from all world without afterworld [l'exilé de tout monde sans arrière-monde], and who finally does not even have language as his abode [comme demeure], no more than he may be said to have language to speak in either the affirmative or negative mode.
Responding to Levinas's challenge about the ethical demand to which literature must give voice, Blanchot, it seems, accepts: literature is not other than philosophy, and thus cannot announce the death of philosophy or its abandonment in favour of those mystical effusions or solipsistic enthusiasms of which Blanchot, like Levinas, is so intensely wary. Literature remains, like philosophy, as a thinking which demands confrontation with evil, and if its fate is bound to the philosophy that is friendship, then what this suggests is that for literature, too, the question is one of responsibility. This indeed is what Blanchot himself underlines, at the end of 'Notre compagne clandestine', albeit less obliquely than Levinas in 1956, by invoking the name not of Amalek, but Auschwitz, citing as he does so Levinas's dedication of _Autrement qu'être_ to the victims, Jews and non-Jews alike, of the Nazi camps.
The accord here between Blanchot and Levinas is real and unmistakable. But the friendship between them inscribes within that accord a necessary dissymmetry. As philosophical friends Blanchot and Levinas agree on many philosophical issues. But friendship cannot be reduced to that which is purely philosophical, for it is a response to what lies beyond philosophy in the singular address of friendship itself. It is here that the differences between Blanchot and Levinas emerge, as indeed they must if the relation between them is to occur as a relation of friendship at all. In Levinas's terms, the Other is what escapes thematisation; the Other can be named therefore only provisionally, by recourse to a glorious Saying – what Levinas calls 'le Dire' – that constantly erases what is said in order constantly to renew itself as always other than what it was. This is why all speaking to the Other, as Derrida has often argued, is irreducibly idiomatic. It is therefore not only unsurprising, but absolutely necessary that the thought of the Other, like the thought of the neutre, should give rise to irreducibly dissymmetrical thematisations. The possibility of language is itself on this condition. To respond faithfully to the Other means refusing – and refusing even in adopting – the language of the Other in the nameless name of the infinity and otherness of language.
That which is beyond philosophy – philosophy still, or literature one more time, in the refusal of the name that constitutes the necessity of both – is possible perhaps only as a singular writing that displaces, transforms, suspends, ignores, travesties, and erases conceptuality. On one level, such an enterprise may be criticised as philosophically naive, as Blanchot readily concedes on Levinas's behalf. But naivety here is better seen as a provisional, and unpredictable name for philosophical inventiveness. What counts is not the conceptuality of the Said, but the infinite movement of Saying that always insists beyond the Said. Blanchot and Levinas write differently of difference, and speak otherwise of the Other; but this is because the ground of their accord is also necessarily a ground of discord, their contemporaneity with each other a time of irreducible diachrony, and their undoubted community of thought a curved, dissymmetrical space where, as Blanchot puts it in _L'Entretien infini,_ the distance from point A to point B is not isomorphic with that between point B and point A, 'a distance', adds Blanchot, paraphrasing Levinas, 'excluding reciprocity and presenting a curvature whose irregularity extends to the point of discontinuity' ( _EI,_ 104; 73).
## Transcendence and the Other
The _il y a_ is one of Levinas's most fascinating propositions: his temptation, too, the reverse side of transcendence, so to speak, and thus indistinguishable from it, which may be described in terms of being, but as the _impossibility_ of not being, the incessant insistence of the neuter, the nocturnal murmur of the anonymous, that which never begins (and therefore an-archic, because perpetually eluding the decisiveness of a beginning), the absolute but as absolute indeterminacy: all this is bewitching, that is, it draws us towards the uncertain outside, speaking infinitely outside truth, like some Other we could not get rid of simply by calling him deceitful (the evil demon) nor because he might be said to be a mere mockery, since this speaking, which is only a laugh perfidiously suppressed, giving meaning while yet eluding all interpretation, neither gratuitous nor cheerful, grave, just as much, and the illusion of seriousness, so to speak, and thereby what disturbs us most, is also the movement most apt to deny us the resources of being as place and light: the gift perhaps of literature, without one knowing whether it enchants by disenchanting or whether its words that please and disgust do not ultimately attract us because it is a promise (a promise it keeps and does not keep) to illuminate what is obscure in all speech, what in speech eludes revelation and manifestation: the trace again of non-presence, the opaqueness of transparency.
'Notre compagne clandestine' (86)
Blanchot's explicit engagement with Levinas passes through a number of distinct phases. Shortly after publication in 1961, as already indicated, Blanchot devoted to _Totalité et infini_ a series of three dialogues between unnamed interlocutors that were subsequently taken up, in modified form, in _L'Entretien infini._ These texts, however, should not be seen in isolation. The impact of Levinas's thinking on Blanchot extends well beyond the explicit mention of Levinas's own work, and is inseparable from the way in which _L'Entretien infini_ is structured as a whole. Indeed, by adopting the expression 'l'entretien infini' as his title, and by presenting the three discussions of _Totalité et infini,_ alongside the opening narrative, as the first texts in which the reader encounters the dialogue form in the book, Blanchot inscribes _L'Entretien infini_ as a whole – composed of many diverse fragmentary forms as well as many seemingly traditional, discursive essays – under the sign of that peculiar dissymmetry between self and other that Blanchot, following Levinas, describes in the three chapters specifically concerned with _Totalité et infini._ In this respect _L'Entretien infini_ is not a random collection of essays, but a work, painstakingly revised and rewritten, that in its complex compositional structure seeks to respond to the following question that Blanchot formulates from the outset and couches in largely Levinasian terms:
How might one speak in such a way that speaking is essentially plural? How may one affirm the search for a mode of plural speaking no longer founded on equality and inequality, hierarchy and subordination, or reciprocal mutuality, but rather on dissymmetry and irreversibility, so that, between two instances of speech, a relation of infinity would always obtain as the movement of signification itself?
Alternatively, how may one write so that the continuity of the movement of writing allows the fundamental intervention of interruption as meaning and rupture as form?
( _EI,_ 9; 8)
In the terms of this project, the infinite conversation dramatised in Blanchot's book is far from resembling what is usually understood as philosophical dialogue. First of all, no single textual form predominates: fragment, essay, and conversation constantly delimit each other and challenge their relative priority. Furthermore, even within the dialogue forms themselves, there is no clear distinction between the two – perhaps at times even more than two – voices involved, and no progress towards synthesis or conclusion. Blanchot's interlocutors do not seek unity or homogeneity, but, in their intermingled voices, embody difference, separation, and withdrawal. Turning to _Totalité et infini,_ they register on one level, perhaps unsurprisingly, a large measure of unanimity with Levinas. But this is not to say the voices necessarily agree with one another, and there is much in their conversation that remains unresolved or open to question, or is deferred necessarily till later.
In Levinas's book, we are told by one interlocutor, perhaps recalling the challenge addressed to Blanchot by Levinas in 1956, 'there is here a new departure in philosophy and a leap that it and we are being urged, so to speak, to perform' ( _EI,_ 74; 52). Accepting the invitation, Blanchot quickly makes his own the main articulations of Levinas's account of the irreducibility of the Other to the Same; and it is clear from this point on that Levinas's work provides Blanchot with the opportunity for addressing in very different terms than before some of the issues left in suspense by him at the end of _L'Espace littéraire._ Prominent among these, alongside the question of the relation of art to work, world, and truth, that Blanchot by the early 1960s had already begun to rearticulate by recourse to the thought of the neutre, was the demand of what Levinas insisted on calling ethics and to which Blanchot himself, by that time, had already begun to give a decisive and distinctive political inflection. Indeed, it would be misleading to conclude from Levinas's questioning of Blanchot in his 1956 review of _L'Espace littéraire_ that during the 1950s Blanchot had somehow remained deaf to the question of ethics in its relation to literature. In some respects, nothing could be further from the truth. As the essays collected in 1959 in _Le Livre à venir_ testify, Blanchot throughout the 1950s was seeking in his critical writing – while never formulating his project explicitly in these terms – precisely to address the ethical dimension of literature as such. In doing so, however, he refused to refer the demand of literature to the jurisdiction of morality or law and subordinate the act of writing to conventional concepts of duty and choice, freedom and responsibility, insisting instead, to the extent that the term of ethics was appropriate at all, that the only injunction to which literature fell subject was one based on its own impossibility and dispossessed anonymity and on the indigent challenge to universality that for Blanchot were the hallmark of writing as such.
By the early 1960s it does seem that _Totalité et infini_ supplied Blanchot with a fresh philosophical framework for taking this argument further. A rapid transformation takes place in Blanchot's critical writing with the result that what, for instance, had figured earlier in his texts under the rubric of impossibility, is henceforth reformulated in largely Levinasian language as an infinite relation to the Other, a relation beyond power or possibility and beyond being or non-being. The account Levinas gives, in _Totalité et infini,_ of the relation of non-relation that obtains between the Same and the Other thus provides Blanchot with the means of re-inventing or re-articulating as the question of the Other the question of the impossibility of dying that is so crucial in his earlier work. Till now, for Blanchot, as _L'Espace littéraire_ had made clear ( _EL,_ 106–7; 106), it had been the impossible embodied in death itself – its irreducibility to mastery by any subject present to itself – that had provided, in exemplary fashion, the model (beyond all possible models and models of possibility) of the relation of non-relation that for Levinas constituted the radicality of the Other's challenge to the Same and the Self. The impossibility of dying and the challenge from the Other become here, in Blanchot, the two sides of a radical rethinking of the logic of relation and of its implications both for the question of being and the question of ethics. For if death was alterity and alterity death, it followed that the key ethical question, perhaps indeed the only ethical question at all, was the question (beyond power or possibility) of the death of the Other, on which alone, as Blanchot put it in 1983 in _La Communauté inavouable_ ( _CI,_ 21–2; 9), community – and thereby ethics as such – may be founded.
The question of the possibility or impossibility of death, in Blanchot, is never far from the question of language; indeed, the two in Blanchot are largely indissociable, if not in fact synonymous. In Blanchot's second dialogue on _Totalité et infini_ one of the voices makes the connection explicit by claiming in turn that the question of the Other is in fact the question of language. Ethics here in Blanchot, as in Levinas, is reconfigured at the furthest extremity possible from moral legislation as a fundamental and inescapable dimension of language itself. 'The Other speaks to me [Autrui me parle]', one of Blanchot's interlocutors remarks pointedly, and adds:
The revelation of the Other [la révélation d'autrui] that does not occur within the luminous space of forms is wholly speech. The Other expresses himself [Autrui s'exprime], and in this act of speaking puts himself forward as other [comme autre]. If there is a relation in which the other and the same [l'autre et le même], even as they hold each other in relation, _absolve themselves_ from that relation, remaining as terms that are _absolute_ within the relation itself, as Levinas strikingly puts it, that relation is language.
( _EI,_ 79; 55)
Blanchot's interlocutors explain, in a passage added in 1969 that rehearses much that will be familiar to readers of _Totalité et infini,_ there are at least three distinct ways of configuring the relation between the Same and the Other. According to the first type, one voice proposes, the law is that of the Same, and the aim of the process the incorporation of the Other within the unity and totality of the whole: this is the realm of the historical dialectic, the purpose of which is to assimilate the Other to the Same through the mediation of labour, by a process of negation, reduction, adequation, and identification. As far as the second type is concerned, rejoins his partner, none of this struggle is necessary. Unity here is achieved immediately, and the relation between the Same and the Other is one of ecstatic fusion; the Same, as it were, is magically absorbed into the Other, and unity and totality again prevail, with the only difference that here the law is that of the Other, but the Other functioning only as a mirror image of the Same that is always One.
Both voices wonder, obstinately though hesitantly, whether there is not in addition a third type of relation, and this is what Blanchot goes on to explore in the ensuing discussion. In this relation of the third kind, the Other is thought not as another Self, but as radically different, irreducible to the One or to the Same. This type of relation occurs, so to speak, beyond the horizon of world and being; it is relation without ratio, adequation, equality, symmetry, or reciprocity. This is relation without relation, relation in the form of a pure interval belonging neither to being nor non-being, irreducible to all thought of truth, visibility, veiling or unveiling, and figurable only as non-reversible dissymmetry, as a strange space in which the distance from me to the Other is not the same as the distance from the Other to me. Here, all topographical continuity is abolished. The Other is precisely not another 'I', but absolutely Other, beyond all unity or duality; as one of Blanchot's speakers confirms:
When the Other speaks to me [Quand Autrui me parle], he does not speak to me as an 'I' [il ne me parle pas comme moi]. When I appeal to the Other [l'Autre], I respond to that which speaks to me from no place, and am separated from him by a caesura such that the Other forms with me neither duality nor unity. It is this fissure – this relation with the other [ce rapport avec l'autre] – that we ventured earlier to characterise as an interruption of being; let us now add: between each human and the next, there is an interval that may be said to belong neither to being nor to non-being, an interval borne by the Difference of speech, a difference that precedes all that is different or single.
( _EI,_ 99; 69)
This third relation, Blanchot, some pages later, in another subsequent gloss, describes as a 'rapport neutre', a neutral relation ( _EI,_ 104; 73), and, despite their irreducible accord, the word signals here, between Blanchot and Levinas, just as it had in 1957 and 1958, the existence of a fundamental divergence. This is not simply a relic from that earlier period. Indeed, despite Levinas's explicit rejection of the neuter, and the effacement of difference it is taken to connote, Blanchot himself persists with the word; and, when revising his texts on Levinas for inclusion in _L'Entretien infini,_ Blanchot in fact far extends its currency in his writing. For instance, in the original version of the chapter 'Tenir parole' ('Keeping to Words'), Blanchot had described the face of the Other, much in keeping with Levinas's own account, as being by essence that which was naked or bare ('nu', in Levinas's idiom); reworking the passage in 1969, Blanchot substitutes for this the word: 'neutre', so that instead of having, as in 1962: 'un rapport nu ou la nudité même du rapport' Blanchot's text now reads: 'un rapport neutre ou la neutralité même du rapport' ( _EI,_ 84; 59).
This diffident effacement of a important term in Levinas's conceptual vocabulary is not an isolated gesture. Throughout the discussion of _Totalité et infini,_ Blanchot's interlocutors repeatedly cast suspicion on Levinas's attempt to formulate a language specifically his own, adequate to the project of grounding the possibility of ethics. At one point, for instance, the conversation turns to Levinas's use of the face ('le visage'), beyond all reference to the manifestation of Being in Heideggerian terms, as the most urgent embodiment of the presence of the Other as Other; and though the two interlocutors eventually agree that the face in Levinas is precisely that which is beyond representation, they do so despite the word rather than because of it, leading the reader to conclude that the word itself would be best withdrawn, precisely because it falls victim to the very logic of representation it seeks to transcend ( _EI,_ 77; 54). Elsewhere, in similar manner, on two separate occasions, it is the word ethics that is queried on the grounds of its generality and derivative character ( _EI,_ 78, 89; 55, 63); and in much the same way Levinas's concept of exteriority, which gives _Totalité et infini_ its subtitle, is effaced in passing by one of the speakers only to be replaced instead by a reference to the pre-conceptual outside: 'le dehors' ( _EI,_ 78; 55). Later, it is the turn of the privilege Levinas accords to the spoken word to be called into doubt also ( _EI,_ 80; 56), until finally it is the crucial term 'Autrui' (meaning: Others in general), that it is proposed be abandoned owing to its moralistic – 'altruistic' – overtones ( _EI,_ 100; 70).
As these examples suggest, Blanchot treats much of the conceptual language of _Totalité et infini_ with a discreet but rigorous scepticism. Of all the minor disputes that arise as a result, perhaps the most revealing is the deep reservation expressed in _L'Entretien infini_ by Blanchot about Levinas's persistent recourse in _Totalité et infini_ to the name of God. Levinas's use of the word, of course, is not gratuitous; it is a direct and unrepentant consequence of the project of rethinking the philosophical significance of Judaism. As such, its importance within Levinas's philosophical writing is difficult to underestimate. Indeed, as Blanchot reminds us, 'All true discourse, Levinas solemnly declares, is discourse with God, not a conversation between equals' ( _EI,_ 80; 56). 'The Other [Autrui]', for Levinas, he adds, 'must always be considered by me as closer to God than myself' ( _EI,_ 82; 57). But despite the prominent part played by the name of God in Levinas's text, Blanchot makes equally clear his own uncompromising refusal of that name. As one of the voices in the discussion insists, 'Let us leave God to one side, the name is too imposing' ('Laissons Dieu de côté, nom trop imposant') ( _El,_ 71; 50). The disagreement here is plain. For his part, Blanchot could well argue that it is he who is being more consistent than Levinas; for if it is the case, as Levinas contends, that the relation with the Other is without mediating concept or intermediary of any kind, so it would follow from this, in the compelling words of one of Blanchot's speakers, that henceforth 'there is therefore neither god, nor values, nor nature between man and man' ( _EI,_ 84; 59). To reinforce the point further, Blanchot leaves little doubt elsewhere in _L'Entretien infini_ as to the resilience of his commitment to atheism.
But more is at issue here than the difference between Blanchot's rigorous atheologism and Levinas's Judaism. For this debate on the question of God is not about religion as such. On numerous occasions, Levinas shows himself to be acutely aware of the fundamental differences between philosophy, religion, and theology, and is insistent that when he invokes God in his philosophical writings he does so not as a believer or theologian, but as a philosopher. Moreover, in the exchanges between Blanchot and Levinas on the subject of Judaism itself, it is clear that, despite their considerably different relationship with Judaic tradition, both writers display a broadly similar understanding of its philosophical significance. This is perhaps not surprising. On his own admission, Blanchot's access to Judaism was at the very least profoundly influenced by his friendship with Levinas, a friendship he described in 1968, in dedicating the text 'Parole de fragment' ('The Fragment Word') to Levinas, as being 'in a relation of invisibility with Judaism' ('en rapport d'invisibilité avec le judaïsme'). And this fundamental accord regarding the philosophical significance of Judaism was no doubt what allowed Levinas, in 1969, in the course of a lecture subsequently published in a volume of Talmudic readings, to claim confirmation from Blanchot, the non-Jew (who was, interestingly, left nameless by Levinas at the time) for the view that what was distinctive about Judaism was not only its universalism, but the fact that in Judaism there was also something else that was beyond universality, not in the guise of ethnic particularism, Levinas explains, but, transcending society and politics, what he describes as the holy burden of infinite responsibility for the Other.
But despite this relative unanimity regarding the philosophical importance of Judaism, one central issue remains: the question of how to address the transcendent as such. Or as one of Blanchot's interlocutors formulates it in _L'Entretien infini:_ 'Who is "the Other"?' 'Qui est "Autrui"?' ( _EI,_ 99; 70). As his partner immediately acknowledges, the question is an improper one, in that it risks substantialising the Other rather than responding to the unmediated challenge of his or her presence. However, Blanchot's text goes on to reiterate the importance of locating transcendence exclusively in the social field, outside of all theology. As one speaker had announced from the outset, speculatively, but with uncompromising seriousness: 'it may be that everything that may be affirmed of the relation of transcendence – the relation of God to his creature – ought primarily (for my part, I would say exclusively) to be applied to the social relation' ( _EI,_ 77; 54). In this way, while endorsing virtually without reservation Levinas's account of the transcendence of the Other and the relation of non-relation between the Same and the Other which it implies, Blanchot endeavoured to transpose into the political or social arena what Levinas articulates for his part as an ethical relation. The implicit question, then, that is put to _Totalité et infini_ by Blanchot, and which Blanchot himself formulates only obliquely in 'Notre compagne clandestine' in 1980, is to ask whether, by thematising transcendence by the use of religious vocabulary and thus equating transcendence with the name of God – albeit a God who, as Levinas insists, is not the God of onto-theology – Levinas does not run the risk of falling back into a notion of transcendence that, instead of being absolutely Other, is ultimately no more than the reverse side of humanistic immanence, and to that degree not absolutely transcendent at all but ultimately reliant on a travestied version of immanence.
As Blanchot and Levinas are both aware, much here turns on the referent of the word God: species or genus of supreme being, or proper name of that which is beyond being? The Christian God is traditionally seen as the former, while God in the Talmud, according to Levinas, is unmistakably the latter; and it is generally the case that the God Levinas invokes in his philosophical work is understood precisely not as (a) being at all, but as a name beyond being or non-being. The name of God is fundamental for Levinas in this respect for it is what allows him to address his philosophical work not to the concept of the Other but to the Other as absolute singularity. To the extent that God is a proper name, it is the proper name par excellence; the invocation of God, therefore, is not a theological velleity on Levinas's part, but a philosophical necessity, one without which the whole edifice of Levinas's thinking would collapse, leaving behind only the necessary fixity of a conceptual framework that can do no more than reinforce being as representation.
But at the same time, the absolute transcendence of God, this alterity that, as Levinas writes, is always other than alterity itself, necessarily tends towards an extreme point which is that of its own absence, and the invocation of God must always be threatened by the possibility of this divine comedy of God's disappearance. Indeed, as Levinas admits in 'Dieu et la philosophie' ('God and Philosophy') in 1975, the reference to God must itself always be exposed to this danger, and the risk must always exist that it may turn into a mere doublet of the _il y a_ and the immanent impossibility of not being. The result is an ambiguity that is doubly worrying, for not only does it call Levinas's project into doubt, it also has the potential to transform Levinas's account of the transcendence of the Other into precisely one of those philosophies of the neuter (in Levinas's sense of the term) in which transcendence is never more than a mirror image of immanence, and vice versa. Which is to say perhaps no more than what Derrida suggests in 'Violence et métaphysique' ('Violence and Metaphysics') when he demonstrates how fraught with risks Levinas's project is, and necessarily not as sharply differentiated from that of Hegel as Levinas hoped or wished.
From Blanchot's perspective, it seems that God in Levinas is both impossible to accept for reasons of Blanchot's atheism, and yet, because of its importance in singularising Levinas's whole conceptuality, impossible to refuse. Faced with these difficulties, Blanchot adopts a characteristically bold and complex strategy. In _L'Entretien infini,_ responding still to _Totalité et infini,_ Blanchot replaces the question: who is the Other?, which he had withdrawn some pages before, with another question, bearing instead on what Blanchot describes, within neutralising quotation marks, as the problem of 'community'. The question Levinas's work leads one of Blanchot's voices to ask is therefore this:
What is the implication for human 'community' [Qu'en est-il de la 'commtmauté' humaine], once it is necessary for it to respond to that relation of strangeness between each man and the next which the experience of language leads one to sense, a relation that is both exorbitant and without common measure?
( _EI,_ 101; 71)
To address this question, Blanchot implies, it is necessary to rethink transcendence not according to the model (which is necessarily anything but a model) of the relation between man and God, but solely within the realm of what, albeit misleadingly, is usually described by philosophy as intersubjective space, which is a space occupied, for Blanchot, contrary to what the name suggests, not by two subjects, nor by a subject and his or her other, but by what Blanchot terms a doubly dissymmetrical relation between two (or more) existents, irreducible to all dialectical reciprocity and equalisation, and without continuity or hierarchy.
In a remarkable shift, what throughout _Totalité et infini_ Levinas addresses under the rubric of the asymmetry between the Same and the Other is reconfigured, in Blanchot's exposition, as the double dissymmetry between Self and Other ( _EI,_ 100; 70–1). For the privative prefix 'a-', connoting absence, Blanchot substitutes the prefix 'dis-', meaning: two, and always at least double. In other words, what Blanchot proposes is that the relation of non-relation between the Same and the Other be redoubled by a second relation of non-relation passing from the Other to the Same, with the complication that now it is the Same – 'I', so to speak, albeit henceforth an 'I' without 'I' and without name – who, for the Other, is now nothing other than the Other. The hierarchical relation of non-relation between the Same and the Other described by Levinas is presented by Blanchot in such a way that the relation, beyond reversibility or reciprocity, is now seen always to be running in at least two distinct directions. The effect is to compromise the verticality of the relation to the Other as the Most High, and thus both secularise and multiply the relation as such.
The move, however, is not without its risks, and it is perhaps unsurprising that by some of Levinas's readers Blanchot has been severely taken to task for what is allegedly a gross misunderstanding of Levinas. Indeed, as _L'Entretien infini_ readily concedes, there is a danger that by redoubling the relation of non-relation one thereby allows the dissymmetry of 'community' to be recuperated precisely within the reversible logic of a dialectic of the Same for which the Other is only just another version of the Self. If that were the case, the only community to which Blanchot's double dissymmetry would give rise would be community founded on the immanence of communion and closure, that is, on the effacement of alterity and the assimilation of the Other within the Same, with all the dire – totalitarian – political implications, as Blanchot argues in _La Communauté inavouable,_ that would follow. Blanchot insists, however, that his own redoubling of the Levinasian relation of asymmetry must be understood, according to the logic of the neutre, to be fundamentally and necessarily irreducible to any such equalising dialectic of reciprocity. On the contrary. Blanchot explains,
this redoubling of irreciprocity – the reversal that makes me apparently the other of the other [qui fait de moi apparemment l'autre de l'autre] – cannot, at the level we are describing, be incorporated into any dialectic [être pris en charge par la dialectique], for it does not tend to re-establish any equality whatsoever; on the contrary, it signifies a double dissymmetry [une double dissymétrie], a double discontinuity, as though the emptiness between the one and the other were not homogeneous but polarised, as though it constituted a non-isomorphic field, bearing a double distortion, both infinitely negative and infinitely positive, and such that one should call it neuter, so long as it is clear that the neuter does not annul or neutralise this double-signed infinity, but bears it in the manner of an enigma.
( _EI,_ 100–1; 70–1)
As Blanchot recasts Levinas's account of transcendence, he draws once more then on the conceptual resources of the neutre. Its intervention is twofold. First, it turns aside the threat of homogeneity implied by dialectical reciprocity and, exceeding the double bind of being and non-being, assertion and negation, imparts to the relation between humans an infinity that, beyond continuity or unity, radically reconfigures the possibility of social relations as such. Second, and more provocatively, the neutre allows Blanchot to suspend, though without negating or neutralising it, the vicious circle of transcendence and immanence within which philosophy as such is enclosed; and by both stepping beyond and not stepping beyond that fatal alternative, Blanchot's writing begins to address the human other, no longer in terms of either transcendence or immanence, but absolutely otherwise: that is, writes Blanchot, beyond the province of any name at all, 'not other as God, or other as nature, but as "man", more Other than all other' ('non pas autre comme Dieu ou autre comme nature, mais, en tant qu' "homme", plus Autre que tout ce qu'il y a d'autre') ( _EI,_ 102; 72). It is, so to speak, to this human other, other than all other other, to another than alterity itself, that the neutre in Blanchot, with its distinctive diacritical signature, makes its silent appeal.
What Blanchot stages here is not an attempt to confound Levinas, or refute his work in any way; Blanchot's efforts are directed rather at producing a reading of Levinas that responds to the radical namelessness of the Other more readily than it does to the Other's proximity to God. This allows Blanchot, here and elsewhere, both in his writing and in the political domain itself, to broach a set of issues that, in Levinas's philosophical work, are for the most part either postponed or put to one side, and which bear on the relationship, within history and beyond history, not between humanity and the name of God, but among humans themselves, within what, for want of a better word, and with all due precautions, Blanchot, like Bataille and Jean-Luc Nancy, refers to as 'community'. Here, Blanchot's reading of Levinas has to be seen closely in relation with an exigency which is that of political activism, albeit an activism of an original and distinctive kind which, as Blanchot demonstrates in his own explicit political involvements in France, in 1958 to 1960, 1968, and since, turns aside from the pursuit of power as such to voice instead the infinite demand of refusal, of refusal maintained in the name – without name – of the Other.
This question of the political is not a marginal concern for Blanchot in his reading of Levinas. On the contrary, it continues to obsess Blanchot, and is a persistent preoccupation in the fragments making up the 'Discours sur la patience' ('Discourse on Patience') that Blanchot devoted in 1975 to Levinas's then recent books, _Humanisme de l'autre homme_ and _Autrement qu'être._ In the 'Discours', as in _L'Ecriture du désastre_ as a whole, Blanchot subscribes virtually without reservation to Levinas's account of the infinite responsibility for the Other he had been elaborating philosophically since the late 1960s. Blanchot, however, also does something more, which is to exploit his own (non-)position, writing in the margins of Levinas's text (which is how Blanchot presents his reading of Levinas in 1975) to ask, with infinite fidelity, but beyond all piety, a number of sceptical and challenging questions of Levinas's work. And Blanchot does so, as a marginal commentator, almost entirely within the terms of argument offered by Levinas, demurring only intermittently at Levinas's conceptual vocabulary, well aware that Levinas himself is struggling against the limitations of philosophical language as such, limitations which mean, for instance, that when Levinas speaks of passivity – of a passivity beyond all passivity –s as that which precedes all being or non-being, the endeavour is inevitably dogged by the fact that philosophy accepts the term only by inscribing it within a binary paradigm that has always already privileged being as activity and thereby transforms passivity from the outset into not even the reverse, but more like the ghost of itself. And this is what Blanchot means when he remarks: 'Passivity as opposed to activity: such is the always restricted field of our thoughts' ( _ED,_ 30; 15).
The predicament is not new as far as Levinas's work is concerned. Blanchot does not use it as a pretext to relinquish his fidelity to Levinas; rather, by always seeking the more difficult and more exacting reading, as, for instance, when he accepts the invitation to think the notion of the immediate in Levinas in relation not to the present tense but the past ( _ED,_ 44; 24), Blanchot allows himself to be challenged by the text, obligated to it, and infinitely responsive to its strangeness. To read here is not to retain concepts, but, beyond conceptuality, to respond to the dissymmetry or interval that comes between the so-called letter of the text and the invincible scepticism of the writing. Indeed, the awareness of the gap between what is said in Levinas's text and the address to the Other enacted in the saying of Levinas's language – the distinction that Levinas himself thematises as the difference between 'le Dire' and 'le Dit', the Saying and the Said – does not relieve the reader of the responsibility of reading sceptically; on the contrary, it requires of the reader that he or she read with an ever increasing degree of vigilance: in other words, with infinite regard for the ever persistent namelessness of the neutre.
While allowing himself to be pressed by Levinas, as it were, Blanchot presses Levinas in return. In so doing, his responsibility is to scrutinise – as well as radicalise – the political implications of Levinas's thinking. And this is why, to the surprise of some readers, Blanchot asks of Levinas a question such as the following: if I am obsessed, persecuted by the Other, whose hostage I am, and to whom I owe infinite responsibility, to the extent of being separated from myself, abandoned to a passivity without present and burdened with a charge that may never be fulfilled or relinquished, as Levinas's work both claims and affirms, how do I guard against the possibility I may simply be dealing with the paranoid will of an oppressive, sadistic Master ( _ED,_ 36–8; 20)? For if that were to be the case, the obligation that would fall to 'me' would not only be an obligation of infinite responsibility for and to the Other, but also the obligation to resist and struggle against the Other persecuting me and to do so by adopting all necessary and available means. What here is the relationship between infinite passivity (beyond passivity) and the highly determinate injunction to act? In Levinas's terms, it is clear that I am without power before the face of the Other; but it is precisely for that very reason, as Blanchot puts it at one stage, that 'justice demands refusal, resistance, even the violence whose purpose is to repel violence' ( _ED,_ 187; 121). Levinas himself points out that he is not putting forward in his work a prescriptive moral code but, more importantly, endeavouring to establish the ethical basis for the possibility of an ethics. He plainly insists that passivity beyond passivity does not mean passivity before evil; but if so, what then is the relation between that which belongs to being and that which is otherwise than being, between the dialectics of history, on the one hand, and, on the other, the burden of infinite responsibility for the Other? And if the second has priority over the first, how are these demands to be reconciled or mediated?
The questions are abrupt, even brutal in their implications; and the suggestion has been made that they are based on an error of reading, or are rendered inadmissible by the very spirit of Levinas's work. This may be true. But to disregard Blanchot's obstinate question in this way is to pay insufficient attention to Blanchot's unfaithful fidelity to Levinas; and it is to do violence to the suspensive motion of a fragmentary mode of writing in which every statement, assertion, or question in Blanchot's text is interrupted by the boundlessness of the neutre and its strange (il)logic of continuity and discontinuity, silence and infinity. Though on one level they may appear illegitimate, then, Blanchot's questions are compelling ones; and they address issues that are crucial if Levinas is to be taken seriously at all. For his part, Blanchot's own position with regard to the questions he raises is very clear. As Levinas's own work is the first to testify, i: is that no single language or conceptual system can lay claim to total authority and that in the ethical or political domain, as in all others, there must be, simultaneously but dissymmetrically, always at least two languages, two voices, two demands in operation:
one dialectical, the other not dialectical, one where negativity is the task, the other where the neuter stands out beyond both being and non-being [où le neutre tranche sur l'être et le non-être], in the same way that it is essential both to be a free, speaking subject, and to disappear as the passive-patient traversed by dying and hidden from view [comme le patient-passif que traverse le mourir et qui ne se montre pas].
( _ED,_ 38; 20)
What is essential in this relation, Blanchot adds, is that the ambiguity between these two languages be not resolved or equalised, and that the interval of dissymmetry and discord that passes – without passing – between these two demands be itself maintained and respected, in the manner of a void whose only obedience is to the neutre, indeed as that within thinking which infinitely exceeds the limits of thinking itself. As Blanchot puts it some pages later:
The necessity to live and die by this double language, and in the ambiguity of a time without present _and_ of a history capable of exhausting all temporal possibilities (in order to accede to the satisfaction of presence): this is the irreparable decision, the unavoidable madness, which is not the content of thought, for thought will not contain it, any more than consciousness or unconsciousness provides a status by which to determine it.
( _ED,_ 48; 27)
The lesson of the neutre, then, for Blanchot is that there is always an excess within language that is beyond language, and thus always an other otherness, so to speak, that not only exceeds all possible representations of the Other, but also effaces all possible names for the Other. To this extent, Levinas, whose own invocation of God could be described in much the same terms, is clearly right to conclude that the neutre in Blanchot is simply another name for transcendence; indeed, as Levinas observes in 1971, in an interview with André Dalmas, on the question of the neutre in Blanchot: 'this Neuter is not somebody nor even a something. It is only an excluded _middle_ [un _tiers_ exclu] which, properly speaking, not even _is._ Yet there is more transcendence to it than was ever disclosed by any after-world [plus de transcendance qu'aucun arrière-monde n'a jamais entr'ouvert]'. But if the neutre is but another name for transcendence, this serves only to reinforce the fact that the neutre, neither negating transcendence nor affirming it, is both more and less than transcendence, and that, if the neutre evokes a movement of thought at all, it is a movement irreducible to any dialectic of limitation and transgression, if only because the only law to which it bears witness is the law of its own limitlessness and reiterated effacement.
Radically side-stepping all literary and philosophical classifications enforced by convention or by law, it is arguably as a lengthy commentary – and dramatisation – of this strange movement that _Le Pas au-delà,_ published in 1973, may best be read. So much is already announced in the book's title. Embodied in the fragmentary mode that _Le Pas au-delà_ explores as it returns and advances, here is an act of transgression that, turning aside from the temptation to sacralise the limit of thought and become locked in dangerous fascination with the figure of the law, suspends the law by suspending its own movement of transgression, with the result that the relation with the limit, like the relation with death, becomes an infinite relation of non-relation, one that gives rise not to another law, but the limitlessness of the outside as such, an outside that is thematised henceforth not as transcendence, but as the impossibility that (is) the neutre. As Blanchot writes in one fragment:
Empty transgression, an image of the movement of all transgression that is preceded by no taboo, but posits no limit either by crossing beyond what cannot be crossed. No before, during, or after. It is as in another sphere, the other of any sphere. Within the realm of the day, what rules is the law, the prohibition it pronounces, the possible and the language of justification. Within the space of the night, are infringements of the law, the violence that breaches prohibition, the non-possible, the silence that refuses what is just. Transgression belongs to neither day nor night. Never does it encounter the law that is however everywhere. Transgression: the unavoidable accomplishment of what is _impossible_ to accomplish – which might be called dying itself.
( _PA ,_ 146–7; 106–7)
Transgression and transcendence are hereby effaced, pluralised, made radically other than the theological concepts that for Blanchot they once were and arguably still remain. Rewriting them as instantiations of the neutre, Blanchot reminds us that the neutre itself is in fact anything other than a concept. For if it is deployed by Blanchot, in _Le Pas au-delà,_ in the guise of the fragmentary, the multiple, and the infinitely (dis)continuous, it is that the neutre, as the infinite movement of withdrawal and re-inscription, is already a form of writing, and an antidote, so to speak, to both the fragility and the fixity of words. As such, its infinite scepticism imperiously recalls what Levinas thematises in his own texts as Saying, 'le Dire', which is a term Blanchot at times borrows from Levinas as a version of what, in earlier texts, he had addressed as the neutre of the narrative voice. This explains why the thought of the neutre in Blanchot is also the thought of literature, and why literature or writing is what responds most exactingly to the passivity beyond passivity that is the sign in Levinas of the infinite responsibility to and for the Other. The point is made in _L'Écriture du désastre_ :
If there is some relation between writing and passivity, it is that both suppose the effacement, the extenuation of the subject: suppose a change of time: suppose that between being and non-being something which is not accomplished happens nonetheless as having long since already occurred – the worklessness of the neuter [le désœuvrement du neutre], the silent rupture of the fragmentary.
( _ED,_ 29–30; 14)
In many of his texts of the late 1960s and 1970s, Levinas, perhaps responding in his turn to the strictures voiced by Blanchot the atheist, radicalises the terms in which the appeal is made in his work to the name of God. In _Le Pas au-delà,_ belatedly, Blanchot provides what may be read as an oblique commentary on Levinas's recourse to a non-theological, non-ontological God, suggesting that, in so far as the name of God names nothing, the name is an empty name, a non-name, a pure name naming only what is without name; but if this is so, it also implies that the name of God, by bearing witness to the sickness of language, is what allows language never to be cured of the malady that, as Blanchot once rejoined to Sartre, is in fact the best available proof of language's unassuming vitality. The name of God becomes here Blanchot's preferred – homeopathic – antidote to all endeavours to cure language of its supposed sickness, an antidote that, by maintaining the sickness of language, maintains the impossibility of a word such as that of God, an impossibility that is of course the very condition of possibility of the name of God itself ( _PA ,_ 69–70; 48). Under the protection (without protection) of the neutre, it is as though Blanchot is able here to rewrite the name of God not as theological transcendence, but as the name for that which in language always escapes language, which is one way of describing the neutre itself. This rewriting of the name of God as an instantiation of the neutre explains perhaps how in a series of later texts, the name of God finds its way back into Blanchot's writing. It does so, however, not as evidence of a tardy abandonment of atheism, but on two conditions: first, that what it names without naming is not the God of Christianity but the God of the Talmud, and that the name of God, no sooner written, is immediately retracted and re-(de-)named as that of an Other. Which, on Blanchot's part, is a way of affirming that, in the face of the absolute alterity of the Other, the only beginning of response, in the impossibility of its possibility and its withdrawal both of what is and of what is not, beyond immanence and transcendence alike, is what Mallarmé used to call: 'le seul acte d'écrire', the sole act of writing.
## The demand of writing
(A primal scene?) You who live later, near to a heart that beats no more, suppose, suppose this: the child – aged seven, eight perhaps? – standing by the curtain, drawing it aside and, through the window-pane [à travers la vitre], looking. What he sees: the garden, the wintry trees, the wall of a house: yet as he looks on, no doubt as children do, at where he usually plays, he grows weary and slowly looks up towards the ordinary sky, its clouds and grey light, the dull day without depth.
What happens next: the sky, the _same_ sky, suddenly open, absolutely black and absolutely empty, revealing (as though through the broken window-pane [la vitre brisée]) an absence such that everything has always and forever been lost in it, so much so that affirmed and dissipated within it is the dizzying knowledge that nothing is what there is and, to begin with, nothing beyond [rien est ce qu'il y a, et d'abord rien au-delà]. What is unexpected about the scene (its interminable trait) is the feeling of happiness that immediately engulfs the child, the ravaging joy to which he can bear witness only in tears, an unending stream of tears. People think he is just having a childish upset and try to console him. He says nothing. He will live hence forth with that secret. He will weep no more.
_L'Écriture du désastre_ ( _ED,_ 117; 72)
At the end of _L'Entretien infini,_ Blanchot summarises the arguments put forward in the volume as a whole by enumerating nineteen propositions of varying length and complexity under the general heading: 'L'Absence de livre' ('The Absence of the Book'). The title immediately recalls the phrase 'absence d'œuvre' (absence of the work) which, increasingly in his later writing, Blanchot comes to substitute for the concept of 'désœuvrement'. As with the expression 'absence d'œuvre', the syntax adopted for 'absence de livre' is not without some significance. For by depriving the book of any grammatical article, definite or indefinite, the phrase refers to the idea of the book as in effect always already under erasure; Blanchot's writing thereby endows the word absence with dynamic qualities that change it from a substantive into something more nearly resembling a verbal form. The term, accordingly, comes to function like a new coinage. Absence does not imply nostalgia for an entity formerly present and now lost, but functions rather as a force of effacement and re-inscription that from the outset inhabits the book as its abiding, innermost principle (or absence of principle), albeit as a principle that in fact lies outside the book and presents itself to writing as the necessity of the book's past and future erasure. Absence here, then, is proffered as an exteriority beyond the book, and which the book, by definition, cannot incorporate within itself; and in this sense, once more, with regard to the book itself, the absence of the book, in Blanchot, functions as a simultaneous condition of possibility and impossibility.
In this closing chapter, which acts a moment of eschatological invocation, bringing a certain past to an end, and manifesting the infinite promise of the future, a gesture the writer redoubles by addressing his text to the absent books of anonymous friends, Blanchot begins with both an ending and a beginning: the one, falling at the end of history, bears the name of Mallarmé, while the other, at the very beginning of time, carries instead the mark of that mythic original man who first put tool to surface, tracing an inscription and inaugurating what was to become, as Blanchot puts it, the possibility of presence, meaning, and culture, in short, the history of the book as such in its threefold guise: as empirical object, precondition of all knowledge, and self-identical totality. At either end of the millennia, in Blanchot's account, two absent events seem to call to one other, opening and closing a history: first, an initial act of inscription, coming at the dawn of intelligibility itself, prior to all thought of the book, but giving rise to the possibility of the book, even as the act itself, preceding both the visible and the invisible, neither adding an object to the world nor removing one from it, installs within the horizon of the world – which it thus suspends – a gap in the universe, a universal void, a 'vide d'univers' ( _EI,_ 620; 422) that precedes and exceeds, enables and disables the book as such; then, many centuries later, a moment of completion in which the act of writing, bringing the order of the Book to a close, spills at last, once again, beyond closure, in order to enact, under the auspices of the absence of the book, its own double movement of erasure and re-inscription as a process of infinite dissemination, dispersion, and fragmentation. And between these two turning points, according to Blanchot, enabled but also undone by the sole act of writing: the theological history of the book, having now reached its end though enduring still, founded on the assurance of self-presence and temporal continuity, embodying in its dialectical unity and closure the arrogant self-certainty of a culture for which otherness is only ever an occasion for the demonstration of mastery.
While the law of the book demands of language that it remain subject to the dialectic of discourse – and the discourse of dialectics – writing is what is always already beyond the power and possibility of that law. Writing, so to speak, in refusing all posing, is radically opposed (or preposed) to any book. 'To write', argues Blanchot, 'is to produce the absence of the work (worklessness)' ( _EI,_ 622; 424); and he adds, some pages later, by way of explanation:
The absence of the book: the prior deterioration of the book, its dissident play with regard to the space in which it is inscribed; the prior dying of the book [le mourir préalable du livre]. Writing, the relation to the _other_ of every book, to what in the book would be de-scription [dé-scription], a scriptuary exigency outside discourse, outside language. Writing on the edge of the book, outside the book.
( _EI,_ 626; 427)
Though it always interrupts the rule of the book, writing is not simply a name for negativity, nor is it merely a function of the book's absence from itself. Writing is called forth not by the book, which is bound always to seek to re-incorporate writing within itself as presence, discourse, and sense, but by the attraction exerted on writing by the pure exteriority of the outside in its irreducible alterity and disseminated plurality. The outside in Blanchot, in that it precedes, exceeds, and traverses the book, is beyond presence. Lacking any principle of self-identity or permanence, it cannot as a result ever be construed 'as such'. Like the neutre, it resists the law of naming, which is why, if writing is a trace inscribed on the outside, what it traces, as it were, is not a trace of itself, but a trace that is always already other, always already multiple, and thus always already effaced. To the extent that it is of the outside, then, writing remains absolutely bereft of origin or beginning. Irreducible to the continuity or temporality of narrative, writing in Blanchot is thought only as irremediable repetition, infinite fragmentation, ineliminable excess, impossible alterity, which in turn explains why the figure of absence of the book, though it may work as a possible name for writing, cannot be read as a concept of writing, since what it seeks to name is prior to all concepts; it functions instead – impossibly – as a pre-conceptual trace that falls beyond the jurisdiction of all philosophical or critical discourses to the extent that all such discourses are necessarily founded on the self-presence and permanence of the law.
Crucially, if it is to be articulated or promulgated at all, the law itself must be written down; and it is for this reason that the possibility of writing necessarily precedes the law and cannot be coerced by it. This is also why writing, which nothing precedes, is dangerous and violent, and why, as Blanchot had learnt from the example of Hölderlin, the relation to writing, necessarily a relation of non-relation, must always be articulated obliquely and by means of a detour: the detour that, for Blanchot, constitutes literature itself. So while it is true, as Blanchot affirms from the outset, that writing is the greatest violence of all, because it exceeds both the law and its own law ( _EI,_ viii; xii), it is also the case that one of the functions of the law is to rescue human order from the violence of writing by affirming the necessity of the limit in the form of a second writing turning away from the outside, and which is the writing of the law itself ( _EI,_ 634; 432). But inevitably the limit instituted by the writing of the law has always already been transgressed; indeed, without such prior violence no law would be possible at all. Whence the law's awareness of its own dual status, for while it seeks to impose the limit that it values, properly, as its own, the law is also bound to acknowledge its fragility and impropriety in that, if it is to be written at all, it must always already have been breached.
In this sense, far from being beyond the law, transgression already belongs to the law, and in this perspective it is evident transgression is a necessary moment in the proper deployment of the law as such. But this solidarity between law and transgression, though it necessarily confirms the supremacy of the law, does not mean the law has no outside. On the contrary, what it implies is that the law and transgression have themselves already been transgressed, so to speak, in a more radical sense, by that which is already prior to the law because it is that upon which the law itself always already depends. Such, Blanchot argues, is the case of writing, which is why writing stands with regard to the law in a relation of exteriority, dissymmetry, and dissidence. The relation between writing and the law is a relation of non-relation; and it is from this that derives the 'transgressive' demand – the 'pas au-delà' – of writing, writing that is always other than the law, not because of its superior power or legitimacy in the face of the law, but rather because of its refusal, on grounds of impotence, impossibility, and weakness, to allow itself to be addressed by the language of the law and constituted by the law as a subject or hostage of that law. In other words, the transgressive extremity of writing is not the result of its ability to assert, against established law, and in competition with the law, the authority of a counter-law, but because, preceding the discourse of law, lacking all unity, presence, and identity, writing is a challenge to any authority whatsoever, including of course its own.
For Blanchot, this is not to claim, against the threat of external censorship, that writing enjoys (or should enjoy) special privileges on the grounds that it is a vehicle for the all-conquering creativity of the imagination. It is to argue instead that writing must be thought as a response to a more exacting demand than that of the law, a demand which is that of limitless contestation; writing here is a response to the impossibility and infinite alterity of the outside. The ethics of writing, therefore, for Blanchot, have nothing to do with enforcing workaday societal norms, however desirable or just these may be; and it is this that founds Blanchot's consistent opposition to all forms of censorship, whether religious, moral, or political. With respect to all forms of authority – and this includes the principle of its own so-called aesthetic autonomy – writing is in a position of irreducible dissidence. Since literature does not and cannot obey external moral injunctions of whatever sort, it has no alternative but to affirm its essential dispersion and insubordination to all. What follows from this for Blanchot is not a retreat into aestheticism or art for art's sake. For it is precisely because writing is irreducible to any form of moral, political, or aesthetic authority, that it is able to constitute – or, better, de-stitute – itself as a response to the infinite demand that is the demand of the Other. Writing here for Blanchot does not simply have an ethical dimension; it has become absolutely inseparable from the exigency of the ethical as such. As Blanchot commented in 1983 in answer to a questionnaire:
Writing is admittedly a form of work, but a perfectly insane one, which asks nothing, does not justify itself, and which no reward might possibly satisfy. Writing: a singular demand (let us call it bizarre), more ethical than aesthetic, since that to which it responds is an impersonal injunction without obligation or sanction [un _'il faut'_ sans obligation ni sanction].
The argument presented by Blanchot in 'L'Absence de livre' is not a purely philosophical one; for it comes in the form of an impossible narrative that is itself in part a myth of origins. But while the text presents to the law of the book a narrative dealing with the donation and withdrawal of the law, it makes clear that its own narrative is a necessary concession to the detour of writing as such. The word writing in Blanchot is never simple, but always multiple; and here it becomes a name for that in writing which permits the formation of a pact, or alliance, between writing and the law and gives rise to the possibility of literature. Yet while Blanchot's essay provides for this narrative structure by narrating the genesis of that structure, it also neutralises and withdraws narrative from itself, with the result that narrative is maintained as a necessary detour but ultimately inscribed within a space that narrative itself cannot enclose and whose borders pass beyond and transgress the possibility of narrative. Narrativity is thereby suspended and put under erasure. So if 'L'Absence de livre' is in the form of a narrative, it is also at the same time an interruption of narrative; and while on one level it adopts a tabular structure reminiscent of the articles of the law itself, on another level it also fragments the law, pressing, within the fissures of the law, the claims of a demand that, not benefiting from the law's mediation, is in fact more rigorous still than the law itself.
This double movement of obliqueness and suspension, self-contestation and dissimulation is a structure that perseveres throughout the whole of Blanchot's writing. It recurs in especially persistent form in those narrative texts or _récits_ that, alongside his more familiar books of literary criticism and philosophical analysis, Blanchot, since the early 1950s, had continued to write and publish: _Au moment voulu_ ( _When the Time Comes_ ) _, Celui qui ne m'accompagnait pas_ ( _The One Who Was Standing Apart From Me_ ) _, Le Dernier Homme_ ( _The Last Man_ ) _,_ and _L'Attente L'Oubli._ Admittedly, as noted before, this last title, published in 1962, was Blanchot's last free-standing _récit;_ and, in any case, the book owed perhaps as much to metafictional essay as it did to conventional narrative. But this apparent falling silent of Blanchot's fictional writing after 1962 is deceptive, first because that silence was in fact not a silence, giving way as it did to a series of texts, notably _L'Entretien infini, Le Pas au-delà,_ and _L'Écriture du désastre,_ which, though not fictional works in any received sense, nonetheless contain extensive fictional passages that serve, at the very least, to challenge the ease with which these texts are usually categorised as works of philosophy and, moreover, to delimit the ambitions of philosophy by practising a mode of textuality for which the boundaries that pass between philosophy and its other(s) are neutralised and deprived of their title and identity. There is a second reason why the apparent coming to an end of Blanchot's fictional works in 1962 is deceptive, which is that for Blanchot, as _L'Arrêt de mort_ testifies, the ending of fiction had in a sense always already taken place before that date, in that the endlessness of the end for Blanchot was inherent in fiction itself as a version of the limitless interruption that coincides with the inscription of the limit itself. What lies withdrawn in Blanchot's apparent abandonment of fiction after 1962, then, is not the closure of fiction, but its always prior responsiveness to the limitlessness that is the reverse side of finitude and is inscribed most insistently by Blanchot in the heterogeneous, fragmentary manner adopted by him in his books from _L'Attente L'Oubli_ onwards. Here, the ending of fiction is as much a case of repeated return as perpetual effacement, and indeed it is both: effacement as return, repetition as perpetuity, impossibility as affirmation.
Like their predecessors, Blanchot's _récits_ of the 1950s entertain complex, paradoxical relations with narrative structure. Though they submit in part to the constraints of narrative, these fictions follow, as they do so, a detour of dissimulation with the result that Blanchot's writing is constantly responding in narrative to that which is infinitely other than narrative. If the absence of the book is a token of the infinity of writing, so the limitations imposed on narrative in Blanchot's _récits_ are like a tribute paid to the limitlessness of writing itself. This tension between finitude and the infinite, between possibility and impossibility is one Blanchot explores in all the _récits,_ and it is what gives rise, clearest of all perhaps, to the peculiar oscillation between movement and paralysis, eventfulness and suspension that is characteristic of them, and which Derrida neatly sums up by punning on the closing sentence from _La Folie du jour,_ reading Blanchot's 'pas de récit' both as that which is no narrative at all, and as that which is a step towards, along, or even beyond the path of narrative as such. The effects of this stumbling rhythm of advance and return, reprise and interruption, perseverance and paralysis, are everywhere to be found in Blanchot's texts; but such is the irregularity of the movement produced, in relation not only to the law of narrative in general but the structure of Blanchot's own writing in particular, that there are good reasons, ethical as well as literary critical ones, as Derrida has shown, for hesitating before applying to Blanchot's _récits_ any all-purpose interpretative grid. Writing in Blanchot, one may say, is never the rule, always the exception.
But the singularity of writing, while it may contest the law of exemplification, is not solipsistic closure, and to read Blanchot's _récits,_ as any text, is necessarily first to become entangled in a complex set of conventions and expectations governing the treatment of fictional space and narrative time. The paradox applies no doubt by definition to probably all literary texts; it is at any event what gives Blanchot's _récits_ their peculiar status, as Derrida has argued, as examples without example, paradigms without paradigm, cases without law. In this regard, _Au moment voulu,_ published in 1951, if not exemplary, is arguably no exception. Like the author's other fictions, the book is set almost entirely in an anonymous interior location – here, an apartment – whose most salient characteristic is its labyrinthine emptiness, and its strange spatial configuration that seems to accommodate both boundless immensity and sparse confinement. And though the text begins abruptly and incisively, with the unexpected surprise of the narrator's encounter with Judith, a woman of his acquaintance, on the threshold of her flat, the narrative barely proceeds much beyond this liminary metatextual event, about which much in the way of information is deferred or withheld, save to evoke a constantly suspended, forgotten, or abandoned visit to fetch a glass of water from the kitchen, with the result that the fiction as a whole remains suspended on an interminable scene that refuses to develop beyond this allusive beginning. Indeed, two years later, this request for a glass of water was still being rehearsed, or repeated, in identical words – 'Give me a glass of water ['Donnez-moi un verre d'eau'] – by the narrator of Blanchot's next _récit, Celui qui ne m'accompagnait pas_ ( _MV,_ 12; 3; CQ, 96; 51).
'Nobody here desires to bind themselves to a story' ('Personne ici ne désire se lier à une histoire') ( _MV,_ 108; 47), declares Claudia, Judith's friend and companion, to the narrator at one stage, and the protagonist comments as follows:
This sentence made a great impression on me. I thought I could see a light radiating forth from it, I had touched a point of startling luminosity. A sentence? more a slippage, a picture not yet framed, a movement of sparkling brightness that shone in quick dazzling bursts, and this was no calm light, but a sumptuous and capricious chance occurrence, the moodiness of light itself.
( _MV,_ 108–9; 47)
Claudia's pronouncements are endowed here with a powerful emblematic force, and Blanchot confirms as much when he re-cites them, eleven years later, in _L'Attente L'Oubli,_ using them again to allude, as here, to the relation of non-relation passing – or not passing – between two sexually differentiated bodies or voices. At this climactic point in the text, like the phenomenon of the windowpane – what Blanchot in _Au moment voulu,_ as elsewhere, refers to as 'le phénomène de la vitre' ( _MV,_ 82; 35) – that separates inside from outside while also connecting them, transforming presence into absence and infinite distance into boundless proximity, Claudia's words interrupt the narrative to mark a suspension in the story that is also a suspension of the story. In the process, what is revealed in Blanchot's protagonists, beyond narrative, is an avid commitment to desire that requires not the binding of narrative events, but rather their effacement, and an openness to capricious chance rather than stern necessity, in the knowledge that the former is no more than an ecstatic version of the latter, albeit with the risk that the narrator, for one, will find himself deprived of the protection afforded by narrative, and bound henceforth, as he puts it, not to story but to the poverty of dispossession that lies beyond and conditions it.
Shortly before _Au moment voulu_ was published by Gallimard in 1951, Blanchot brought out a fragment of the text, corresponding to the first twenty-five pages of the book version, under the title 'Le Retour' ('The Return'). The term did not specify, of course, whether it was a case of the narrator returning to earlier, strangely familiar haunts, as the first paragraph of the text suggests, or, more disturbingly, whether the return more closely resembled the ghostly recurrence of another time, come to haunt the narrator's memory and confront him with what, beyond memory, constitutes the condition of possibility of recollection as such. At any event, and in either case, the term provides some clue as to how, in its suspension of narrative, Blanchot's _récit_ may be approached; it also gives an indication of the nature of the book's reliance on a series of events that constantly resist designation as events, and which, if one concedes the term, are in fact more like absent events, events without event, in which case they demand most plausibly to be read as a form of all-pervasive metatextual commentary on the event without event that is the _récit_ itself. Residual though it is, it is certainly true that the narrative structure of _Au moment voulu_ is dominated by this figure of return, which has its own recurrent motto in the form of the oftrepeated phrase: 'À, nouveau! à nouveau!' ( _MV,_ 14; 4), meaning: again, again, once more, once more: once more as though for the very first time. The phrase, which is to be found in many other places in Blanchot's writings, is one that the author glosses himself in 'L'Expérience originelle' ('The Original Experience') from _L'Espace littéraire,_ a text he was writing shortly after _Au moment voulu,_ and in which he suggests: ' "Again, again!" ["A nouveau, à nouveau!"], is the cry of anguish grappling with the irremediable, with being [le cri de l'angoisse aux prises avec l'irrémédiable, avec l'être]' ( _EL,_ 256; 243). And Blanchot explains:
It is in this sense that in the precincts of art there is a pact concluded with death, repetition, and failure. Beginning again, repetition, the inescapability of return, all the things evoked by those experiences in which eeriness is allied with the strangely familiar, where the irremediable takes the form of endless repetition, where the same is given in the dizziness of replication, in which there is no cognition but only _re_ cognition [où nous ne pouvons pas connaître mais reconnaître], all this alludes to that initial error which may be expressed as follows: what is primary is not beginning, but rebeginning, and being is precisely the _impossibility_ of being for the first time.
( _EL,_ 255; 243)
In other words, what occurs without occurring, once more, for the first time, repeatedly but always differently, throughout _Au moment voulu_ as well as Blanchot's other _récits,_ is that which is irremediable: what in 'L'Expérience originelle' Blanchot describes as the impossibility of not being, but which in later writings is more frequently thematised by him as the impossibility not only of non-being, but of being as well, a radicalisation of Blanchot's thinking that belongs to the transformation in his thought brought about by the intervention of the neutre. The shift – arguably more one of degree than substance – was no doubt prompted not only by Blanchot's reading of Levinas during the latter part of the 1950s, but also by the experience of writing, for example, without example, a text such as _Au moment voulu._ For in that text, as the title suggests, the figure of return itself repeats – once more, as though for the very first time – Nietzsche's experience of eternal recurrence, an experience that is itself indeed the experience of 'once more, as though for the very first time', and which as such, as Pierre Klossowski was to remind Blanchot some years later, is in fact founded on an impossible aporia, in that, having first posited the possibility of revelation, it is then forced to negate that possibility by the very content of the revelation it imparts. Understood in this way, the experience of return is properly unthinkable within the horizon of being; what it affirms, beyond being, is the impossibility of beginning, ending, meaning, presence, even finitude itself. Depriving the narrator of responsibility for his own story, it exposes him to the lack of origin and arbitrariness of a wheel of chance which is none other than the wheel of writing itself, for which all past is future and all future past, in a radical effacement of all possibility of presence.
The figure of return functions here as a paradoxical, but fatal and fateful exigency, one that reveals in the inevitability of death the impossibility of dying, in the law of repetition the excess of difference, in reason madness, and in the temptation to conclude, the necessity of infinite incompletion. Without the promise or memory of the end, writing, like life or speech, cannot in fact begin, and to speak at all, in the words of _Celui qui ne m'accompagnait pas,_ is always to be nailed to the spot; the end however is always out of reach, it can only ever be announced in advance of itself, and to write till the end and in view of the end is to submit forcibly to an endless circle where the end has always already taken place without taking place and remains infinitely deferred as the limit that gives rise only to its own limitlessness. Here, Blanchot's stumbling narrative leads narrator and reader into an experience where each step forward is necessarily a step back; where omnipotence is weakness and remembering forgetting; where the only illumination is darkness, and the only truth violence and passion; where the plenitude of the world gives way to emptiness beyond the world; where the only revelation lies in the infinite impossibility of revelation; and where the suspension of time embodies the demand of a unique moment that, across infinite distance, is both a thought to be affirmed and an Other to be addressed.
In this sense, the figure of return is not a figure at all, but more an evanescent trace, always already effaced, that, like the pre-conceptual singularity it is, addresses itself to thought as the trace of thought itself: thought, that is, not as self-identity, but as what one might call, with Blanchot, the giving withdrawing of the neutre itself, thought as both the necessity of death and the impossibility of dying, thought in the sense it has for Blanchot when he writes, in _Le Pas au-delà,_ of its inadequacy in the face of death, an inadequacy that implies that thought itself is already a manner of dying, already a way of approaching the impossibility of dying which is but a name for the limitless impossibility of thought itself: 'Death', Blanchot writes,
being that to which we are not accustomed, we approach it either as the unaccustomed that astonishes or as the unfamiliar that horrifies. The thought of death does not help us to think death, does not give us death as something to think. Dying, thinking, so close to one another that thinking, we die, if, dying, we dispense with thinking: each thought might be termed mortal; each thought a final thought.
( _PA ,_ 7; 1)
Alongside the neutre, the outside, and disaster, Blanchot writes in _L'Écriture du désastre,_ the word return is one of the four names without name for what Blanchot describes as 'thought, whenever it allows itself to be unbound, by writing, to the point of the fragmentary [lorsque celle-ci se laisse, par l'écriture _,_ délier jusqu'au fragmentaire]' ( _ED,_ 95; 57). As such, it is a vestige of what might be termed the language of the absence of the book, a language lost and never uttered, a language of infinite alterity. Eschatology here, if it is the language of the limit, is also the language of the limitlessness of the limit. This dual, ethical responsiveness to both the end and the endlessness of the end is what sustains the intensity with which Blanchot's _récit_ challenges narrative order and pushes meaning to the brink of a catastrophe that is not an erasure of thought, but irremediable testimony of that which in thought always escapes thought. This in turn is what gives Blanchot's writing in _Au moment voulu_ what Derrida has called its apocalyptic tone, which, as Derrida maintains, rather than a moment of ultimate truth, is what turns Blanchot's version of apocalypse into an apocalypse without apocalypse, an apocalypse of apocalypse. For what it brings, beyond truth, beyond falsity, is the experience of thought itself as an encounter with the extremity of the limit and with the limitlessness of the limit. In the absence of the book, then, writing for Blanchot turns aside from morality, law, being, and sense, to measure up to the immeasurable extremity of the outside. As it does so, what it affirms is its own infinite obligation to alterity, which requires of writing that it challenge and contest all law, all power, all sense, all identity, all stability of thought of whatever kind.
## An uninterrupted questioning
Suppose a past, a future, without anything which would make it possible to pass from one to the other, in such a way that the line of demarcation, by remaining invisible, would demarcate them all the more: the hope of a past, the completion of a future. All, then, that would remain of time would be this line to be crossed, which, being always already crossed, is nevertheless uncrossable and, in relation to 'me', impossible to situate. It is perhaps only the impossibility of situating this line that we presumably call the 'present'.
The law of return, in supposing that 'all' may be said to recur, appears to posit time as closed: the circle beyond circulation of all circles; but in so far as it breaks the cycle in its middle, it puts forward a time that is not incomplete, but, on the contrary, finite, except at that point which is now, and which alone we believe we possess and which, lacking, introduces the break of infinity, obliging us to live as though in a state of perpetual death.
_Le Pas au-delà_ ( _PA ,_ 22; 12)
Writing, Blanchot maintains, though solitary, is not a solipsistic enterprise; indeed solipsists, he implies, much like pessimists, probably do not write at all ( _ED,_ 174; 113). For Blanchot, to write is instead always to address the unknown other: the other in so far as it is unknown, the unknown in so far as it is radically other. Writing, then, necessarily implies communication, community, even communism; and such were the terms, borrowed from Bataille, Jean-Luc Nancy, and others, with which Blanchot, in 1983, began the book that has most obvious claim to being the author's ultimate political testament: _La Communauté inavouable._ But while invoking these words, Blanchot also made it clear that they were themselves sufficient to the task ('"convenables"', as he put it, redoubling his point by the silent intervention of quotation marks ( _CI,_ 10; 1–2)) only to the extent that they were treated as being already under erasure: discredited and emptied of meaning as a result of their own past history, but for that very reason the site of an inescapable demand, one to which it is possible to respond only by what Blanchot calls their proper–improper abandonment ( _CI,_ 10; 2).
Bound as it is not only to the thought of the neutre but also to the absolute exteriority, beyond any world whatever, that Blanchot terms disaster, such a gesture of refusal yet submission, effacement yet re-inscription, gives a clear indication of the temporality and purpose of Blanchot's book. For while in _La Communauté inavouable_ Blanchot was drawing lessons from the historical past, he did so by evoking, beyond presence, the very futurity that, perhaps surprisingly for some of his contemporaries, a certain, now distant past still contained. In that way, the suggestion on Blanchot's part was evidently also that those who hoped politically the future might prove to be different from the past might first have to relinquish any residual piety they might have in order to confront the radicality of what Blanchot invokes throughout _Le Pas au-delà,_ in a sustained meditation on the lessons of May 1968, as a future without presence, self-identity, continuity, or _telos;_ for only such a future, in Blanchot's view, founded on repetition, and intervening within time as infinite interruption rather than a manner of deferred presence, would paradoxically prevent the past from repeating itself, in time-honoured manner, as tragedy or farce: a verdict or injunction that in 1983 was not without its discreet relevance for a France already embarked, in conscious recollection of the lost opportunities of the Popular Front nearly a half-century earlier, on the first seven years of the socialist presidency of François Mitterrand.
Like any testament, _La Communauté inavouable_ has a prospective as well as retrospective dimension; and while on the one hand Blanchot takes the opportunity of recalling the pre-war political itinerary of Bataille and his own later experience of the campaign against the Algerian war and the _événements_ of May 1968, his more urgent concern is to respond in the book to a series of more recent writings, the most important of which, though in some ways the least visible, is arguably a paper by Derrida, first presented at the 1981 Cerisy conference on 'The Ends of Man', and itself addressed in part to Blanchot, entitled: 'D'un ton apocalyptique adopté naguère en philosophie' ('Of An Apocalyptic Tone Recently Adopted in Philosophy'). In that paper, at the end of a long discussion on the indissociability of enlightenment and eschatology, philosophical reason and apocalypse, Derrida concludes by evoking, as he had in an earlier text on Blanchot, the irreducible urgency and affirmative tone of a certain 'viens', meaning: come!, of which Derrida writes that, like some necessary but impossible pre-conceptual event that is precisely not an event, it resists appropriation by ontology, theology, rhetoric, grammar, even eschatology, precisely because it is the trace, or gesture, which gives rise to the conditions on which such determinations themselves depend. As such, writes Derrida, 'in this _affirmative_ tone, "Come" marks in itself neither a desire, nor an order, nor a prayer, nor a request [ni un désir, ni un ordre, ni une prière, ni une demande]'.
Blanchot immediately concurs, and glosses his accord by citing in modified form a passage from _Le Pas au-delà,_ given three pages before the end of the book, that Derrida himself had cited in the course of his essay 'Pas' in 1976: 'Come, come, each and all, you for whom any injunction, prayer, or expectation is inappropriate' ('Viens, viens, venez, vous ou toi auquel ne saurait convenir l'injonction, la prière, l'attente') ( _CI,_ 26; 12). Such closing words, Blanchot comments, and the writing of which they are the ruinous enactment, are intimately bound to the possibility of existence of community itself in that they address the unknown other; but if that is so, it is a necessary aspect of their address that they function as a disastrous or catastrophic invocation falling – coming – before the identification of any sender, or addressee, or language, or message, or judgement. Which is why, in a disastrous destruction of revealed truth, 'viens', in Derrida's terms, effects a kind of apocalypse without apocalypse, an apocalypse of apocalypse, so to speak, and why for Blanchot – 'perhaps', as the writer puts it – this apocalyptic voice also constitutes the condition of possibility – or impossibility – of community as such ( _CI,_ 26; 57).
Throughout _La Communauté inavouable,_ with the sole exception of this particular – and particularly consonant – exchange, the reference to Derrida remains muted; elsewhere in the book, however, Blanchot makes a number of discreet gestures in the direction of this logic of redoubled apocalypse by insisting on the irreducible – and self-questioning – excess of ethics or politics over ontology, and for instance by casting humankind as a whole at one stage, in its messianic dispersion and worklessness, in the role of the people of God failing even to leave Egypt in their quest for the promised land ( _CI,_ 57; 33). Turning aside from the debate with Derrida, Blanchot concerns himself instead with two more recent texts, Jean-Luc Nancy's essay 'La Communauté désœuvrée', and the story _La Maladie de la mort b_ y Marguerite Duras. But, while ostensibly dealing with Nancy and Duras in turn, in the two halves of the book, namely 'La Communauté négative' ('The Negative Community') and 'La Communauté des amants' ('The Community of Lovers'), Blanchot's more urgent concern, in continuation of his earlier reading of Bataille and Levinas set out in _L'Entretien infini,_ was to explore further the implications of the notion of double dissymmetry introduced there.
As a result of these various turns, _La Communauté inavouable_ comes to be organised not as a philosophical treatise, but as an unavowable community of slippages between texts and between at times seemingly incompatible proper names. And as one name supplements or is substituted for another, what is described in Blanchot's text, between Derrida, Bataille, Levinas, Nancy, and Duras (not to mention Edgar Morin, Sade, Sartre, Lacoue-Labarthe, Heidegger, Leiris, Char, Kierkegaard, Plato, the Old Testament, Dante, Tsetaieva, Proust, Wittgenstein, Blanchot himself, and others too), is a kind of textual drift dramatising how the writer, far from adopting a philosophical position exclusive of all alterity, is instead, in writing, constantly being exposed to questioning by a proliferating number of different literary, philosophical, political discourses and events. In this, of course, Blanchot is doing nothing extraordinary, save perhaps to verify the plural, iterative structure of 'viens' itself. For as Derrida remarks at the end of his Cerisy paper:
'Come' is not addressed to an identity determinable in advance. It is a drifting [dérive] underivable from the identity of a determination.'Come' is _only_ derivable, absolutely derivable, but only from the other, from nothing that is an origin or a verifiable, decidable, presentable, appropriable identity, from nothing that is not already derivable and arrivable without demarcation [dérivable et arrivable sans rive].
In addition to the exchange with Derrida, _La Communauté inavouable_ was also written in response to a more localised occasion. This was the founding, at Derrida's suggestion, after the 1981 Cerisy conference, by Philippe Lacoue-Labarthe and Jean-Luc Nancy, of a 'Centre de recherches philosophiques sur le politique', or Centre for Philosophical Research into the Political. The purpose of the Centre, Lacoue-Labarthe and Nancy explained, was to engage in the specifically philosophical enterprise of questioning not politics in its epiphenomenality (' _la_ politique'), but the logic of the political as such (' _le_ politique'). Aside from political science or political philosophy, they argued, there was an urgent need to pose the more radical question of the very essence of the political. For Lacoue-Labarthe and Nancy, this meant ultimately attempting to think the question of relation outside of any concept of the subject and of subjective identification; and to this end they placed high on the agenda of their own work the project of examining the political itineraries of Heidegger and Bataille, who, within the general context of totalitarianism, were thought to have touched on the twin limits of the political not only by their reframing of community but also by their failure fully to dismantle their reliance on the concept of the subject. Thus, according to Lacoue-Labarthe, Heidegger's reliance on the German Volk as a philosophical and political subject of destiny, which was what lay behind the endorsement of National-Socialism. Thus, too, according to Nancy, Bataille's forlorn search for sacrificial community, giving rise, he explains, to 'the paradox of a thinking drawn towards community, yet governed by the theme of the sovereignty of a _subject'_.
Evidently, there was much in the analysis put forward by Lacoue-Labarthe and Nancy that could not fail to be of major interest to Blanchot. Given his own political itinerary since the early 1930s, and his own reading of Heidegger and Bataille since that period, it is not hard to see how Blanchot may have felt addressed by the work of the Centre. Indeed, much of his own thinking of the question of the political, ever since the late 1950s, as the texts collected in _L'Entretien infini_ show, was closely focused on the need to dismantle the concept of the subject in order to construe otherwise, with the help of Levinas, the very question of relation that Lacoue-Labarthe and Nancy sought to address in their work for the Centre. In this respect, _La Communauté inavouable_ may best be read as constituting Blanchot's own contribution, presented so to speak in absentia, to the deliberations of the Centre. But while, by its very publication, the book was evidently supportive of the initiative of the Centre's foundation, it was also obliquely critical of at least some of the analysis put forward within the Centre; and while Blanchot entirely endorsed Lacoue-Labarthe's account of Heidegger's politics, as proposed in 1981 in the paper 'La Transcendance finit dans la politique', his response to Nancy's work on Bataille, presented mainly in 'La Communauté désœuvrée', was much more nuanced.
As always, evidence for this divergence of view is largely implicit within Blanchot's text. At first, in 'La Communanté négative' Blanchot closely follows Nancy's own exposition, even to the point of reproducing several of Nancy's quotations from Bataille from 'La Communauté désœuvrée', with the result that the opening ten or so pages of Blanchot's book read almost like an exact restatement of the corresponding sections of Nancy's original essay. In common with Nancy, for instance, Blanchot identifies hitherto existing communism as resting on a totalising conception of human immanence – what Nancy, followed by Blanchot, terms the immanence of man to man ('l'immanence de l'homme à l'homme') or the status of man as absolutely immanent being ('l'homme comme l'être absolument immanent') – and therefore on an understanding of man as constituted by the totality of his works, among which Blanchot counts humanity, nature, and God (and characteristically enough, in making this last point, Blanchot silently paraphrases the quotation from Herder provided by Nancy ( _CI,_ 11; 2)). Similarly, Blanchot goes on to endorse Nancy's view of totalitarianism and individualism as representing the two sides of a single coin, with overarching unity on the one hand mimicking atomised closure on the other. And despite their responsiveness to the word communism, Nancy and Blanchot are agreed in their decisive rejection of any conception of community founded on the nostalgia for fusional, eucharistic communion and on the integrative, ultimately totalitarian power of nationalist or communitarian myth; community, instead, writes Blanchot, citing Nancy, is 'the presentation of finitude and of excess without return that founds finite-being [la présentation de la finitude et de l'excès sans retour qui fonde l'être-fini]' ( _CI,_ 24; 11).
Rather than a choice between contending ideologies, such as liberalism or communism, the question of politics, for Blanchot, may perhaps best be described as the question of the alternative between the legacy of a society founded in totalising myth and in the image of its own self-identity and the futural prospect of a community exposed to exteriority and interrupted by alterity. In other words, between what in _La Communauté inavouable_ Blanchot terms communion and substitution ( _CI,_ 24; 11), or between what he describes elsewhere, after Levinas, as the choice between the sacred and the saintly, or holy, that is, between effusion and election. Up to a certain point, of course, much the same might be said of Nancy, were it not that Blanchot's slippage into the vocabulary of Levinas at this stage, within the community of thought constituted by Blanchot and Nancy, is evidence of a crucial but necessary divergence between them. For it is apparent that, by rephrasing Nancy's argument in Levinasian terms – with regard to the primacy or originality of which Nancy is clearly sceptical – Blanchot in his turn voices a degree of scepticism with regard to Nancy's endeavour to rethink 'community' on the basis of a re-articulation of Heideggerian 'Mitsein' (or 'l'être-avec', being-with, as Nancy phrases it) and thus within the framework of an ontology, albeit one that, as Nancy's subsequent work demonstrates, despite its Heideggerian starting point, it is difficult, if not impossible, to reconcile with Heidegger's own.
Such at any rate is the implication behind Blanchot's insistence, expressed in 'La Communauté des amants', on the Levinasian principle of the priority of ethics over ontology ( _CI,_ 73; 43). Nancy's scepticism towards Levinas is discreetly rebuffed, so to speak, by Blanchot's infinite scepticism towards Nancy. But infinite scepticism must not be confused here with dogmatic rejection; indeed, as Blanchot argues in his readings of Levinas, infinite scepticism is one of the chief characteristics of the movement of withdrawal and re-inscription traced by the thought of the neutre itself. And it is clear, as witnessed by the notion of abandonment used from the outset, that Blanchot's thinking of community is decisively oriented by the (non-)concept of the neutre. This explains Blanchot's strategy of acceptance and displacement, affirmation and transformation that governs his relation to Nancy's reading of Bataille. And this is also why Blanchot at one moment is able to adopt Nancy's vocabulary as his own, while at other times underlining his differences with Nancy and in the process changing much of the emphasis of Nancy's analysis. At the very moment Blanchot, unlike Nancy, affirms in 'La Communauté des amants' the excess of ethics over ontology, he provides the reader with an intriguing emblem of this double strategy by placing at the head of that essay, taken from an earlier paper by Nancy, 'L'Être abandonné', the following epigraph, which offers a kind of philosophical rejoinder on Nancy's part to Blanchot's earlier scepticism regarding the commitment to ontology, and provides a discreet clue to the divergence of view at stake in the exchange: 'The only law of abandonment, like that of love, is to be without return and without recourse [d'être sans retour et sans recours]' ( _CI,_ 51; 29).
This epigraph functions as more than a decoration or homage. It highlights one of the most pervasive conceptual motifs running through _La Communauté inavouable;_ at the same time, it marks both the accord and discord between Blanchot and Nancy embodied in the book as a whole. Throughout, Blanchot's debate with Nancy is organised to a large extent by the different inflections of the word 'abandonner', taken in the first instance by Blanchot from Bataille, and doubled up in such a way as to signify both abandonment ('abandon') and gift ('don'), that is: gift as abandonment and abandonment as gift. '"To sacrifice"', writes Blanchot, quoting Bataille, '"is not to kill, but abandon and give [Sacrifier n'est pas tuer, mais abandonner et donner]." To be bound to Acéphale is to abandon oneself and to give oneself: _to give oneself without return to limitless abandonment_ [ _se donner sans retour à l'abandon sans limite_ ]' ( _CI,_ 30; 15). As Blanchot explains:
The gift or abandonment [le don ou l'abandon] is such that, at the limit, there is nothing to give or abandon and that time itself is only one of the ways in which this nothing to give is offered and withdrawn like the whim of the absolute which leaves itself behind by giving rise to other than itself, in the form of an absence.
( _CI,_ 30–1; 15)
The absence of community to which Bataille testifies in his writing, Blanchot concludes, is not the failure of community, but community's most extreme moment: of both possibility and impossibility.
According to Blanchot, this infinite movement of abandonment and giving is what makes the project of human sacrifice associated with Acéphale irreducible to the logic of sacrificial communion; and it is here, most visibly, that Blanchot in his reading of Bataille diverges from Nancy. For it was precisely, Nancy writes, Bataille's realisation of the 'total absurdity', not to say 'puerile character', of that project of sacrifice and the nostalgia for communion it represented that lay behind Bataille's abandonment of all communitarian endeavours after Acéphale. Blanchot rejoins:
Was it absurd? Yes, but not only that, for it meant breaking with the law of the group, the law that had constituted it by exposing it to that which transcended it, without that transcendence being other than that of the group, i.e. to the outside which was the intimacy of the group's singularity. In other words, the community, by organising and by taking on as a project the execution of a sacrificial death, may be said to have renounced its renunciation of creating a _work,_ be it a work of death, or even the simulation of death. The impossibility of death in its most naked possibility (the knife meant to cut the victim's throat and which, with the same movement, would cut off the head of the 'executioner'), suspended until the end of time the illicit action in which the exaltation of the most passive passivity would have asserted itself.
( _CI,_ 29–30; 14)
On Nancy's reading, the project of Acéphale shows a Bataille imprisoned in a nostalgia for sacrificial communion deriving from the unthought legacy of a dialectic of the subject. Blanchot for his part, though subscribing still to much of Nancy's general argument, at this point takes a very different view. At issue in this exchange is the question of the relationship between the community of Acéphale and death itself. In Blanchot's view, in its (self-)exposure to the impossibility of dying, Acéphale was already precisely not a community founded on the possibility and the subjective truth of death as a moment of dialectical closure. Indeed, the law of sacrifice Acéphale adopted as its apparently defining principle of identification or identity, Blanchot insists, was in fact scarcely a principle at all, to the extent that it was something Acéphale had already suspended in the very movement of its simultaneous constitution and dissolution.
Blanchot's claim, then, contrary to Nancy's assessment, is that Acéphale was not a community founded on immanence and on the work of death, but on the limitlessness of the alterity of dying. As such, it already belonged, he argues, to the impossibility of being and to that nameless exteriority Blanchot terms disaster, and in response to which community is instituted only for the duration of an apocalyptic 'viens'. For Blanchot, Bataille's thought of the community cannot be attributed to a logic of sacrificial subjectivity; rather, Blanchot maintains, the question of the death of the other, in Bataille, like friendship or love, is precisely not thought in relation to any concept of the subject, but rather what he calls 'the slippage beyond limits', 'le glissement hors des limites' ( _CI,_ 33; 16). '"Inner experience"', Blanchot adds,
says the opposite of what it seems to say: it is a movement of contestation that, coming from the subject, ravages it, but has a deeper origin in that relation with the other which is community itself – community that would be nothing if it did not open whoever is exposed to it to the infinity of alterity – at the same time that it decides its inexorable finitude.
( _CI,_ 33; 16–17)
Between Nancy and Blanchot the debate at this point turns on perhaps two differing interpretations of the term 'abandonment'. For Blanchot, according to the thought of disaster, abandonment necessarily implies all abandonment of being; while for Nancy abandonment is still thought by him as the being of the abandonment of being. The outcome, from Blanchot's perspective, is a fatal retention of thought within the orbit of ontology, which results in the fact, from Blanchot's perspective, that Nancy hypostatises Bataille's demand for community by situating his writing within the closure of sacrificial subjectivity and by analysing Acéphale exclusively as a moment of 'désœuvrement' within that closure ( _CI,_ 43; 23). To do so, however, is to be in some danger, in Blanchot's eyes, of disregarding the extent to which Bataille in his writing already exceeds the bounds of any received concept of the political; and this in turn would be to underestimate the suspensive, neutre force of the impossible alterity on which Bataille's search for community is in fact premised, and thereby miss the unavowable futurity – the unspeakable risk and chance – announced, albeit already in the historical past, by a project such as that of Bataille. It was at any rate to make that claim, in the name both of a politics of the future and of justice for the friend that was Bataille, that Blanchot addressed his text to Nancy (and others), appealing to their friendship, as Bataille once had it, as to that friendship which is 'friendship for the unknown without friends', and 'friendship for the exigency of writing that excludes all friendship' ( _CI,_ 44; 24).
'La Communauté négative' concludes with this oblique dedication or demand, to which it adds a closing reference to the heart (or law) of fraternity that for Bataille – and Blanchot – is proposed here as one of the fundamental secrets of community, thereby prompting at least one reader – Jacques Derrida – to question what is at stake in this final invocation of brotherhood in a text explicitly dedicated to community without communion. But at this very point, as though in advance of that objection, Blanchot interrupts his account of Bataille and Nancy in order to supplement it, in a manner he claims was unforeseen and unplanned, with an essay devoted to Duras's _La Maladie de la mort,_ in which at least two important further slippages beyond established limits take place: the shift to a literary (rather than philosophical) text and to a female-authored (rather than male-authored) work.
This twofold displacement does not however represent a retreat from the demand of community or from the question of the political. On the contrary, the name of Duras, like that of Bataille, is also that of a former political ally and personal friend, and Blanchot records as much by evoking their common past involvement in the campaign against the Algerian war and in the _événements_ of May 1968 ( _CI,_ 51–8; 29–34). Moreover, on each of the three previous occasions when he had written about Duras's work, Blanchot had also shown himself to be one of the few readers to realise that her texts were all stories of the impossible reciprocity of love and of the incommensurability of sexual and social relations. To introduce the name of Duras into the debate with Bataille and Nancy was thus not only to raise the issue of community and communication with regard to literature, but also to pose the question of the place of desire and sexual difference within community as such.
As he turns to _La Maladie de la mort,_ Blanchot exposes his account of Bataille and thoughts on fraternity to a question of sexual difference. With respect to community, his text asks, is sexual difference primary or secondary? Is it internal or external to community? As Blanchot reminds the reader, the Biblical evidence on this point is confused, with Genesis providing two mutually incompatible strategies for configuring the origin of the sexes, one that posits duality before unity, the other that first posits unity, then duality ( _CI,_ 68; 40). As many of Duras's own texts suggest, what this indicates is a deep-seated resistance to all narrative teleology or linearity on the part of sexuality, and it is as though the conundrum of sexual difference – separation within the One or infinite non-complementarity of the more than One? – throws into question the possibility of constructing any coherent narrative of community at all, and thus strikes, so to speak, at the very possibility of community as such. Sexual difference on this reading would seem to belong not to community as an entity present to itself, but to disaster, the outside, and irreducible and incommensurable alterity. During May 1968, on the other hand, Blanchot reports, in a rare convergence of presence and non-presence, differences of sex and gender, like distinctions of class or age, if not annulled, were nonetheless suspended; and the result was what Blanchot describes as explosive communication: 'that openness that allowed each and every one, without distinction of class, age, sex, or culture, to mix with the first comer, as if with a person already loved, precisely because they were the unknown-familiar [le familier-inconnu]' ( _CI,_ 52; 30–1). What such moments serve to illustrate, however, Blanchot argues, is not the precedence of community over social or sexual differences, but rather that a community such as that formed, briefly, during the _événements_ of May 1968, was a necessary response to the dissymmetry between Same and Other and that, for that dissymmetry to be realised, community itself must be affirmed, even as, at the very same time, it must also be put under erasure, maintained as a suspended moment of non-presence rather than being absorbed once again within the closure of sacrificial subjectivity.
Sexual difference in Blanchot inscribes a twofold process: it exposes community to alterity and exposes alterity to community. It is thus a condition both of possibility and of impossibility. Indeed, without sexual difference it is difficult to see how any community is conceivable at all; but it is clearly also one of the fundamental properties of sexual difference at the same time to threaten the existence of any such possibility of community. The heterogeneity embodied in sexual difference effects what Bataille calls an insidious loosening of the social bond, and this is why for Blanchot, as for Duras's story, sexual difference is seen as corresponding to a logic that is not only an interruption of logic, but also an interruption of the possibility of narrative and of the homogeneity that threatens in all social organisations. Blanchot explains the point by contrasting here two relations to death that are also two logics of sexual relation and two sharply opposed types of social or political structure: the one, embodied in homogeneity, immanence, and the enforcement of law, finding expression for instance in the homosexual exclusion of the sexual Other by the group (the example given is the classically paradigmatic case of the Hitlerite SA); and the other dedicated to heterogeneity, alterity, and the suspension of law, in which, according to Blanchot, 'woman then becomes the "outsider" [l' "intruse"] who perturbs the untroubled continuity of the social bond and does not recognise prohibitions [l'interdit]. She is hand in glove with the unavowable [Elle a partie liée avec l'inavouable]' ( _CI,_ 70; 59).
The logic of sexual relations in _La Maladie de la mort,_ as Blanchot demonstrates, is a logic of disarticulation before it is one of articulation; it affirms disjunction, non-complementarity, and the impossibility of satisfaction as the groundless ground for the encounter – if any such word may be deemed adequate – between the book's two protagonists. The only relation between them is a relation founded on non-contemporaneity and dissymmetry, and thus on the very absence of relation. From the perspective of Duras's text, sexual difference belongs to no common, shared world, but rather to an absent horizon and an always already prior absence of world; as a result, the reputed worldliness of love relations gives way in her story to the radical worldlessness of desire itself. This possibility of speaking the absence of world in its very impossibility is the only scant privilege Blanchot confers here on literature, raising Duras's story – problematically, as Blanchot is the first to admit ( _CI,_ 88; 54) – to the status of original or pre-original myth by giving to Duras's unnamed female protagonist the name of Eve, or Lilith – Adam's first wife, according to Rabbinical tradition, before the creation of Eve – or by casting her in the part of chthonian Aphrodite, whose pact is not with the world but the underworld, and whose relation is not with life but rather with death ( _CI,_ 77; 46). But what is precisely at stake here, in Blanchot's thinking of community, with respect not only to sexual difference, but literature too, is this question of death, that is, not the sacrificial sublation of death as work, project, or monument, but rather that relation of non-relation for which death is only ever a sign of the impossibility of dying and, beyond finitude, of the infinity that is desire, and the demand that is community.
It is for this reason, as Blanchot puts it, that Duras's lovers in _La Maladie de la mort_ are not so much separated, or divided from each other, as infinitely inaccessible to one another. The other face of finitude, Blanchot insists, is the infinite relation with the Other that finitude necessarily implies. The worldlessness experienced by Duras's lovers is thus not a basis for romantic nostalgia, but a kind of violent contestation of the closure implied by any world as such. It is in this sense that love in Duras may be seen as an apocalyptic indictment of the finite character of the world; but if love is to be the source of catastrophic revelation, as far as Blanchot is concerned, it is not because love offers its derelict participants the hope of communion within some higher unity, but because inherent in finitude itself is the knowledge that any limit is necessarily accompanied, as by its invisible shadow, by the limitlessness that is both cause and effect of the limit, and that the absence of relation with the Other is therefore only another name for the infinity of that relation. Love here, for Blanchot, acts as a kind of reprise of the ethical relation. As such, it insists not on the necessity of obeying worldly laws, but on the need to contest all such laws; but if it chooses to defy these laws, it can only be in the name of the higher law that comes before all codes of law or morality, and which is the law of that from which law derives, which is responsibility towards the Other, and to which responsibility is owed not because it is the law but because, like passion, it is what precedes freedom, choice, deliberation as such.
In _La Communauté inavouable,_ under the rubric of the ethical, what Blanchot endeavours to address, as in his readings of Levinas, is an alterity that is irreducible not only to immanence but, Blanchot puts it, any received form of transcendence also. Such alterity without name, transcending all transcendence, Blanchot in _La Communauté inavouable_ calls disaster. Disaster here is another name for the infinite contestation or anonymity that in earlier texts Blanchot formulates as the neutre; at any event, as an inassimilable force of infinite withdrawal, separation, and displacement, it is that which requires community, if it is to be affirmed at all, to remain unavowable. The word is one Blanchot takes, like others in this context, from Bataille. Twice before explicitly introducing the phrase: 'la communanté inavouable' in the book, Blanchot uses it by inverting subject and adjective as 'l'inavouable communauté'. This is no doubt done for stylistic effect, but it also leaves it uncertain whether it is community that gives rise to the unavowable secret or the secret to community. In either case, the secret itself is left unexplained, save that, like the allusion to an unspeakable pre-conceptual singularity, it is what, in language, nonetheless stands separate from language. The secret of community, like the secret to which the narrator of _L'Arrêt de mort_ remains attached as to a compelling absence from words, is a secret that is beyond possibility; to the extent it is a secret to be revealed at all, it is a secret without secret, a revelation without revelation. An unthematisable interval separates community from itself as the ground without ground of the futurity of community and its openness to the Other.
As Blanchot writes elsewhere, apocalypse here disappoints ( _A_ , 118–27). The invocation of the end is a tribute necessarily paid to infinity, as Blanchot makes clear by pointing at the end to the self-defeating paradox of Wittgenstein's famous conclusion to the _Tractatus._ To speak of the end is always to defer the end; no sooner is it pronounced than the finality of an apocalyptic 'come!' in fact suspends the end as a moment of perpetual (re)beginning. Apocalypse in Blanchot is therefore not an apocalypse, for it is apocalypse without end, truth, or finality. It is, as Derrida had predicted, apocalypse without apocalypse. Blanchot says as much himself at the end of _La Communauté inavouable,_ as he translates into something resembling a project the 'viens' that precedes and exceeds his text and which he addresses to his reader as a sign of the unavowability of community as that which in language is always other than language. This explains why perhaps, in _La Communauté inavouable_ as a whole, beyond literature itself, the only moments of community which Blanchot is able to affirm – Acéphale, the events of May 1968, the demonstrations of 13 February 1962 in protest at the Charonne killings – are all so many (non-)events of uncertain status. Like the love enacted in _La Maladie de la mort,_ their main effect seems to have been to put the world into parentheses by virtue of the deconstructive force of the neutre, and in that interruption of world and power to have affirmed the infinity of responsibility to alterity and the Other. This is why Blanchot refuses, vigorously and incisively, all modes of political representation and all forms of politics premised on the pursuit of power. The politics to which Blanchot's writing gives voice instead is an eschatology – eschatology beyond eschatology – which addresses the future not as power but as judgement, not as imminent presence but as infinite promise. The hope is not for more, or better representation, but rather for the destruction of the present as such and thus for a revolution that would open time itself to the otherness that presence always excludes: what Blanchot in May 1968 invoked as
a revolution ... destroying all without anything destructive, destroying, rather than the past, the very _present_ in which it took place and not attempting to provide a future, extremely indifferent to any possible future (judged as success or failure), as though the time it sought to open up was already beyond these standard determinations. (Révolution ... détruisant tout sans rien de destructeur, détruisant, plutôt que le passé, le _présent_ même où elle s'accomplissait et ne cherchant pas à donner un avenir, extrêmement indifférent à l'avenir possible (la réussite ou l'échec), comme si le temps qu'elle cherchait à ouvrir fut déjà au-delà de ces déterminations usuelles.)
One might conclude that Blanchot's politics, like those of Duras, are to this extent a politics of disaster. But this is not the same as a politics of despair. Quite the reverse.
## The unexpected word
The example of Marx helps us to understand that the language of writing [la parole d'écriture], which is a language of ceaseless contestation, must constantly be developed and interrupted in _multiple_ ways. The language of communism is always _at one and the same time_ tacit and violent, political and wise, direct, indirect, total and fragmentary, lengthy and nearly instantaneous. Marx does not live comfortably with this plurality of languages constantly colliding and being forced apart within him [qui toujours se heurtent et se disjoignent en lui]. Even if these languages seem to converge towards the same end, they remain untranslatable into one another, and their heterogeneity, and the distance and interval that decentre them, make them non-contemporaneous with each other [non contemporains], such that, giving rise to an effect of irreducible distortion, they oblige those who have to withstand the challenge of reading (or executing) them to submit to a process of ceaseless readjustment.
'Les Trois Paroles de Marx' ( _A_ , 117)
There are, Blanchot maintains, always at least two languages, two voices, two demands. Such multiplicity of tone, according to Derrida, is a trait common to all eschatology; but Blanchot raises it to the status of a philosophical, literary, and political strategy. Writing, he argues, falls subject to an injunction which by definition cannot be satisfied and is capable of supplying neither sanction nor recompense. But for every demand addressed to the act of writing by virtue of its own absence of worldly foundation or justification, there is always another demand requiring that justice be done without delay in the world. These two demands, the one requiring the obliqueness of infinite patience, the other demanding urgent and decisive action, function according to different rhythms, different temporalities, and different logics of possibility and of impossibility. While not necessarily opposed to one other, and not homogeneous even within themselves, they are nonetheless radically disjoined, and it is essential, in the nameless name of the neutre, Blanchot argues, that the dissymmetry arising from this disjunction be affirmed, respected and obeyed.
The two demands, most crucially of all, must not therefore be mistaken the one for the other, or homogenised with each other, which is also why absolute dedication to the one does not constitute grounds for release from the urgency of the other. While the language of politics, as Blanchot argues in _La Communauté inavouable,_ must allow itself to be interrupted by the alterity of writing, so the commitment to the dissymmetry and infinity of relation that writing introduces into community does not thereby dispense the writer – not as a citizen, but as writer – from the challenge of meeting politics on its own terms: indeed, this is, according to Blanchot, precisely what writing demands, even if the endeavour to respond to that demand may imply abandoning, provisionally, the indirect discourse of the literary in order to contend with the immediacy of the political as such. Commitment here depends not on the figure of the intellectual as superior consciousness or exemplary conscience, addressing the community at large with the authority of universal values, but rather on the solitude and alterity of writing itself, in whose name the writer is unavoidably bound to affirm freedom as an absolute, irreducible condition of language and community as such. Its emblem, therefore, for Blanchot, is not, say, the image of Sartre, whom even de Gaulle mistook for Voltaire, writing plays so as not to foment despair in the hearts of the Renault car workers of Billancourt, but the gesture of Hölderlin, in a letter to his brother on New Year's Day 1799, declaring himself ready to throw his pen under the table in order to go wherever the need was greatest to defend the cause of liberty against the threat of darkness.
Blanchot's relationship as a writer with the political is, in this way, both singular and far-reaching; for that reason alone it is worth examining here that thinking and questioning of the political put to the test by Blanchot in a remarkably heterogeneous series of political writings – including statements of political principle, polemical texts, open letters, petitions, journalism, and responses to questionnaires – written, intermittently and irregularly, alone or in collaboration with others, during the period following his return to Paris in 1957 and in response to several important and challenging political events, each of which, Blanchot wrote in 1986, for one reason or another, was greater than its own actual meaning. As Blanchot has indicated in various retrospective accounts of his own, these moments, so many turning points, so to speak, were essentially three in number: the resistance to the Algerian War from 1958 till 1962; the _événements_ of May 1968; and the memory and witness to the Nazi death camps that came to dominate much of Blanchot's political thinking after 1971.
Blanchot's first explicitly political text to be published since before the Second World War appeared in October 1958 in _Le 14 Juillet._ Called 'Le Refus', ('Refusal') ( _A_ , 130–1), it was the first of two contributions by Blanchot to the campaign, launched earlier that year by Mascolo and Schuster, to challenge the legitimacy of de Gaulle's return to power in May. The title of the piece was no doubt symbolic. Importantly, however, it also mobilised, with undiminished virulence, but in the service of a different kind of political project, one of the few terms in Blanchot's political lexicon to have survived from his pre-war activist past. This shows how far Blanchot's return to political commitment, whatever some have charged, was not premised on a culpable repudiation of his pre-war involvements, which in any case remained largely unknown to the majority of his new political associates. Equally, however, this return to politics can hardly be seen, on Blanchot's part, as constituting an uncritical continuation of his pre-war political endeavours, and it is indisputable that throughout the 1950s, 1960s and 1970s, though at times obliquely, Blanchot in all his political thinking was constantly reflecting on the events that had led up to 1938. This is demonstrated by the frequency with which questions of nationalism and internationalism, militarism and freedom of speech, anti-semitism and colonialism recur in his later political writings.
To refuse the present, as Blanchot wrote to Mascolo in 1958, also meant refusing the past; and in both cases what Blanchot had uppermost in his mind was the need to refuse – more radically still, as I have suggested, than he had done during the 1930s – not only all complicity with the politics of representation, but also any reliance on the self-presence of the nation as subject of history. In 1958, then, refusal in Blanchot was not a simple posture of negation or revolt, and it would be wrong to see Blanchot's opposition to de Gaulle and to the Algerian War as relying on a simple gesture of protest. Refusal for Blanchot was not an isolated or token matter, but an act, as he put it in 'Le Refus', that was absolute and categorical in itself and as a result potent enough, albeit in negative terms, to call forth a community, a community founded, so to speak, on what Blanchot calls 'the friendship of this No, certain, unshakable, and exacting [l'amitié de ce Non, certain, inébranlable, rigoureux], which holds men united in solidarity' ( _A,_ 130).
The refusal to sanction de Gaulle's return to power was not on Blanchot's part a passing concession to the demands of the moment. It was a natural extension of much of Blanchot's thinking about literature during the previous decade; indeed what it evoked most clearly of all, on one level, was the same logic of all or nothing, without mediating concept, that Blanchot had explored in relation to writing in 'La Littérature et le droit à la mort', and which, in so far as it corresponded to an act of absolute refusal of the world and power, was also to be understood, implicitly, as an act of absolute affirmation of that which was radically other than world or power. And it was in much the same way, over and beyond a profound objection to de Gaulle's militarism and self-important nationalism, that what Blanchot's act of political refusal affirmed was the necessity of rupture: of a break in continuity in politics, so to speak, and one that put at the centre of political discourse an interval and a disjunction – beyond being and non-being, so to speak – that not only called into question de Gaulle's endeavour to substitute his own personalised pseudo-religious authority for democratic government, but also entirely discredited the opportunism of those who were content to accept the new ideological order in the name of so-called political realism and which did no more in fact than emphasise the esssential complicity between politics in the received sense and the established order itself.
This was not all. In 1958 Blanchot was clearly aware that it was not enough merely to attempt to create an interval in the process of political representation. It was equally important, if not indeed more so, within that interval to refuse the lure of the nation's demand for self-presence and self-identification, which would always risk reinforcing the rule of representation as such, and to respond instead to the demand of the Other. This is why, as the campaign developed, it became a struggle whose prime goal was not to defend the constitutional, democratic traditions of the French republic, but rather to end the oppressive rule of French colonialism in Algeria (which during the 1950s democratically elected governments in France had struggled to defend as proper and legitimate). And this was one of the ways in which, in the specifically political domain, Blanchot endeavoured to avoid all the dangerous ambiguity of his revolutionary pronouncements of 1936 and 1937, and draw the necessary conclusions from that (for many) largely forgotten pre-war history.
As the struggle against de Gaulle turned into the campaign against the Algerian War, the logic of interruption inherent in the act of political refusal, as Blanchot saw it, underwent a further inflection. This was most clearly visible in the full title – devised, it seems, by Blanchot himself – adopted by Mascolo, Schuster, Blanchot, Nadeau, and the document's other co-authors, to head the text that, distributed clandestinely in September 1960, quickly came to be known by the more familiar name of the 'Manifeste des 121'. The document's full title was: 'Déclaration sur le droit à l'insoumission dans la guerre d'Algérie' ('Declaration on the Right to Insubordination in the Algerian War'). As Marguerite Duras, in whose flat in the rue Saint-Benoît the Declaration was first drafted, later commented, on the occasion of the twenty-fifth anniversary of the publication of the text, the document was not a pacifist one, did not amount to a refusal of war in general, if only because of the deep association of that position, in French eyes, with the appeasement of Hitler at Munich in 1938. But nor did the Declaration appeal to any sense of obligation or duty on the part of its readers. What it did instead, Duras continued, was to place those to whom it was addressed before an essential and solitary responsibility, which was the responsibility to decide both for themselves and in relation to themselves; this was why the text of the Declaration, while referring to a set of specific circumstances, invoked not an obligation, but an absolute, fundamental, and inalienable right. The distinction was an important one. As Blanchot explained in his interview with Madeleine Chapsal:
I believe that the whole force of the Declaration, its whole power of disturbance [tout son pouvoir d'ébranlement], comes from the authority with which it utters the single word insubordination, a solemn word, signifying utmost refusal: the Right to insubordination. I say Right and not Duty [Droit et non pas Devoir], as some, in an ill-considered way, would have liked the Declaration to say, no doubt in the belief that the formulation of a duty goes further than that of a right. But not so: an obligation refers to a prior moral code that shields, guarantees, and justifies it; wherever there is duty, all that is necessary is to close one's eyes and carry it out blindly; everything then is quite straightforward. The right to do something, on the other hand, refers only to itself, to the exercise of that freedom of which it is the expression; a right is a free power for which each individual, for himself and with regard to himself, is responsible and which binds him completely and freely: nothing is stronger, nothing is more serious. That is why it is essential to say: the right to insubordination; each person takes their own sovereign decision [chacun en décide souverainement].
In its opposition to the war, the Declaration did not invoke a moral duty, based on a universalising code of laws, principles, values, and obligations. As Blanchot points out, it was this that distinguished it from an act of commitment in the Sartrian sense; indeed it might be argued in this respect that the Declaration was one of the first texts, in France, to contest and rethink the figure of the intellectual as universal conscience, as Foucault and others were increasingly to do after May 1968. Instead of appealing to morality, and thus necessarily to some institutionalised code that had disquieting similarities with the very authority of the state it sought to challenge, the Declaration reaffirmed each signatory's inalienable right of refusal, a right that was absolute to the extent that it logically preceded any form whatsoever of recognition of the power of the state and any complicity in its decisions.
In its own way, of course, the affirmation of the right to insubordination was an act of political violence, and Blanchot was well aware that, implicit though it was, such violence was liable to provoke a violent response from the state itself. But to acknowledge the violence of refusal arguably also made the Declaration politically more effective, for the signatories to the text found themselves not in the position of seeking to impose upon others the obligation to act in conformity with a given set of universalising values, but in the position of taking their own share of responsibility, with the modest means at their disposal, for the violence being committed in their name by the French Republic in its prosecution of the Algerian War, a war which, crucially, the French state even declined to acknowledge as such, claiming it to be a purely domestic matter of internal security.
The reaction of the state to the Declaration was immediate and uncompromising; the document was suppressed and each of its co-authors became the object of criminal proceedings. On one level this was hardly surprising; but it served to make clear, if clarity was needed, that the conflict between the state and the resistance to the war was itself a form of warfare, in which the state sought to defeat the actions undertaken against it by deploying against its opponents greater, or more effective, violence than they themselves were able to muster in their defence. The analogy here, already evoked in the text of the Declaration itself, was between the regime installed by the French military in May 1958 and that set up under the Occupation by Pétain in 1940. In both cases, the legal machine of government had lost democratic legitimacy and, though the state pretended otherwise, was in the hands of an oppressive, authoritarian regime. So much was evident from the Jeanson trial of September 1960, the smooth running of which it had been the initial purpose of the Declaration to disrupt. But what the trial in its turn demonstrated, and as the Declaration itself confirmed, was that, alongside the war of arms being waged between the state and the – illegal – opposition, another, equally important conflict was taking place; and that, as Blanchot was to discover during the court proceedings against him, this other war was a war of language and discourse.
It is here that the logic of absolute, categorical refusal which Blanchot develops in his political texts of the late 1950s and 1960s demonstrates its particular originality and political significance. For if refusal was a right, as Blanchot maintained, it was in the first instance essentially always a right to refuse a certain kind of language: authoritarian, self-assured, peremptory, repetitive, oppressive. Such language, however, was at best only a hollow parody of the language of literature and writing, an attempt to escape rather than confront the emptiness at the heart of language; this at any rate was what Blanchot had argued in a 1955 essay entitled 'Mort du dernier écrivain' ('The Death of the Last Writer') ( _LV,_ 265–70; _BR,_ 151–6), in which he puts forward a portrait of the dictator as a spectacular kind of failed writer. If the analogy was correct, it explained not only the liking for great art expressed by countless emperors and dictators down the ages and why the national-aestheticism of many of Blanchot's contemporaries had led them in the 1930s to endorse a host of authoritarian political systems, but also why, more importantly, for Blanchot the language of power was ultimately always vulnerable: vulnerable not to the challenge that might come from a rival code of values, equally authoritative in its own terms as that embodied in dictatorship, but vulnerable instead to the infinite scepticism affirmed by the language of writing itself.
In that case, it was, Blanchot argued, in its very refusal of the language of politics in the received sense that literature might be seen to be the most profoundly political mode of thinking of all. 'That is why', suggested Blanchot in 1955, referring to a writer not very different from himself, 'when speaking of politics, he is already speaking of something else: of ethics; when speaking of ethics, of ontology; when speaking of ontology, of poetry; when speaking, last of all, of literature, "his sole passion", it is to revert again to politics, "his sole passion"' ( _LV,_ 302; _GO,_ 119). Here, what was most important of all about the right of refusal – like the right to death that it mirrored – was that, like literature itself, it placed at the centre of political discourse a disjunction of languages, an infinite gap or interval that was a space both of silence and endless contestation. In so doing, what it put at the top of the political agenda was the need not simply to bring about a change in the content of politics, but a radical transformation, too, in the very language of politics; and as the 'Entretien sur un changement d'époque' ('Conversation on a Change of Epoch') ( _EI,_ 394–404; 264–71) in April 1960 had testified, it was to the imminent possibility of such a radical transformation, affecting the way in which politics and community might henceforth be addressed, that Blanchot sought to contribute in his ill-fated efforts to realise the project of an International Review.
If all Blanchot's renewed political activities of the late 1950s and early 1960s may be placed under the rubric of what one might call a politics of refusal, the same is true – if not indeed more so – of his involvement in the events of May 1968. There was of course much that was new and unexpected about May; but most striking of all was that, for the vast majority of those who took part in the _événements,_ it was politics itself that demanded an absolute rejection of the politics of representation embodied in the French parliamentary system and in all established political parties. The real political arena of the movement of May was in this sense the street. This corresponded suggestively to the whole question of the place – or non-place – of the outside that Blanchot had been articulating in relation to literature since the early 1950s; and indeed, in a text entitled 'Tracts, affiches, bulletin' ('Handbills, Posters, Bulletin'), which appeared unsigned in _Comité_ in October 1968, Blanchot was suggesting that one of the most distinctive and far-reaching manifestations of May were the many different, evanescent modes of writing – named in Blanchot's title – to which the movement gave rise and that, like so many nomadic, anonymous inscriptions or graffiti, belonged to the infinite exteriority of a time that was the time of the absence of time and a place that was the place of the absence of place. Of such texts, Blanchot wrote:
They do not say everything, but on the contrary ruin everything, are outside everything [hors de tout]. They act, reflect in fragmentary fashion. They leave no traces behind: traits without trace [Ils ne laissent pas de traces: trait sans trace]. Like the words on the walls, they are written in insecurity, received under threat, are themselves the bearers of danger, then pass with the passer-by who passes them on, loses or forgets them.
Blanchot's particular involvement in the _événements,_ as mentioned earlier, was largely as a member of the Comité d'action étudiants-écrivains (the Students-Writers Action Committee). As a mode of revolutionary organisation – or, rather, non-organisation – the action committee was a distinctive, intensely anti-authoritarian form of political activity and one that, in the course of the _événements,_ quickly emerged as the key embodiment of the movement of May itself. Such committees all functioned on an ad hoc basis; they fulfilled no expressive or representative function; the purpose was not the pursuit of power, but the contestation of all power in whatever form, and this explains why, as Blanchot put it in 'Tracts, affiches, bulletin', as far as the action committees were concerned, all notions of success or failure were at best irrelevant, at worst pernicious. Meetings as a result would last as long as they lasted, and committees would exist beyond themselves only as what they were in themselves, during which time, so to speak, the revolution itself became like an immediate, if evanescent, presence (without present). 'This is why', Blanchot wrote to Mascolo on the subject of such committees in December 1968, 'they are nothing outside of the presence constituted by each meeting, a presence that is their whole existence, and in which it goes without saying that the Revolution, by that very fact, is present: in much the same way as in seances when a Spirit shows itself'. As Blanchot's letter went on to suggest, the limitations of the action committee as a mode of political activity were self-evident; but in themselves these were not important, for what such committees created, by their very existence, was potentially much more subversive than it seemed; for what they effected was a radical hiatus in the political order itself.
As far as the Students-Writers Action Committee was concerned, what went on in meetings was that texts were proposed, contested, withdrawn, and rewritten, until by collective decision they were finally either affirmed or annulled. The political activity of the committee was essentially, therefore, a linguistic or textual one; and what it placed highest on the political agenda, therefore, as _Le Très-Haut_ had predicted twenty years before, beyond all economism, reformism, or concern with party political organisation, was the need to suspend the dialectical closure of representational politics, alongside the essential complicity of government and legal opposition deriving from it, in order to affirm a different kind of politics, no longer dependent on the law of possibility, and, as _La Communauté inavouable_ suggests, beyond the reliance on received political concepts such as those of project or subject. This was why the anonymous production of texts, literary as well as non-, extra- and anti-literary, by doing away with such concepts, was such a crucial political touchstone. Making an obvious comparison at one stage, in _Comité,_ between May 1968 in France and the Prague Spring, Blanchot suggested that what was at stake in both movements was far more vital than a call for greater dialogue between government and governed:
Something quite different is at issue: a movement beyond measure, irrepressible, incessant, the impetus of _outraged_ speech [l'élan d'une parole d' _outrage_ ], speaking always beyond, transcending, overwhelming and thereby threatening all that confines and limits [parlant toujours au-delà, dépassant, débordant et ainsi menaçant tout ce qui borde et tout ce qui limite]; the transgressive act of speech itself.
In the demand for greater freedom and plurality in speech, the community Blanchot sought to address and call into existence was a community beyond nationalism or patriotism, even beyond culture itself. It was a community that, being essentially beyond community, and in relation to itself always already under erasure, coincided only with a demand for community that was a demand for exteriority, and that demand put it beyond the self-presence or self-identity of any existing community whatever. This demand for alterity was what in 1968 Blanchot named as: communism, and which he went on to define – while refusing to define it – in _Comité,_ in a celebrated text entitled 'Le Communisme sans héritage', as 'that which excludes (and excludes itself from) all already constituted community'. Yet despite the radicality of this refusal of the past, Blanchot in May cannot be termed an anarchist. For while Blanchot insisted on the necessity for speech or writing to challenge all laws, he did so in the name of a rupture or break that, far from being a refusal of all constraint, is in the form of an absolute submission to the law of infinite contestation, which is also the law of infinite commitment to alterity. Such a law was necessarily opposed to the established political and moral order, and to this extent May embodied for Blanchot a moment of necessary political violence. That violence, however, he wrote, was infinite; like the law it obeyed, it was both calm and terrible. The most violent moment of May, he suggests, was in this respect more a moment of extreme non-violence, and occurred when demonstrators – workers, students, and revolutionaries alike – responded to the deportation of Daniel Cohn-Bendit and to anti-semitic jibes by government ministers against him by launching the famous slogan: 'Nous sommes tous des juifs allemands' ('We are all German Jews'). This repudiation of anti-semitism, writes Blanchot, was an exemplary moment in the entire unfolding of May: 'Never', he claims, 'had this previously been said anywhere, never at any time: it was an inaugural moment of speech, opening and overturning borders, opening, overthrowing the future [ouvrant et renversant les frontières, ouvrant, bouleversant l'avenir]'.
In another text from _Comité,_ entitled 'Rupture du temps: révolution', Blanchot suggested, after Walter Benjamin, that the interruption of history effected by a revolution such as May necessarily ushered in a moment of innocence. Such innocence, however, proved to be short-lived. For it was precisely such innocence, together with the redemption from history it blindly assumed to be already at hand, that, according to Blanchot, was the cause of the decision of those groups on the extreme left which had emerged from May, in the name of anti-imperialism and anti-Zionism, to formulate their explicit opposition to the existence of the State of Israel. Fearing this to be anti-semitism under another name, Blanchot withdrew at this point from all active involvement in the aftermath of May. This turning aside from politics obviously raises numerous issues. But perhaps most importantly of all it poses the question of the relationship between the interruption of history and of political discourse embodied in May and the question of ethics. Did transgression of the established order also suspend ethics, or, in the suspense of history, did the claim of the ethical become more compelling still? Was the end of the political also the end of ethics or, more radically perhaps, the possibility of its very beginning? And did the step beyond history take for granted the possibility of beginning anew, or was it not rather a sign of the impossibility of beginning, and of the return of all that history had hitherto excluded and sought to forget and still needed to be confronted again?
Such were the questions that the outcome of May bequeathed to Blanchot; and it is not difficult, reading Blanchot's texts of the 1970s and 1980s, to see how these political issues, though they may have rarely been named as such, remained at the heart of all his writing as the fundamental interrogation with which those late texts are concerned. That this is so is shown by Blanchot's explicitly political actions of the period; for the aftermath of May left him, like many others in France at the time, writers of fiction and philosophers alike, returning obsessively, and far more explicitly than ever before, to the question that was to dominate all of Blanchot's thinking for the next twenty-five years, and which is the question of responsibility for the Nazi death camps. If anti-semitism, as Blanchot wrote to Levinas in 1969, was the limit he was not prepared to cross, what this suggested to him was that anti-semitism, to the extent it constituted what one might call the limit of the political as such, was the moment when political power, founded as it was on the self-identity of the national community and on the exclusion of all nomadic alterity, showed itself in its true identity. This is what confers on the name Auschwitz its potency as an emblematic event for Blanchot, albeit an emblematic event that is the result of a specific history, and an event to which grave disservice is done once it becomes merely a symbol, albeit a symbol only of itself. But despite the difficulty of naming Auschwitz, the question remains: for if Auschwitz represents the unthinkable, extreme point of the political in the West in the modern age, how is it possible to think politics after Auschwitz, and think a politics which follows that injunction of Adorno's which requires of us that we act in such a way that Auschwitz does not repeat itself?
Responding to Auschwitz, Blanchot writes, can only mean responding to the double bind – the aporia and impossibility – contained in two mutually exclusive injunctions: 'Know what happened, do not forget, and at the same time never will you know' ( _ED,_ 131; 82). Responding to Auschwitz is impossible, since Auschwitz outstrips all limits, of knowledge, understanding, experience, or empathy. But precisely because it is impossible to know, one must rejoin, so it is essential to remember, even if what is being remembered is the impossibility of remembering. To remember the impossibility of memory, though, is to acknowledge the limits of remembering and to respect the limitlessness of forgetting which alone makes memory possible. Auschwitz here becomes the name without name for an event with which no politics can ever be commensurate, and to which the only response, in the certain knowledge of its inadequacy, is one of infinite responsibility. The quest for a politics that would be adequate to the Holocaust can only be in the form of infinite memory and attention to alterity; and this has been what has motivated, since 1971, Blanchot's repeated commitment in his writing to the impossible necessity of bearing witness in the absence of all witness to the event of the camps, to the necessity for the survival of Israel, for human rights, for Salman Rushdie, against apartheid.
Like that of many other men and women of his generation, Blanchot's political thinking since 1971 is traversed by the sombre legacy of the camps; and to the extent that the memory of Auschwitz remains as a necessary burden to be confronted by each and every future political project, proposal, or venture, his own political legacy may seem to some readers pessimistic and despairing. On one level this may be true, and to pretend otherwise would be to ignore the fact that the political history of the twentieth century has largely been a tale of alienation, oppression, totalitarianism, and genocide committed in the name of this or that political code or dogma; but in remembering this, Blanchot enjoins us also not to forget that no invocation of the end is without also necessarily invoking the limitlessness of the end; and therefore the fact that the impossible demand of the Other leaves us always already without recourse is itself testimony to the infinite alterity of language itself.
# 5: Extreme contemporary
No finality where finitude reigns.
_La Communauté inavouable_ ( _CI,_ 38; 20)
– You've left out rather a lot.
**–** But is it ever possible to avoid gaps and deficiencies in a work of this kind? I know some readers will think I've gone on about Blanchot for far too long as it is. Leaving things out is surely a condition of saying anything. You start out trying to speak of everything, then you find this can't be done, for reasons of fatigue, lack of time or space, ignorance, laziness, or plain stupidity; then you realise it's more serious than that, and that what's been left out is what makes it possible to say anything at all. The more you try to reach the end, the more the end becomes impossible to reach. I know this paradox has become dreadfully familiar. But like the one about the Cretan liar which Foucault recalls in his essay on Blanchot, it's rigorously inescapable. Blanchot shows how it ruins all attempts at closure or totalisation and implies that at the very outset of speech or writing there is an otherness at work, or, more accurately, not at work, beyond all thought of work, at the non-existent origin of work and non-work, which escapes the whole and is irreducible to it. In these circumstances, it is necessarily the case that there isn't a discourse which will frame Blanchot's texts and that all the discourses that attempt this turn out to be riddled with gaps. This is unavoidable. The only alternative is to say nothing, but, as Blanchot keeps reminding us, to say nothing is always already to have said something.
– I want to come to your fondness for pastiche in a moment. In the meantime, I was hoping you might say something about the title. Isn't it another quotation?
**–** Yes, it's another quotation. Let me own up to an absurd hypothesis. What I had in mind was this. Take all the major French philosophers or literary critics that have become associated in Britain or America with the preposterous term of post-structuralism, thinkers like Foucault, Derrida, Deleuze, Bataille, Klossowski, Levinas, Lacoue-Labarthe, Lyotard, Nancy, Barthes, Kristeva, others I've forgotten. Then draw up a list of the most important or influential writers in French over the last fifty years: Beckett, Duras, Perec, Laporte, Antelme, Des Forêts, Char, Paulhan, Leiris, Robbe-Grillet, Jabès, who else? Finally, make up a third list including all the key figures of our modernity: Sade, Kafka, Sartre, Musil, Heidegger, Nietzsche, Mallarmé, Broch, Ponge, Hegel, Woolf, Hölderlin, Rilke, Freud, Henry James, Breton, Marx, Artaud, Celan. You're allowed to add other names too if you wish. But if you then look at where these different lists connect up or intersect, you will find that they all do in the place – but can it be called a place? – occupied by the name of Blanchot. This tells us, I think, two things: that the name Blanchot is not a stable identity consistently present to itself, but more like a field of forces, an empty name that coincides with its own multiplicity and endless fragmentation; but also that, wherever you look across the vast network of texts that go to make up what might be described as contemporary thought or the thought of the present in general, what is most striking of all is this insistent reference to the text of Blanchot.
**–** I'd like to pause here. I appreciate what you're saying, but isn't it more important to challenge this kind of foolish totalisation? And shouldn't we be infinitely sceptical about this idea of contemporaneity? I'm sure we all remember those passages where Blanchot specifically throws into doubt the very concept of contemporaneity, invoking instead, alongside Levinas, the inescapable diachrony of the relation of non-relation I entertain with the Other. This surely disqualifies the notion that the present is somehow contemporary with itself. In any case, to think of yourself as your own contemporary would be to think of the future in metaphysical terms as a kind of deferred present. And, even then.
**–** Allow me to interrupt. I think it was Mallarmé who pointed out how uninformed one would have to be to believe oneself one's own contemporary. But isn't it possible that we can take extreme contemporary in a slightly different sense? One might say that the contemporary, to be contemporary at all, has also to be that which is beyond the contemporary, and which addresses the present only to the extent that the present is understood, here and now, as that which is without presence and offers – Mallarmé again! – only a false appearance of the present. In that sense, the present would be more like the future, the future not as deferred presence but as the outside, that absolute alterity which is chance, which we cannot address precisely because it is what addresses us.
– In that sense, if I follow what you're proposing, extreme contemporary would be both what addresses us in the present most radically, and in that very gesture addresses us from some other place or time which does not belong to the contemporary, but creates a fissure or caesura in temporality itself. That would seem to imply that the extreme contemporary is neither extreme nor contemporary at all; but that the two words, being vacated by their own meaning, are given to the reader as a kind of infinite absence.
**–** Wouldn't that also suggest that to be contemporary would mean to have exhausted all the discourses it is possible today for a writer to employ, and thus to have reached a point of absolute unreadability? Is that why people find Blanchot so hard to read?
**–** Reading is never easy. But it's perhaps more that Blanchot's texts demand infinitely close attention. But there is another point here, too. I'm aware that to say anything of the change of epoch Blanchot promises is to pre-empt the future by reducing it to the horizon of the already known. But what future do you see for Blanchot's texts?
– The future of Blanchot's texts depends on the future within those texts, and in writing this book that future has really been my sole preoccupation. I certainly haven't wanted to imply there is only one possible way of reading Blanchot. There are obviously an infinite number and this particular book has only had space for one or two. In any case, the future is not down to me, but Blanchot's texts and their readers. Other than that, the future is surely incalculable.
– I wonder if you don't somehow still think it possible to say everything there is about Blanchot? Wouldn't that miss the point and turn reading Blanchot into a sterile academic exercise, whose fidelity to what is at stake in Blanchot's writing would be seriously in jeopardy?
**–** Blanchot shows that all commentary is necessarily unfaithful. It's one of the things that distinguish him from Heidegger. I'm sure we agree that each one of us has a duty to justice and that we're all obliged to avoid misrepresentation, distortion, and falsehood wherever possible. But one is also required, it seems, to respond to Blanchot's writing according to the singularity of one's own proper name, in the knowledge that any response will necessarily be inadequate. The other condition – in fact it's the only condition – is that one takes responsibility for what one has written by pursuing it to the limit, which is where the approach adopted breaks down of its own accord anyway.
– As here.
– Yes.
– My turn, then?
– Yes.
– Yes.
# Notes
## Chapter 1: An intellectual itinerary
### Why Blanchot?
See Bataille's two reviews of Blanchot's _récits, Au moment voulu_ and Le _Dernier Homme,_ in: _Œuvres complètes,_ 12 vols (Paris, Gallimard, 1970–88), XII (1988), 173–8 and 457–66; and Jacques Derrida, _Parages_ (Paris, Galilée, 1986).
On the importance of literary journalism in France during the period 1930 to 1960, see Régis Debray, _Le Pouvoir intellectuel en France_ (Paris, Ramsay, 1979).
I am drawing here on Derrida's (oral) contribution to Didier Cahen and Jean-Claude Loiseau, _Sur les traces de Maurice Blanchot,_ broadcast by France-Culture, 17 September 1994.
Jacques Derrida, _Parages,_ 55. In his turn, Blanchot has offered his own thanks to Derrida in: 'Grâce (soit rendue) à Jacques Derrida', _Revue philosophique,_ 2, April–June 1990, 167–73 ('Thanks [Be Given] to Jacques Derrida', _BR,_ 317–23).
See Pierre Madaule, _Une tâche sérieuse?_ (Paris, Gallimard, 1973). Madaule's title is itself, of course, a reprise of the closing words of the now deleted final paragraph of Blanchot's _L'Arrêt de mort_ ( _Death Sentence_ ) of 1948.
### An ethics of discretion
Maurice Blanchot, 'Les Intellectuels en question', _Le Débat,_ 29, March 1984, 3–28 (17) ( _BR,_ 206–27 [217]).
Maurice Blanchot, 'Sur Edmond Jabès', _Les Nouveaux Cahiers,_ 31, Winter 1972–73, 51–2. One of these fragments had in fact appeared even earlier in: 'Fragmentaires', _L'Ephémère,_ 16, January 1971, 376–99 (388–9).
Paul Auster, in his English translation of this text, renders this first clause as: 'History does not withhold meaning'. Unfortunately, as the context will confirm, this is almost exactly the opposite of what Blanchot originally wrote. Readers of Blanchot in English need sometimes to note: Caveat lector!
Maurice Blanchot, 'Les Intellectuels en question', _Le Débat,_ 29, March 1984, 3–24 ( _BR,_ 206–27).
Maurice Blanchot, 'Les Intellectuels en question', 13 ( _BR,_ 213).
The phrase is used again by Blanchot as the title of the two following pieces: 'N'oubliez pas!', _La Quinzaine littéraire,_ 459, 16–31 March 1986, 11–12; and '"N'oubliez pas"', _L'Arche,_ May 1988, 68– 71 ( _BR,_ 244–9). For a detailed assessment of Blanchot's references to Auschwitz during the 1970s and 1980s, see Michel Lisse, 'Écrire "après Auschwitz"? Maurice Blanchot et les camps de la mort', La Littérature des camps, Vincent Engel (ed.), _Les Lettres romanes,_ special issue, 1995, 121–38.
### A share of biography
Maurice Blanchot, 'Les Rencontres', _Le Nouvel Observateur,_ 1045, special issue, November 1984, 84.
Maurice Blanchot, 'Pour l'amitié', in: Dionys Mascolo, _A la recherche d'un communisme de pensée, entêtements_ (Paris, Éditions Fourbis), 1993, 5–16 (16). Among Blanchot's other memories of their encounter, see the letter reproduced in _Exercices de la patience, 1,_ 1980, 67; 'Notre compagne clandestine', in _Textes pour Emmanuel Lévinas,_ François Laruelle (ed.) (Paris, Jean-Michel Place, 1980), 79–87; and '"N'oubliez pas"', _L'Arche,_ May 1988, 68–71 (BR, 244–9). Levinas, for his part, recalls their relationship in: François Poirié, _Emmanuel Lévinas: Qui êtes-vous?_ (Lyon, La Manufacture, 1987), 70–1.
See Maurice Blanchot, 'Penser l'apocalypse', _Le Nouvel Observateur,_ 22–8 January 1988, 77–9.
On Aminadab Levinas (whose name, in Hebrew, is said to signify 'my people is generous'), see Marie-Anne Lescourret's otherwise disappointingly unreliable biography of Levinas, _Emmanuel Levinas_ (Paris, Flammarion, 1994); for an account of Blanchot's action on behalf of Levinas's family, see 121– 2. More recently, Levinas's daughter, Simone, has added her own personal testimony regarding Blanchot's actions on her behalf and that of her mother and grandmother during the Occupation; see Simone Hansel, née Levinas, 'Pour Maurice Blanchot', _Le Monde,_ 1–2 December 1996, 10.
See Michel Surya, _Georges Bataille, la mort à l'œuvre,_ 2nd edn (Paris, Gallimard, 1992), 378–84; and Pierre Prévost, _Pierre Prévost rencontre Georges Bataille_ (Paris, Jean-Michel Place, 1987), 86. Drawing up a curriculum vitae of his own in 1958, Bataille, referring to himself in the third person, wrote as follows: 'By the end of 1940, he meets Maurice Blanchot with whom in admiration and agreement he forms an immediate bond of friendship', _Œuvres complètes,_ VII (1976), 462.
This is the view proposed by Pierre Klossowski in a fragmentary memoir of his own entitled: 'De "Contre-Attaque" à "Acéphale"', _Change_ , 7, July 1970, 103–7 (107).
Beyond the unpredictability of what was necessarily a chance event, the name and text of Nietzsche provide perhaps a clue to the reason for the immediate friendship between Blanchot and Bataille. Although Blanchot is unlikely to have been aware of it at the time, in January 1937 Bataille brought out an issue of _Acéphale_ devoted to the theme: 'Nietzsche et les fascistes'. In the first of his contributions to the issue (now reproduced in his _Œuvres complètes,_ I (1970), 447–65), Bataille cites a press cutting, taken from the front page of the daily paper _Le Temps_ for Saturday, 4 November 1933, which recounts the gift to Hitler by Elisabeth Foerster-Nietzsche of a sword-stick having previously belonged to her brother (448). Bataille adds a reference to a photograph showing Hitler in Weimar standing alongside Nietzsche's bust, a photograph which was used by Nietzsche's cousin Richard Oehler as the frontispiece for his 1935 book _Friedrich Nietzsche und die deutsche Zukunft,_ a book in which Oehler attempted to portray the author of _Mein Kampf_ as Nietzsche's direct philosophical heir. Bataille's point, in 1937, was to underline the unscrupulous 'treachery' of Nietzsche's immediate family and their willingness to falsify Nietzsche's work for their own crude anti-semitic and fascist ends. Thirty-two years later, in 1969, in a long footnote to his essay, 'Nietzsche, aujourd'hui' ( _EI,_ 204–5; 449), wishing to make a similar point about the Nazi appropriation of Nietzsche, though without mentioning the name of Bataille, Blanchot included a reference to the same photograph and the same book by Oehler, and even went as far as to cite the self-same press cutting from 4 November 1933. Oddly enough, while Bataille reproduces the cutting from _Le Temps_ accurately, Blanchot modifies the text slightly, though without altering its sense in any way. Nevertheless, it is somehow as though what is being recalled here, despite or even because of these minor stylistic changes, discreetly and obliquely, in belated homage to his late friend, is an encounter between Blanchot and Bataille around the name of Nietzsche, and around the affirmation of the irreducibility of Nietzsche's texts to fascism. (Throughout the period of their first meeting, Bataille was still actively reading Nietzsche, and from February to August 1944 was busy working at his _Sur Nietzsche,_ which appeared the following year; see _Œuvres complètes,_ VI (1973).)
On the dating of Blanchot's various manuscripts during that period, see the letter from Blanchot to Bataille, dated 13 January [1948], BN, Mss, N. a. fr. 15853/295. The sluggishness of Gaston Gallimard in publishing some of these texts – as was notably the case with _Le Très-Haut_ – seems to have given Blanchot some grounds for complaint; as a result, his contractual obligations to Gallimard were the source of some friction at the time (and at one point, for instance, Blanchot even explored the possibility of having the Editions de Minuit, which had recently taken over responsibility for publishing _Critique,_ buy back the rights for his books from Gallimard).
The text was reissued in 1973 under the now more familiar title of _La Folie du jour._ On the problematic character of the title, see Derrida, _Parages,_ 131–5.
See the letter from Blanchot to Bataille, undated, BN, Mss, N. a. fr. 15835/252. Something of this tension is apparent in the fact that when the second part of Blanchot's essay 'Rilke et l'exigence de la mort' ('Rilke and Death's Demand') appeared in _Critique_ for May 1953, the editors felt obliged to append a postscript offering the reader some account of the volume purported to be under review! The article was to be Blanchot's last regular contribution to the journal.
For the quotation from Mallarmé, see Stéphane Mallarmé, _Œuvres complètes,_ Henri Mondor and G. Jean-Aubry (eds) (Paris, Gallimard, 1945), 645–6.
'Un itinéraire politique', interview with Dionys Mascolo by Aliette Armel, _Le Magazine littéraire,_ 278, June 1990, 36–40 (40). The three issues of _Le 14 Juillet_ are now available in a facsimile reprint as: _Le 14 Juillet_ (Paris, Séguier-Lignes, 1990).
On the campaign of intellectuals against the war, see Hervé Hamon and Patrick Rotman, _Les Porteurs de valise: la résistance française à la guerre d'Algérie_ (Paris, Seuil, revised edn, 1982), where the text of the Manifesto is reproduced (393–6). Attempts at wider dissemination of the document led to Blanchot's only ever recorded interview, with Madeleine Chapsal, for _L'Express,_ which, however, refused to carry it; the text appeared some time later in: _Le Droit à l'insoumission: 'le dossier des 121'_ (Paris, François Maspero: Cahiers libres, 14, 1961), 90–3 ( _BR,_ 196–9).
On the history and background of the project, see the collection of preparatory texts and editorial correspondence in: _Lignes,_ 11, September 1990, 179–301. Only one issue of the 'revue internationale' in fact ever appeared, which it did in 1964 under the auspices of the Italian journal _Il Menabò._ As Blanchot commented in 1991: 'The failure of our project did not show it to be a utopian one. What does not succeed remains necessary. This is still my concern.' ('L'échec de notre projet n'a pas démontré que c'était une utopie. Ce qui ne réussit pas reste nécessaire. C'est toujours notre souci.') See Maurice Blanchot, 'Sur le nationalisme', _La Règle du jeu,_ 3, January 1991, 221–2.
Robert Antelme, _L'Espèce humaine_ (Paris, Gallimard, [1947] 1957). From 1939 to 1945 Antelme was also married to Marguerite Duras, who recounts her experience of Antelme's deportation in _La Douleur_ (Paris, POL, 1985), translated by Barbara Bray as _La Douleur_ (London, Collins, 1986). Dionys Mascolo, a close friend of both Antelme and Duras atthe time, records his own version of those events in his homage to Antelme, _Autour d'un effort de mémoire : sur une lettre de Robert Antelme_ (Paris, Maurice Nadeau, 1986). Antelme left an indelible impression on all who knew him; see for instance the description by Claude Roy in: _Nous_ (Paris, Gallimard, 1972), 102–9, and the obituary of Antelme written by Edgar Morin in _Le Monde,_ 2 November 1990. Blanchot himself writes further of Antelme in a brief contribution to the special issue of _Lignes,_ 21, January 1994, devoted to Antelme's memory, now republished, with additional material, as Robert Antelme, _Textes inédits, sur 'L'Espèce humaines', essais et témoignages_ (Paris, Gallimard, 1996).
On Blanchot's experience of May 1968, see his remarks in _La Communauté inavouable_ ( _CI,_ 52–5;29– 33) and in _Michel Foucault tel que je l'imagine_ ( _MF,_ 9–10; 63–4). Of the thirty or so original texts included in this first (and only) issue of _Comité,_ Dionys Mascolo attributes as many as eighteen to Blanchot (personal communication, 18 April 1994). Of these only one, entitled 'Lire Marx', has ever been claimed explicitly by Blanchot and is reproduced as 'Les Trois Paroles de Marx' in: _L'Amitié_ (A, 115–17); two more were attributed to Blanchot by _Gramma in_ 1976; and in 1984 Mascolo reproduced a selection of material from the issue under the title: 'Mots de désordre', _Liberation,_ 28–9 January 1984, 23, now available in a translation by Michael Holland in The _Blanchot Reader_ ( _BR,_ 200–5).
Blanchot's letter is cited by Levinas in: _Du sacré au saint_ (Paris, Minuit, 1977), 48–9. Blanchot makes a similar point about the extreme left's commitment to the Palestinian cause in: 'N'oubliez pas!', _La Quinzaine littéraire,_ 459, 16–31 March 1986, 11–12. On Blanchot's continuing support for Israel (and for Shimon Peres rather than Menachem Begin), see 'Cequim'est le plus proche...', _Globe,_ 30, July– August 1988, 56.
### Pas de récit?
See the bibliography by Peter Hoy and Michael Holland in _Gramma,_ 3–4 (1976), 224– 45, and 5 (1976), 125–32. For an influential (though not always reliable) account of Blanchot's involvement with the Jeune Droite in the 1930s, see Jean-Louis Loubet del Bayle, _Les Non-conformistes des années 30, une tentative de renouvellement de la pensée politique française_ (Paris, Seuil, 1969).
See for instance his response to Jeffrey Mehlman, reproduced in Mehlman's essay, 'Blanchot at _Combat:_ Of Literature and Terror', _Modern Language Notes,_ 95, 1980, 808– 29 (819); to Diane Rubenstein, reproduced in her _What's Left: the Ecole Normale Supérieure and the Right_ (Madison, University of Wisconsin Press,1990), 187; and to RogerLaporte, in _Blanchot: the Demand of Writing,_ Carolyn Bailey Gill (ed.) (London, Routledge, 1996), 209–11.
For the charge of historical evasiveness, see Jeffrey Mehlman, _Genealogies of the Text: Literature, Psychoanalysis, and Politics in Modern France_ (Cambridge, Cambridge University Press, 1995), 194; and, more generally, Steven Ungar, _Scandal and Aftereffect: Blanchot and France since 1930_ (Minneapolis and London, University of Minnesota Press, 1995). For a vigorously argued counterview, based on a reading of Blanchot's _L'Instant de ma mort,_ see Jacques Derrida, 'Demeure : fiction et témoignage', in _Passions de la littérature_ : _avec Jacques Derrida,_ Michael Lisse (ed.) (Paris, Galilée, 1996). Often, while claiming to react against an excessive mythologisation of Blanchot as literary recluse and archproponent of the self-deconstructing, minimalist modern text, critics such as Mehlman or Ungar have in fact done litttle more than propose an alternative mythology, which privileges above all else one part of Blanchot's early journalism and, detaching it from its proper philosophical or historical context, construes the writer's subsequent intellectual itinerary solely in terms of the supposed psychological or moral legacy of these early political involvements. The historical and other distortions produced by this strangely reductive approach are many, and are perhaps best illustrated, in the wake of Mehlman and Ungar, by the unconvincing and at times almost wilfully extravagant attempt at remythologising Blanchot's intellectual and political career provided by Philippe Mesnard in his _Maurice Blanchot: Le Sujet de l'engagement_ (Paris, L'Harmattan, 1996).
One of the most striking examples of Blanchot's determination to draw a discreet veil over the detail of his political past is the reference, in _La Communauté inavouable,_ while discussing Bataille's prewar involvement in groups such as Contre-Attaque and Acéphale, to what Blanchot calls 'our history' ('notre histoire', _CI,_ 14; 5), meaning by that the parallel lives followed during the 1930s by Bataille and himself, even though the pair only in fact met in 1940. In so doing, Blanchot boldly side-steps the explicit record of his own pre-war nationalist political affiliations, which were very different from those of Bataille; indeed, despite some surface similarities with those of Blanchot, Bataille's political activities during the 1930s all expressed his implacable opposition to nationalism; see, for instance, the texts for 'Contre-Attaque' written by Bataille in 1935 and 1936, in: _Œuvres complètes,_ I (1970), 379– 428.
Letter from Maurice Blanchot to Roger Laporte, 9 December 1984, in: _Blanchot: the Demand of Writing,_ Carolyn Bailey Gill (ed.), 9.
### The acts of the day (1)
By the end of the decade, in 1939, circulation of the _Journal des débats_ had fallen to a modest 25,000 copies. This compares, in the same year, with 1,800,000 for the popular Paris evening paper, _Paris-Soir,_ and 320,000 for _L'Humanité,_ the Communist Party daily; _Le Temps,_ one of the main rivals to the _Journal des débats,_ was by then selling 90,000 copies; these sales figures and those of the other newspapers of the time are provided in: Raymond Manévy, _La Presse de la III e République_ (Paris, Joseph Foret, 1955), 243–4.
Georges Bataille, _Œuvres complètes,_ V (1973), 256.
'Ses discours étaient to ujours des triomphes et ses actes des échecs', wrote Blanchot in the piece, which is entitled simply 'M. Briand', _Journal des débats,_ 9 March 1932, 1. The article is attributed to Blanchot by Levinas (who slightly misquotes the phrase); see Marie-Anne Lescourret, _Emmanuel Levinas,_ 112.
For the review of Malaparte, see Blanchot, 'Comment s'emparer du pouvoir?', _Journal des débats,_ 18 August 1931.
On the Jeune Droite and the various journals and other groups it spawned, see Jean-Louis Loubet del Bayle, _Les Non-conformistes des années 30._
See Maurice Blanchot, 'Les Années tournantes', _Journal des débats,_ 21 March 1933, and 'Le Monde sans âme', _La Revue française,_ 27e année, 3, 25 August 1932, 460–70.
See Maurice Blanchot, 'Le Marxisme contre la révolution', _La Revue française,_ 28e année, 4, 25 April 1933, 506–17.
On the political career of Georges Mandel (born Louis Rothschild), see John M. Sherwood, _Georges Mandel and the Third Republic_ (Stanford, Stanford University Press, 1970) and Bertrand Favreau, _Georges Mandel ou la passion de la République 1885–1944_ (Paris, Fayard, 1996). Lévy himself had formerly worked with Mandel, some thirty years before, as theatre correspondent on the newspaper _L'Aurore._ After the war, Lévy assembled the copious evidence for his own consistently anti-German and anti-Nazi stance in a volume of collected journalism entitled _Au temps des grimaces_ (Paris, Nagel, 1948).
See, respectively, 'La Démocratie et les relations franco–allemandes', _La Revue du XX e siècle,_ 4, February 1935, 56–9 (58); 'La Vraie Menace du Troisième Reich', _Le Rempart,_ 69, 29 June 1933, 3; 'M. de Monzie, émule de Mussolini et de Hitler', _Le Rempart,_ 32, 23 May 1933, 1–2; and 'Des violences antisémites à l'apothéose du travail', _Le Rempart,_ 10, 1 May 1933, 3. The paper's prescient and consistent condemnation of Hitler is clear and unambiguous in every issue of _Le Rempart_ I have been able to trace; indeed it was claimed later, by Paul Lévy, at the time of Munich, that the very purpose behind setting up the paper in 1933 was to alert French opinion to the Nazi threat (see Paul Lévy, 'Le Sedan diplomatique', _Aux écoutes,_ 24 September 1938, 19). All the more puzzling therefore is Steven Ungar's claim in his _Scandal and Aftereffect_ that Blanchot's articles manifest 'a certain degree of admiration' for Hitler (97). Ungar misreads as advocacy for Hitler the endeavour on Blanchot's part, as early as 1933, to understand the reasons for Hitler's successful rise to power. Though Blanchot at the time was unconvinced that the Western parliamentary democracies had the will or strength to dispose of the threat to peace that Hitler represented (and events were of course unfortunately to prove Blanchot right), there is not the slightest evidence that Blanchot ever expressed a desire to see Nazism emulated in France either in its German form or in any corresponding French incarnation; it is difficult to resist the conclusion that Ungar's analysis is both superficial and tendentious.
See Maurice Blanchot, 'La Révolution nécessaire', _Le Rempart,_ 62, 22 June 1933, 2. Reading these words, it is important to give due weight to the date when they were written. Ideological and political debate in the 1930s rarely stood still; all too often, perhaps, commentators of these texts have failed to see them in their proper historical context.
Maurice Blanchot, 'M. de Monzie, émule de Mussolini et de Hitler', _Le Rempart,_ 32, 23 May 1933, 1– 2.
De Fabrègues, though much the older, was perhaps the less important of the two editors, and it is the name of Thierry Maulnier that, throughout the 1930s, most often recurs in relation to that of Blanchot. A prolific literary critic and political journalist, Maulnier was the author of books such as _La Crise est dans l'homme_ (1932), _Nietzsche_ (1933), _Racine_ (1935), _Le Nationalisme au-delà du nationalisme_ (1938), _Lecture de Phèdre_ (1943), and many others. Originally viewed as the natural successor to Maurras at the head of _Action française,_ Maulnier, to Maurras's dismay, adopted during the 1930s a more radical, anti-capitalist stance; during 1936–39, he was the driving force behind _Combat,_ and in 1937 became one of the founder editors of _L'Insurgé._ During the early years of the Occupation, he wrote regularly for _Action française_ and other nationalist papers. After the war, Maulnier reverted to a more traditional conservatism and was eventually elected to the Académie française in 1964. On Maulnier's career, see Paul Sérant, _Les Dissidents de l'Action française_ (Paris, Éditions Copernic, 1978), 211–44, and, for an account of his political and literary writings, David Carroll, _French Literary Fascism: Nationalism, Anti-Semitism, and the Ideology of Culture_ (Princeton, Princeton University Press, 1995), 222–47. For an authorised (Action Française) view of Maurras's disagreements with Maulnier in 1937, at the time of the launch of _L'Insurgé_ (which took place while Maurras was in prison), see Henri Massis, _Maurras et notre temps,_ 2 vols (Paris-Geneva, La Palatine, 1951), II, 88–91.
See, for instance, Zeev Sternhell, _Ni droite ni gauche: l'idéologie fasciste en France,_ new edn (Brussels, Éditions Complexe, 1987), 263–85.
See Robert Brasillach, _Notre avant-guerre_ (Paris, Plon, 1941), 185–6. There is little doubt that one of the contributors Brasillach had in mind here was Blanchot; indeed the antagonism between the two men was mutual. Nevertheless, during the first year of _Combat's_ existence, Blanchot and Brasillach often found themselves contributing to the same issues (between February and December 1936 this occurred on six separate occasions). Eccentrically enough, according to Claude Roy, Brasillach thought Maulnier at _Combat_ to be 'too much of a Marxist' (see Claude Roy, _Moi je_ (Paris, Gallimard: folio, 1969), 244–6). On the disagreements between _Combat_ and _Je Suis Partout_ (which Brasillach edited from 1937), which stemmed mainly from the latter's virulent anti-semitism and enthusiasm for Hitler, see Pierre-Marie Dioudonnat, _Je suis partout 1930–44, les maurrassiens devant la tentation fasciste_ (Paris, La Table ronde, 1973), 190. It is worth noting that when _Thomas l'Obscur_ appeared in 1941, it was reviewed in _Je suis partout_ by an unnamed critic (Lucien Rebatet?), whose verdict was brutally simple: here was a book, he wrote, 'as outdated as the Jewish art from which it takes its inspiration' ('aussi démodé que l'art juif dont il se réclame', _Je suis partout,_ 18 October 1941, 8).
See Pierre-Marie Dioudonnat, _Je suis partout 1930–44, les maurrassiens devant la tentation fasciste,_ 89.
Anti-semitism was not a consistent or even dominant feature of _L'Insurgé;_ but in the work of several of its collaborators, including both writers and cartoonists, it was nevertheless quite explicit. Witness for instance the following remark which arises in the course of an attack on Blum's Jewishness: 'The invasion gets bigger day by day; one after the other the public services are jewified; in every ministry all you see are hook noses and frizzy hair' ('L'invasion va chaque jour en augmentant; tous les services publics l'un après l'autre sont enjuivés; dans tous les ministères, on ne rencontre que nez busqués et cheveux crépus'); see Robert Castille, '"Nous sommes les fils du peuple de ce pays... " (Léon Blum)', _L'Insurgé,_ 7, 24 February 1937.
### A question of responsibility
See Tzvetan Todorov, _Critique de la critique_ (Paris, Seuil, 1984), 73 ( _Literature and its Theorists,_ translated by Catherine Porter (London, Routledge and Kegan Paul, 1987), 61). In an equally expeditious move, Todorov extends his criticism to refer more widely to what he describes, bizarrely enough, as Blanchot's 'nihilism' (72); this appears to be directed at Blanchot's refusal to subordinate literature to what, with disarming naivety, Todorov calls 'truth and values'.
These are the texts whose genesis Blanchot describes in his letter to Roger Laporte of 9 December 1984 in the following terms:
It must be said that the émigré community which found in Paul Lévy a constant source of support constituted at the time almost my natural environment; the truth about the extreme danger represented by Hitler was plain to see there, but amidst other, fanciful rumours (that Hitler was seriously ill, or mad – and how could one avoid thinking that some kind of madness was behind his horrifying political projects: the Reichstag fire, _Kristallnacht,_ or the annihilation of his close companions). Others, who were the most numerous, kept saying on the other hand: let's not exaggerate, it's essential to be careful, reserved, and to warn the Jews against themselves. This is how those texts came about for which I am reproached today, and rightly so. But it would be odious to displace on to others the responsibility which is mine alone. In addition, there was the distrust felt by assimilated French Jews towards Zionism. Levinas had taught me the importance and the meaning of the Diaspora, that unfortunate wandering which had as its counterpart the 'dissemination' of Jewish singularity, its exclusion from any form of nationalism as final truth, its participation in history in an entirely other manner. This is what caused me to say something – too much about the 'new doctrine' of Israel.
('Il faut dire que l'émigration qui trouvait auprès de Paul Lévy un appui constant, constituait alors presque mon milieu naturel; la vérité sur l'extrême danger que représentait Hitler se faisait jour là clairement, mais parmi aussi des rumeurs fantaisistes (que Hitler était gravement malade, qu'il était fou – et comment ne pas assimiler à une sorte de folie ses desseins politiques horrifiants: l'incendie du Reichstag, la Nuit de Cristal, l'annihilation de ses proches compagnons). D'autres, les plus nombreux, disaient au contraire : n'exagérons rien, il faut être prudent, réservé, mettre en garde les Juifs contre eux-mêmes. C'est de là que sont venus les textes que, avec raison, on me reproche. Mais il serait odieux de rejeter sur d'autres une responsabilité qui est la mienne. A cela s'ajoutait la méfiance des Juifs français assimilés à l'égard du sionisme. Levinas m'avait appris l'importance et la signification de la Diaspora, l'errance malheureuse qui avait comme contrepartie la 'dissémination' de la singularité juive, son exclusion de tout nationalisme comme vérité dernière, sa participation à l'histoire sous une forme tout à fait autre. C'est pourquoi j'ai été amené à dire un mot (un mot de trop) sur la 'doctrine nouvelle' d'Israël'.)
See my 'Introduction', in _Blanchot: the Demand of Writing,_ Carolyn Bailey Gill (ed.), 9–10. It is perhaps also worth adding at this stage that to write at all in _Combat_ was to submit, willy-nilly, to certain constraints, and it is not out of the question, as Blanchot has been known to reveal on at least one occasion, that his texts in the paper were also the subject to editorial interference from those in control of its political line.
Maurice Blanchot, 'Blum, notre chance de salut . . .', _L'Insurgé,_ 3, 27 January 1937, 4.
See Diane Rubenstein, _What's Left: the Ecole Normale Supérieure and the Right,_ 187. It is impossible to say of course what form this assurance took; evidently, if Blanchot's condition was ever enforced, this did not happen until at least fifteen months after _Combat_ began publishing, in March 1937, the date of Brasillach's last contribution to the magazine; interestingly, after Brasillach's own last piece in _Combat,_ Blanchot himself was to contribute only two more articles of his own.
See Charles Deleuze, 'La Santé de nos enfants est en de bonnes mains . . .', _L'Insurgé,_ 42, 27 October 1937, 5. Some months earlier, in an article published alongside a piece by Blanchot on the coronation of George VI, Deleuze had launched a bitter attack on the two teams playing in the French Cup Final for fielding between them eleven players who were not 'of pure French origin', a fact that, for the journalist, was proof that French football too had fallen victim to what he called 'a gang of Jews and wogs' ('une bande de juifs et de métèques'); see Charles Deleuze, '. . . et les métèques l'envahissent', _L'Insurgé,_ 19, 19 May 1937, 4.
_Aux écoutes_ was founded by Lévy in 1922; but apart from his own regular column, the vast majority of articles in the magazine were left unsigned. It therefore cannot be said with any certainty when Blanchot became its _rédacteur en chef;_ internal evidence suggests, however, he was playing a full part in the paper by June 1937 at the latest. In subsequent years, Blanchot's association with Lévy was to be one of the reasons for the violent hostility of the anti-semitic, collaborationist press towards Blanchot. This is explicit in the review of _Thomas l'Obscur in Je suis partout_ mentioned earlier; and also, for instance, in a brief item reacting to the (unfounded) rumour that Blanchot was about to take over the _Nouvelle Revue française_ from Drieu, under the title 'Videz Thomas', that appeared in _L'Appel,_ 28 May 1942, 4.
See for example Maurice Blanchot, 'Le Caravansérail', _Combat,_ 10, December 1936. One of the readers of Blanchot most troubled by what he sees as the writer's 'doctrine of violence' in these early texts is Michael Holland; see his 'A Wound to Thought', in _Blanchot: the Demand of Writing,_ Carolyn Bailey Gill (ed.), 174–89. That Blanchot was aware of the explicit and violent language used by some contributors to _L'Insurgé,_ and also prepared to go to some lengths to endorse that violence, is plain from the various court cases in which, as one of the founding editors of the paper, he became involved. The most notorious of these occurred in reaction to a vitriolic attack made in the paper on Léon Blum on grounds of his alleged responsibility for the deaths of two right-wing political activists, the one a member of the fascist Parti Populaire Français (PPF) and the other a member of the rival Parti Social Français (PSF). The offending article, 'Communistes assassins', signed 'L'Insurgé' and published in _L'Insurgé,_ 8, 3 March 1937, led to the paper's seizure by the authorities and to a charge of provocation to murder and incitement to violence against Guy Richelet, the managing editor of the paper who, under French law, was legally responsible for it. In a letter of 4 March 1937, printed the following week, the five other founding editors (Thierry Maulnier, Jean-Pierre Maxence, Ralph Soupault, Kléber Haedens, and Maurice Blanchot) wrote to the examining magistrate in the case to claim collective responsibility for the article, a ploy that led to considerable procedural delays, and in due course, when the paper reiterated its attack on Blum, to further charges. The tactic was employed again by _L'Insurgé_ later in the year. (It was of course also used twenty-three years later, in 1960, by the various co-signatories to the 'Manifeste des 121', who similarly claimed, before the examining magistrate, that they were each responsible for drafting the declaration.)
Maurice Blanchot, 'On demande des dissidents', _Combat,_ 20, December 1937, [155].
Zeev Sternhell, _Ni droite ni gauche,_ 257; and Jeffrey Mehlman, _Genealogies of the Text,_ 87–8.
Maurice Blanchot, 'Le Marxisme contre la révolution', _La Revue française,_ 28e année, 4, 25 April 1933, 506–17 (516).
See Jacques Derrida, _Force de loi_ (Paris, Galilée, 1994).
This is clear for instance from the various articles published in _Le Rempart_ throughout May 1933, recommending a taxpayers' strike as a means of protesting against what Blanchot saw as the financial incompetence of the government of the day.
### The acts of the day (2)
See Thierry Maulnier, 'Il ne fallait pas faire cette guerre', _Combat,_ 28, October 1938, [3–4], and 'Les Nouvelles Conditions imposées à l'action politique en France', _Combat,_ 29, November 1938, [3–4]. Writing some months earlier, Louis Salleron, another regular contributor, had struck a more strident isolationist note in an article entitled 'La France doit-elle se battre pour la Tchécoslovaquie?', in _Combat,_ 27, July 1938, [8–9], which concluded thus: 'Let us not carry at arm's length this stupid Europe that hates us. Let us go back into our own back yard and barricade ourselves inside it. It's at home we need to do some cleaning up. Not anywhere else.'
_Aux écoutes,_ 24 September 1938, 19 and 9. Churchill's phrase is reported also in Hugh Dalton, _The Fateful Years_ (London, Frederick Muller, 1957), 198.
On the paper's withdrawal to Clermont-Ferrand, see Jean-Noël Jeanneney, _François de Wendel en république: l'argent et le pouvoir 1914–1940,_ 3 vols, thèse Université Paris X, 1976, II, 846–7 and III, 1374; according to a note in his diary, de Wendel met Nalèche for the last time in Vichy in May 1941, and attempted, in vain, to convince Nalèche 'how ignominious it is for the hired rag the _Journal des débats_ has become to worship, every day, everything that, for ten years, it strove to destroy' (II, 847).
On Lévy's activities during the Occupation, see his _Journal d'un exilé_ (Paris, Grasset, 1949), which recounts, in particular, how in November 1940 Lévy was warned by Blanchot's sister of his imminent arrest and was thus able to make good his escape (29–30).
_Aux écoutes,_ 15 June–13 July 1940, 10.
For Blanchot's own account of his involvement in Jeune France, and in the negotiations with Paulhan and Drieu, see Maurice Blanchot, 'Pour l'amitié', in Dionys Mascolo, _A la recherche d'un communisme de pensée, entêtements_ (Paris, Éditions fourbis, 1993), 5–16. The association Jeune France was set up at the end of 1940, with the financial support of the Vichy government, in order to promote various cultural initiatives across the country. As its president, Pierre Schaeffer, recounts in his autobiography, _Les Antennes de Jéricho_ (Paris, Stock, 1978), the association was largely split between those who, in a populist, humanistic perspective, were in favour of providing leisure activities for the community at large, and those who, like Blanchot, insisted on the prime importance of creative autonomy and artistic integrity (276–7). Blanchot and those of his friends who shared this view soon left the organisation, which, in any case, was itself dissolved by March 1942 as a result of its unpopularity with the Vichy government. In these early years of the Occupation there was obviously some debate among opposition intellectuals in Paris as to whether it was better to be active politically in some way and run the risk of being compromised, or preferable to stand aloof and demonstrate total rejection of the situation; some echo of the debate may be found in an entry in Michel Leiris's recently published diary of the period: see, for instance, the entry for 16 February 1941 in his _Journal 1922– 1989,_ Jean Jamin (ed.) (Paris, Gallimard, 1992), 336–8. In 1941 Leiris was a proponent of the second of these courses of action, and had little patience with those friends – like Bataille and Blanchot – who initially favoured the opposite; though Leiris in later years was to be a signatory to the Déclaration of the 121, his annoyance at Blanchot's politics extended to the latter's actions during May 1968, which he found overblown and, as he puts it in his diary, 'ridiculous' (628).
Maurice Blanchot, 'Pour l'amitié', in Dionys Mascolo, _A la recherche d'un communisme de pensée, entêtements,_ 6.
### A temptation?
Maurice Blanchot, 'De la révolution à la littérature', _L'Insurgé,_ 1, 13 January 1937, 3.
See Maurice Blanchot, 'Nietzsche, aujourd'hui', _La Nouvelle Revue française,_ 68, August 1958, 284–95 ( _EI,_ 201–15; 136–43). As Michael Holland has pointed out, Blanchot's text underwent considerable revision before being republished in _L'Entretien infini;_ see his paper, 'A Wound to Thought', _Blanchot: the Demand of Writing,_ Carolyn Bailey Gill (ed.), 174–89. Holland also draws attention to the fact that, in a quite unusual gesture, Blanchot the following month added a postscript to the article 'Passage de la ligne' ( _EI,_ 215–27; 143–51 ), explicitly stating his opposition to de Gaulle (and to the support for de Gaulle shown by Paulhan).
In a note to her translation of Blanchot's essay in _The Infinite Conversation_ (451), Susan Hanson unfortunately mis-identifies the text by Heidegger to which Blanchot is referring here: it is in fact _not_ the notorious Rectorship Address of 27 May 1933, 'Die Selbstbehauptung der deutschen Universität', which Hanson cites, but the far more explicit address Heidegger gave in Leipzig, on the anniversary of the armistice, 11 November 1933, in which, using much of the same language as he had in his properly philosophical writings, he publicly committed himself to Hitler and to the National-Socialist state; see Guido Schneeberger, _Nachlese zu Heidegger_ (Bern, Suhr, 1962), 148–50, and _The Heidegger Controversy: a Critical Reader,_ Richard Wolin (ed.) (Cambridge, Mass., MIT Press, 1993), 49–52.
## Chapter 2: The (im)possibility of literature
### Founding fictions
This is not to say there are not a number of useful studies of these early novels available for consultation. See, in particular, Jean-Paul Sartre, _'Aminadab_ ou du fantastique considéré comme un langage', in _Situations,_ I (Paris, Gallimard, 1947), 148–73 ( _Literary and Philosophical Essays,_ translated by Annette Michelson (London, Hutchinson, 1955), 56–72); Jean Starobinski, _'Thomas l'obscur:_ chapitre premier', _Critique,_ 229, June 1966, 498–513; Michel Foucault, 'La Pensée du dehors', _Critique,_ 229, June 1965, 523–46 ('The Thought from Outside', translated by Brian Massumi in _Foucault–Blanchot_ (New York, Zone Books, 1987), 9–58); Daniel Wilhem, _Maurice Blanchot: la voix narrative_ (Paris, Union générale d'éditions, 1974); and Ann Smock, '"Où est la loi?": Law and Sovereignty in _Aminadab_ and _Le Très-Haut', Sub-stance,_ 14, 1976, 99–116. In this section, I refer primarily to the 1941 version of _Thomas l'Obscur,_ using the abbreviation _TO1;_ wherever possible, I have also indicated the corresponding passages in the far shorter 1950 text _(TO2)_ on which Robert Lamberton's 1973 English translation is based. On the two versions of the novel, see Rainer Stillers, _Maurice Blanchot: Thomas l'Obscur: Erst-und Zweitfassung als Paradigmen des Gesamtwerks_ (Frankfurt a. M., Peter Lang, 1979).
The paradox is one Blanchot himself formulates elsewhere. Witness for instance, in a review of _Le Voyeur_ by Alain Robbe-Grillet, the reference to the peculiarly uniform light characterising events in that novel, which Blanchot describes as: 'a brightness that makes all clear, and since it reveals all, except for itself, is the most secret thing of all' ( _LV,_ 195; _SS,_ 207).
Emmanuel Levinas, _De l'évasion_ (Montpellier, Fata morgana, 1982), 87. In the essay, first published in 1935, Levinas gives himself the task of drawing out the philosophical consequences of what he describes as contemporary literature's radical condemnation of the philosophy of being (69–70). As Blanchot was to remark some forty years later, referring to the similarly abrupt invocation of shame that occurs at the end of Kafka's _Trial,_ 'one might think the death scene represents forgiveness, the termination of the interminable; only there is no end, since Kafka makes it clear that shame survives, which is to say, infinity itself, the mockery of life as life's beyond' ( _ED,_ 89; 53).
On the compelling figure of the _other_ night in Blanchot, see _L'Espace littéraire_ ( _EL,_ 169– 84; 163–76).
See Martin Heidegger, 'Vom Wesen des Grundes' (1929) in _Wegmarken, Gesamtausgabe,_ vol. 9 (Frankfurt a. M., Klostermann, 1976), 123–75. As Heidegger explains in his 1949 preface to the paper, 'The Nothing is the Not of that which is and is thus Being experienced from the stand point of that which is' ('Das Nichts ist das Nicht des Seienden und so das vom Seienden her erfahrene Sein') (123).
Maurice Blanchot, 'L'Ébauche d'un roman', _Aux écoutes,_ 30 July 1938, 31 ( _BR,_ 33–4); the recently translated volume of texts by Heidegger to which Blanchot is referring was _Qu'est-ce que la métaphysique?,_ translated by Henry Corbin (Paris, Gallimard, 1938). In addition to the title essay, the volume also contained complete versions of 'Vom Wesen des Grundes' and Heidegger's 1936 Rome lecture, 'Hölderlin und das Wesen der Dichtung', as well as §§ 46–3 and 72–6 of _Sein und Zeit,_ and §§ 42–5 of _Kant und das Problem der Metaphysik._ See Tom Rockmore, _Heidegger and French Philosophy: Humanism, antihumanism and being_ (London, Routledge, 1995), 217.
For Levinas's own early thinking on the topic, see _De l'évasion,_ 70; and _De l'existence à l'existant_ (Paris,Vrin [1947] 1990), 83–105 ( _Existence and Existents,_ translated by Alphonso Lingis (Dordrecht, Kluwer, 1978), 52–64). Levinas refers explicitly to _Thomas l'Obscur_ in the discussion ( _De l'existence à l'existant,_ 103; _Existence and Existents,_ 63). On the relationship between _es gibt and il y a,_ see Levinas' preface to the second edition of the book, _De l'existence à l'existant,_ 10–13; Heidegger himself comments on the difference in the 'Brief über den Humanismus' (1946), _Wegmarken,_ 334–7 (Martin Heidegger, _Basic Writings,_ David Farrell Krell (ed.) (London, Routledge, 1993), 237– 40).
Levinas, _De l'existence à l'existant,_ 100; _Existence and Existents,_ 61.
Paul de Man, _Allegories of Reading_ (New Haven and London, Yale University Press, 1979), 205.
### How is literature possible?
See Philippe Lacoue-Labarthe, _L'Imitation des modernes_ (Paris, Galilée, 1986), 175–200; and _La Fiction du politique_ (Paris, Bourgois, 1987) ( _Heidegger, Art and Politics,_ translated by Chris Turner (Oxford, Blackwell, 1990)). On 'national-aestheticism' among the ideologues of the pre-war nationalist right in general, see David Carroll, _French Literary Fascism: Nationalism, Anti-Semitism, and the Ideology of Culture_ (Princeton, Princeton University Press, 1995).
Maurice Blanchot, 'De la révolution à la littérature', _L'Insurgé,_ 1, 13 January 1937, 3.
See, for instance, Maurice Blanchot, 'Le Silence des écrivains', _Journal des débats,_ 19 April 1941, 3 ( _BR,_ 25–8), and 'Recherche de la tradition', _Journal des débats,_ 16–17 June 1941, 3 ( _BR,_ 29–32).
Charles Maurras, _Romantisme et révolution_ (Paris, Nouvelle Librairie Nationale, 1922), 272; Robert Brasillach, 'A propos de Mallarmé', _Je suis partout,_ 7 June 1940, 3; Pierre Drieu La Rochelle, _Notes pour comprendre le siècle_ (Paris, Gallimard, 1941 ), 100.On the reception of Mallarmé by the extreme right in general, see David Carroll, _French Literary Fascism: Nationalism, Anti-Semitism, and the Ideology of Culture._
See Albert Thibaudet, _La Poésie de Stéphane Mallarmé_ (Paris, Gallimard, 1926), 454–68; and Thierry Maulnier, _Introduction à la poésie française_ (Paris, Gallimard, 1939), 100–1.
See for instance Paul Valéry, 'Je disais quelquefois à Stéphane Mallarmé', _Œuvres,_ 2 vols, Jean Hytier (ed.) (Paris, Gallimard, 1957), I, 644–60. In numerous later texts, Blanchot is significantly more critical of Valéry's intellectualism, and writes for instance in December 1942 that 'The mind of Mallarmé worked with absolute rigour towards the poem he wanted to produce. The mind of Valéry works with equal rigour towards the work of which the poem is the effect'; see 'Les _Mauvaises Pensées_ de Paul Valéry', _Journal des débats,_ 16 December 1942, 3. I have examined Blanchot's reading of Mallarméat greater length else where; see my 'Blanchot and Mallarmé', _MLN,_ 105, 5, December 1990, 889–913.
In Blanchot's literary criticism for the _Journal des débats,_ there is surprisingly little explicit evidence of the writer's pre-war nationalism; this is probably explained by Blanchot's distaste at the possibility of being too closely associated by his readership with Vichy ideology. However, at critical moments, the question of French national literary tradition does arise and, with it, if only indirectly, the question of the current state of France. Often, as in a piece like 'Le Jeune Roman', _Journal des débats,_ 14 May 1941, 3 (FP, 209–12), the literary analysis offers a transposed version of current political debate, with Blanchot, in this instance, upbraiding contemporary French novelists for their complacent and mistaken reliance on a dubious tradition of literary realism; elsewhere, Blanchot – with discreet provocation – elects as the embodiment of all that is most essential in the French literary tradition a lineage of writers running from Scève to Éluard via Mallarmé See 'La France et la civilisation contemporaine', _Journal des débats,_ 26–27 May 1941, 3.
See Jean Paulhan, _Les Fleurs de Tarbes, ou la terreur dans les lettres,_ Jean-Claude Zylberstein (ed.) (Paris, Gallimard: idéés, [1941] 1973). Blanchot wrote three pieces in all in response to Paulhan's book: 'La Terreur dans les lettres', _Journal des débats,_ 21 October 1941, 3; 'Comment la littérature estelle possible?' (I), _Journal des débats,_ 25 November 1941, 3; and 'Comment la littérature est-elle possible?' (II), _Journal des débats,_ 2 December 1941, 3. All three articles were collected in 1942 in a slim volume entitled _Comment la littérature est-elle possible?_ ( _BR,_ 49–60); the second two were reproduced again in _Faux Pas_ ( _FP,_ 92–101). Blanchot returns to his reading of Paulhan in three other essays: 'Le Mystère dans les lettres' and 'Le Paradoxe d'Aytré' ( _PF,_ 49–78; 43–73), and 'La Facilité de mourir' (A, 172–91; _BR,_ 301–16). For a succinct summary of the early debate between the two, see Michael Syrotinski, 'How Is Literature Possible?', _A New History of French Literature,_ Denis Hollier (ed.) (Cambridge, Mass., Harvard University Press, 1989), 953–8.
Michael Syrotinski, 'How Is Literature Possible?', 955.
This circular conclusion is not present as such in the 1941 version of Paulhan's book, to which he planned to add a second volume; though this never in fact appeared, there is good evidence, based on the earlier, 1936 version of Paulhan's text (which Blanchot may have read when it was serialised in _La Nouvelle Revue française_ ), that this was the ending Paulhan originally intended. See _Les Fleurs de Tarbes,_ 217–49.
On Hölderlin's theory of tragedy in this context, see Philippe Lacoue-Labarthe, 'La Césure du spéculatif', _L'Imitation des modernes,_ 39–69 ('The Caesura of the Speculative', translated by Robert Eisenhauer, in Philippe Lacoue-Labarthe, _Typography: Mimesis, Philosophy, Politics_ (Cambridge, Mass., Harvard University Press, 1989), 208–35). It falls to Blanchot, in a later essay on Paulhan, to make the important point that, whatever Paulhan's commitment to unity in _Les Fleurs de Tarbes_ and elsewhere, his account of paradox, inversion, and logical circularity was far removed from the duplicities of speculative dialectics (A, 175; _BR,_ 303).
This is the view presented by Jeffrey Mehlman in his _Legacies of Anti-Semitism in France,_ 13, and by Steven Ungar in _Scandal and Aftereffect,_ 120–1. But while Mehlman reads Blanchot's essay on Paulhan as a 'coded farewell to plans for a French fascism', Ungar claims the very opposite, admittedly with slightly more justification, suggesting that in 1941 Blanchot was 'displacing the violence and radicality of political revolution onto the concept of literature itself'.
### 'Naming the gods': from Heidegger to Hölderlin
Corbin's preface justifying this term is reproduced in Martin Heidegger, _Questions I_ (Paris, Gallimard, 1968), 13–20. Jacques Derrida comments on the philosophical consequences of this – now abandoned – translation of 'Dasein' in 'Les Fins de l'homme', _Marges de la philosophie_ (Paris, Minuit, 1972), 135–9 ('The Ends of Man', _Margins of Philosophy,_ translated by Alan Bass (Brighton, Harvester, 1982), 114–17).
Martin Heidegger, 'Hölderlin und das Wesen der Dichtung', in _Erläuterungen zu Hölderlins Dichtung, Gesamtausgabe,_ vol. 4 (Frankfurt a. M., Klostermann, 1981), 33– 48. All further references to Heidegger's texts on Hölderlin will be to this edition and given in the text, preceded by the abbreviation _E;_ unless otherwise indicated, all translations from Heidegger's German are my own. (A version of Heidegger's 1936 lecture, translated by Douglas Scott, is available in English as 'Hölderlin and the Essence of Poetry', in _Existence and Being_ (London, Vision Press, 1949), 293–315.)
See Friedrich Hölderlin, _Werke und Briefe,_ Friedrich Beißner and Jochen Schmidt (eds), 3 vols (Frankfurt a. M., Insel Verlag, 1969), I, 163; and Friedrich Hölderlin, _Poems and Fragments,_ translated by Michael Hamburger (London, Routledge and Kegan Paul, 1966), 428–9.
There are a number of other echoes, both implicit and explicit, of Heidegger elsewhere in Blanchot's critical essays of the time; an almost identical passage to the one cited here, for instance, appears in 'La Poésie de Mallarmé est-elle obscure?', _Journal des débats,_ 24 February 1942, 3 ( _FP_ , 128–9); on that occasion, interestingly, Blanchot deleted his original reference to phenomenology ('les phénoménologues') when revising the paper for inclusion in _Faux Pas;_ he similarly also removed Heidegger's name from the essay on Meister Eckhart first published in _Journal des débats_ for 4 November 1942, 3 (FP, 31–6).
In a fascinating account of the relation between Blanchot, Heidegger and Levinas, Paul Davies reads the phrase _à peu près_ (meaning 'more or less', but which Davies, like Charlotte Mandell in _The Work of Fire,_ renders, somewhat inaccurately in the circumstances, as 'almost') as constituting the exact measure of the distance that separates and joins the thinking of Blanchot and Heidegger on the topic of poetry; see Paul Davies, 'A Linear Narrative? Blanchot with Heidegger in the Work of Levinas', in _Philosophers' Poets,_ David Wood (ed.) (London, Routledge, 1990), 37–69 (59). In what follows, I want to suggest that in Blanchot's account of Heidegger there is clear evidence of a more decisive break than is allowed by Davies's argument, a break which is not just poetological but philosophical and political, too.
On Heidegger's political involvement with Nazism, see Hugo Ott's now authoritative _Martin Heidegger: A Political Life,_ translated by Allan Blunden (London, HarperCollins, 1993). Heidegger's first seminar course on Hölderlin (for 1934–5) appears in vol. 39 of the _Gesamtausgabe_ (Frankfurt a. M., Klostermann, 1980); a number of elements from that course are incorporated into 'Hölderlin und das Wesen der Dichtung'.
Blanchot's rejection of Heidegger's Nazism is perhaps the most likely motive behind the unexpectedly disparaging reference to 'the dross produced by German philosophy, particularly that of Heidegger' ('les produits de rebut de la philosophie allemande, en particulier celle de Heidegger') that appears in Blanchot's review of Denis de Rougemont's _Penser avec les mains_ in _L'Insurgé,_ 3, 27 January 1937, 5. The remark is all the more curious in that nowhere in Rougemont's book is the name of Heidegger in fact ever mentioned! In later texts, Blanchot lingers at much greater length on the implications of Heidegger's politics; see, for instance, the long note in _L'Entretien infini_ to which I have already referred ( _EI,_ 208–10; 449–51 ), and Blanchot's contributions to the so-called Heidegger affair of 1988: 'Penser l'apocalypse', _Le Nouvel Observateur,_ 22– 8 January 1988, 77–9; and '"N'oubliez pas"', _L'Arche, May_ 1988, 68–71.
Geert Lernout, _The Poet as Thinker: Hölderlin in France_ (Columbia, Camden House, 1994), 10.
Friedrich Hölderlin, _Werke und Briefe,_ I, 135–7; _Poems and Fragments,_ 372–7. Heidegger's paper, entitled: '"Wie wenn am Feiertage . . ."', is collected in: _Erläuterungen zu Hölderlins Dichtung,_ 49– 77; the piece was given as a lecture in 1939 and 1940 and first published in 1941. The secondary literature devoted to Heidegger's readings of Hölderlin is of course huge; for a discussion of some of the issues, see Beda Allemann, _Hölderlin und Heidegger_ (Zurich and Freiburg i. Br, Atlantis Verlag, 1954); Paul de Man, _Blindness and Insight,_ second edn (London, Routledge, 1983); Philippe Lacoue-Labarthe, _L'Imitation des modernes_ and _La Fiction du politique;_ Andrzej Warminski, _Readings in Interpretation: Hölderlin, Hegel, Heidegger_ (Minneapolis, University of Minnesota Press, 1987); Véronique M. Fóti, _Heidegger and the Poets: Poésis/Sophia/Techné_ (New Jersey and London, Humanities Press, 1992); Christopher Fynsk, _Heidegger: Thought and Historicity,_ expanded Ithaca, NY ( Ithhaca, NY, Cornell University Press, 1993); and Marc Froment-Meurice, _C'est à dire_ (Paris, Galilée, 1996).
On the dating of the article, see Blanchot's letter to Bataille of 17 October [1946], BN, Mss, N. a. fr. 15853/232, in which Blanchot writes: 'At any event, I would be tempted by your proposal about Hölderlin, although I've not seen the _Fontaine_ special issue, despite the fact that poetry is beginning to whisper discreetly in my ear that I'm wearing it out.' ('De toute manière, je serais tenté par votre proposition sur Hölderlin, bien que je n'aie pas vu l'ensemble de _Fontaine_ et que la poésie commence à me dire discrètement, à l'oreille, que je la fatigue.') In a very different way, the years immediately following the end of the war were crucial ones for Heidegger as well. As Hugo Ott describes, Heidegger was able to achieve a degree of philosophical rehabilitation at the time only as a result of the interest shown in his work by the French; indeed in the very month that Blanchot's essay appeared, Heidegger was finally relieved of his post at Freiburg and banned from all further university teaching. In France itself, the appearance of Blanchot's article also coincided with some vigorous discussion of Heidegger's politics, with the attempt being made, by some, to disengage Heidegger's thought from the political choices with which it was identified before 1945, a move that, perhaps surprisingly, found a willing audience on the intellectual left, notably in Sartre's _Les Temps modernes._ See Maurice de Gandillac, 'Entretien avec Martin Heidegger' and Alfred de Towarnicki, 'Visite à Martin Heidegger', _Les Temps modernes,_ January 1946, 713–24; an informed rebuttal of these attempts to defend Heidegger against the charge of Nazism followed later the same year in the form of Karl Löwith's 'Les Implications politiques de la philosophie de l'existence chez Heidegger', _Les Temps modernes,_ November 1946, 343–60 (the article is reproduced in _The Heidegger Controversy,_ Richard Wolin (ed.), 167–85).
Herman Rapaport, _Heidegger and Derrida, Reflections on Time and Language_ (Lincoln, University of Nebraska Press, 1989), 112. In fact, the motifs in Blanchot's essay that Rapaport claims to be the result of Blanchot's intuitive anticipation of later work by Heidegger are already to be found in Heidegger's two Hölderlin papers of 1936 and 1941.
Blanchot returns to the debate, albeit obliquely, on a number of later occasions; see, for instance, 'La Folie par excellence', _Critique,_ 45, February 1951, 99–118 ( _BR,_ 110–28); 'L'Itinéraire de Hölderlin' ( _EL,_ 283 92; 269–76); 'Le Livre à venir' ( _LV_ , 271–97); and 'Le Grand Refus' ( _EI,_ 46–69; 33–48).
Friedrich Hölderlin, _Werke und Briefe,_ I, 143; _Poems and Fragments,_ 389.
The gesture is of course crucial to Heidegger's account of the work of art as such; compare for instance the remarks on the translation of Greek into Latin in the opening section of 'Der Ursprung des Kunstwerks', _Holzwege_ (Frankfurt a. M., Klostermann, 1950), 7–8 ('The Origin of the Work of Art', _Basic Writings,_ 149).
'The thinker speaks Being. The poet names the sacred' ('Der Denker sagt das Sein. Der Dichter nennt das Heilige'), Heidegger puts it; see Martin Heidegger, _Wegmarken, Gesamtausgabe,_ 9 (Frankfurt a. M., Klostermann, 1976), 312.
Friedrich Hölderlin, _Werke und Briefe,_ II, 671.
See Martin Heidegger, _Sein und Zeit_ (Tübingen, Niemeyer, 1986), §§ 50–3, 249–67.
I am quoting here from the revised version of the essay given in _La Part du feu;_ in the original 1946 text, instead of 'il se laisserait apaiser', Blanchot had written: 'il se _laisse_ apaiser' (see 'La Parole "sacrée" de Hölderlin', _Critique,_ 7, December 1946, 579–96 (592), emphasis mine). This shift from indicative to conditional, from the constative to the alleged, which is echoed elsewhere in the text, denotes an unmistakable hardening of Blanchot's critical stance towards Heidegger. Sad to say, like much else in the text, the nuance receives no acknowledgement from Blanchot's translator on this occasion, Charlotte Mandell.
Interestingly enough, Blanchot treats Heidegger's sentence in this way on each occasion he cites it, which he does at least twice more in his work. See 'La Folie par excellence', _Critique,_ 45, February 1951, 106 ( _BR,_ 115–16); and 'Le Grand Refus' ( _EI,_ 51–2; 37). In this last instance, Blanchot is evidently quoting from memory or from his own previous text.
Martin Heidegger, _Holzwege,_ 48, _Basic Writings,_ 186–7.
Blanchot, most likely quoting from his own earlier work, attributes the lines to Hölderlin for a second time in 'Énigme', _Yale French Studies,_ 79, 1991, 5–7 (7).
Friedrich Hölderlin, _Werke und Briefe,_ I, 148; Michael Hamburger translates the line somewhat differently as: 'A mystery are those of pure origin', _Poems and Fragments,_ 411.
The gesture is the same as in 1959 when Blanchot, invited to contribute to a Festschrift for Heidegger, supplied a fragment of what was later to become _L'Attente L'Oubli;_ see 'L'Attente', _Martin Heidegger zum siebzigsten Geburtstag_ (Pfullingen, Verlag Günter Neske, 1959), 217–24 ( _BR,_ 272–8).
### The limitlessness of the limit
For a later formulation of the structure of the limitless of the limit, see Maurice Blanchot, 'La Voix narrative (le "il", le neutre)' ('The Narrative Voice (the "he", the neutral)'), _EI,_ 556–67; 379–87.
On the title(s) of Blanchot's _récit,_ see Jacques Derrida, _Parages_ (Paris, Galilée, 1986) ('Living On', translated by James Hulbert, in Harold Bloom _et al., Deconstruction and Criticism_ (London, Routledge and Kegan Paul, 1979), 75–176). Derrida reproduces the original cover, table of contents, and first page of the story in _Parages,_ 132–4.
Jacques Derrida, _Parages,_ 143; _Deconstruction and Criticism,_ 97–8.
Philippe Lacoue-Labarthe, _L'Imitation des modernes,_ 64–5; _Typography: Mimesis, Philosophy, Politics, 231–2._
It is now known, of course, from the account given in _L'Instant de ma mort,_ that the scene of confrontation with death that features in _La Folie du jour_ had a particular autobiographical relevance for Blanchot. Interestingly, the same word – 'allégresse', or gaiety – is used in both texts to describe the experience ( _FJ,_ 12; 7; and IM, 10).
See Friedrich Hölderlin, 'Anmerkungen zum Oedipus', _Werke und Briefe,_ II, 736 ('Remarks on _Oedipus', Essays and Letters on Theory,_ translated by Thomas Pfau (Albany, NY, State University of New York Press, 1988), 108). Hölderlin describes the principle of this idea of return in a famous letter to Böhlendorff (4 December 1801) in _Werke und Briefe,_ II, 940–2; _Essays and Letters on Theory,_ 149–51. Blanchot's own commentary, which draws extensively on the work of Beda Allemann, appears in _L'Espace littéraire_ ( _EL,_ 284–7; 270–2).
On the Hölderlinian 'caesura', see Philippe Lacoue-Labarthe, _L'Imitation des modernes,_ 39–69, _Typography: Mimesis, Philosophy, Politics,_ 208–35. As Hölderlin writes in explanation, in the 'Remarks on _Oedipus',_ in a passage that is already almost a reading of _La Folie du jour:_
Indeed, tragic _transport_ is properly empty, and it is the most unbound. As a result, in the rhythmic sequence of representations in which _transport_ presents itself, there becomes necessary _what in prosody is called a caesura,_ the pure word, the counter-rhythmic interruption [das reine Wort, die gegenrhythmische Unterbrechung], so as to meet the surging alternation of representations at their apogee, with the result that what then appears is not the alternation of representation, but representation itself [nicht mehr der Wechsel der Vorstellung, sondern die Vorstellung selber].
( _Werke und Briefe,_ II, 730; E _ssays and Letters on Theory,_ 101–2, translation modified)
## Chapter 3: Writing the neuter
### From work to worklessness
In his opening speech in the first scene of the fragmentary third version of the play, Empedocles affirms: 'For death is what I seek. It is my right' ('Denn sterben will ja ich. Mein Recht ist dies'). See Friedrich Hölderlin, _Werke und Briefe,_ II, 551; _Poems and Fragments,_ 327.
See G. W. F. Hegel, _Werke,_ Eva Moldenhauer and Karl Markus Michel (eds), 20 vols (Frankfurt a. M., Suhrkamp, 1970), III, 294–311; _Phenomenology of Spirit,_ translated by A. V. Miller (Oxford, Clarendon Press, 1977), 237–52.
See Alexandre Kojève, _Introduction à la lecture de Hegel,_ Raymond Queneau (ed.) (Paris, Gallimard, 1947). Kojève writes that 'Hegel's "dialectical" or anthropological philosophy is, in the last analysis, _a philosophy of death_ (or what amounts to the same thing: of atheism)' (539). For a detailed account of Blanchot's use of Kojève, see Anne-Lise Schulte Nordholt, _Maurice Blanchot: l'écriture comme expérience du dehors_ (Geneva, Droz, 1995), 31–57. Like several other commentators, however, Schulte Nordholt falls into the trap of attempting to systematise Blanchot's text. This leads to some distortion; despite claims to the contrary, for instance, Blanchot in 'La Littérature et le droit à la mort', does _not_ hold to the view that language may be divided into two modes, functional and essential; indeed, no sooner is the dichotomy introduced than Blanchot demonstrates its profound chiasmic instability.
For Kojève's commentary on the 'spiritual animal kingdom', see Alexandre Kojève, _Introduction à la lecture de Hegel,_ 90–4; Bataille's letter to Kojève, including further unpublished material is in Georges Bataille, _Œuvres complètes V_ (1973), 369–71 and 562–5. Bataille refers explicitly to the concept of the Tierreich in a passage from the letter omitted from the published version (see 564).
Alexandre Kojève, _Introduction à la lecture de Hegel,_ 91.
Alexandre Kojève, _Introduction à la lecture de Hegel,_ 93–4.
Georges Bataille, _Œuvres complètes,_ VI (1973), 416.
See Georges Bataille, _Œuvres complètes,_ V, 289, 369–71, 562–3. Blanchot returns to the concept of 'négativité sans emploi' in a later essay on Bataille. See his 'L'Affirmation et la passion de la pensée négative' ( _EI,_ 300–13; 202–11).
Jean-Paul Sartre, _Qu'est-ce que la littérature?_ (Paris, Gallimard: folio, 1948); _What is Literature?,_ translated by Bernard Frechtman (London, Methuen, 1950). The essays collected in the volume were first published in _Les Temps modernes_ between February and July 1947.
Jean-Paul Sartre, _Qu'est-ce que la littérature?,_ 27; _What is Literature?,_ 12.
It was largely in these terms, of course, that Sartre had attacked Bataille in his 1943 review of _L'Expérience intérieure._ See Jean-Paul Sartre, 'Un nouveau mystique', _Situations,_ I, 174–229.
See G. W. F. Hegel, _Werke,_ III, 296–8; _Phenomenology of Spirit,_ 239–40. The passage, to which Blanchot returns on a number of other occasions, is paraphrased towards the beginning of 'La Littérature et le droit à la mort' ( _PF,_ 295–7; 302–5).
See Rodolphe Gasché, 'The Felicities of Paradox. Blanchot on the Null-Space of Literature', in _Blanchot: the Demand of Writing,_ 34–69. Blanchot gives his own explanation of the strategy behind his deconstructive exacerbation of the Hegelian dialectic in a series of fragments published in _L'Écriture du désastre_ ( _ED,_ 100–1, 118–20; 61, 72–4). On Blanchot's reading of the _Phenomenology,_ see also Andzej Warminski, _Readings in Interpretation: Hölderlin, Hegel, Heidegger,_ 183–91. As Blanchot observes in _L'Écriture du désastre,_ in a discussion of Hegel's two deaths, conceptual and natural:
Indeed, let us recall Hegel's earliest work. He too, even before what is termed his early philosophy, considered that the two deaths could not be dissociated, and that only the fact of confronting death, not merely facing it or exposing oneself to its danger (which is what characterises heroic courage), but of entering into its space, of undergoing it as an infinite death and, also, mere death, 'natural death', could found sovereignty and mastery: spirit and its prerogatives. The result was perhaps absurdly that what set the dialectic in motion, the inexperienceable experience of death, was what immediately brought it to a halt, a halt of which the entire subsequent process retained a sort of memory, as of an aporia that was necessarily always part of the equation.
( _ED,_ 111–12; 68)
The gesture is not without recalling Blanchot's robust rebuttal of Hegel's anti-Judaism, even anti-semitism, in _L'Écriture du désastre_ ( _ED,_ 104; 63).
Interestingly, while Blanchot in his footnote attributes the description of the _il y a_ to Levinas and refers the reader to _De l'existence à l'existant,_ Levinas, for his part, in a footnote in that self-same book (103; _Existence and Existents,_ 63), attributes the description of the _il y a_ to Blanchot and refers the reader to the second chapter of _Thomas l'Obscur._ To speak of the _il y a_ at all, it seems, is always already to subscribe to a principle not of priority, originality, or propriety, but of repetition, endebtedness, alterity, and return.
For a wide-ranging account of some of the implications of Blanchot's use of the _il y a,_ see Simon Critchley, ' _Il y a_ – A Dying Stronger than Death (Blanchot with Levinas)', _Oxford Literary Review,_ 15, 1–2, 1993, 81–131.
It is true that later accounts by both Levinas and Blanchot claim that the thought of the _il y a_ arose entirely independently of their reading of Heidegger. My claim here is not a historical one; it is simply to emphasise, whatever its origins, to what extent the _il y a_ necessarily comes to function as a reworking and displacement of Heideggerian Being.
The passage from the Preface to the _Phenomenology_ to which Blanchot refers here is as follows:
Death, if that is what we want to call this non-actuality [jene Unwirklichkeit], is of all things the most dreadful [das Furchtbarste], and to hold fast what is dead requires the greatest strength. Lacking strength, Beauty hates the Understanding for asking of her what it cannot do. But the life of Spirit is not the life that shrinks from death and keeps itself untouched by devastation, but rather the life that endures and maintains itself in it [Aber nicht das Leben, das sich vor dem Tode scheut und vor der Verwüstung rein bewahrt, sondern das ihn erträgt und in ihm sich erhält, ist das Leben des Geistes]. It wins its truth only when, in utter dismemberment [in der absoluten Zerrissenheit, what Blanchot translates as 'la déchirure de l'homme'], it finds itself. It is this power [Macht], not as something positive, which closes its eyes to the negative, as when we say of something that it is nothing or is false, and then, having done with it, turn away and pass on to something else; on the contrary, spirit is this power only by looking the negative in the face, and tarrying with it. This tarrying with the negative is the magic power [die Zauberkraft] that converts it into being [in das Sein]. This power is identical with what we earlier called the Subject [das Subjekt], which by giving determinateness an existence in its own element supersedes [aufhebt] abstract immediacy, i.e. the immediacy which barely is, and thus is authentic substance: that being or immediacy whose mediation is not outside of it but which is this mediation itself.
(G. W. F. Hegel, _Werke_ , III, 36; _Phenomenology of Spirit,_ 19)
In this respect, dying in Blanchot has many similarities – as well as important differences – with ecstasy as articulated by Bataille: i.e. not a fusion of subject with object, but rather an interval or interruption in subjectivity, language, time itself; as Blanchot observes:
One can write the word (ecstasy) only by putting it carefully between quotation marks, because nobody can know what this is, and first whether it ever took place: going beyond knowledge, implying non-knowledge, it refuses to be affirmed other than through chance words that cannot in any way guarantee it. Its decisive trait is that the one who experiences it is no longer there when he experiences it, is thus no longer there to experience it.
( _CI,_ 36–7; 19)
I examine the question of quotation marks and their relationship with the neutre in Blanchot later in this chapter.
### The worklessness of being
On the relation between the _il y a_ and hypostasis, see Levinas, _De l'existence à l'existant_ , 107–45 ( _Existence and Existants,_ 65–85); and _Le Temps et l'autre_ (Paris, Presses universitaires de France [1948], 1983), 31–4; also John Llewelyn, _Emmanuel Levinas, the Genealogy of Ethics_ (London, Routledge, 1995), 21–30.
### Outside
G. W. F. Hegel, _Werke,_ XIII, 25.
See Heidegger, 'Der Ursprung des Kunstwerks', in _Holzwege,_ 1–72 (the Afterword appears on 65–7), _Basic Writings,_ 143–212 (204–12). Heidegger's essay, though first published in 1950, was completed in 1936, making it exactly contemporary with the first of Heidegger's published essays on Hölderlin discussed earlier. Also worth mentioning here, and contained in the same volume ( _Holzwege,_ 265– 316), is Heidegger's essay of 1946 on Rilke, 'Wozu Dichter?' ('What are Poets For?') in _Poetry, Language, Thought,_ translated by Albert Hofstadter (London, Harper and Row, 1971), 91–142, which takes its title from Hölderlin's famous line in the elegy 'Brot und Wein': 'wozu Dichter in dürftiger Zeit?' ('Wherefore poets in time of distress?') which Heidegger had already cited, in 1936, in conclusion to 'Hölderlin und das Wesen der Dichtung' ( _Erläuterungen zu Hölderlins The Truth in Painting, Dichtung,_ 47–8; _Existence and Being,_ 315). For convenience, I follow Blanchot (and Ann Smock in _The Space of Literature_ ) in translating Hölderlin's 'in dürftiger Zeit' as 'in time of distress'. There are many other possible translations of 'dürftig', including: destitute, needy, wretched, miserable, meagre, etc.
Reliance on the earth is of course a central theme in 'Der Ursprung des Kunstwerks', and it is one that, as Derrida has shown, has some surprising implications for Heidegger's treatment of van Gogh's paintings of 'a pair of peasant shoes'. See Jacques Derrida, _La Vérité en peinture_ (Paris, Flammarion, 1977), 291–436 ( _The Truth in Painting_ , translated by Geoffrey Bennington and Ian McLeod (Chicago, University of Chicago Press, 1987), 255–382). As long ago as 1940, if not indeed before, disquiet at the political implications of Heidegger's peasant reliance on the earth was one major reason why Levinas's thinking was governed, as he puts it, by 'a profound need to leave the climate of that philosophy' ( _De l'existence à l'existant,_ 19; _Existence and Existents,_ 19); and it is arguably to a similarly disturbing valorisation of motifs such as that of the earth that Blanchot is alluding here.
See 'Hölderlin und das Wesen der Dichtung', _Erläuterungen zu Hölderlins Dichtung,_ 47, _Existence and Being,_ 313.
'Der Ursprung des Kunstwerks', _Holzwege,_ 66, _Basic Writings,_ 205.
See '"Wie wenn am Feiertage . . ."', _Erläuterungen zu Hölderlins Dichtung,_ 51.
As Heidegger puts it in his opening remarks in the essay 'Wozu Dichter?':
The nearer it reaches midnight in the night of the world, the more exclusively that which is distressing [das Dürftige] prevails in that it withdraws its essence. Not only does the sacred as the trace [die Spur] of divinity go missing, but even the traces of this lost trace [die Spuren zu dieser verlorenen Spur] have become virtually extinguished [ausgelöscht].
( _Holzwege,_ 268; _Poetry, Language, Thought,_ 94–5).
As Blanchot would rejoin, a quarter of a century later: 'All must be effaced, all will be effaced. Writing takes place and takes its place in accordance with the infinite demand of effacement' ('Tout doit s'effacer, tout s'effacera. C'est en accord avec l'exigence infinie de l'effacement qu'écrire a lieu eta son lieu') ( _PA,_ 76; 53).
Compare Heidegger's assertion that: 'Each answer remains in force as an answer only so long as it is rooted in questioning' ('Jede Antwort bleibt nur als Antwort in Kraft, solange sie im Fragen verwurzelt ist') ( _Holzwege,_ 57; _Basic Writings,_ 195).
As Blanchot puts it in a paper belonging to the same period, which retraces much of the same argument:
Whoever affirms literature in itself affirms nothing. Whoever seeks it seeks only that which slips away; whoever finds it finds only what falls short of literature or, even worse, what lies beyond it. This is why, in the end, it is non-literature that each book pursues as the essence of what it loves and yearns passionately to discover.
('La Disparition de la littérature', _LV,_ 244; _BR,_ 142).
'Die Wahrheit ist in ihrem Wesen Un-wahrheit' ('Truth, in its essence, is un-truth'), ( _Holzwege,_ 40, _Basic Writings,_ 179).
'Der Ursprung des Kunstwerks', _Holzwege,_ 61, _Basic Writings,_ 200. Heidegger's word 'das Un-geheure', which Albert Hofstadter translates here as 'awesome', is the term lying hidden in Blanchot's 'monstrueux' (monstrous) in the passage that follows.
Levinas's account of the differences between Heidegger and Blanchot is characteristically astute: 'the move that dominates the later philosophy of Heidegger', he writes in a 1956 review of _L'Espace littéraire:_
consists in interpreting the essential forms of human activity – art, technology, science, economy – as modes of truth (or its forgetting). That the march towards the truth, the response to the call, involves for Heidegger taking paths that lead one astray, and that error is contemporary with truth, and that the revelation of being is also its concealment, all points to the existence of a very close relationship between the Heideggerian notion of being and the making real of the unreal, the presence of absence, the existence of nothingness that is what, in Blanchot's view, the work of art or the poem voices. But for Heidegger the truth – a primordial unveiling – is the condition of all wandering, which is why the whole of the human can be finally spoken of in terms of truth and be described as an 'unveiling of being'. In Blanchot, _the work discloses, in a disclosure that is not truth,_ a darkness. A disclosure that is not truth! What a strange way of disclosing and seeing the 'content' which its formal structure determines: a darkness that is absolutely exterior, on which no purchase is possible. Like in a desert one finds no residence. From the depths of sedentary existence a nomadic memory arises. Nomadism is not an approach of the sedentary state. It is an irreducible relationship with the earth: a residence without _place._ Faced with the night to which art calls it back, as in the face of death, the 'I', as support of power, dissolves into the anonymous 'one', across a land of pilgrimage. The Self of a timeless Nomad, grasping itself in its displacement and not its place, on the borders of non-truth, the realm that spreads further than the true.
(Emmanuel Levinas, _Sur Maurice Blanchot_ (Montpellier, Fata morgana, 1975), 21–2)
Blanchot develops his criticism of Heidegger's negative view of anonymity and impersonality in a remarkable reading of Louis-René Des Forêts's novel, _Le Bavard,_ 'La Parole vaine', in _L'Amitié_ (A, 137–49, esp. 145–6).
'Philosophy', claimed Heidegger in 1966,
will not be able to effect an immediate transformation of the present condition of the world. This is not only true of philosophy, but of all merely human thought and endeavour. Only a god can still save us. The sole possibility of salvation that is left to us is to propose a sort of readiness, through thinking and poeticizing, for the appearance of the god or for the absence of the god in the time of decline; so that, crudely put, we do not simply rot away [daß wir nicht, grob gesagt, 'verrecken'] but, if we do perish, do so in the face of the absent god [im Angesicht des abwesenden Gottes].
( _Antwort. Martin Heidegger im Gespräch,_ Günther Neske and Emil Kettering (eds) (Neske, Pfullingen, 1988), 99–100; _The Heidegger Controversy: a Critical Reader,_ Richard Wolin (ed.), 107, translation modified)
Some pages later, Heidegger glosses the readiness to which he refers here, as ever, with an invocation of Hölderlin as 'the poet who points into the future, expects god and thus ought not to remain solely an object of Hölderlin Studies in the literary historical sense' (106; 112). Though Blanchot might plausibly concur with this last sentiment, it is clear from these lines that in every other respect he develops a radically different understanding of the future to that proposed by Heidegger.
### Neutre
'René Char et la pensée du neutre', _L'Arc,_ 22, Summer 1963, 9–14 (11–12). The covert citation contained in the last sentence of this passage, as readers of the chapter 'Parler, ce n'est pas voir' will recall ( _EI,_ 44; 31), is from Heraclitus, Fr. 93: 'The lord whose oracle is in Delphi neither speaks out nor conceals, but gives a sign', according to the translation proposed by G. S. Kirk, J. E. Raven, and M. Schofield, _The Presocratic Philosophers,_ second edn (Cambridge, Cambridge University Press, 1983), 209.
These particular changes were first brought to attention in 1973 by Roger Laporte in his essay 'Une Passion', in Roger Laporte and Bernard Noël, _Deux lectures de Maurice Blanchot,_ 142–3. Prefacing a later, revised version of the essay, Laporte tells how he came to regret the tone of his earlier remarks on Blanchot's rewriting and has since refused to allow the 1973 version of the essay to be reprinted. It was at Blanchot's suggestion, however, Laporte states, that in 1983, in his turn, like Blanchot with _Thomas l'Obscur,_ he came to rewrite the essay under the title: 'Une Passion (nouvelle version)'; see Roger Laporte, _A l'extrême pointe: Bataille et Blanchot_ (Montpellier, Fata morgana, 1994), 34–53. To write at all, it seems, for both Blanchot and Laporte, is to be condemned to the inescapable necessity of rewriting; which is simply to say, of course, that writing itself is always already a rewriting, a perpetual redrafting of what is necessarily without beginning or end. As Blanchot writes in _Le Pas au-delà,_ 'to write, in this sense, is first always to rewrite, and to rewrite does not refer to any previous writing, any more than to an anteriority of speech or of presence or of signification' ( _PA ,_ 48; 32).
See Roger Laporte, _A l'extrême pointe: Bataille et Blanchot,_ 41. As far as the early work of Derrida is concerned, the key date is 1967, which saw the original publication of _Grammatology, Speech and Phenomena,_ and _Writing and Difference._ Blanchot himself writes of his debt to Derrida in the essay 'Grâce (soit rendue) à Jacques Derrida', _Revue philosophique,_ 2, April–June 1990, 167–73 _(BR,_ 317– 23). Derrida's influence on (at least) Blanchot's vocabulary is evident from the changes made to some of the articles of the late 1950s or early to mid-1960s in the versions published in _L'Entretien infini_ and _L'Amitié;_ alongside the withdrawal of terms like 'présence', for instance, Blanchot also has more extensive recourse to words such as 'écriture' or 'différence', using them in their largely Derridean sense. One example may suffice: in the initial version of 'La Voix narrative' (first published October 1964) the following sentence appeared: 'And narrative may be said to be nothing other than an allusion to the initial detour that is borne by speech, that carries it away and causes us, in speaking, to yield to a sort of perpetual turning aside'; but when the passage reappeared in _L'Entretien infini,_ the two references to speech and speaking, 'la parole' and 'parlant', had been deleted, replaced, respectively, by 'l'écriture' and 'écrivant' ( _EI,_ 564; 385). Admittedly, as Derrida points out, the phenomenon was a general one in France at the time, and Blanchot was far from the only writer to revise his texts in this way; see Jacques Derrida, _Résistances de la psychanalyse_ (Paris, Galilée, 1996), 80. While the responsiveness of Blanchot to Derrida is beyond question, Derrida's own debt to Blanchot, as Derrida has been the first to make clear, is difficult to underestimate; indeed, much of the principle – if not the detail – of Derrida's account of phonocentrism in _De la grammatologie_ in 1967 is already contained in the opening pages of Blanchot's essay on Char, 'La Bête de Lascaux', first published in 1953.
Elsewhere in Blanchot's writing there is ample evidence of a similar endeavour to reinvent the implications of the term 'presence'. Witness for instance in _Le Dernier Homme_ of 1957 the following description of the narrator's friend and associate:
He was present in such a strange manner: so completely and so incompletely. When he was there, I could not help coming up against his self-effacement, which made his approach even more clumsy, cruelly disproportionate: maybe insignificant, maybe dominating. As though his presence was all there was of him and did not allow him to be present: it was an immense presence, and even he did not seem able to fill it, as though he had disappeared into it and had been absorbed by it slowly, endlessly – a presence without anyone there perhaps [une présence sans personne peut-être]? . . . His presence and not the idea of his presence. It seemed to me this presence destroyed all idea of itself, that I couldn't even have a false idea of it.
( _DH,_ 50–1; 28)
'Neutre' in Blanchot's text is not an easy word to translate. Of the possibilities available in English, the term 'neuter' is normally taken to mean: neither masculine nor feminine; as for the word 'neutral' (the translation preferred by Susan Hanson in _The Infinite Conversation_ ), this is mainly used to refer to impartiality or the refusal to engage in conflict (neutrality, in addition, has the disadvantage of being a traditional Latinate concept). The difficulty with Blanchot's 'neutre' is that while it may encounter some of these meanings, it is irreducible to them all; it is a term that, while not single in itself, nevertheless resists generality, which is why, for my part, in this discussion and subsequently in this book, as though using a strange proper name, I have preferred to call it: the neutre.
See Emmanuel Levinas, _Totalité et infini_ (Paris, Le Livre de poche, 1990), 332 ( _Totality and Infinity,_ translated by Alphonso Lingis (Dordrecht, Kluwer, 1991), 298). On Levinas's (mis)reading of Heideggerian Sein as neutrality, see Jacques Derrida, 'Violence et métaphysique' in _L'Écriture et la différence_ (Paris, Seuil, 1967), 200–6 ( _Writing and Difference,_ 136–40).
See Maurice Blanchot, 'L'Étrange et l'étranger', _La Nouvelle Revue française,_ 70, October 1958, 673–83 (681). Unusually, this article by Blanchot is one of the few not to have been incorporated by him in a subsequent collection. First published in _Revue de métaphysique et de morale,_ 3, 1957, 241–53, 'La Philosophie et l'idée de l'infini' is reproduced in Emmanuel Levinas, _En découvrant l'existence avec Husserl et Heidegger,_ third edn (Paris, Vrin, 1974), 165–78 ( _Collected Philosophical Papers,_ translated by Alphonso Lingis (Dordrecht, Martinus Nijhoff, 1987), 47–59).
As Levinas himself concedes in 1971, in an interview with André Dalmas, in answer to a question about the difference between Heidegger and Blanchot on the subject of the neuter:
As the distance of After death 'with respect to being', which cannot be arrived at by simple negation, Blanchot's Neutre is foreign to the world – with a foreignness beyond all foreignness, with an 'exponent', so to speak. Further than any God. Quite different to Heidegger's _Sein_ – which is anonymous, too – which (different from beings, as you rightly point out) is the very luminosity of the world, place, landscape, peace.
(Emmanuel Levinas, _Sur Maurice Blanchot,_ 49)
Emmanuel Levinas, _Sur Maurice Blanchot,_ 52.
'Comment découvrir l'obscur?', _La Nouvelle Revue française,_ 83, November 1959, 867–79 (876).
Martin Heidegger, 'Moira (Parmenides VIII, 34–41)', in: _Vorträge und Aufsätze_ (Pfullingen, Neske, 1954), 243–4.
The question arises here as to the relationship between the _il y a_ and the neutre in Blanchot, and whether this is best portrayed as one of surreptitious continuity or decisive opposition. It is true, on the one hand, that the _il y a_ is formulated primarily by Blanchot (and Levinas) as the thought of the presence of the absence of being and of the inescapability of being or its irreducibility to negation, all of which would seem to put the _il y a_ decidedly within the realm of ontology and at some distance from the neutre. But, on the other hand, to the extent that the _il y a_ ultimately deprives being of all proper foundation and undermines the distinction between being and non-being, it would seem that the _il y a_ is in fact already pointing beyond being, and thus not only towards the thought of the Other, but towards the thought of literature as well, and in that case it may be argued that the _il y a,_ as an instantiation of the neutre, has already begun to suspend the whole question of being or non-being. It is at any rate in terms such as these that Blanchot, in a number of later retrospective texts, tends to describe the _il y a._ 'Above all', he writes for instance in _L'Écriture du désastre,_ 'the _there is_ [l' _il y a_ ], as neuter [en tant que neutre], eludes the question that relates to it: if asked, it ironically absorbs the asking, which has no priority over it' ( _ED,_ 108; 65).
Much the same emphasis is of course found in Levinas; compare for instance, _Totalité et infini,_ 13 ( _Totality and Infinity,_ 28). It might be argued that Blanchot's criticism of the optical metaphor can find an easy target in much of Blanchot's own earlier work, where, as we have seen, the figure of the light of day in particular plays a major role. The question is one Blanchot raises himself, albeit obliquely, in an inconclusive exchange between two of the interlocutors in 'Parler, ce n'est pas voir', when one speaker voices his faith in the idea that 'in the speaking that responds to waiting, there is a manifest presence that does not belong to the day, a disclosure that discloses before any _fiat lux,_ disclosing the darkness by the detour that is the essence of darkness'. To which his partner replies: 'In spite of all your efforts, when speaking of the darkness, to avoid us having to refer to the light, I cannot help but relate everything you say back to the day as the sole measure' ( _EI,_ 43; 31).
In the version of this sentence given by Susan Hanson in _The Infinite Conversation,_ the translator unfortunately misconstrues this last clause which she renders as: 'a possibility of saying that would say being without saying it, and yet without denying it either'. This is to have the phrase say the reverse of what it means; oddly, when faced with an identical statement elsewhere in _L'Entretien infini_ ( _EI,_ 104; 73), Hanson translates it accurately!
### July 1948: writing, dying; dying, writing
The appearance of the two volumes was accompanied by two notices in Gallimard's NRF Bulletin for July 1948 (No. 13, 1–2); the second of these, devoted to _Le Très-Haut,_ began as follows:
'These two books,' writes Maurice Blanchot, ' _Le Très-Haut_ and _L'Arrêt de mort,_ that have appeared simultaneously have no doubt nothing in common. But it seems to me, having written them, that one is present so to speak behind the other, not like two texts implicated within each other, but like two divergent and yet concordant versions of a same reality that is equally absent from both.'
('Il n'y a,' écrit Maurice Blanchot, 'sans doute rien de commun entre ces deux livres, _Le Très-Haut, L'Arrêt de mort,_ qui paraissent en même temps. Mais, à moi qui les ai écrits, il me semble que l'un est comme présent derrière l'autre, non comme deux textes impliqués, mais comme les deux versions inconciliables et cependant concordantes d'une même réalité, également absente de toutes deux.')
Maurice Blanchot, 'Le Chant des sirènes', _La Nouvelle Revue française,_ 19, July 1954, 95–104. Among the most influential earlier proponents of dais distinction between _roman_ and _récit_ in France was André Gide, for whom a _roman_ required a large cast of characters, a wealth of incident, a multiplicity of different narrative viewpoints, and a third-person narrative voice, while a _récit_ entailed instead a limited number of characters, few events, restricted narrative point of view, and narration in the first person.
This is by way of the phrase, 'la reine-mère' ('the Queen Mother'), used in _Le Très-Haut_ by the narrator's sister, Louise, to refer to their mother ( _TH,_ 59; 55), and repeated in _L'Arrêt de mort_ by J.'s sister, also called Louise, to refer also to their mother ( _AM,_ 12; 4). It may be remembered that the main female character of 'L'Idylle' is also named Louise.
The phrase, used in _Le Très-Haut_ to describe the image in the dilapidated tapestry in Louise's room, is not without recalling the phrase, 'l'ancien, l'effroyablement ancien', referring to the temporality of narrative and myth as such, that, as Roger Laporte has recorded, recurs no less than seven times in Blanchot's essays and _récits;_ see Roger Laporte, _Études_ (Paris, POL, 1990), 11–50. It is worth noting perhaps that the scene with Louise ends with the narrator saying to her: 'Viens donc. . . . Viens!' ( _TH_ , 56; 53).
Among the more striking or controversial contributions to the debate, see Pierre Madaule, _Une tâche sérieuse?_ (Paris, Gallimard, 1973) and _Véronique et les chastes_ (Dijon, Ulysse fin de siècle, 1988); Bernard Noël, 'D'une main obscure', in _Deux lectures de Maurice Blanchot_ (Montpellier, Fata morgana, 1973), 9–47; Jacques Derrida, 'Living On', in: _Deconstruction and Criticism,_ 75–176, and in _Parages,_ 119–218; Jeffrey Mehlman, _Genealogies of the Text,_ 82–96; Edmond Jabès, _Dans la double dépendance du dit_ (Montpellier, Fata morgana, 1984), 65–6; J. Hillis Miller, _Versions of Pygmalion_ (London, Harvard University Press, 1990), 179–210; Marie-Claire Ropars-Wuilleumier, _Écraniques: le film du texte_ (Paris, Presses universitaires de Lille, 1990), 33–55; and Michael Holland, 'Rencontre piégée: "Nadja" dans _L'Arrêt de mort',_ in _Violence, théorie, surréalisme,_ Jacqueline Chénieux-Gendron and Timothy Mathews (eds) (Pleine marge, 3, Paris, Lachenal & Ritter, 1994), 117–38. It is worth noting that in 1971 _L'Arrêt de mort,_ like many other titles by Blanchot, was reissued by Gallimard; as well as adopting a new format and different pagination, this reprint also deleted the appellation _récit and_ the short third section of the narrative (which is retained in Lydia Davis's 1978 English translation). Unless otherwise indicated, all references to the French text will be to this 1971 edition.
The ellipses regarding proper names in _L'Arrêt de mort_ are perhaps what are most reminiscent of Constant's _Adolphe,_ to which Blanchot had devoted an essay in October 1946, in which one also reads:
The key element in Constant's tragedy is that he is exposed, in its pure state, with all the acuteness of a singular sensitivity, to the following paradox: that we have relations with others only if we are not as one with others, and communicate fully with someone only by possessing not what they are but what separates us from them, their absence rather than their presence, in other words, the infinite movement towards overcoming and thus regenerating this absence.
( _PF,_ 229; 234–5)
The words, of course, clearly have some bearing on not only _L'Arrêt de mort_ but Blanchot's later _récits_ as well.
The references to Munich are of course what has prompted Jeffrey Mehlman, in _Genealogies of the Text_ (82–96), to jump to the conclusion that _L'A rrêt de mort_ is an allegorical portrayal of Blanchot's renunciation of his alleged commitment to French fascism. Mehlman supports this contention by reminding the reader that Munich represented an important turning point for the French extreme right, which was now forced to choose between its commitment to French nationalism and its enthusiasm for German fascism. It is indeed true that for Blanchot, as for many of his contemporaries, the year 1938 was a crucial moment, and in Blanchot's case it does seem to have marked the moment of his abandonment of revolutionary nationalism, and his decision, in spite of all, to rally to the cause of the Republic. That particular shift, however, to judge from the evidence of Blanchot's published texts, took place much earlier in the year, well before the Autumn of 1938; in any case, Blanchot had long been a fierce opponent of the policy of appeasement that culminated in Munich, and his own opposition to Munich, when it came, was robustly unambiguous. As for the temptation of fascism, it is clear that on this score, too, Blanchot's mind had been made up for several years, since at least 1933; but, regrettably, this is a feature of the historical record Jeffrey Mehlman in his reading of _L'Arrêt de mort_ is plainly unwilling to acknowledge.
On the dating of this scene, see Pierre Madaule, _Véronique et les chastes,_ 119.
Michael Holland, 'Rencontre piégée: "Nadja" dans _L'Arrêt de mort',_ 137.
On Laure's dying words, see Georges Bataille, _Œuvres complètes,_ V, 512. Bataille's account of Laure's death was originally intended to be part of _Le Coupable,_ but was deleted by Bataille shortly before publication; Colette Peignot's own writings are now collected in Laure, _Écrits, fragments, lettres,_ Jérôme Peignot (ed.) (Paris, Pauvert, 1977). The recurrence of the 12 October in _L'Arrêt de mort_ and Bataille's account of Laure's death, together with the repeated motif of the rose, are among the elements discussed by Derrida in his reading of _L'Arrêt de mort;_ see _Parages,_ 192–208, _Deconstruction and Criticism,_ 148–64.
See Pierre Madaule, _Véronique et les chastes,_ 119. The narrator comments: 'These comings and goings [ces péripéties], at a time like this, seem to me like the last grimace of the world' ( _AM,_ 118; 74). The words are significant ones; what they suggest is that personal actions are not reducible to world historical ones, and that if _L'Arrêt de mort_ does have a historical or political message, it is that in spite of history men and women can still perform honourable acts, and that in the face of a catastrophe such as Munich and the events that followed, the only viable response is to invoke the impossible, to refuse history, and affirm the limitlessness of life itself even into death. Witness the closing homage, paid to J., after her death, by the narrator: 'For myself,' he concludes, 'I see nothing important in the fact that this young woman, who was dead, returned to the living in response to my call, but I see an astounding miracle in her fortitude, and in her energy, which was great enough to make death impotent as long as she wanted' ( _AM,_ 52–3; 30).
The publisher's blurb for _L'Arrêt de mort_ given in Gallimard's NRF Bulletin for July 1948 (1–2), which Blanchot is more than likely to have written himself, confirms this date as follows: 'This [hi]story isn't a dream, and doesn't take place in any dream world; it began some years ago, on Wednesday 13 October; it happened amongst us, and it is possible it is still not over.' ('Cette histoire n'est pas un rêve, elle n'a pas lieu dans un monde de rêve; elle a commencé il y a peu d'années, le mercredi 13 octobre; elle s'est déroulée parmi nous, et il se peut qu'elle ne soit pas encore finie.')
Interestingly enough, perhaps, on 13 October 1943, Blanchot published in the _Journal des débats_ a book review, devoted mainly to Roger Caillois's _Puissances du roman and_ entitled 'Récits autobiographiques' ('Autobiographical Narratives'), the main contention of which was the uncertain character of the borderline between autobiography and fiction, indeed the merging of the two in many cases. 'Why', asks Blanchot, 'does one generally hesitate in considering autobiographical works as novels . . . ?'
Referring to this unspecified living proof of the events in the story, Blanchot's narrator first writes: 'preuve "vivante"' ( _AM,_ 9; 3), then, on the following page: 'cette "preuve"' ( _AM,_ 10; 3), and it is as though the use of quotation marks, like a silent intervention of the neutre, applied now to one word, then to the next, functions to dissipate the enigma by making it more enigmatic still: for here is 'living' 'proof' that is not living, nor is it proof of anything at all.
In this respect, the irremediable is perhaps simply another name for finitude (or death itself); this, at any rate, is what is implied by the story of the squirrel Blanchot's narrator uses to explain the word; he writes:
I once saw a squirrel caught in a cage hanging from a tree: he was about to cross the threshold with all the enthusiasm of his happiest of lives, but hardly had he landed on the board inside when the trigger clapped the door shut, and though he had not been hurt, and was still free, since the cage was huge, with a small pile of nuts inside, his jump froze at a stroke, and he stood there paralysed, struck in the back by the certainty that the trap had closed upon him.
( _AM,_ 65–6; 38–9)
A threshold, a confined space, leading to immobility, paralysis, and a sense of entrapment, together with a boundless appetite for freedom: these are the motifs with which one might perhaps begin to articulate a poetics of Blanchot's _récits;_ such a poetics, however, would be impossible, for there is no place from which to construe it: the reader, too, like the narrator of _Celui qui ne m'accompagnait pas,_ is condemned to remain perpetually 'nailed to the spot', 'cloué sur place' (CQ, 46; 23).
Blanchot makes use of this figure of the window pane, which is like the figure (without figure) of figurality itself, both in _Au moment voulu_ and in _Celui qui ne m'accompagnait pas,_ where it features as a way of suspending narrative and problematising questions of space and time, identity and alterity, repetition and singularity. The topos of the 'vitre' also recurs in a semi-autobiographical context in _L'Écriture du désastre_ ( _ED,_ 117; 72). On Blanchot's use of the motif in _L'Arrêt de mort,_ see Jacques Derrida, _Parages,_ 183–7, _Deconstruction and Criticism,_ 139–44; and J. Hillis Miller, _Versions of Pygmalion,_ 179–210.
## Chapter 4: The absence of the book
### Beyond philosophy?
Blanchot addresses _Totalité et infini_ explicitly in three consecutive chapters in _L'Entretien infini:_ 'Connaissance de l'inconnu' ('Knowledge of the Unknown'), 'Tenir parole' ('Keeping to Words') and 'Le Rapport du troisième genre' ('The Relation of the Third Kind') ( _EI,_ 70–105; 49–74). Of these, the first two essays originally appeared in the _Nouvelle Revue française_ for December 1961 and February 1962 respectively; albeit with numerous minor changes, both are reproduced in _L'Entretien infini_ in essentially the same form in which they were first published. In April 1962, Blanchot brought out a third, related essay, 'L'Indestructible', devoted in part to _L'Espèce humaine_ by Robert Antelme, most of the matter which now appears under the title 'L'Espèce humaine' ('Humankind') in a later section of _L'Entretien infini_ ( _EI,_ 191–200; 130–5). In place of the original third article on Levinas from April 1962, Blanchot incorporated into _L'Entretien infini,_ under the title 'Le Rapport du troisième genre', a chapter made up of two passages retained from 'L'Indestructible' and a significant amount of new material, presumably written shortly before publication in 1969 ( _EI,_ 94–9, 100–2, 103–5; 66–70, 70– 1, 72–4). Six years later, following the publication of Levinas's _Humanisme de l'autre homme_ (1972) and _Autrement qu'être, ou au-delà de l'essence_ ( _Otherwise Than Being,_ 1974), Blanchot also published a series of fragments on Levinas, under the title 'Discours sur la patience', in _Le Nouveau Commerce,_ 30–1, 1975, 19–44; these were later incorporated, with modifications, into _L'Écriture du désastre._ Blanchot's last essay, 'Notre compagne clandestine', appeared in a Festschrift for Levinas, _Textes pour Emmanuel Lévinas,_ François Laruelle (ed.) (Paris, Jean-Michel Place, 1980), 79–87 (translated by David B. Allison as 'Our Clandestine Companion', in _Face to Face with Levinas,_ Richard A. Cohen (ed.) (Albany, NY, State University of New York Press, 1986), 41–50). There are numerous other references to Levinas in Blanchot's texts of the 1970s and 1980s that are too extensive to list here.
See, for instance, Blanchot's personal tribute published in the special Levinas issue of _Exercices de la patience,_ 1, 1980, 67. On the philosophical relationship between the two, see Françoise Collin, _Maurice Blanchot et la question de l'écriture_ (Paris, Gallimard, 1971), and 'La Peur: Emmanuel Lévinas et Maurice Blanchot', _Cahiers de l'Herne: Emmanuel Levinas,_ Catherine Chalier and Miguel Abensour (eds) (Paris, L'Herne, 1991), 313–27; Joseph Libertson, _Proximity: Levinas, Blanchot, Bataille and Communication_ (The Hague, Martinus Nijhoff, 1982); Paul Davies, 'Difficult Friendship', _Research in Phenomenology,_ XVIII, 1988, 149–72; and 'A Fine Risk: Reading Blanchot Reading Levinas', in _Re-Reading Levinas,_ Robert Bernasconi and Simon Critchley (eds) (Bloomington and Indianapolis, Indiana University Press, 1991), 201–26; and Simon Critchley, _'Il y a –_ Holding Levinas's Hand to Blanchot's Fire', in _Maurice Blanchot: the Demand of Writing,_ Carolyn Bailey Gill (ed.), 108–22.
But if proximity is derived from distance in this way, Blanchot writes in _Le Pas au-delà,_ it still risks being thought within being as deferred presence; this is why, Blanchot contends, proximity is in fact perhaps the greatest distance of all, the true infinity beyond both presence and absence. If so, proximity and distance both need to be thought outside of being or non-being; Blanchot writes: 'Far and near are dimensions of what escapes presence as well as absence under the attraction of the [impersonal] "it" [sous l'attrait du "il"]. It draws away, draws close, the same ghostly affirmation, the same premises of non-presence' ( _PA,_ 100; 71).
Emmanuel Levinas, _Sur Maurice Blanchot,_ 9–10.
For the allusion to Amalek, see Emmanuel Levinas, _Sur Maurice Blanchot,_ 26. Amalek, in the Bible, is eventually defeated by the Jews; however, as God says to Moses: 'Write this for a memorial in a book, and rehearse it in the ears of Joshua' (Exodus, 17: 14). Or, as Blanchot phrases it many years later, 'N'oubliez pas!', 'Do not forget'. Blanchot provides his own pertinent gloss on the figure of Amalek in two subsequent texts: 'L'Écriture consacrée au silence', _Instants,_ 1, 1989, 239–41; and 'Grâce (soit rendue) à Jacques Derrida', _Revue philosophique,_ 2, April–June 1990, 167–73 ( _BR,_ 317–23 [322]).
Emmanuel Levinas, _Sur Maurice Blanchot,_ 23. On Heidegger's refusal of 'ethics', see the famous passage in the 'Brief über den Humanismus', in _Wegmarken,_ 352–7, _Basic Writings,_ 254–9. Heidegger's position on ethics is still of course the subject of some controversy, as Joanna Hodge has shown; see her _Heidegger and Ethics_ (London, Routledge, 1995). By way of explanation in his own case, Blanchot in 1962, in a dialogue on Levinas collected in _L'Entretien infini,_ has both his interlocutors each announce their distrust of the term 'ethics': 'I find in this word', says one, 'only secondary meanings [que des sens derivés]' ( _EI,_ 89; 63).
Emmanuel Levinas, _Sur Maurice Blanchot,_ 24.
See Maurice Blanchot, 'Notre compagne clandestine', 79, 'Our Clandestine Companion', 41.
Maurice Blanchot, 'Notre compagne clandestine', 80, 'Our Clandestine Companion', 42. As readers of Levinas will be aware, the motif of wakefulness – and insomnia – is another borrowing from Levinas; it is also worth noting at this stage to what extent the relation with philosophy for Blanchot, like that with the (other) law in _La Folie du jour,_ is dramatised in relation to the question of sexual difference.
See Maurice Blanchot, 'Notre compagne clandestine', 83, 'Our Clandestine Companion', 45. Blanchot makes a similar point in relation to Georges Bataille in _L'Amitié_ ( _A,_ 328).
See Maurice Blanchot, 'Pour l'amitié', in Dionys Mascolo, _A la recherche d'un communisme de pensée,_ 16.
Maurice Blanchot, 'Notre compagne clandestine', 84, 'Our Clandestine Companion', 47. The phrase from Levinas given just before – that Blanchot slightly misquotes in 'Notre compagne clandestine' – comes from: _Autrement qu'être ou au-delà de l'essence_ (The Hague, Martinus Nijhoff, [1974] 1978), 8 ( _Otherwise Than Being or Beyond Essence,_ translated by Alphonso Lingis (The Hague, Martinus Nijhoff, 1981), p. 7). Blanchot provides further commentary on the phrase in _L'Écriture du désastre_ ( _ED,_ 176; 114).
### Transcendence and the Other
Blanchot's growing reservations regarding the term 'dialogue'–on the grounds that it subordinates the multiple to the One – are clearly visible if one compares the two versions of an essay on psychoanalysis first published in September 1956 and republished in 1969, with extensive revisions, as 'La Parole analytique' ('The Speech of Analysis') ( _EI,_ 343 54; 230–7). In the first version, Blanchot, following Lacan's celebrated 'Rome Discourse' of 1953, had grudgingly accepted that the analytic encounter might be described in terms of a dialectical relation between analyst and analysand. Thirteen years later, this reference to dialectics is the occasion for a series of trenchant and detailed qualifications. Blanchot accordingly withdraws the word dialogue by surrounding it with quotation marks, much as he does with 'présence' elsewhere; these diacritical marks serve, as it were, both to interrupt the text and to displace the whole concept of dialogue by silently invoking the infinity of language and the otherness of the neutre.
In the essay '"Il ne saurait être question de bien finir"' ('A happy end is out of the question'), published under the title 'Quand la morale se tait' as early as January 1954, Blanchot writes for instance of art's demand as follows:
What kind of demand is being announced here, of a sort that may not be assimilated to any current morality, does not make whoever transgresses it guilty, nor innocent whoever believes they are carrying it out, that releases us from all the injunctions of 'I must', all the pretentions of 'I want' and all the resources of 'I can', in order to leave us free? yet not free, not deprived of freedom either, as though it were drawing us forward to a point where, the air of the possible having been exhausted, what is held out to us is the bare relation that is not power, which precedes even all possibility of relation.
( _LV,_ 38; SS, 46)
Compare for instance Emmanuel Levinas, _Totalité et infini,_ 69–75 ( _Totality and Infinity,_ 72–7).
Witness for instance in the following passage from Levinas's conclusion:
Man as Other comes to us from the outside, a separated – or holy – face. His exteriority, that is, his appeal to me, is his truth. My response is not added as an accident to a 'nucleus' of his objectivity, but first _produces_ his truth (which his 'point of view' upon me cannot nullify). This surplus of truth over being and over its idea, which we suggest by the metaphor of the 'curvature of intersubjective space', signifies the divine intention of all truth. This 'curvature of space' is, perhaps, the very presence of God.
( _Totalité et infini,_ 324; _Totality and Infinity,_ 291)
Both statements are in fact loose quotations: the first paraphrases Levinas's comment, based on Plato's _Phaedrus,_ that 'Le Discours est discours avec Dieu et non pas avec les égaux' ( _Totalité et infini,_ 330; 'Discourse is discourse with God and not with equals', _Totality and Infinity,_ 297); while the second (which Blanchot also cites in _La Communauté inavouable_ ( _CI,_ 67–8; 40)) is based on a sentence from 'La Philosophic et l'idée de l'Infini' of 1957, which reads: 'Il faut qu'Autrui soit plus près de Dieu que Moi' ( _En découvrantl' existence avec Husserl et Heidegger,_ 174; 'The other must be closer to God than _I', Collected Philosophical Papers,_ 56).
See for instance the 1967 essay, 'L'Athéisme et l'écriture. L'Humanisme et le cri' ('Atheism and Writing. Humanism and the Cry') ( _El,_ 367–93; 246–63).
See Emmanuel Levinas, _Humanisme de l'autre homme_ (Paris, Le Livre de poche, [1972] 1990), 89.
Maurice Blanchot, 'Parole de fragment', in _L'Endurance de la pensée,_ (Paris, Plon, 1968), 103–8 (103); a modified version of the essay, without the dedication, is reprinted in _EI,_ 451–5; 307–10. On the fraught circumstances surrounding the dedication, see Geert Lernout, _The Poet as Thinker: Hölderlin in France,_ 38.
See Emmanuel Levinas, _Du sacré au saint,_ 46–9. Describing the philosophical singularity of Judaism himself in 1962, Blanchot, for his part, did so in largely secular, linguistic terms:
let me say, brutally, that what we owe to Jewish monotheism is not the revelation of the one God, but the revelation of speech as the place where men hold themselves in relation with what excludes all relation: the infinitely Distant, the absolutely Foreign. God speaks, and man speaks to him. This is the great achievement of Israel.
( _EI,_ 187; 127–8)
Blanchot re-emphasises the view expressed in his 1969 letter in a number of later texts; as well as the footnote added to the text of _L'Écriture du désastre in_ 1980 ( _ED,_ 45; 148–9), see for instance 'Le Bienfait le plus lourd', _Le Nouvel Observateur,_ 31 May–6 June 1985, 79; and '"N'oubliez pas"', _L'Arche,_ May 1988, 68–71.
The move is a problematic one for many reasons, not least of which is the irreducibility, as Levinas himself construes it, of the ethical to the socio-political. Already in 1964, in 'Violence et métaphysique' in _L'Écriture et la différence_ (152), Derrida, by querying its very possibility, took issue with Blanchot's contention, expressed a few pages later, to the effect that Levinas's account of language as constituting the transcendent relation as such must be maintained independently of the 'theological context' in which it appears ( _EI,_ 80; 56). Blanchot, in a subsequent note, accepts the objection; at the same time he also reaffirms his original point by adding elsewhere to the 1969 version of 'Connaissance de l'inconnu' the qualification: 'for my part exclusively', that now appears within parentheses in the passage cited. The more general question arises here: can there be a politics that is faithful to Levinas's thinking? In _The Ethics of Deconstruction_ (Oxford, Blackwell, 1992), Simon Critchley for instance claims that for Levinas 'ethics is ethical for the sake of politics' (223), but this subordination of the ethical to the political raises perhaps more questions than it answers. Levinas, for his part, consigns the properly political to the realm of history and thus to the province of being; as he puts it in the title given to one of the pieces collected in _L'Au-delà du verset_ (Paris, Minuit, 1982, 221–8; _Beyond the Verse,_ translated by Gary D. Mole (London, Athlone, 1994), 188–95), 'Politique après!' ('Politics Last!'). Ethics, on the other hand, as addressed by Levinas, is what disrupts being and history, and by effacing the present opens it to the infinite claims of eschatology. The issue then is this: how to maintain together – but without joining or articulating them by recourse to any prior conceptuality – the 'ethical' and the 'political'? The question sets the scene, one might argue, for Blanchot's own later political writings, the very motto of which, in their refusal to embrace any single totalising political discourse or strategy, and in their endeavours always to contest the possibility of power in the name of the impossibility beyond power, might be: politics or eschatology? or even: politics _and_ eschatology?
See Emmanuel Levinas, _L'Au-delà du verset,_ 148, _Beyond the Verse,_ 120. If less explicitly or vigorously than in many later texts, Levinas is nonetheless already arguing in favour of a non-theological and non-ontological understanding of God in _Totalité et infini;_ see for example, 325–6 ( _Totality and Infinity,_ 293–4). As far as Blanchot's own later texts are concerned, the key issue remains the need rigorously to divorce the transcendence of the Other from all ontology, in which requirement Blanchot is of course still following Levinas. As Blanchot enjoins in _L'Écriture du désastre:_
Abandon the futile hope of finding in being support for the separation, the rupture, the revolt that could be achieved, and thus _verified._ For this would mean you still need the truth and still need to raise it above 'error', just as you want to distinguish death from life and death from death, remaining loyal to the absoluteness of a faith which dares not acknowledge its own emptiness and is content with a transcendence of which _being would still be the measure._
( _ED,_ 140; 88–9)
See Emmanuel Levinas, _De Dieu qui vient à l'idée_ (Paris, Vrin, 1986), 115 ( _Collected Philosophical Papers,_ 165–6). Interestingly, this particular paper by Levinas appeared for the first time alongside Blanchot's 'Discours sur la patience' in 1975; and the passage cited is one Blanchot himself paraphrases at length in the quotation from 'Une compagne clandestine' that is given as an epigraph for this section.
On asymmetry in Levinas, see, for instance, the section 'Asymétrie de l'interpersonnel' ('The Asymmetry of the Interpersonal'), _Totalité et infini,_ 236–8 ( _Totality and Infinity,_ 215–16). It is worth noting that the passage in 'Le Rapport du troisième genre' which develops this notion of double dissymmetry is a later addition from 1969. A fuller account of the exchanges between Blanchot and Levinas would of course have to take into consideration Levinas's later texts of the 1960s, written while on the way to _Autrement qu'être,_ and examine the impact of Blanchot's work on that of Levinas during that whole period. For reasons of space, the task is unfortunately one that cannot be undertaken here.
This is the charge levelled at Blanchot's reading of Levinas in _L'Entretien infini_ by Joseph Libertson, in his _Proximity: Levinas, Blanchot, Bataille and Communication,_ 274–88. Libertson's main contention, already in fact pre-empted by Blanchot, is that, in appealing to the notion of double dissymmetry, Blanchot lapses into an uncritical and ultimately Hegelian concept of intersubjectivity.
Blanchot's use of '"l'homme"', in this context, albeit within the displacement effected by quotation marks, and though clearly intended to be taken not as _vir,_ but as _homo,_ raises nevertheless a question about sexual difference in Blanchot's writing. Does '"l'homme"' include or exclude 'la femme'? And, assuming it might be possible to answer this question satisfactorily in either one way or the other, what then would be the implications? The issue of sexual difference is not one Blanchot addresses at all explicitly in his critical or philosophical work; it remains nonetheless crucial in many of his fictional texts, notably, as we have seen, _Le Très-Haut, L'Arrêt de mort,_ and _La Folie du jour,_ not to mention such later narratives as _Au moment voulu_ and _L'Attente L'Oubli._
Emmanuel Levinas, _Sur Maurice Blanchot,_ 52. While he acknowledges that Blanchot never in fact uses the term, Levinas may nevertheless regularly be found reading Blanchot as a thinker of transcendence; see, for instance, _Sur Maurice Blanchot,_ 13.
It is of course consistently in these terms that the neutre is presented by Blanchot. Using it at one point as the basis for a reading of Kafka's _Castle,_ in the essay 'Le Pont de bois' ('The Wooden Bridge'), Blanchot comments:
What then is above Transcendence, what is below Transdescendence? . . . Let me choose momentarily to call it by the most modest, effaced, and neutral of names: precisely by calling it the neuter – because to name the neuter is perhaps, is surely to dissipate it, but necessarily still in favour of the neuter.
( _EI,_ 580; 395)
The neutre, one might say, not only designates the neutre, but is itself neutre, which is why it names only namelessness and the infinity of its own withdrawal.
See for instance Blanchot's remark in _Après coup,_ in the course of which he translates the Levinasian concept of Saying precisely into the neutre of the 'narrative voice', which, chronologically, at any rate, predates Levinas's own expression, first introduced in 1968; as Blanchot writes:
prior to all distinction between form and content, signifier and signified, even before the separation between the act of speaking and the utterance, there is the unqualifiable Saying, the glory of a 'narrative voice' which speaks clearly, without it being possible for it to be obscured by the opaqueness or enigma or terrible horror of what is communicated.
( _AC,_ 97–8; 68)
See for instance the two biblical commentaries cited in the preceding section, 'L'Écriture consacrée au silence', in which Blanchot writes, for instance, of '"Dieu" (le nom innommable)' ['"God" (the unnamable name)'], and 'Grâce (soit rendue) à Jacques Derrida', where, similarly, Blanchot writes: '"God" (we name him thus by our incapacity to name him)'. Later in the same piece, Blanchot describes '"God"', as 'the first and last writer', and it is this, finally, that is perhaps Blanchot's clearest response to the challenge of Amalek.
### The demand of writing
This (improper) property is one that the word 'absence' in Blanchot's later texts shares increasingly with terms such as: 'fragment' or 'désastre'; as in Derridean 'différance', what comes to be at stake in such words is the necessity to suspend such philosophical distinctions as between passive and active, presence and absence in order to affirm instead the infinity of the relation between any name and its Other. Blanchot offers his own account of this process in a fragment from _L'Écriture du désastre_ ( _ED,_ 120–1; 74–5).
This position derives from the essential non-essentiality of writing formulated by Blanchot in _L'Espace littéraire._ But it is also possible to find here an echo of Blanchot's dialogue with Bataille during the early 1940s. Bataille, it will be remembered, when formulating the rules of the inner experience, credits Blanchot with advancing the fundamental principle that the inner experience is its own authority, but that this authority must be expiated. In much the same way, writing for Blanchot is its own only authority; but that authority, too, must be challenged, and the price paid.
Maurice Blanchot, 'Nous travaillons dans les ténèbres', _Le Monde,_ 22 July 1983. What this implies too, of course, is that the ethical demand of writing is always multiple, never single, always in the form of an astringent double bind. As Blanchot writes in _L'Écriture du désastre:_
To write is to distrust writing absolutely by trusting in it absolutely [Écrire, c'est se méfier absolument, en s'y confiant absolument, de l'écriture]. Whatever foundation one gives to this double movement, which is not as contradictory as this compressed formulation gives to understand, it remains the rule of every writing practice: 'giving withdrawing' [le 'se donner se retirer'] finds here, not its application or illustration, for these are inadequate terms, but that which, by means of dialectics and outside dialectics, justifies itself by letting itself be said, once there is saying and by which there is saying.
( _ED,_ 170–1; 110–11)
'Giving withdrawing', 'se donner se retirer': one might devise no better way of translating the neutre.
As Michael Holland points out in _The Blanchot Reader_ ( _BR,_ 256–60), though Blanchot showed considerable interest in the literary fragment throughout the 1950s, it was not until after 1958, when the first versions of material belonging to _L'Attente L'Oubli_ began to be published, and in the context of a rereading of Nietzsche, that Blanchot began reconsidering the possibilities of fragmentary writing as such, and which were a crucial part of his plans in connection with the project for the International Review in the early 1960s.
Referring to relations between his two protagonists, in what is also a metatextual gloss on both _L'Attente L'Oubli_ and his own earlier text, Blanchot writes as follows:
From the outside, he would have liked it to be seen more clearly how matters stood: in place of the beginning, a kind of initial void, an energetic refusal to let the story [l'histoire] commence. Story, what does she mean by that? He recalls the words which had one day exploded into his life: 'Nobody here desires to bind themselves to a story.' The memory is almost extinct yet still makes him shudder.
( _AO,_ 22)
See 'Le Retour', _Botteghe Oscure,_ VII, 1951, 416–24.
On the status of the _récit_ as an event in its own right, even in the absence of any recounted events, see 'La Rencontre de l'imaginaire' (LV, 9–17; SS, 59–65) to which reference was made in the previous chapter.
Two further occurrences of 'à nouveau, à nouveau' are worth noting. First, in _Celui qui ne m'accompagnait pas,_ interwoven with the motif of the window-pane:
While searching for it almost at random, I saw in a flash – a flash that was the brilliant, tranquil light of summer – that the figure [cette figure, figure or face] was before my very eyes, a few steps away from me [à quelques pas], the few steps which probably stood between me and the bay-window, and the impression was so deep that it was like a spasm of illumination, a shudder of cold light. I was so struck by this that I could not prevent myself from murmuring: 'Don't move, I believe there is somebody there.' 'Somebody? Here?' 'Somebody is looking at us through the window-pane.' 'Through the window-pane?' The words immediately gave me a sense of terror, of horror, as though the emptiness of the window-pane were itself reflected in them, and as though all that had already taken place, and was doing so again, and again [à nouveau, à nouveau]. I think I cried out, and slipped or fell against what seemed to me to be the table. Yet I could still hear him saying: 'You know, there's nobody there [il n'y a personne].'
( _CQ_ , 36 7; 17–18)
The phrase: 'again, again' evidently here has to do with the transformation of something into nothing, presence into absence, a singular event into empty reflection. For a much later occurrence of the phrase, which imples a similar absence of foundation at the very origin of meaning, see Maurice Blanchot, 'Énigme', _Yale French Studies,_ 79, 1991, 5–10 (5). Of the proposed title for the issue to which he was invited to contribute, 'Literature and the Ethical Question', Blanchot writes: '"Again, again" ["A nouveau, à nouveau"], I kept saying to myself. Not that I have any claim to have exhausted an inexhaustible subject, but on the contrary with the certainty that a subject such as this returns to me [qu'un tel sujet me revient] because it is intractable.' Return, here, in its urgency as an ethical demand, becomes inseparable from literature's refusal of all foundation, its contestation of all morality, its submission to the emptiness and the alterity of the outside.
See Pierre Klossowski, _Nietzsche et le cercle vicieux_ (Paris, Mercure de France, 1969). Blanchot comments at length, albeit obliquely, on Klossowski's book (and on the notion of return implied in it) in 'L'Exigence du retour', first published in _L'Arc_ in 1970 (and given in translation in that original form in _BR,_ 290–7); the essay was subsequently incorporated into the opening sections of _Le Pas au-delà._
### An uninterrupted questioning
As Blanchot puts it in _Le Pas au-delà,_ in lines written, it is worth remembering, shortly after the _événements_ of May 1968:
Know only – in an injunction that does not present itself – that the law of return, valid for the whole of the past and the whole of the future, will never allow you, except by a misunderstanding, to leave a place for yourself in a possible present, nor let any presence reach as far as you.
( _PA ,_ 20–1; 11)
See Jacques Derrida, _D'un ton apocalyptique adopté naguère en philosophie_ (Paris, Galilée, 1983); 'Of an Apocalyptic Tone Recently Adopted in Philosophy', translated by John P. Leavey, Jnr, _The Oxford Literary Review,_ 6, 2 (1984), 3–37.
Jacques Derrida, _D'un ton apocalyptique,_ 93, 'Of an Apocalyptic Tone', 34.
See _Le Pas au-delà_ ( _PA ,_ 185; 135). In his essay 'Pas' as a whole Derrida makes great play of all the many 'instances' of this 'viens' in numerous texts by Blanchot, including notably _L'Arrêt de mort_ and _L'Attente L'Oubli;_ see Jacques Derrida, _Parages,_ 20–116 (the quotation _from Le Pas au-delà_ appears on 50).
See Jean-Luc Nancy, 'La Communauté désœuvrée', _Alea,_ 4, February 1983, 11–49. The essay appeared simultaneously with Nancy's _L'Impératif catégorique_ (Paris, Flammarion, 1983), to which Blanchot also makes reference; it was later republished, in an extended version, as the first chapter of the book _La Communauté désœuvrée_ (Paris, Bourgois, [1986] 1990), translated by Peter Connor _et al._ as _The Inoperative Community_ (Minneapolis, University of Minnesota Press, 1991). The differences between the two texts are significant; for convenience, therefore, I shall be referring here to the 1983 _Alea_ version, to which Blanchot's remarks were originally addressed. Written like Nancy's essay in the latter part of 1982, Duras's _La Maladie de la mort_ (Paris, Minuit, 1983), translated by Barbara Bray as _The Malady of Death_ (New York, Grove Press, 1986), also appeared at almost exactly the same time, prompting from Blanchot a review in _Le Nouveau Commerce_ that was incorporated into _La Communauté inavouable_ later that year. Interestingly, alongside Blanchot, Heidegger, Levinas, and Genet, Duras is one of the few other contemporary names invoked by Derrida in his _D'un ton apocalyptique._ For an account of Duras's work in general and its relationship to that of Blanchot, see my _Marguerite Duras: Apocalyptic Desires_ (London, Routledge, 1993).
Jacques Derrida, _D'un ton apocalyptique,_ 94–5, 'Of an Apocalyptic Tone', 34.
The 'Centre de recherches philosophiques sur le politique' was founded in November 1980; by the time of its dissolution in November 1984, it had given rise to two published volumes of collected papers, _Rejouer le politique_ (Paris, Galilée, 1981), and _Le Retrait du politique_ (Paris, Galilée, 1983).
See Philippe Lacoue-Labarthe and Jean-Luc Nancy, 'Ouverture', _Rejouer le politique,_ 22–3. On the question of relation, see Philippe Lacoue-Labarthe and Jean-Luc Nancy, 'Le "Retrait" du politique', _Le Retrait du politique,_ 196–8.
Jean-Luc Nancy, 'La Communauté désœuvrée', 32; _The Inoperative Community,_ 23.
It is from this period, too, that dates the project by Lacoue-Labarthe and Nancy of a sustained analysis of Blanchot's own political itinerary during the 1930s, a project that, at the time of writing, is regrettably far from reaching completion.
See Philippe Lacoue-Labarthe, 'La Transcendance finit dans la politique', _Rejouer le politique,_ 171– 214; the essay is collected, alongside other papers on Heidegger and Hölderlin, in _L'Imitation des modernes,_ 135–73 ( _Typography: Mimesis, Philosophy, Politics,_ 267–300) and is probably the most immediate source for Blanchot's remark that Heidegger's 'mimetological' formulation of Germany's Hellenic destiny was what lay behind his commitment to National-Socialism ( _CI,_ 27; 13). Lacoue-Labarthe subsequently developed his analysis at greater length in 1987 in _La Fiction du politique_ ( _Heidegger, Art and Politics_ ), a book he dedicated to Blanchot; for Blanchot's own response to the book, see his 'Penser l'apocalypse', _Le Nouvel Observateur,_ 22–28 January 1988, 77–9.
See Jean-Luc Nancy, 'La Communauté désœuvrée', 13; _The Inoperative Community,_ 3.
Compare Jean-Luc Nancy, 'La Communauté désœuvrée', 24; _The Inoperative Community,_ 15. In fact, this is a double misquotation on Blanchot's part; Nancy's original text reads: 'la presentation de la finitude et de l'excès sans _recours_ qui _font_ l'être-fini' ('the presentation of finitude and of excess without _recourse_ that _constitute_ finite-being', emphasis mine). While for Blanchot it is (infinite) excess that founds the (finite) being that is community, for Nancy it seems that it is irreparable excess that, together with finitude, constitutes finite being. What weight do Blanchot's two unavowed interventions carry? One might answer: none, were it not that the difference between these two versions seems to inscribe a parting (or 'partage'), on either side of which one finds, in the form of either infinite giving or finite dereliction, two discreetly divergent readings of Bataille's phrase: 'sacrifier, ce n'est pas tuer, mais donner et abandonner', on which the relation between Blanchot and Nancy in these texts arguably turns.
Compare for instance _L'Écriture du désastre, ED,_ 217–18; 143–4.
See Jean-Luc Nancy, _Être singulier pluriel_ (Paris, Galilée, 1996).
For Nancy's reservations about Levinas, which are not without evoking in part some of the hesitations voiced by Blanchot in _L'Entretien infini,_ see Jean-Luc Nancy, _Une pensée finie_ (Paris, Galilée, 1990), 260–1 ( _The Inoperative Community,_ 104–5), and _Être singulier pluriel,_ 52. In an abruptly polemical account of the debate between Nancy and Blanchot, Robert Bernasconi has argued that Blanchot's text demands in fact to be read as an implicit rejection, from a Levinasian standpoint, of Nancy's whole analysis of community. This is a simplification; indeed to reach this conclusion, Bernasconi is forced into a doubly reductive argument, which by aligning Nancy with Bataille and Blanchot with Levinas, seriously understates the complexity of Blanchot's text and, misleadingly, construes it in terms of a false opposition between the absence of community and the dissymmetry between Same and Other. See Robert Bernasconi, 'On Deconstructing Nostalgia for Community within the West: the Debate between Nancy and Blanchot', _Research in Phenomenology,_ XXIII, 1993, 3–21. Incidentally, in _La Communauté inavouable_ Blanchot for once uses Levinas's own term asymmetry ('asymétrie') to refer to the (non-)relation between Same and Other, rather than his customary dissymmetry ('dissymétrie'); in so doing, as though to concede its impropriety, he conspires to misspell the word as: 'assymétrie' ( _CI,_ 22; 9)!
Jean-Luc Nancy, _L'Impératif catégorique_ , 153.
For the quotation from Bataille, see Georges Bataille, _Œuvres complètes,_ VII (1976), 310. Bataille continues, some pages later:
The sacred is that prodigal ebullience of life which, to endure at all, the order of things domesticates and which domestication turns into an unleashing, in other words, into violence. Without pause it threatens to break down dams, and set against productive activity the precipitate and contagious movement of the pure glory of consumption. The sacred is precisely comparable to the flame that destroys wood by consuming it.
(ibid, 312–13)
In 1959, in a series of notes on 'Le Coupable', Bataille famously declared, with explicit reference to Acéphale: 'I was determined, if not to found a religion, at least to head in that direction' ('j'étais résolu, sinon à fonder une religion, du moins à me diriger dans ce sens') ( _Œuvres complètes,_ VI [1973], 369). Reading lines such as these, it is easy to understand how for so many readers, as Blanchot concedes, Bataille's name is identified with the mystical quest for ecstatic experience ( _CI,_ 18; 7); and how Nancy can find in Bataille much evidence of nostalgia for the sovereign subject. But which of these two readings of Bataille, that of Blanchot or that of Nancy, may be said to be more faithful? The question, no doubt, can be answered in many different ways; what it raises is the whole question of the status or meaning of community itself, starting with the interpretative community made up of the readers of Bataille. For an account of the important issues at stake here, see Geoffrey Bennington, 'Lecture: de Georges Bataille', _Georges Bataille après tout,_ Denis Hollier (ed.) (Paris, Belin, 1995), 11–34.
See Jean-Luc Nancy, 'La Communauté désœuvrée', 26, _The Inoperative Community,_ 17.
Nancy gives a slightly different emphasis to the point in his 1989 text 'L'Insacrifiable' when he writes: 'the thinking of Bataille was, ultimately, less a thinking of sacrifice than a thinking unremittingly strained and riven by the impossibility of renouncing sacrifice' ; see Jean-Luc Nancy, _Une pensée finie,_ 65–106 (86). It is worth noting at this point that, already twenty-one years earlier, Blanchot himself had recorded a rather different view of Acéphale than that given in _La Communauté inavouable,_ more consistent with the criticisms voiced here by Nancy. Indeed, in an article on Bataille first published in 1962, and collected in _L'Entretien infini,_ Blanchot wrote:
the fact is that this act of supreme negation which we have just supposed, and that for a time the enterprise of _Acéphale_ represented for Georges Bataille, still belongs to the possible. Power can do this, it can do everything, even do away with itself as power (the explosion of the atom itself, one of the furthest examples of nihilism). Such an act would not in any way make us accomplish the decisive step, the step that delivers us – in some way in our absence – into the surprise of impossibility, by allowing us to belong to _the non-power that is not simply the negation of power._
( _EI,_ 309–10; 208)
The question again arises: which of these accounts is the more faithful? and more faithful to what? to the possibilities or impossibilities inherent in Bataille's endeavour?
See Jacques Derrida, _Politiques de l'amitié_ (Paris, Galilée, 1994), 56–7, note.
For Blanchot's earlier discussions of Duras, see: 'La Douleur du dialogue' ('The Painfulness of Dialogue') ( _LV_ , 185–94; SS, 199–206); 'La Voix narrative (le "il", le neutre)' ('The Narrative Voice (the "he", the neutral)') ( _EI,_ 556–67; 379–87); and 'Détruire' ( _A_ , 132–6).
In Genesis, I: 27, it is written of man that God created them both male and female; by Genesis, II: 22, however, God is forming Eve from Adam's rib; according to whichever version is believed, it would seem that sexual difference is _both_ primary _and_ secondary as far as humanity as a whole is concerned.
It is worth adding here that this distinction between the homosexual group and the heterosexual community in itself implies no condemnation of homosexuality as such, but rather of the institutionalised and oppressive cohesiveness of certain same-sex social groups, in which commitment to the reproduction of authority or power outweighs responsiveness to the Other, irrespective of whether such groups are made up of individuals who identify themselves as homosexual or heterosexual. (In fact, in May 1996, Blanchot added his name to that of 232 other influential intellectuals in calling on the French government to implement a recommendation from the European Parliament that 'all forms of legal, administrative and social discrimination towards homosexuals' be abolished, and urging it in particular to grant legal recognition in France to all gay and lesbian couples; see 'Pour une reconnaissance légale du couple homosexuel', _Le Nouvel Observateur,_ 9–15 May 1996, 44–5.)
It is worth noting also that, in reading _La Maladie de la mort,_ Blanchot refuses to attribute the incommensurability of relation that is the crux of the story to the non-coincidence of desire between a heterosexual woman and a homosexual man; to do so,in Blanchot's view, by naming or thematising it, would necessarily result in a reduction of the fundamental dissymmetry between Same and Other, and trivialise that relation of non-relation by casting it exclusively in terms of sexual preference; this is not to say, of course, that sexual orientation is unimportant, only that the prior affirmation has to do with the precedence of the Other over the Same. Surprisingly perhaps, in this respect, Duras herself was one of the first to take issue with Blanchot's alleged disregard of the question of homosexuality in _La Maladie de la mort;_ see Marguerite Duras, 'Dans les jardins d'Israël il ne faisait jamais nuit', interview by Pascal Bonitzer, Charles Tesson and Serge Toubiana, _Les Cahiers du cinéma,_ 374, July–August 1985, 5–12.
See Georges Bataille, _Œuvres complètes,_ V, 196, where Bataille writes, in 'Méthode de méditation', under the heading 'L'Essentiel est inavouable' ('The Essential is unavowable'), as follows:
What is not servile is unavowable [inavouable]: a reason to laugh, to . . . : the same is true of ecstacy. What isn't useful must be hidden (beneath a mask). A criminal in his final throes was the first to formulate this commandment, shouting to the crowd: 'Never confess' ['N'avouez jamais'].
On the Charonne killings, see Hervé Hamon and Patrick Rotman, _Les Porteurs de valises,_ 379. On 8 February 1962, during a demonstration in Paris against the OAS, the right-wing terrorist organisation whose aim was to prevent Algerian independence at any cost, eight deaths occurred in the crush at Charonne metro station that was provoked by the intervention of French riot police. Five days later, 500,000 Parisians marched in silent protest at the deaths. On 18 March 1962, the Evian accords were finally signed, giving independence to Algeria.
Maurice Blanchot, 'N'oubliez pas!', _La Quinzaine littéraire,_ 459, 16–31 March 1986, 11–12.
On Duras's own politics in the wake of May 1968, see the first of her projects for the film _Le Camion_ ( _The Truck_ ) in _Le Camion_ (Paris, Minuit, 1977), 73–4. As Duras writes: 'Let the world go to rack and ruin, let it go to rack and ruin, that is the only politics' ('Que le monde aille à sa perte, qu'il aille à sa perte, c'est la seule politique'). On Duras's politics more generally, see my _Marguerite Duras: Apocalyptic Desires,_ 1–39.
### The unexpected word
On the relation between disjunction and injunction in Blanchot, see Jacques Derrida, _Spectres de Marx_ (Paris, Galilée, 1993), 21–85.
Friedrich Hölderlin, _Werke,_ II, 893; Blanchot refers to the letter on a number of other occasions ( _EL,_ 221; 213; _ED,_ 191; 124).
See Maurice Blanchot, 'N'oubliez pas!', _La Quinzaine littéraire,_ 459, 16–31 March 1986, 11–12.
As a way of gauging the degree of continuity or discontinuity in at least some of Blanchot's political thinking from the early 1930s to the late 1960s, it is perhaps worth comparing here a piece such as 'Le Marxisme contre la révolution', published in the right-wing _Revue française_ for 25 April 1933, with an unsigned article, subsequently attributed to Blanchot, such as 'Affirmer la rupture' from _Comité_ in October 1968. In the first, at one point, Blanchot supports the call for radical (spiritual) revolution by arguing:
Refusal is absolutely foreign to all true negation, all absence, all _nothing._ The act of opposition and destruction that represents it, represents also, at the highest point of its force, some desperate affirmation. Rejecting all the negations of consent and the constraints of acceptance, casting aside what abolishes it including even a part of itself, the rebellious spirit searches obstinately, amidst these defeats and deaths, for something proper to him and expressive of him. . . . His act of refusal rids him of everything that is not his own _person_ [tout ce qui n'est pas sa _personne_ ], manifests him as a personal existence whose realisation is the final object and the safeguard of refusal itself.
('Le Marxisme contre la révolution', 516)
Thirty-five years later, in the second text, one reads:
To carry the break [porter la rupture] is not only to disengage or attempt to disengage from their integration within established society those forces straining towards the break, it is to act in such a way that in reality and each time it takes place, without ceasing to be active refusal, refusal is not a _solely negative moment._ Politically and philosophically, that is one of the most powerful characteristics [traits] of the movement. In that sense, radical refusal, as borne by the movement and as we are enjoined to bear it too, far exceeds simple negativity, to the extent that it is the very negation of what has yet to be posited and affirmed. To clarify the singular character [le trait singulier] of this refusal is one of the theoretical tasks of the new political thinking.
('Affirmer la rupture', 4–5)
From one piece to the other, one movement to the other, there is evidently in Blanchot's thinking something of the same logic of absolute refusal and radical contestation; but, at the same time, there are several marked and essential differences, which affect in particular not only the agent of refusal as such – impersonal and anonymous here, personal and self-present there – but, as a result, the logic of refusal itself, which in 1933 is retained within the orbit of a purely self-identical subject and an essentially autonomous person, but which in 1968 is described as corresponding to a far more radical movement of affirmation and rupture, irreducible to any subjective dialectic. To differentiate more fully between these two political moments in Blanchot's itinerary would require, of course, a more detailed analysis than can be attempted here. At any event, there seems little foundation to the view, put forward among others by Steven Ungar in his _Scandal and Aftereffect_ (124–36), that Blanchot's post-war political activities were at best an uncritical reprise of the so-called 'abject dissidence' of his pre-war nationalism.
See Marguerite Duras, 'Écrit pour tousles temps, tous les carêmes', _L'Autre Journal,_ 9, November 1985, 73; the piece is now collected in _Le Monde extérieur_ (Paris, POL, 1993), 78–80. Incidentally, with respect to the memory of Munich among post-war French intellectuals, it is worth recalling here that Duras in the latter years of the 1930s found herself working as an archivist at the French Colonial Office during the period Georges Mandel spent as minister of that government department; indeed, it was in collaboration with a close associate of Mandel, Philippe Roques, who was killed soon after while on a mission to London, that Duras, shortly before the outbreak of war, published under her own family name, Marguerite Donnadieu, her first book, which was a volume of nationalist propaganda largely inspired by the anti-German policies of Mandel and entitled _L'Empire français_ (Paris, Gallimard, 1940).
'Interview de certains signataires', interview with Maurice Blanchot by Madeleine Chapsal, in _Le Droit à l'insoumission: 'le dossier des 121',_ 90–3 (90–1) ( _BR,_ 196–99 (196–7)). Blanchot's distinction between duty and right in this context echoes a similiar point made in relation to Kafka in _L'Espace littéraire,_ where Blanchot describes suicide (and writing) as resting on
an absolute right [un droit absolu], the only one that is not the corollary of a duty [le seul qui ne soit pas l'envers d'un devoir], and yet a right which no real power doubles or reinforces, which arches over like an endless footbridge [une passerelle infinie] that at the decisive moment simply stops, and becomes as unreal as a dream, but over which in reality one still has to pass – a right, then, without power and without duty [un droit donc sans pouvoir et sans devoir], a folly necessary to the integrity of reason, which, moreover, quite often seems to succeed: what is striking about all these traits is that they apply equally well to another experience, apparently less dangerous, but perhaps no less mad, which is that of the artist.
( _EL,_ 106; 105)
The experience of art that allows Blanchot to write these lines explains how, as Blanchot points out in the interview with Chapsal, the decision to put his signature to the Declaration was the act not of a political writer nor a politically active citizen, but of 'a non-political writer impelled to state publicly his position regarding problems that concern him in an essential way' (90).
As Blanchot noted of 'the 121' in 1984:
those who signed the Declaration made no claim to be announcing a universal truth (insubordination for its own sake and in all circumstances), but were doing no more than support decisions they had not taken, acknowledging themselves to be responsible for these decisions and, in so doing, _identifying_ with those who had been forced to take them.
('LesIntellectuels en question', _Le Débat,_ 29,
March 1984, 3–24 (27), _BR,_ 206–27 (225))
In his account of the events surrounding the Declaration in 'Pour l'amitié', Blanchot recounts how at one stage he was summoned by the examining magistrate in the case to make a formal statement; once Blanchot had finished speaking, the magistrate, according to legal convention – the practice is one that has its counterpart in many other European countries – began dictating Blanchot's statement to the clerk of the court. Blanchot however protested, insisting that his own words be recorded, not those of the examining magistrate. As Blanchot comments:
There is a seriously deficient point in this affair, which is the debate between a man with a wealth of legal expertise at his fingertips and another who has perhaps few words and does not even know the sovereign value of speech, of _his_ speech. Why is it that the judge has the right to be sole master of language, dictating (in what is already a _diktat_ ) the words of another, as seems appropriate to him, reproducing them not as they were said, stuttering, meagre and unsure, but made worse, because finer, more consistent with the classical ideal, and, most of all, more definitive.
(Maurice Blanchot, 'Pour l'amitié', in Dionys Mascolo, _A la recherche d'un communisme de pensée, entêtements,_ 5–16 (11))
[Maurice Blanchot], 'Tracts, affiches, bulletin', _Comité,_ 1, October 1968, 16 (BR, 205).
See [Maurice Blanchot], 'Sur les Comités d'action', _Les Lettres nouvelles,_ June–July 1969, 184–5 (184). First published anonymously, the letter is attributed to Blanchot and reproduced by Dionys Mascolo in: _A la recherche d'un communisme de pensée,_ 359–60. It implies much the same view of May as that given in _La Communauté inavouable_ ( _CI,_ 52–6; 29–33). For a less flattering and, by some, contested account of what actually went on in the Students–Writers Action Committee, see [Marguerite Duras], 'Naissance d'un Comité', _Les Lettres nouvelles,_ June–July 1969, 144–50, attributed to Duras and reproduced in _A la recherche d'un communisme de pensée,_ 324–30.
On the relation between _Le Très-Haut_ and the _événements,_ see Georges Préli, _La Force du dehors: extériorité, limite et non-pouvoir à partir de Maurice Blanchot_ (Paris, Recherches, 1977). In his endeavours to rethink the political Blanchot would seem here to be acting on the warning issued to the rebellious Henri Sorge, in _Le Très-Haut,_ by his stepfather – the Egisthus figure in this partial reworking of the Orestes story – to the effect that
the State will know how to use your insubordination, and not only will it take advantage of it, but you, in opposition and revolt, will be its delegate and representative as fully as you might have been in your office, following the law. The only change is that you want change and there won't be any. What you would like to call the destruction of the State will always appear to you really as service to the State. What you will do to escape the law will still have the force of law for you.
( _TH,_ 134; 137)
[Maurice Blanchot], 'La Clandestinité à ciel ouvert', _Comité,_ 1, October 1968, 23.
[Maurice Blanchot], 'Le Communisme sans héritage', _Comité,_ 1, October 1968, 13 ( _BR,_ 203).
[Maurice Blanchot], 'Les Actions exemplaires', _Comité,_ 1, October 1968, 17–18 (18).
[Maurice Blanchot], 'Rupture du temps : révolution', _Comité,_ 1, October 1968, 18 ( _BR,_ 205).
# Texts by Blanchot
## Essays and fic tion
_Thomas l'Obscur,_ Paris, Gallimard, 1941.
_Comment la littérature est-elle possible?,_ Paris, José Corti, 1942.
_Aminadab,_ Paris, Gallimard, 1942.
_Faux Pas,_ Paris, Gallimard, 1943.
_Le Très-Haut,_ Paris, Gallimard, 1948.
_L'Arrêt de mort,_ Paris, Gallimard, 1948.
_La Part du feu,_ Paris, Gallimard, 1949.
_Lautréamont et Sade,_ Paris, Minuit, 1949, revised edn 1963.
_Thomas l'Obscur_ (nouvelle version), Paris, Gallimard, 1950.
_Au moment voulu,_ Paris, Gallimard, 1951.
_Le Ressassement éternel,_ Paris, Minuit, 1951.
_Celui qui ne m'accompagnait pas,_ Paris, Gallimard, 1953.
_L'Espace littéraire,_ Paris, Gallimard, 1955.
_Le Dernier Homme,_ Paris, Gallimard, 1957, new version 1977.
_Le Livre à venir,_ Paris, Gallimard, 1959.
_L'Attente L'Oubli,_ Paris, Gallimard, 1962.
_L'Entretien infini,_ Paris, Gallimard, 1969.
_L'Amitié,_ Paris, Gallimard, 1971.
_Le Pas au-delà,_ Paris, Gallimard, 1973.
_La Folie du jour,_ Montpellier, Fata morgana, 1973.
_L'Ecriture du désastre,_ Paris, Gallimard, 1980.
_De Kafka à Kafka_ , Paris, Gallimard: folio, 1981.
_La Bête de Lascaux,_ Montpetlier, Fata morgana, 1982.
_Après coup, précédé par Le Ressassement éternel,_ Paris, Minuit, 1983.
_La Communauté inavouable,_ Paris, Minuit, 1983.
_Le Dernier à parler,_ Montpellier, Fata morgana, 1984.
_Michel Foucault tel que je l'imagine,_ Montpellier, Fata morgana, 1986.
_Sade et Restif de la Bretonne,_ Brussels, Éditions Complexe, 1986.
_Une voix venue d'ailleurs,_ Paris, Ulysse-fin de siècle, 1992.
_L'Instant de ma mort,_ Montpellier, Fata morgana, 1994.
## Articles, journalism, uncollected texts
This aims to be an exhaustive bibliography, though inevitably there will be some unforeseen gaps. It lists, in chronological order, all Blanchot's signed journalism, literary criticism, and other texts I have been able to trace, together with a number of pieces first published anonymously and subsequently attributed to Blanchot. For the period up to 1976, I am deeply indebted to Peter Hoy and Michael Holland for their work in drawing up the detailed bibliography published in _Gramma,_ 3/4, and 5, which I have verified and amended where necessary. Except in the case of texts available in English only in _The Sirens' Song_ and _The Blanchot Reader,_ to which reference is made separately, page references to collected volumes given here are solely to the standard French editions listed in the preceding section; these are identified by the abbreviations listed at the begining of this volume. These should, however, enable English-speaking readers to find the corresponding English-language volume of any specific text they wish to locate.
'François Mauriac et ceux qui étaient perdus', _La Revue française,_ 26e année, 28 June 1931, 610–11.
'Mahatma Ghandi', _Cahiers mensuels,_ 3, 7 July 1931, 10–17.
'Comment s'emparer du pouvoir?', _Journal des débats,_ 18 August 1931, 1.
_'Deux hommes en moi,_ par Daniel-Rops', _La Revue universelle,_ XLIV, 21, February 1931, 367–8.
'La Culture française vue par un Allemand', _La Revue française,_ 10, 27 March 1932, 363–5.
'Paul Morand : _Flèche d'Orient', Réaction,_ 10, March 1932, 58–9.
'Nouvelle querelle des anciens et des modernes', _Réaction,_ 11, April–May 1932, 11–16.
_'La Guerre à sept ans_ [by Jean Maxence]', _La Revue universelle,_ XLIX, 7, 1 July 1932, 112–14.
'L'Histoire désarmée', _Journal des débats,_ 21 July 1932, 1.
'Les Écrivains et la politique', _Journal des débats,_ 27 July 1932, 1.
'Le Monde sans âme', _La Revue française,_ 27e année, 3, 25 August 1932, 460–70 (republished in _Gramma,_ 5, 1976, 44–52).
_'Ames et visages du XX e siècle_ [by André Rousseaux]', _La Revue universelle,_ 18, December 1932, 742–5.
'Les Faux-Semblants du savoir', _Journal des débats,_ 27 February 1933, 1.
'Les Années tournantes', _Journal des débats,_ 21 March 1933, 1.
'Les Communistes gardiens de la culture', _Journal des débats,_ 25 March 1933, 1.
'Le Marxisme contre la révolution', _La Revue française,_ 28e année, 4, 25 April 1933, 506–17 (republished in _Gramma,_ 5, 1976, 53–61).
'Menaces d'inflation', _Le Rempart,_ 6, 27 April 1933, 1.
'Le Quai d'Orsay contre la France', _Le Rempart,_ 7, 28 April 1933, 3.
'Crise d'autorité', _Journal des débats,_ 29 April 1933, 1.
'Quand l'État est révolutionnaire . . .', _Le Rempart,_ 8, 29 April 1933, 2.
'La Commission sénatoriale exige du gouvernement un budget en équilibre. Elle repousse le monopole des pétroles', _Le Rempart,_ 9, 30 April 1933, 3.
'Des violences antisémites à l'apothéose du travail', _Le Rempart,_ 10, 1 May 1933, 3.
'Le Défaitisme à l'école', _Le Rempart,_ 13, 4 May 1933, 3.
'Psychose de revendication', _Le Rempart,_ 16, 7 May 1933, 2.
'"En vue d'une action." L'Allemagne nouvelle ou le triomphe de la Prusse', _Le Rempart,_ 17, 8 May 1933, 2.
'Une apothéose scandaleuse', _Le Rempart,_ 19, 10 May 1933, 2.
'Les Illusions dangereuses', _Le Rempart,_ 21, 12 May 1933, 2.
'La Levée en masse de l'Allemagne', _Le Rempart,_ 22, 13 May 1933, 2.
'M. de Monzie, grand maître en anarchie', _Le Rempart,_ 23, 14 May 1933, 2.
'L'Avenir ministériel et le sort du pays', _Le Rempart,_ 24, 15 May 1933, 2.
'Crise de l'État', _Le Rempart,_ 29, 20 May 1933, 1.
'Politique des chiffres. Le Budget ou l'inutile conscience du mal', _Le Rempart,_ 30, 21 May 1933, 1.
'Le Club des conflits. La France à l'aventure', _Le Rempart,_ 31, 22 May 1933, 2.
'Les Émeutes de Genève. Bienfaits de la force', _Le Rempart,_ 31, 22 May 1933, 6.
'M. de Monzie, émule de Mussolini et de Hitler. Triturons la jeunesse', _Le Rempart,_ 32, 23 May 1933, 1–2.
'Le Budget à l'encan. Si les contribuables connaissaient leur force . . .', _Le Rempart_ , 33, 24 May 1933, 2.
'Le Gouvernement Daladier au carrefour', _Le Rempart,_ 35, 26 May 1933, 2.
'La Révolte des contribuables contre le Parlement', _Le Rempart,_ 36, 27 May 1933, 1.
'Les Leçons d'une manifestation', _Le Rempart,_ 37, 28 May 1933, 2.
'Le Pacte mortel pour la Paix. L'Europe abandonnée à l'Italie et à l'Allemagne', _Le Rempart,_ 38, 29 May 1933, 1.
'La Révolte contre le Pouvoir', _Le Rempart,_ 40, 31 May 1933, 1.
'Morale et politique', _La Revue du siècle,_ 2, May 1933, 60–5.
'Pour séparer la Petite-Entente de la France, et la Pologne de la Petite-Entente', _Le Rempart,_ 41, 1 June 1933, 2.
'Socialisme bourgeois et nationalisme révolutionnaire', _Le Rempart,_ 44, 4 June 1933, 1.
'Le Choix entre deux capitulations', _Le Rempart,_ 45, 5 June 1933, 1.
'L'Obstination dans la défaite', _Le Rempart,_ 46, 6 June 1933, 2.
'La Politique. Les Suprêmes Défaillances', _Le Rempart,_ 47, 7 June 1933, 3.
'Les Entretiens de Paris. Les Premiers Effets du Pacte', _Le Rempart,_ 48, 8 June 1933, 3.
'Le Pacte. Le Parlement ratifiera-t-il le Pacte?', _Le Rempart,_ 49, 9 June 1933, 3.
'La Nouvelle Faillite allemande', _Le Rempart,_ 50, 10 June 1933, 1.
'L'Abaissement de la Chambre', _Le Rempart,_ 51, 11 June 1933, 2.
'La Conférence de Londres. Il n'y a pas de problèmes des dettes', _Le Rempart,_ 52, 12 June 1933, 3.
'Une intervention du Pape. Offrande au paganisme hitlérien des sacrifices de la chrétienté', _Le Rempart,_ 54, 14 June 1933, 2.
'Une entrevue Daladier-Hitler. La logique dans l'abaissement', _Le Rempart,_ 55, 15 June 1933, 2.
'Les Dettes. La réponse de l'Europe à l'Amérique', _Le Rempart,_ 55, 15 June 1933, 3.
'L'Avantage de la résistance. La Leçon des événements autrichiens', _Le Rempart,_ 56, 16 June 1933, 3.
'Réponse à _La Croix', Le Rempart,_ 57, 17 June 1933, 5.
'La Conférence économique. Un centre de manœuvres et de marchandages', _Le Rempart,_ 58, 18 June 1933, 3.
'La Conférence dans une impasse', _Le Rempart,_ 59, 19 June 1933, 3.
'La Révolte contre le pouvoir', _Le Rempart,_ 60, 20 June 1933, 2.
'La Révolte contre le pouvoir. Un projet de loi du Gouvernement menace de prison les contribuables qui veulent se défendre', _Le Rempart,_ 61, 21 June 1933, 1.
'Quand l'Europe et l'Amérique collaborent', _Le Rempart,_ 61, 21 June 1933, 3.
'La Révolution nécessaire', _Le Rempart,_ 62, 22 June 1933, 2.
'Que signifie pour la France l'union de l'Autriche et de la Hongrie. Pouvons-nous accepter que la Hongrie germanophile et révisionniste domine l'Autriche?', _Le Rempart,_ 63, 23 June 1933, 1.
'Après les incidents de Bray-sur-Somme. La leçon d'une condamnation', _Le Rempart,_ 64, 24 June 1933, 2.
'Le Problème de l'Europe centrale. Le choix entre deux politiques', _Le Rempart,_ 65, 25 June 1933, 3.
'De Londres à Genève. La décadence des Assemblées Internationales', _Le Rempart,_ 66, 26 June 1933, 3.
'Les Quatre Vérités. La jeunesse française devant le monde', _Le Rempart,_ 67, 27 June 1933, 2.
'Le Parlement contre l'économie nationale. La faillite du libéralisme', _Le Rempart,_ 68, 28 June 1933, 2.
'La Vraie Menace du Troisième Reich', _Le Rempart,_ 69, 29 June 1933, 3.
'Le Quatorzième Anniversaire de la paix', _Le Rempart,_ 70, 30 June 1933, 3.
'Le Souvenir et la leçon de Mangin', _Le Rempart,_ 81, 11 July 1933, 3.
'La Crise du socialisme', _Le Rempart,_ 82, 12 July 1933, 3.
'L'Accord de la France et de l'Italie', _Le Rempart,_ 86, 16 July 1933, 3.
'Après la signature du Pacte à Quatre. Le nouveau destin de l'Europe', _Le Rempart,_ 87, 17 July 1933, 3.
'Le Socialisme marxiste s'effondre. Un socialisme national se tourne vers de voies nouvelles', _Le Rempart_ , 88, 18 July 1933, 3.
'Ce qui menace le socialisme international. Les nouveaux opportunistes', _Le Rempart,_ 89, 19 July 1933, 3.
'Tandis que Hitler cherche à se concilier la Reichswehr, les manifestations inconvenantes de M. Henderson', _Le Rempart,_ 90, 20 July 1933, 3.
'Le Socialisme national osera-t-il être un mouvement révolutionnaire?', _Le Rempart,_ 93, 23 July 1933, 1–2.
'La Révolution est condamnée mais l'État devient révolutionnaire. L'hitlérisme contre Hitler', _Le Rempart,_ 94, 24 July 1933, 3.
'Le Bilan de la Conférence de Londres', _Le Rempart,_ 95, 25 July 1933, 3.
'Les Chances du néo-socialisme', _Le Rempart,_ 130, 29 August 1933, 3.
'M. de Neurath expose les principes de la politique allemande. Un discours logique', _Journal des débats,_ 17 September 1933, 1.
_'Positions_ [by Jean-Pierre Maxence]', _La Revue du siècle,_ 6, October 1933, 75–7.
'Ceux qui ignorent', _Journal des débats,_ 12 December 1933, 1.
'La Démocratie et les relations franco-allemandes', _La Revue du XX e siècle,_ 4, February 1935, 56–9.
'Le Dérèglement de la diplomatie française', _La Revue du XX e siècle,_ 6, May–June 1935, 53–7.
'La Fin du 6 février', _Combat,_ 2, February 1936, 26.
'La Guerre pour rien', _Combat,_ 3, March 1936, 42–3.
'Après le coup de force germanique', _Combat,_ 4, April 1936, 59.
'La Peur des efforts', _Journal des débats,_ 29 July 1936, 1.
'Le Terrorisme, méthode de salut public', _Combat,_ 7, July 1936, 106 (republished in _Gramma,_ 5, 1976, 61–3).
'La Grande Passion des modérés', _Combat,_ 9, November 1936, 147.
'Le Caravansérail', _Combat,_ 10, December 1936, 171.
'De la révolution à la littérature', _L'Insurgé,_ 1, 13 January 1937, 3.
'Réquisitoire contre la France', _L'Insurgé,_ 1, 13 January 1937, 4.
'Nous, les complices de Blum . . .', _L'Insurgé,_ 2, 20 January 1937, 4.
_'Sangs,_ par Louise Hervieu', _L'Insurgé,_ 2, 20 January 1937, 5.
'Blum, notre chance de salut . . .', _L'Insurgé,_ 3, 27 January 1937, 4.
_'Penser avec les mains,_ par Denis de Rougemont', _L'Insurgé,_ 3, 27 January 1937, 5.
'Notre première ennemie, la France', _L'Insurgé,_ 4, 3 February 1937, 4.
_'L'Été 1914 (Les Thibault, 7 e partie),_ par Roger Martin du Gard', _L'Insurgé,_ 4, 3 February 1937, 5.
'La Crise qui va s'ouvrir', _L'Insurgé,_ 5, 10 February 1937, 4.
_'La Jeunesse d'un clerc,_ par Julien Benda', _L'Insurgé,_ 5, 10 February 1937, 5.
'L'Impasse', _L'Insurgé,_ 6, 17 February 1937, 4.
_'Le Magasin de travestis,_ par Georges Reyer, _Zobain,_ par Raymond Guérin', _L'Insurgé,_ 6, 17 February 1937, 5.
_'La Dentelle du rempart,_ par C. Maurras', _L'Insurgé,_ 7, 24 February 1937, 5.
'Le Déshonneur français', _L'Insurgé,_ 7, 24 February 1937, 6.
'Ce qu'ils appellent patriotisme', _L'Insurgé,_ 8, 3 March 1937, 4.
_'Le Saladier,_ par Marcel Jouhandeau', _L'Insurgé,_ 8, 3 March 1937, 5 (republished in part as 'Chaminadour', FP, 260–1).
'Joyeuse Mi-Carême ou l'histoire d'une perquisition', editorial signed by Jean-Pierre Maxence, Thierry Maulnier, Ralph Soupault, Maurice Blanchot, Kléber Haedens, and Guy Richelet, _L'Insurgé,_ 9, 10 March 1937, 3.
'M. Delbos paiera', _L'Insurgé,_ 9, 10 March 1937, 4.
_'Romanesques,_ par Jacques Chardonne', _L'Insurgé,_ 9, 10 March 1937, 6.
'Le Temps de la guerre', _L'Insurgé,_ 10, 17 March 1937, 4.
_'Réflexions sur la force,_ par Alphonse Séché', _L'Insurgé,_ 10, 17 March 1937, 5.
'Léon Blum, vous étiez prévenu', editorial signed by Jean-Pierre Maxence, Thierry Maulnier, Ralph Soupault, Maurice Blanchot, Kléber Haedens, and Guy Richelet, _L'Insurgé,_ 10 bis, special issue, 17 March 1937, 2 (reprinted in _L'Insurgé,_ 11, 24 March 1937, 3).
_'La Rue courte,_ par Thyde Monnier, _Bêtafeu,_ par Guy Mazeline', _L'Insurgé,_ 11, 24 March 1937, 5.
'Préparons la vengeance', _L'Insurgé,_ 11, 24 March 1937, 7.
'Blum provoque à la guerre', _L'Insurgé,_ 12, 31 March 1937, 4.
_'Ce qui meurt et ce qui naît,_ par Daniet-Rops', _L'Insurgé,_ 12, 31 March 1937, 5.
'L'Effondrement de la France', _L'Insurgé,_ 13, 7 April 1937, 4.
_'Maldagne,_ par Hubert Chatelion', _L'Insurgé,_ 13, 7 April 1937, 5.
'Les Mystères de Moscou', _L'Insurgé,_ 14, 14 April 1937, 4.
_'Joseph et ses frères,_ par Thomas Mann', _L'Insurgé,_ 14, 14 April 1937, 5.
'M. Delbos a raison', _L'Insurgé,_ 15, 21 April 1937, 4.
_'Destin d'une révolution,_ par Victor Serge', _L'Insurgé,_ 15, 21 April 1937, 5.
'Les Deux Trahisons?', _L'Insurgé,_ 16, 28 April 1937, 10, 12.
'Vos vies sont menacées', _L'Insurgé,_ 16 bis, special issue, 1 May 1937, 3.
'Demain la guerre', _L'Insurgé,_ 17, 5 May 1937, 4.
_'Rêveuse bourgeoisie,_ par Drieu La Rochelle', _L'Insurgé,_ 17, 5 May 1937, 5.
'La France condamnée à avoir tort', _L'Insurgé,_ 18, 12 May 1937, 4.
_'Un homme veut rester vivant,_ par Henri Petit', _L'Insurgé,_ 18, 12 May 1937, 5.
'Les Français et le couronnement', _L'Insurgé,_ 19, 19 May 1937, 4.
_'La Maison au bord du monde,_ par Jean Guirec, _La Rue du Chat-qui-pêche,_ par Jolan Fœldes', _L'Insurgé,_ 19, 19 May 1937, 5.
'La Guerre de M. Blum et de M. Litvinoff', _L'Insurgé,_ 20, 26 May 1937, 4.
_'Journal,_ par François Mauriac', _L'Insurgé,_ 20, 26 May 1937, 5.
'Le Complot de Genève', _L'Insurgé,_ 21, 2 June 1937, 4.
_'Pain de soldat,_ par Henry Poulaille, _El Requete,_ par Lucien Maulvault', _L'Insurgé,_ 21, 2 June 1937, 5.
'Le Chantage à l'antihitlérisme', _L'Insurgé,_ 22, 9 June 1937, 4.
_'Les Plus Beaux de nos jours,_ par Marcel Arland', _L'Insurgé,_ 22, 9 June 1937, 5.
'La Seule Manière d'être français', _L'Insurgé,_ 23, 16 June 1937, 4.
_'Nouvelle Histoire de Mouchette,_ par Georges Bernanos', _L'Insurgé,_ 23, 16 June 1937, 5.
'L'URSS a de plus en plus besoin de la guerre', _L'Insurgé,_ 24, 23 June 1937, 4.
_'Souvenirs de guerre,_ par Alain', _L'Insurgé,_ 24, 23 June 1937, 5.
'Hommage à Claude Séverac', _Aux écoutes,_ 26 June 1937, 11.
'Il ne suffit pas de dire : ni Berlin, ni Moscou', _L'Insurgé,_ 25, 30 June 1937, 4.
_'Les Aventures de Sophie,_ par Paul Claudel', _L'Insurgé,_ 25, 30 June 1937, 5.
'Pour combattre l'Allemagne, il faut soutenir Franco', _L'Insurgé,_ 26, 7 July 1937, 4.
_'L'Honneur de servir,_ par Henri Massis', _L'Insurgé,_ 26, 7 July 1937, 5.
_'Faux passeports,_ par Charles Plisnier', _L'Insurgé,_ 27, 14 July 1937, 5.
_'Le Démon du bien,_ par Henry de Montherlant', _L'Insurgé,_ 28, 21 July 1937, 5.
_'Les Vergers sur la mer, Mes idées politiques,_ par Charles Maurras', _L'Insurgé,_ 29, 28 July 1937, 5.
_'L'École du rénégat,_ par Jean Fontenoy', _L'Insurgé,_ 30, 4 August 1937, 5.
_'La Pêche miraculeuse,_ par Guy de Pourtalès, _Camp volant_ par André Fraigneau', _L'Insurgé,_ 31, 11 August 1937, 5.
_'Journal d'un intellectuel en chômage,_ par Denis de Rougemont', _L'Insurgé,_ 32, 18 August 1937, 4.
_'Lettres à un jeune poète,_ par R. M. Rilke, _Gérard de Nerval,_ par Albert Béguin', _L'Insurgé,_ 33, 25 August 1937, 4.
_'Les Hommes gris,_ par Ettore Settanni [sic]', _L'Insurgé,_ 34, 1 September 1937, 4 (republished as 'Le Monologue intérieur', FP, 278–81).
_'L'Opéra politique,_ par Henri Pollès', _L'Insurgé,_ 35, 8 September 1937, 4.
_'La Paix des profondeurs,_ par Aldous Huxley', _L'Insurgé,_ 36, 15 September 1937, 5.
_'Le Garçon savoyard,_ par C.-F. Ramuz', _L'Insurgé,_ 37, 22 September 1937, 6.
_'Les Vagues,_ par Virginia Woolf', _L'Insurgé,_ 38, 29 September 1937, 5 (republished as 'Le Temps et le roman', FP 282–6).
'Pour une diplomatie révolutionnaire', _L'Insurgé,_ 42, 27 October 1937, 6.
'La France, nation à venir', _Combat,_ 19, November 1937, 131–2.
'On demande des dissidents', _Combat,_ 20, December 1937, 154–5 (republished in _Gramma,_ 5, 1976, 63–5).
'Procès-verbal de carence' (30.12.37), signed by Henri Israël and Maurice Blanchot, in their role as seconds to Paul Lévy, _Aux écoutes,_ 1 January 1938, 17.
'Après une année', from a tribute to Claude Séverac on the anniversary of her death, _Aux écoutes,_ 11 June 1938, 25.
'L'Ébauche d'un roman' (on Sartre's _La Nausée), Aux écoutes,_ 30 July 1938, 31 ( _BR,_ 33–4).
'Lautréamont', _La Revue française des idées et des œuvres,_ 1, April 1940, 67–72 (FP, 197–202).
'Chronique de la vie intellectuelle', _Journal des débats,_ 16 April 1941, 3.
'Le Silence des écrivains', _Journal des débats,_ 19 April 1941, 3 ( _BR,_ 25–8).
'Le Biographe connaît le "génie" et ignore l'"homme"', _Journal des débats,_ 23 April 1941, 3 (republished as 'Le Silence de Mallarmé', FP, 117–20).
'Chronique de la vie intellectuelle', _Journal des débats,_ 4 May 1941, 3.
'Le Jeune Roman', _Journal des débats,_ 14 May 1941, 3 ( _FP,_ 209–12; _BR,_ 35–7).
_'L'Herbe pousse dans la prairie_ par Raymond Dumay; _Baragne_ par C.-F. Landry', _Journal des débats,_ 22 May 1941, 3.
'La France et la civilisation contemporaine', _Journal des débats,_ 26–27 May 1941, 3.
'L'Art de Montesquieu', _Journal des débats,_ 2–3 June 1941, 3.
'Une œuvre à sauver', _Journal des débats,_ 5 June 1941, 1.
'La Naissance d'un mythe', _Journal des débats,_ 9–10 June 1941, 3 ( _FP,_ 219–23).
'Recherche de la tradition', _Journal des débats,_ 16–17 June 1941, 3 ( _BR,_ 29–32).
'La Critique d'Albert Thibaudet', _Journal des débats,_ 23–24 June 1941, 3 ( _FP,_ 323–7).
'La Solitude de Péguy', _Journal des débats,_ 30 June–1 July 1941, 3 ( _FP,_ 318–22).
'Roman et poésie', _Journal des débats,_ 7–8 July 1941, 3.
'Roman et morale', _Journal des débats,_ 14–15 July 1941, 3 ( _FP,_ 268–72).
'L'Art du roman chez Balzac', _Journal des débats,_ 21–22 July 1941, 3 ( _FP,_ 203–8).
'Culture et civilisation', _Journal des débats,_ 31 July 1941, 3.
'Éloge de la rhétorique', _Journal des débats,_ 1 August 1941, 1.
'Chaminadour', _Journal des débats,_ 4–5 August 1941, 3 ( _FP,_ 262–5).
'Une vue de Descartes', _Journal des débats,_ 11–12 August 1941, 3.
'Un roman de M. Mauriac', _Journal des débats,_ 18–19 August 1941, 3.
'Léon-Paul Fargue et la création poétique', _Journal des débats,_ 25–26 August 1941, 3 ( _FP,_ 170–4).
'Le Secret de Melville', _Journal des débats,_ 1–2 September 1941, 3 ( _FP,_ 273–7).
'La Pensée d'Alain', _Journal des débats,_ 11 September 1941, 3 ( _FP,_ 343–8).
'Jeunes Romanciers', _Journal des débats,_ 18 September 1941, 3.
'Le Théâtre et le public', _Journal des débats,_ 23 September 1941, 3.
'Inspirations méditerranéennes', _Journal des débats,_ 30 September 1941, 3.
'L'Ange du bizarre', _Journal des débats,_ 7 October 1941, 3 ( _FP,_ 256–9).
'Auteurs inconnus ou méconnus', _Journal des débats,_ 14 October 1941, 3.
'La Terreur dans les _lettres', Journal des débats,_ 21 October 1941, 3 ( _CLP,_ 9–15; _BR,_ 49–53).
'L'Écrivain et le public', J _ournal des débats,_ 4 November 1941, 3.
'Goethe et Eckermann', _Journal des débats,_ 11 November 1941, 3 (FP, 306–10).
'Le Secret de J.-K. Huysmans', _Journal des débats,_ 18 November 1941, 3.
'Comment la littérature est-elle possible?' (I), _Journal des débats,_ 25 November 1941, 3 ( _CLP,_ 16–20; FP, 92–7; _BR,_ 53–6).
'Comment la littérature est-elle possible?' (II), _Journal des débats,_ 2 December 1941, 3 ( _CLP,_ 21–7; FP, 97–101; _BR,_ 56–60).
'L'Homme pressé', _Journal des débats,_ 9 December 1941, 3.
'Le Roman de la Sorbonne', _Journal des débats,_ 16 December 1941, 3.
'Le _Journal_ de Kierkegaard', _Journal des débats,_ 23 December 1941, 3 ( _FP,_ 25–30).
'Paradoxes sur le roman', _Journal des débats,_ 30 December 1941, 3.
'De l'insolence considérée comme l'un des Beaux-Arts', _Journal des débats,_ 6 January 1942, 3 ( _FP,_ 349–52).
'Du Moyen Age au Symbolisme', _Journal des débats,_ 15 January 1942, 3.
'Littérature', _Journal des débats,_ 20 January 1942, 3 ( _FP,_ 109–14).
'Littérature', _Journal des débats,_ 27 January 1942, 3 (republished as 'Romans mythologiques', _FP,_ 224–8).
'Un roman de Madame Colette', _Journal des débats,_ 3 February 1942, 3.
'Bergson et le symbolisme', _Journal des débats,_ 10 February 1942, 3 (republished in part in: _FP,_ 132–5).
'Autour de la pensée hindoue', _Journal des débats,_ 17 February 1942, 3 ( _FP,_ 42–6).
'La Poésie de Mallarmé est-elle obscure?', _Journal des débats,_ 24 February 1942, 3 ( _FP,_ 126–31).
'Contes et récits', _Journal des débats,_ 3 March 1942, 3.
'La Politique de Sainte-Beuve', _Journal des débats,_ 10 March 1942, 1–2.
'Le Mariage du ciel et de l'enfer', _Journal des débats,_ 25 March 1942, 3 ( _FP,_ 37–41).
'Le Silence de Mallarmé', _Journal des débats,_ 1 April 1942, 3 ( _FP,_ 121–5).
'L'Énigme du roman', _Journal des débats,_ 8 April 1942, 3 ( _FP,_ 213–18).
'Récits d'enfance', _Journal des débats,_ 15 April 1942, 3.
'Roman et poésie', _Journal des débats,_ 22 April 1942, 3 ( _FP,_ 232–6).
'Une œuvre de M. Paul Claudel', _Journal des débats,_ 29 April 1942, 3 (republished as 'Une œuvre de Paul Claudel', _FP,_ 328–32, 335–6).
'Le Destin de M. Jean Giono', _Journal des débats,_ 6 May 1942, 3.
'La Révélation de Dante', _Journal des débats,_ 13 May 1942, 3.
'Les Trois Romans', _Journal des débats,_ 20 May 1942, 3.
'Réflexions sur la jeune poésie', _Journal des débats,_ 27 May 1942, 3 ( _FP,_ 149–52).
'Les Poètes baroques du XVIIe siècle', _Journal des débats,_ 3 June 1942, 3 ( _FP,_ 143–8).
'Molière', _Journal des débats,_ 10 June 1942, 3 ( _FP,_ 295–9).
'Après _Les Liaisons dangereuses', Journal des débats,_ 17 June 1942, 3.
'Les Malheurs de Duranty', _Journal des débats,_ 24 June 1942, 3.
'Les Chances du réalisme', _Journal des débats,_ 1 July 1942, 3.
'Jupiter, Mars, Quirinus', _Journal des débats,_ 8 July 1942, 3.
'Au pays de la magie', _Journal des débats,_ 15 July 1942, 3.
'Situation de Lamartine', _Journal des débats,_ 22 July 1942, 3 ( _FP,_ 175–9).
'Histoire de fantôme', _Journal des débats,_ 29 July 1942, 3.
'La Poétique', _Journal des débats,_ 5 August 1942, 3 ( _FP,_ 136–42).
'Pour le bon usage de Montherlant', _Journal des débats,_ 12 August 1942, 3.
'Le Roman de l'étranger', _Journal des débats,_ 19 August 1942, 3 ( _FP,_ 248–53).
'Stendhal et les âmes sensibles', _Journal des débats,_ 26 August 1942, 3 ( _FP,_ 300–5).
'De l'humour romanesque', _Journal des débats,_ 2 September 1942, 3 (republished as 'Romans mythologiques', _FP,_ 228–31 ).
'Considérations sur le héros', _Journal des débats,_ 9 September 1942, 3.
"'Le Plus Beau Livre du romantisme"', _Journal des débats,_ 16 September 1942, 3.
'Cette affaire infernale', _Journal des débats,_ 23 September 1942, 3.
'Roman et souvenirs', _Journal des débats,_ 30 September 1942, 3 (republished as 'Poésie et roman', _FP,_ 237–41).
'Une édition critique des _Fleurs du mal', Journal des débats,_ 7 October 1942, 3 (republished as 'Une édition des _Fleurs du mal', FP,_ 180–6).
'Vigiles de l'esprit', _Journal des débats,_ 14 October 1942, 3.
'Le Feu, l'eau et les rêves', _Journal des débats,_ 21 October 1942, 3.
'Le Souvenir de Maupassant', _Journal des débats,_ 28 October 1942, 3.
'Maître Eckhart', _Journal des débats,_ 4 November 1942, 3 ( _FP,_ 31–6).
'Au sujet des _Nourritures terrestres', Journal des débats,_ 11 November 1942, 3 ( _FP,_ 337–42).
'Les Inconnus du romantisme', _Journal des débats,_ 18 November 1942, 3.
'Le Mythe de Sisyphe', _Journal des débats,_ 25 November 1942, 3 ( _FP,_ 65–71).
_'Refuges_ de Léon-Paul Fargue', _Journal des débats,_ 2 December 1942, 3.
'Œuvres poétiques', _Journal des débats,_ 9 December 1942, 3 (republished in part as 'Poésie involontaire', _FP,_ 154–6).
'Les _Mauvaises Pensées_ de Paul Valéry', _Journal des débats,_ 16 December 1942, 3.
'Romans nouveaux', _Journal des débats,_ 23 December 1942, 3.
'De Taine à M. de Pesquidoux', _Journal des débats,_ 30 December 1942, 3.
'Nicolas de Cues', _Journal des débats,_ 6 January 1943, 3.
'La Correspondance de Madame de Lafayette', _Journal des débats,_ 13 January 1943, 3.
'Le Livre', _Journal des débats,_ 20 January 1943, 3.
'Roman et récit de guerre', _Journal des débats,_ 17 January 1943, 3.
'André Gide et Goethe', _Journal des débats,_ 3 February 1943, 3 ( _FP,_ 311–17).
'Recherches sur le langage', _Journal des débats,_ 10 February 1943, 3 ( _FP,_ 102–8).
'Charles-Louis Philippe', _Journal des débats,_ 17 February 1943, 3.
'Romans de la terre', _Journal des débats,_ 24 February 1943, 3.
'Les Souvenirs de Tocqueville', _Journal des débats,_ 3 March 1943, 3.
'Rilke', _Journal des débats,_ 10 March 1943, 3 ( _FP,_ 59–64).
'Le Symbolisme et les poètes d'aujourd'hui', _Journal des débats,_ 17 March 1943, 3.
'Les Carnets de Léonard de Vinci', _Journal des débats,_ 24 March 1943, 3 ( _FP,_ 86–91).
'Sur la pièce de M. de Montherlant', _Journal des débats,_ 31 March 1943, 3.
'Le Roman de Marie Dorval et de Vigny', _Journal des débats,_ 8 April 1943, 3.
'Romans', _Journal des débats,_ 14 April 1943, 3.
'Machiavel', _Journal des débats,_ 21 April 1943, 3.
'L'Éloquence et la littérature', _Journal des débats,_ 28 April 1943, 3.
'L'Expérience intérieure', _Journal des débats,_ 5 May 1943, 3 ( _FP,_ 47–52).
'L'Expérience de Proust', _Journal des débats,_ 12 May 1943, 3 ( _FP,_ 52–8).
'De l'œuvre de M. Jouhandeau', _Journal des débats,_ 19 May 1943, 3 (republished in part as 'Chaminadour', _FP,_ 265–7).
'Les Treize Formes d'un roman', _Journal des débats,_ 26 May 1943, 3.
'De la louange à la souveraineté', _Journal des débats,_ 2 June 1943, 3.
'La Poésie religieuse', _Journal des débats,_ 9 June 1943, 3.
'Romans', _Journal des débats,_ 14–15 June 1943, 3.
'Suite française', _Journal des débats,_ 23 June 1943, 3.
'Le Fantastique de Hoffmann', _Journal des débats,_ 30 June 1943, 3.
'Sur la Chanson de Roland', _Journal des débats,_ 7 July 1943, 3.
'Kierkegaard et l'esthétique', _Journal des débats,_ 13 July 1943, 3.
'L'Art de la nouvelle', _Journal des débats,_ 21 July 1943, 3.
'Le Mythe d'Oreste', _Journal des débats,_ 27 July 1943, 3 ( _FP,_ 72–8).
'La Religion de Rabelais', _Journal des débats,_ 4 August 1943, 3.
'Études sur Paul Claudel', _Journal des débats,_ 11 August 1943, 3 (republished in part as 'Une œuvre de Paul Claudel', _FP,_ 332–5).
'Lecture de Phèdre', _Journal des débats,_ 18 August 1943, 3 (republished as 'Le Mythe de Phèdre', _FP,_ 79–85).
'Romancières d'aujourd'hui', _Journal des débats,_ 25 August 1943, 3.
'Poésie et langage', _Journal des débats,_ 1 September 1943, 3 ( _FP,_ 157–62).
'Voyages de Montesquieu', _Journal des débats,_ 8 September 1943, 3.
'Après Rimbaud', _Journal des débats,_ 15 September 1943, 2–3 ( _FP,_ 163–9).
'Une histoire de la littérature française', _Journal des débats,_ 22 September 1943, 3.
'L'Influence du roman américain', _Journal des débats,_ 29 September 1943, 3.
'La Mystique d'Angelus Silesius', _Journal des débats,_ 6 October 1943, 3.
'Récits autobiographiques', _Journal des débats,_ 13 October 1943, 3.
'L'Histoire et les chefs-d'œuvre', _Journal des débats,_ 20 October 1943, 3.
'Mallarmé et l'art du roman', _Journal des débats,_ 27 October 1943, 3 ( _FP,_ 189–96; _BR,_ 43–8).
'Une étude sur l'apocalypse', _Journal des débats,_ 3 November 1943, 3.
'La Fontaine sans les Fables', _Journal des débats,_ 13–14 November 1943, 2.
'Le Baron d'Holbach', _Journal des débats,_ 27–28 November 1943, 3.
'Le Roman pur', _Journal des débats,_ 4–5 December 1943, 2 ( _BR,_ 38–42).
'Les Plaintes de l'ombre', _Journal des débats,_ 10 December 1943, 1.
'Le Roman du regard', _Journal des débats,_ 18–19 December 1943, 2.
'Tradition et surréalisme', _Journal des débats,_ 23 December 1943, 1.
'Sur un monde en ruines', _Journal des débats,_ 30 December 1943, 1.
'Le Mystère de la critique', _Journal des débats,_ 6 January 1944, 2–3.
'Le Pèlerinage aux sources', _Journal des débats,_ 13 January 1944, 2–3.
'D'un roman à l'autre', _Journal des débats,_ 20 January 1944, 2–3.
'Les Quatre Évangiles , Journal des débats, 27 January 1944, 2–3.
'De Jean-Paul à Giraudoux', _Journal des débats,_ 3 February 1944, 2–3.
'Journal sans épisode', _Journal des débats,_ 10 February 1944, 2–3.
'Autour du langage', _Journal des débats,_ 17 February 1944, 2–3.
'Le Roman d'Aïssé', _Journal des débats,_ 24 February 1944, 2–3.
'Le Bonheur de conter', _Journal des débats,_ 2 March 1944, 2–3.
'Les Idoles hors la loi', _Journal des débats,_ 9 March 1944, 2–3.
'L'Art d'André Dhotel', _Journal des débats,_ 16 March 1944, 2–3.
'Le Travail de Balzac', _Journal des débats,_ 23 March 1944, 2–3.
'Le Roman noir', _Journal des débats,_ 30 March 1944, 2–3.
'Les Secrets du rêve', _Journal des débats,_ 6 April 1944, 2–3.
'Un roman de Jarry', _Journal des débats,_ 13 April 1944, 2–3.
'Nouvelles et récits', _Journal des débats,_ 20 April 1944, 2–3.
'Le Secret de Chateaubriand', _Journal des débats,_ 27 April 1944, 2–3.
'Romans fantastiques', _Journal des débats,_ 4 May 1944, 2–3.
'L'Air et les songes', _Journal des débats,_ 11 May 1944, 2–3.
'Le Premier Roman de Joyce', _Journal des débats,_ 18 May 1944, 2–3.
'L'Accent du secret', _Journal des débats,_ 25 May 1944, 2–3.
'Le "Je" littéraire', _Journal des débats,_ 1 June 1944, 2–3.
'Charles Cros', _Journal des débats,_ 8 June 1944, 2–3.
'Naissance de Rome', _Journal des débats,_ 15 June 1944, 2–3.
'William Blake', _Journal des débats,_ 22 June 1944, 2–3.
'Des diverses façons de mourir', _Journal des débats,_ 29 June 1944, 2–3.
'Pages de Paul Claudel', _Journal des débats,_ 6 July 1944, 2–3.
'Récits', _Journal des débats,_ 13–14 July 1944, 2–3.
'Léon Bloy', _Journal des débats,_ 20 July 1944, 2–3.
'Poèmes', _Journal des débats,_ 27 July 1944, 2–3.
'Le Souci de sincérité', _Journal des débats,_ 3 August 1944, 2–3.
'Fils de personne', _Journal des débats,_ 10 August 1944, 2–3.
'L'Expérience magique d'Henri Michaux', _Journal des débats,_ 17 August 1944, 2–3.
'Le Tout-Puissant', _La Table Ronde,_ 3, July 1945, 189–98 (republished, in a much revised version, in _TH,_ 221–4).
'Quelques réflexions sur le surréalisme', _L'Arche,_ 8, August 1945, 93–104 (republished as 'Réflexions sur le surréalisme', _PF,_ 90–102).
'Autour du roman', _L'Arche,_ 9, September 1945, 105–14.
'Les Romans de Sartre', _L'Arche,_ 10, October 1945, 120–34 ( _PF,_ 188–203).
'Le Mythe Giraudoux', _Paysage Dimanche,_ 17, 7 October 1945, 5.
'Le Roman de Jean-Paul Sartre', _Paysage Dimanche,_ 19, 21 October 1945, 5.
'La Lecture de Kafka', _L'Arche,_ 11, November 1945, 107–16 ( _PF,_ 9–19; _K_ , 62–74).
'Les Malheurs de _Peau d'âne', Paysage Dimanche,_ 21, 4 November 1945, 5.
'A l'ombre du romanesque', _Paysage Dimanche,_ 23, 18 November 1945, 3.
'La Critique de Charles du Bos', _Paysage Dimanche,_ 25, 2 December 1945, 3.
'L'Enchantement de Melville', _Paysage Dimanche,_ 27, 16 December 1945, 3.
'Sur André Malraux', _Paysage Dimanche,_ 29, 30 December 1945, 3 (republished as 'Note sur Malraux', _PF,_ 204–7).
'Du côté de Nietzsche', _L'Arche,_ 12, December 1945–January 1946, 103–12 ( _PF,_ 278–89).
_'L'Espoir_ d'André Malraux', in _L'Espagne libre,_ edited by Georges Bataille, Paris, Calmann-Lévy (collection 'Actualité'), 1946, 106–11.
'L'Énigme de la critique', _Carrefour,_ 75, 24 January 1946, 6.
'L'Homme noir du XVIIe siècle', _Saisons,_ 2, Spring 1946, 69–79.
'Mallarmé et le langage', _L'Arche,_ 14, March–April 1946, 134–46 (republished as 'Le Mythe de Mallarmé', _PF,_ 35–48).
'En bonne voie', _Cahiers de la Pléiade,_ 1, April 1946, 143–51 (republished, in a revised version, in _TH,_ 233–8).
'Le Mystère dans les lettres', _L'Arche,_ 15, May 1946, 95–111 ( _PF,_ 49–65).
'Le Paradoxe d'Aytré', _Les Temps modernes,_ 9, June 1946, 1576–93 ( _PF,_ 66–78).
'De Lautréamont à Miller', _L'Arche,_ 16, June 1946, 129–39 ( _PF,_ 160–72).
'Traduit de . . .', _L'Arche,_ 17, July 1946, 114–28 ( _PF,_ 173–87).
'Quelques remarques sur Sade', _Critique,_ 3–4, August–September 1946, 239–46.
'L'Honneur des poètes', _L'Arche,_ 18–19, August–September 1946, 162–74.
'René Char', _Critique,_ 5, October 1946, 387–99 ( _PF,_ 103–14).
'Adolphe ou le malheur des sentiments vrais', _L'Arche,_ 20, October 1946, 82–94 ( _PF,_ 221–37).
'Valéry et Faust', _L'Arche,_ 21, November 1946, 92–102 ( _PF,_ 263–77).
'La Parole "sacrée" de Hölderlin', _Critique,_ 7, December 1946, 579–96 ( _PF,_ 115–32).
'Gide et la littérature d'expérience', _L'Arche,_ 23, January 1947, 87–98 ( _PF,_ 208–20).
'L'Échec de Baudelaire' (I), _L'Arche,_ 24, February 1947, 80–91 ( _PF,_ 133–42).
'L'Échec de Baudelaire' (II), _L'Arche,_ 25, March 1947, 97–107 ( _PF,_ 142–51).
'Le Sommeil de Rimbaud', _Critique,_ 10, March 1947, 195–202 ( _PF,_ 152–9).
'L'Idylle' (July 1936), _La Licorne,_ 1, March 1947, 33–58 ( _RE_ , 9–97; _AC,_ 9–56).
'Note sur Pascal', _L'Arche,_ 26, April 1947, 107–21 (republished as 'La Main de Pascal', _PF,_ 249–62).
'Grève désolée, obscur malaise', _Cahiers de la Pléiade,_ 2, April 1947, 134–7.
'Regards d'outre-tombe', _Critique,_ 11, April 1947, 291–301 ( _PF,_ 238–48).
'Le Roman, œuvre de manvaise foi', _Les Temps modernes,_ 19, April 1947, 1304–17 ( _BR,_ 61–73).
'Du merveilleux', _L'Arche,_ 27, May 1947, 120–33.
_Le Dernier Mot_ (1935), Éditions de la revue Fontaine, Collection 'L'Age d'or' (45), 1947 ( _RE,_ 99–146; _AC,_ 57–81).
'A la rencontre de Sade', _Les Temps modernes,_ 25, October 1947, 577–612 ( _LS,_ 17–49; _BR,_ 74–99).
'Le Règne animal de l'esprit', _Critique,_ 18, November 1947, 387–405 (republished as the first part of 'La Littérature et le droit à la mort', _PF,_ 293–311; _K,_ 11–34).
'La Littérature et le droit à la mort', _Critique, 20,_ January 1948, 30–47 ( _PF,_ 312–31; _K,_ 35–61).
"'Un livre vivant"', _Critique,_ 22, March 1948, 195–205 (republished in Restif de la Bretonne, _Sara,_ Paris, Stock, 1949; and in Maurice Blanchot, _Sade et Restif de la Bretonne,_ Brussels, Éditions Complexe, 1986, 142–56).
'Lautréamont ou l'espérance d'une tête', _Cahiers d'Art,_ 1, 1948, 69–71 ( _LS,_ 58–9, 84–92).
'Lautréamont et le mirage des sources', _Critique,_ 25, June 1948, 483–98 ( _LS,_ 55–8, 60–70).
'Les Plaisirs de la vertu', _Cahiers de la Pléiade,_ 4, June 1948, 71–85 (republished in Restif de la Bretonne, _Sara,_ Paris, Stock, 1949; and in Maurice Blanchot, _Sade et Restif de la Bretonne,_ Brussels, Éditions Complexe, 1986, 105–42).
'Un récit', _Empédocle,_ 2, May 1949, 13–22 (republished as _La Folie du jour,_ Montpellier, Fata morgana, 1973).
'Kafka et la littérature', _Cahiers de la Pléiade,_ 7, July 1949, 93–105 ( _PF,_ 20–34; _K,_ 75–93).
'Lautréamont ou l'espérance d'une tête', in Lautréamont, _Les Chants de Maldoror,_ Paris, Le Club français du livre, 1950, xi–xxvi ( _LS,_ 58–9, 84–92, 164–5, 166, 185–8; this is an extended version of the text published under the same title in 1948).
'La Condition critique', _L'Observateur,_ 6, 15 May 1950, 18.
'Le Docteur Faustus', _L'Observateur,_ 8, 1 June 1950, 18.
'Le Compagnon de route', _L'Observateur,_ 22 June 1950, 17.
'Au-dessous du volcan', _L'Observateur,_ 13, 6 July 1950, 18.
'Les Justes', _L'Observateur,_ 15, 20 July 1950, 17.
'Hölderlin', _L'Observateur,_ 17, 3 August 1950, 19.
'Le Destin de l'œuvre', L _'Observateur,_ 19, 17 August 1950, 19.
'Thomas Mann et le mythe de Faust', _Critique,_ 41, October 1950, 3–21.
'Le Musée, l'art et le temps' (I), _Critique,_ 43, December 1950, 195–208 ( _A,_ 21–36).
'Le Musée, l'art et le temps' (II), _Critique,_ 44, January 1951, 30–42 ( _A,_ 36–51).
'La Folie par excellence', _Critique,_ 45, February 1951, 99–118 (republished, with a supplementary note in [1970] Karl Jaspers, _Strindberg et Van Gogh, Swedenborg–Hölderlin,_ Paris, Minuit, [1953] 1970, 9–32; _BR,_ 110–28).
'Les Deux Versions de l'imaginaire', _Cahiers de la Pléiade,_ 12, Spring–Summer 1951, 115–25 ( _EL,_ 266–74).
'Le Retour', _Botteghe Oscure,_ VII, 1951, 416–24 ( _MV,_ 7–25).
'Kafka et l'exigence de l'œuvre', _Critique,_ 58, March 1952, 195–221 ( _K_ , 94–131; republished as 'L'Espace et l'exigence de l'œuvre', _EL,_ 45–81).
'L'Art, la littérature et l'expérience originelle' (I), _Les Temps modernes,_ 79, May 1952, 1921 –51 (republished as 'La Littérature et l'expérience originelle, I: L'Avenir et la question de l'art' and 'La Littérature et l'expérience originelle, II: Les Caractères de l'œuvre d'art', _EL,_ 219–44).
'L'Art, la littérature et l'expérience originelle' (II), _Les Temps modernes,_ 80, June 1952, 2195–212 (republished as 'La Littérature et l'expérience originelle, III: L'Expérience originelle', _EL,_ 245–60).
'Mallarmé et l'expérience littéraire', _Critique,_ 62, July 1952, 597–91 (republished as 'L'Expérience de Mallarmé', _EL,_ 30–41).
'La Mort possible', _Critique,_ 66, November 1952, 915–33 ( _EL,_ 88–107; and, under the title 'La Mort contente', _K_ , 132–9).
'Le Compagnon de route', _Botteghe Oscure,_ X, 1952, 39–53 ( _CQ,_ 7–30, 42–6, 77–8).
'La Solitude essentielle', _La Nouvelle Revue française,_ 1, January 1953, 75–90 ( _EL,_ 11–25).
'Continuez autant qu'il vous plaira', _La Nouvelle Revue française,_ 2, February 1953, 308–14 (republished as 'L'Inspiration, le manque d'inspiration', _EL,_ 185–91).
'L'Écriture automatique, l'inspiration', _La Nouvelle Revue française,_ 3, March 1953, 485–92 (republished as 'L'Inspiration, le manque d'inspiration', _EL,_ 191–6).
'La Bête de Lascaux', _La Nouvelle Revue française,_ 4, April 1953, 684–93 (republished as a separate volume by GLM in 1958, and again by Fata morgana in 1986; the essay also appears in _L'Herne,_ special issue on René Char, 1971, 71–7).
'Rilke et l'exigence de la mort' (I), _Critique,_ 71, April 1953, 291–304 (republished as 'Rilke et l'exigence de la mort, 2: L'Espace de la mort', _EL,_ 135–50).
'Lire', _La Nouvelle Revue française,_ 5, May 1953, 876–83 ( _EL,_ 199–206).
'Rilke et l'exigence de la mort' (II), _Critique,_ 72, May 1953, 387–99 (republished as 'Rilke et l'exigence de la mort, 3: Transmutation de la mort', _EL,_ 150–64).
'Le Regard d'Orphée', _Cahiers d'Art,_ 28, 1, June 1953, 73–5 ( _EL,_ 179–84).
'L'Expérience d'Igitur', _La Nouvelle Revue française,_ 6, June 1953, 1075–86 ( _EL,_ 108–20).
'Où va la littérature?' (I), _La Nouvelle Revue française,_ 7, July 1953, 98–107 (republished as 'La Disparition de la littérature', _LV,_ 237–45; _BR,_ 136–42).
'Où va la littérature?' (II), _La Nouvelle Revue française,_ 8, August 1953, 291–303.
'Plus loin que le degré zéro', _La Nouvelle Revue française,_ 9, September 1953, 485–94 (republished as 'La Recherche du point zéro', _LV,_ 246–55; _BR,_ 143–50).
'Où maintenant? Qui maintenant?', _La Nouvelle Revue française,_ 10, October 1953, 678–86 ( _LV,_ 256–64; _SS,_ 192–8).
'Le Dehors, la nuit', _La Nouvelle Revue française,_ 11, November 1953, 877–85 ( _EL,_ 169–78).
'L'Œuvre et la communication', _La Nouvelle Revue française,_ 12, December 1953, 1064–71 (republished as 'La Communication', _EL,_ 207–16).
'Dionys Mascolo: _Le Communisme', La Nouvelle Revue française,_ 12, December 1953, 1096–9 (republished as 'Sur une approche du communisme (besoins, valeurs)', A, 109–14).
'Quand la morale se tait', _La Nouvelle Revue française,_ 13, January 1954, 96–104 (republished as '"Il ne saurait être question de bien finir"', _LV,_ 37–44; _SS,_ 45–51).
'Orphée, Don Juan, Tristan', _La Nouvelle Revue française,_ 15, March 1954, 492–501 (republished as 'Réflexions sur l'enfer, 4: Orphée, Don Juan, Tristan', _EI,_ 280–8).
'Réflexions sur l'enfer' (I), _La Nouvelle Revue fraçaise,_ 16, April 1954, 677–86 ( _EI,_ 256–64).
'Réflexions sur le nihilisme' (II), _La Nouvelle Revue française,_ 17, May 1954, 850–9 (republished as 'Réflexions sur l'enfer, 2: Victoire logique sur "l'absurde"', _EI,_ 264–71).
'Tu peux tuer cet homme' (III), _La Nouvelle Revue française,_ 18, June 1954, 1059–69 (republished as 'Réflexions sur l'enfer, 3: Tu peux tuer cet homme', _EI,_ 271–80).
'Le Chant des sirènes', _La Nouvelle Revue française,_ 19, July 1954, 95–104 (republished as 'La Rencontre de l'imaginaire', _LV,_ 9–17; _SS,_ 59–65).
'Proust', _La Nouvelle Revue française,_ 20, August 1954, 286–94 (republished as 'L'Expérience de Proust, 1: Le Secret de l'écriture', _LV,_ 18–26; _SS,_ 66–78).
'Jean Santeuil', _La Nouvelle Revue française,_ 21, September 1954, 479–87 (republished as: 'L'Expérience de Proust, 2: L'Étonnante Patience', _LV,_ 26–34).
'Kafka et Brod', _La Nouvelle Revue française,_ 22, October 1954, 695–707 ( _A,_ 272–84; _K,_ 140–54).
'L'Échec de Milena', _La Nouvelle Revue française,_ 23, November 1954, 875–88 ( _K,_ 155–70).
'Le Tour d'écrou', _La Nouvelle Revue française,_ 24, December 1954, 1062–72 ( _LV,_ 155–64; _SS,_ 79–86).
'Le Tournant', _La Nouvelle Revue française,_ 25, January 1955, 110–20 (republished in part as 'L'Itinéraire de Hölderlin', _EL,_ 283–92).
'A toute extrémité', _La Nouvelle Revue française,_ 26, February 1955, 285–93 (republished in part as the first section of 'La Mort possible', _EL,_ 85–8; and also in part in _LV,_ 131–2).
'Mort du dernier écrivain', _La Nouvelle Revue française,_ 27, March 1955, 485–91 ( _LV,_ 265–70; _BR,_ 151–6).
'Sur le journal intime', _La Nouvelle Revue française,_ 28, April 1955, 683–91 (republished in part as 'Le Journal intime et le récit', _LV,_ 224–30).
'Le Secret du Golem', _La Nouvelle Revue française,_ 29, May 1955, 870–8 ( _LV,_ 108–15).
'Notes sur un roman', _La Nouvelle Revue française,_ 31, July 1955, 105–12 (republished as 'La Clarté romanesque', _LV,_ 195–201; _SS,_ 207–12).
'Broch', _La Nouvelle Revue française,_ 32, August 1955, 295–303 (republished as 'Broch, 1: Les Somnambules: le vertige logique', _LV,_ 136–42).
'L'Autre Claudel', _La Nouvelle Revue française,_ 33, September 1955, 404–23 (republished as 'Claudel et l'infini', _LV,_ 83–97).
'La Mort de Virgile', _La Nouvelle Revue française,_ 34, October 1955, 747–59 (republished as 'Broch, 2: La Mort de Virgile: la recherche de l'unité', _LV,_ 143–54).
'Naissance de l'art', _La Nouvelle Revue française,_ 35, November 1955, 923–33 ( _A,_ 9–20).
'Joubert', _La Nouvelle Revue française,_ 36, December 1955, 1127–36 (republished as 'Joubert et l'espace, 1: Auteur sans livre, écrivain sans écrit', _LV,_ 63–71; _SS,_ 52–8).
'Le Calme', _Botteghe Oscure,_ XVI, 1955, 28–36 ( _DH,_ 106–21).
'Joubert et Mallarmé', _La Nouvelle Revue française,_ 37, January 1956, 110–21 (republished as 'Joubert et l'espace, 2: Une première version de Mallarmé', _LV,_ 71–82).
'Combat avec l'ange', _La Nouvelle Revue française,_ 38, February 1956, 288–99 ( _A,_ 150–61).
'La Douleur du dialogue', _La Nouvelle Revue française,_ 39, March 1956, 492–503 ( _LV,_ 185'–94; _SS,_ 199–206).
'L'Homme au point zéro', _La Nouvelle Revue française,_ 40, April 1956, 683–94 ( _A,_ 87–97).
'H. H.', _La Nouvelle Revue française,_ 41, May 1956, 872–83 (republished as 'H. H., 1: La Poursuite de soi-même', _LV,_ 202–12).
'Le Jeu des jeux', _La Nouvelle Revue française,_ 42, June 1956, 1051–62 (republished as 'H. H., 2: Le Jeu des jeux', _LV,_ 212–23).
'La Pensée tragique' (I), _La Nouvelle Revue française,_ 43, July 1956, 113–22 ( _EI,_ 137–45).
'Pierre Angélique: _Madame Edwarda', La Nouvelle Revue française,_ 43, July 1956, 148–50 (republished as 'Le Récit et le scandale', _LV,_ 231–3).
'La Pensée tragique' (II), _La Nouvelle Revue française,_ 44, August 1956, 299–305 ( _EI,_ 146–52).
'Freud', _La Nouvelle Revue française,_ 45, September 1956, 484–96 (republished as 'La Parole analytique', _EI,_ 343–54).
'Le Dernier Homme', _La Nouvelle Revue française,_ 46, October 1956, 653–53 ( _DH,_ 1–23).
'Artaud', _La Nouvelle Revue française,_ 47, November 1956, 873–81 ( _LV,_ 45–52; _BR,_ 129–35).
'La Confession dédaigneuse', _La Nouvelle Revue française,_ 48, December 1956, 1050–6 (republished as 'La Chute : la fuite', _A,_ 228–35).
'Comme un jour de neige', _Botteghe Oscure,_ XVIII, 1956, 11–19 ( _DH,_ 125–7, 134–47).
'La Parole prophétique', _La Nouvelle Revue française,_ 49, January 1957, 101–10 ( _LV,_ 98–107).
'Brecht et le dégoût du théâtre', _La Nouvelle Revue française,_ 50, February 1957, 283–92 (republished as 'L'Effet d'étrangeté', _EI,_ 529–39).
'D'un art sans avenir', _La Nouvelle Revue française,_ 51, March 1957, 488–98 (republished in part under the same title, _LV_ , 132–5).
'Le Mal du musée', _La Nouvelle Revue française,_ 52, April 1957, 687–96 ( _A,_ 52–61).
'Le Temps des encyclopédies', _La Nouvelle Revue française_ , 53, May 1957, 863–74 ( _A,_ 62–8).
'La Grande Tromperie', _La Nouvelle Revue française,_ 54, June 1957, 1061–73 ( _BR,_ 157–66).
'Simone Weil et la certitude', _La Nouvelle Revue française,_ 55, July 1957, 103–14 (republished as 'L'Affirmation (le désir, le malheur)' I, _EI,_ 153–65).
'L'Expérience de Simone Weil', _La Nouvelle Revue française,_ 56, August 1957, 297–310 (republished as 'L'Affirmation (le désir, le malheur)' II, _EI,_ 165–79).
'Ecce liber', _La Nouvelle Revue française,_ 58, October 1957, 726–40 (republished as 'Le Livre à venir, 1: Ecce liber', _LV,_ 271–83; _SS,_ 227–35).
'Le Livre à venir', _La Nouvelle Revue française,_ 59, November 1957, 917–31 (republished as 'Le Livre à venir, 2: Une entente nouvelle de l'espace littéraire', _LV,_ 283–97; _SS,_ 235–48).
'L'Infini et l'infini', _La Nouvelle Revue française,_ 61, January 1958, 98–110 (republished in part as 'L'Infini littéraire: L'Aleph', _LV,_ 116–19, SS, 222–5, and in its entirety, together with a letter to Raymond Bellour, in _L'Herne,_ 1966, special issue on Henri Michaux, 80–8).
'Musil', _La Nouvelle Revue française,_ 62, February 1958, 301–9 (republished as 'Musil, 1: La Passion de l'indifférence', _LV,_ 165–173).
'Musil' (II), _La Nouvelle Revue française,_ 63, March 1958, 479–90 (republished as 'Musil, 2: L'Expérience de "l'autre état"', _LV,_ 173–84).
'La Puissance et la gloire', _La Nouvelle Revue française,_ 64, April 1958, 683–90 ( _LV,_ 298–304).
'La Cruelle Raison poétique', _Cahiers de la compagnie Madeleine Renaud–Jean-Louis Barrault,_ 22–3, May 1958, 66–73 (republished as 'La Cruelle raison poétique: rapace besoin d'envol', _EI,_ 432–8).
'L'Attrait, l'horreur du jeu', _La Nouvelle Revue française,_ 65, May 1958, 856–65.
'Jean-Jacques et la littérature', _La Nouvelle Revue française,_ 66, June 1958, 1057–66 (republished as 'Rousseau', _LV,_ 53–62).
'La Passion de l'indifférence', _La Nouvelle Revue française,_ 67, July 1958, 93–101 (republished as 'La Terreur de l'identification', A, 236–45).
'Nietzsche, aujourd'hui', _La Nouvelle Revue française,_ 68, August 1958, 284–95 (republished as 'Réflexions sur le nihilisme, 1: Nietzsche, aujourd'hui, _EI,_ 201–15).
'Passage de la ligne', _La Nouvelle Revue française,_ 69, September 1958, 468–79 (republished as 'Réflexions sur le nihilisme, 2: Passage de la ligne', _EI,_ 215–27).
'Le Refus', together with an extract from a letter by Blanchot, _Le 14 Juillet,_ 2, 25 October 1958, 3 ( _A,_ 130–1).
'L'Étrange et l'étranger', L _a Nouvelle Revue française,_ 70, October 1958, 673–83.
'La Vocation de Virginia Woolf', _La Nouvelle Revue française,_ 71, November 1958, 865–73 (republished as 'L'Échec du démon: la vocation', _LV,_ 120–8; _SS,_ 87–96).
'L'Attente', _Botteghe Oscure,_ XXII, 1958, 22–33 ( _AO,_ 7–13, 16–18, 19–21, 26–7, 31–4, 38, 44–5, 47–8, 49, 50–1, 52–3).
'Le Bon Usage de la science-fiction', _La Nouvelle Revue française,_ 73, January 1959, 91–100.
'Qu'en est-il de la critique?', _Arguments,_ 12–13, January–February–March 1959, 34–7 ( _LS,_ 9–14).
'Le Dernier Mot de Kafka' (I), _La Nouvelle Revue française,_ 74, February 1959, 294–300 (republished as 'Le Dernier Mot', _A,_ 285–91; _K,_ 202–9).
'Le Dernier Mot de Kafka' (II), _La Nouvelle Revue française,_ March 1959, 481–88 (republished as 'Le Dernier Mot', _A,_ 291–9; _K,_ 209–18).
'"Vaste comme la nuit"', _La Nouvelle Revue française,_ 76, April 1959, 684–95 ( _EI,_ 465–77).
'Enquête auprès d'intellectuels français', signed by Maurice Blanchot, André Breton, Dionys Mascolo, and Jean Schuster, _Le 14 Juillet,_ 10 April 1959 (reprinted in _Le 14 Juillet_ , 3, 18 June 1959, 1).
'La Perversion essentielle', _Le 14 Juillet,_ 3, 18 June 1959, 18–20 ( _BR,_ 167–73).
'Gog et Magog' (I), _La Nouvelle Revue française,_ 78, June 1959, 1068–74 ( _A,_ 259–65).
'Enquête sur la méthode critique d'Henri Guillemin, réponse de Maurice Blanchot', _Les Lettres nouvelles,_ 24 June 1959, 9–10.
'Gog et Magog' (II), _La Nouvelle Revue française,_ 79, July 1959, 101–7 ( _A,_ 265–71).
'La Fin de la philosophie', _La Nouvelle Revue française,_ 80, August 1959, 286–98 (republished as 'Lentes Funérailles', _A,_ 98–108).
'Le Grand Refus', _La Nouvelle Revue française,_ 82, October 1959, 678–89 (republished as 'Le Grand Refus, I', _EI,_ 46–57).
'Comment découvrir l'obscur?', _La Nouvelle Revue française,_ 83, November 1959, 867–79 (republished as 'Le Grand Refus, 2: Comment découvrir l'obscur?', _EI,_ 57–69).
'L'Attente', _Martin Heidegger zum siebzigsten Geburtstag,_ Pfullingen, Verlag Günter Neske, 1959, 217–24 ( _AO,_ passim; _BR,_ 272–8).
'Héraclite', _La Nouvelle Revue française,_ 85, January 1960, 93–106 ( _EI,_ 119–31).
'Albert Camus', _La Nouvelle Revue française,_ 87, March 1960, 403–4 (republished as the opening section of 'Le Détour vers la simplicité', _A,_ 214–15).
'Entretien sur un changement d'époque', _La Nouvelle Revue française,_ 88, April 1960, 724–34 ( _EI,_ 394–404).
'Le Détour vers la simplicité', _La Nouvelle Revue française,_ 89, May 1960, 925–37 ( _A,_ 215–27).
'La Marche de l'écrevisse', _La Nouvelle Revue française,_ 91, July 1960, 90–9 (republished as 'Parler, ce n'est pas voir', _EI,_ 35–45).
'Reprises', _La Nouvelle Revue française,_ 93, September 1960, 475–83 (reproduced in part as the opening sections of 'L'Effet d'étrangeté', _EI,_ 528–9; the remainder reproduced in part as 'Traduire', _A,_ 69–73).
'Déclaration sur le droit à l'insoumission dans la guerre d'Algérie', signed by Maurice Blanchot and 120 others, _Vérité-liberté,_ 4, September–October 1960 (reprinted in _Gramma,_ 3/4, 1976, 27–31).
'Oublieuse mémoire', _La Nouvelle Revue française,_ 94, October 1960, 746–52 ( _EI,_ 459–64).
'La Question la plus profonde' (I), _La Nouvelle Revue française,_ 96, December 1960, 1082–6 ( _EI,_ 12–16).
'La Question la plus profonde' (II), _La Nouvelle Revue française,_ 97, January 1961, 85–9 ( _EI_ , 16–21).
'Interview de certains signataires', interviews by Madeleine Chapsal with Maurice Blanchot, Nathalie Sarraute, Simone Signoret, and Jean Baby, _Le Droit à l'insoumission: 'le dossier des_ 121', Paris, François Maspero: Cahiers libres (14), 1961, 89–99 (the interview with Blanchot is on 90–3; _BR,_ 196–9).
'La Question la plus profonde' (III), _La Nouvelle Revue française,_ 98, February 1961, 282–91 ( _EI,_ 21–32).
'Notre épopée', _La Nouvelle Revue française,_ 100, April 1961, 690–8 (republished as 'Les Paroles doivent cheminer longtemps', _EI,_ 478–86).
'Rêver, écrire', _La Nouvelle Revue française,_ 102, June 1961, 1087–96 ( _A,_ 162–70).
'Rimbaud et l'œuvre finale', _La Nouvelle Revue française,_ 104, August 1961, 293–303 (republished as 'L'Œuvre finale', _EI,_ 421–31).
'L'Oubli, la déraison', _La Nouvelle Revue française,_ 106, October 1961, 676–86 ( _EI,_ 289–99; compare _AO,_ 87).
'Connaissance de l'inconnu', _La Nouvelle Revue française,_ 108, December 1961, 1081–94 ( _EI,_ 70–83).
'Tenir parole', _La Nouvelle Revue française,_ 110, February 1962, 290–8 ( _EI,_ 84–93).
'L'Indestructible', _La Nouvelle Revue française,_ 112, April 1962, 671–80 (republished in part, with extensive additions, as 'Le Rapport du troisième genre: Homme sans horizon', _EI,_ 99–100, 102–3; the remainder republished as 'L'Indestructible, 2: L'Espèce humaine', _EI,_ 191–200).
'Edmond Beaujon: _Le Dieu des Suppliants', La Nouvelle Revue française,_ 113, May 1962, 910–13 (republished as 'La Mesure, le suppliant', _EI,_ 132–6).
'L'Homme de la rue', _La Nouvelle Revue française,_ 114, June 1962, 1070–81 (republished as 'La Parole quotidienne', _EI,_ 355–65).
'Etre juif ' (I), _La Nouvelle Revue française,_ 116, August 1962, 279–85 (republished as 'L'Indestructible, 1: Etre juif', _EI,_ 180–6).
'Etre juif' (II), _La Nouvelle Revue française,_ 117, September 1962, 471–6 (republished as _'L'Indestructible,_ 1: Etre juif', _EI,_ 187–90).
'L'Expérience-limite', _La Nouvelle Revue française,_ 118, October 1962, 577–92 (republished as L'Expérience-limite, 1: L'Affirmation et la passion de la pensée négative', E _I, 3_ 00–13).
'L'Amitié', _Les Lettres nouvelles,_ 29, 1962, 7–12 ( _A,_ 326–30).
'La Littérature encore une fois' (I), _La Nouvelle Revue française,_ 120, December 1962, 1055–61 ( _EI,_ 583–90).
'La Littérature encore une fois' (II), _La Nouvelle Revue française,_ 121, January 1963, 102–7 ( _EI,_ 590–5).
'La Pensée et sa forme' (I), _La Nouvelle Revue française,_ 123, March 1963, 492–6 (republished as 'La Pensée et l'exigence de discontinuité', _EI,_ 1–6).
'La Pensée et sa forme' (II), _La Nouvelle Revue française,_ 124, April 1963, 684–8 (republished as 'La Pensée et l'exigence de discontinuité', _EI,_ 6–11).
'Ars Nova', _La Nouvelle Revue française,_ 125, May 1963, 879–87 ( _EI,_ 506–14).
'A Rose is a rose . . . _', La Nouvelle Revue françai_ se, 127, July 1963, 86–93 ( _EI,_ 498–505).
'René Char et la pensée du neutre', _L'Arc,_ 22, Summer 1963, 9–14 ( _EI,_ 439–46).
'Le Jeu de la pensée', _Critique,_ 195–6, August–September 1963, 734–41 (republished as 'L'Expérience-limite, 2: Le Jeu de la pensée', _EI,_ 313–22).
'Traces', _La Nouvelle Revue française,_ 129, September 1963, 472–80 (republished in part in _A,_ 246–52).
'Le Problème de Wittgenstein', _La Nouvelle Revue française,_ 131, November 1963, 866–75 ( _EI,_ 487–97).
'La Parole vaine', in Louis-René Des Forêts, _Le Bayard,_ Paris, Union générale d'éditions, 1963, 163–84 ( _A,_ 137–49).
'Le Pont de bois', _La Nouvelle Revue française,_ 133, January 1964, 90–103 (republished as 'Le Pont de bois (la répétition, le neutre)', _EI,_ 568–82; _K,_ 185–201).
'L'Apocalypse déçoit', _La Nouvelle Revue française,_ 135, March 1964, 488–96 ( _A,_ 118–27).
'La Conquista dello spazio', translated into Italian by Guido Neri, _Il Menabò,_ 7, 1964, 10–13 ( _BR,_ 269–71).
'Il nome Berlino', translated into Italian by Guido Neri, _Il Menabò,_ 7, 1964, 121–5 (republished, in an English translation by James Cascaito, as 'The Word Berlin', in _Semiotext(e),_ IV, 2, 1982, 60–5; then, as 'Le Nom de Berlin', in a French version reconstructed by Hélène Jelen and Jean-Luc Nancy, in _Der Name Berlin/Le Nom de Berlin,_ Berlin, Merve Verlag, 1983 and in _Café librairie,_ 3, Autumn 1983, 42–6; finally, this last version, under the title 'Berlin', together with parallel English, German, and Russian translations by Aris Fioretos, Werner Hamacher, and Mikhail Yampolsky, is republished in: _MLN,_ 109, 1994, 345–55; _BR,_ 266–8).
'La Parola in arcipelago', translated into Italian by Guido Neri, _Il Menabò,_ 7, 1964, 156–9 (republished in the original French as 'Parole de fragment', in _L'Endurance de la pensée,_ essays in honour of Jean Beaufret, Paris, Plon, 1968, 103–8; and in _EI,_ 451–5).
'Il "Quotidiano"', translated into Italian by Gabriella Zanobetti, _Il Menabò,_ 7, 1964, 260–1 (republished in the original French, in dialogue form, as the closing page of 'La Parole quotidienne', _EI,_ 366).
'L'Interruption', _La Nouvelle Revue française,_ 137, May 1964, 869–81 (republished in part as 'L'Interruption (comme sur une surface de Riemann)', _EI,_ 106–12; the remainder republished as part of the essay 'Traces', A, 252–8).
'L'Athenaeum', _La Nouvelle Revue française,_ 140, August 1964, 301–13 ( _EI,_ 515–27).
'La Voix narrative', _La Nouvelle Revue française,_ 142, October 1964, 675–85 (republished as 'La Voix narrative (le "il", le neutre)', _EI,_ 556–67; K, 171–84).
'Le Héros', _La Nouvelle Revue française,_ 145, January 1965, 90–104 (republished as 'La Fin du héros', _EI,_ 540–55).
'Les Grands Réducteurs', _La Nouvelle Revue française,_ 148, April 1965, 676–86 (A, 74–86).
'Le Rire des dieux', _La Nouvelle Revue francaise,_ 151, July 1965, 91–105 (A, 192–207).
'Français, encore un effort . . . _', La Nouvelle Revue française,_ 154, October 1965, 600–18 (republished as 'L'Expérience-limite, 3: L'Insurrection, la folie d'écrire', _EI,_ 323–42).
'L'Entretien infini', _La Nouvelle Revue française,_ 159, March 1966, 385–401 (republished, without its title, in _EI,_ ix–xxvi).
'Nietzsche et l'écriture fragmentaire' (I), _La Nouvelle Revue française,_ 168, December 1966, 967–83 (republished as 'Réflexions sur le nihilisme, 3: Nietzsche et l'écriture fragmentaire', _EI,_ 227–42).
'Nietzsche et l'écriture fragmentaire' (II), _La Nouvelle Revue française,_ 169, January 1967, 19–32 (republished as 'Réflexions sur le nihilisme, 3: Nietzsche et l'écriture fragmentaire', _EI,_ 242–55).
'Le Demain joueur', _La Nouvelle Revue française,_ 172, April 1967, 863–88 ( _EI,_ 597–619).
'L'Athéisme et l'écriture. L'Humanisme et le cri' (I), _La Nouvelle Revue française,_ 178, October 1967, 586–504 (EI, 367–84).
'L'Athéisme et l'écriture. L'Humanisme et le cri' (II), _La Nouvelle Revue française,_ 179, November 1967, 812–21 ( _EI,_ 384–93).
'Le Tout Dernier Mot', _La Nouvelle Revue française,_ 185, May 1968, 780–808 (A, 300–25; K, 219–48).
'Il est capital que le mouvement des étudiants oppose et maintienne une puissance de refus', statement signed by Maurice Blanchot and 34 others, including Robert Antelme, Maurice Nadeau, Louis-René Des Forêts, Marguerite Duras, Jean Schuster, Michel Leiris, Dionys Mascolo, Jérôme Peignot, Pierre Klossowski, Nathalie Sarraute, Monique Wittig, Jean Ricardou, André Gorz, Jean-Paul Sartre, Jacques Lacan, Henri Lefebvre and François Chôtelet, _Le Monde,_ 10 May 1968, 9.
'La Rue', leaflet distributed 17 July 1968, subsequently published in _Comité,_ 1, October 1968, 11, attributed to Blanchot by Dionys Mascolo.
'Lettre ouverte au Parti Communiste de Cuba', signed by Maurice Blanchot, with Robert Antelme, Marguerite Duras, and Dionys Mascolo, _L'Archibras,_ 5, hors série, 30 September 1968, 9.
'En état de guerre', _Comité,_ 1, October 1968, 3–4 (published anonymously, attributed to Blanchot by Dionys Mascolo).
'Affirmer la rupture', _Comité,_ 1, October 1968, 4–5 (first published anonymously, republished in part and attributed to Blanchot by Dionys Mascolo in 'Mots de désordre', _Libération,_ 28–9 January 1984, 23; _BR,_ 200–1).
'Aujourd'hui . . .', _Comité,_ 1, October 1968, 7 (published anonymously, attributed to Blanchot by Dionys Mascolo).
'La Mort politique', _Comité,_ 1, October 1968, 8 (first published anonymously, republished in part and attributed to Blanchot by Dionys Mascolo in 'Mots de désordre', _Libération,_ 28–9 January 1984, 23; _BR,_ 201–2).
'Le Communisme sans héritage', _Comité,_ 1, October 1968, 13 (first published anonymously, republished and attributed to Blanchot in _Gramma,_ 3/4, 1976, 31–3; _BR,_ 202–4).
'Depuis longtemps, la brutalité . . .', _Comité_ , 1, October 1968, 14 (published anonymously, attributed to Blanchot by Dionys Mascolo).
'Tracts, affiches, bulletin', _Comité,_ 1, October 1968, 16 (first published anonymously, republished and attributed to Blanchot in _Gramma,_ 3/4, 1976, 33–4; _BR,_ 204–5).
'Que l'immense contrainte . . .', _Comité,_ 1, October 1968, 17 (published anonymously, attributed to Blanchot by Dionys Mascolo).
'Les Actions exemplaires', _Comité,_ 1, October 1968, 17–18 (published anonymously, attributed to Blanchot by Dionys Mascolo).
'Deux innovations caractéristiques', _Comité,_ 1, October 1968, 18 (published anonymously, attributed to Blanchot by Dionys Mascolo).
'Rupture du temps : révolution', _Comité,_ 1, October 1968, 18 (first published anonymously, republished in part and attributed to Blanchot by Dionys Mascolo in 'Mots de désordre', _Libération,_ 28–9 January 1984, 23; _BR,_ 205).
'Pour le camarade Castro', _Comité,_ 1, October 1968, 22–3 (published anonymously, attributed to Blanchot by Dionys Mascolo).
'La Reddition idéologique', _Comité,_ 1, October 1968, 23 (published anonymously, attributed to Blanchot by Dionys Mascolo).
'La Clandestinité à ciel ouvert', _Comité,_ 1, October 1968, 23 (published anonymously, attributed to Blanchot by Dionys Mascolo).
'Conseils aux gens de la rue', _Comité,_ 1, October 1968, 27 (published anonymously, attributed to Blanchot by Dionys Mascolo).
'Lire Marx', _Comité,_ 1, October 1968, 31 (first published anonymously, republished as 'Les Trois Paroles de Marx', A, 115–17).
'L'Absence de livre', _L'Éphémère,_ 10, April 1969, 201–18 ( _EI,_ 620–36).
'La Facilité de mourir', _La Nouvelle Revue française,_ special issue devoted to Jean Paulhan, May 1969, 743–64 (A, 172–91; _BR,_ 301–16).
'Sur les Comités d'action', _Les Lettres nouvelles,_ June–July 1969, 184–5 (letter first published anonymously, subsequently attributed to Blanchot by Dionys Mascolo in Dionys Mascolo, _A la recherche d'un communisme de pensée, entêtements,_ Paris, Éditions fourbis, 1993, 359–60).
'Détruire', _L'Éphémère,_ 13, June 1970, 22–6 (A, 132–6).
'L'Exigence du retour', _L'Arc,_ 43, 1970, 48–53 ( _PA ,_ 7, 10–15, 21–7, 33–6, 73).
'En guise d'introduction', letter to Piera Aulagnier, _Topique,_ 4–5, October 1970, 7–10.
'Fragmentaires', _L'Ephémère,_ 16, January 1971, 376–99 ( _PA,_ 121–36, 156, 137–9, 140–52).
'Une nouvetle raison?', _La Nouvelle Revue française,_ 223, July 1971, 94–101.
'Le "Discours philosophique"', _L'Arc,_ 46, 1971, 1–4.
'Le Dernier à parler', _La Revue de Belles-Lettres,_ 96, 2–3, 1972, 171–83 (republished as _Le Dernier à parler,_ Montpellier, Fata morgana, 1984; _ACTS: A Journal of New Writing,_ 8/9, 1988, special issue on 'Translating Tradition: Paul Celan in France', Benjamin Hollander (ed.), 228–39).
'Sur Edmond Jabès', _Les Nouveaux Cahiers,_ 31, Winter 1972–73, 51–2 ( _PA,_ 156, 56–7, 49).
'La Comédie d'avoir de l'ordre', _La Quinzaine littéraire,_ 171, 16–30 September 1973, 3–4.
'Discours sur la patience', _Le Nouveau Commerce,_ 30–1, 1975, 19–4 ( _ED, 7–_ 10, 28–49, 52–5, 220, 15, 16, 17).
'Fragment', _Change,_ 22, February 1975, 223 ( _ED,_ 64).
'Fragmentaire', in Pierre Alechinsky _et al., Celui qui ne peut se servir des mots,_ Montpellier, Fata morgana, 1975, 19–31 ( _ED,_ 23–5, 27, 24, 14, 27, 23, 72, 49, 17, 18, 61, 18–19, 22, 26, 49, 53, 56, 57–8, 65–7, 67–8, 71, 72, 72–4, 78–9, 76–80).
'On tue un enfant', _Le Nouveau Commerce,_ 33–4, Spring 1976, 19–29 ( _ED,_ 108–17).
'Trois lettres à Christian Limousin', _Gramma,_ 3/4, 1976, 5–7 (republished in part in _ED,_ 64).
'Une scène primitive', _Première livraison,_ 4, 1976, 1.
'La Poésie, mesdames, messieurs', _Givre,_ special issue devoted to Bernard Noël, 2–3, 1977, 176–7 ( _ED,_ 143–4).
'Une lettre', fragment of a letter to Emmanuel Levinas (1969), in Emmanuel Levinas, _Du sacré au saint,_ Paris, Minuit, 1977, 48–9.
'Il n'est d'explosion', in Maurice Blanchot _et al, Misère de la littérature,_ Paris, Christian Bourgois, 1978, 11–12 (republished in part in _ED,_ 190–1).
'Une scène primitive', _Le Nouveau Commerce,_ 39–40, Spring 1978, 43–51 ( _ED,_ 191–6, 202–6).
'Ne te retourne pas', _Digraphe_ 18/19, 1979, 160–3.
'Lettre à Jeffrey Mehlman [26 November 1979]', fragment of a letter reproduced in Jeffrey Mehlman,
'Blanchot at _Combat:_ Of Literature and Terror', _MLN,_ 95, 1980, 819.
'Notre compagne clandestine', in François Laruelle (ed.), _Textes pour Emmanuel Lévinas,_ Paris, Jean-Michel Place, 1980, 79–87 ( _Face to Face with Levinas,_ Ralph A. Cohen (ed.), Albany, NY, State University of New York Press, 1986, 41–50).
'L'Ecriture du désastre', _La Nouvelle Revue française,_ 330–1, July–August 1980, 1–33 ( _ED_ , 60–107, 120–1).
'Une lettre [11 février 1980]', _Exercices de la patience,_ 1, 1980, 67.
'Refuser l'ordre établi', response to a questionnaire on literary commitment, _Le Nouvel Observateur,_ special issue, May 1981, 45–6.
'Prière d'insérer', _Exercices de la patience,_ 2, Winter 1981, 104–7.
'La Maladie de la mort (éthique et amour)', _Le Nouveau Commerce,_ 55, Spring 1983, 31–46 _(CI,_ 58–77).
'Nous travaillons dans les ténèbres', _Le Monde,_ 22 July 1983, 9.
'Les Intellectuels en question', _Le Débat,_ 29, March 1984, 3–24 ( _BR,_ 206–27).
'La Parole ascendante, ou: Sommes-nous encore dignes de la poésie? (notes éparses)', in Vadim Kozovoï, _Hors de la colline,_ translated by the author with Michel Deguy and Jacques Dupin, Paris, Hermarm, 1984, 19–27.
'Les Rencontres', response to a questionnaire on the occasion of the magazine's twentieth anniversary, _Le Nouvel Observateur,_ 1045, special issue, November 1984, 84.
'Pourquoi écrivez-vous?', response to a questionnaire, _Libération,_ hors-série, March 1985, 64 (reproduced in _Pourquoi écrivez-vous? 400 écrivains répondent,_ Jean-François Fogel and Daniel Rondeau (eds), Paris, Le Livre de poche, 1988, 188).
'Le Bienfait le plus lourd', _Le Nouvel Observateur,_ 31 May–6 June 1985, 79.
'Blanchot ouvre Laporte', _Libération,_ 6 March 1986, 35.
'N'oubliez pas!', _La Quinzaine littéraire,_ 459, 16–31 March 1986, 11–12.
'Notre responsabilité', in _Pour Nelson Mandela,_ Preface by Dominique Lecoq, Paris, Gallimard, 1986, 215–17.
_'L'Excès-l'usine_ ou l'infini morcelé' (on _L'Excès-l'usine_ by Leslie Kaplan), _Libération,_ 24 February 1987, 35.
'Penser l'apocalypse', _Le Nouvel Observateur,_ 22–28 January 1988, 77–9.
'"N'oubliez pas'", letter to Salomon Malka, _L'Arche,_ May 1988, 68–71 ( _BR,_ 244–9).
'Ce qui m'est le plus proche . . .', _Globe,_ 30, July–August 1988, 56. Author's blurb, _Le Très-Haut,_ Paris, Gallimard: L'Imaginaire, 1988.
'La Poursuite tempérée-éperdue', in Roger Laporte, _Lettre à personne,_ Paris, Plon, 1989, 91–5.
'L'Écriture consacrée au silence', _Instants,_ 1, 1989, 239–41.
'Une voix venue d'ailleurs', _La Quinzaine littéraire,_ 1–15 July 1989, 5–6 (UV, 13–17).
'Qui?', _Cahiers Confrontation,_ 20, Winter 1989, 49–51 ( _Who Comes After the Subject?,_ Eduardo Cadava, Peter Connor and Jean-Luc Nancy (eds), New York, Routledge, 1991, 58–60).
'André Delmas: L'Arrière-Monde', _La Nouvelle Revue française, 440,_ September 1989, 58–60.
'Oui, le silence est nécessaire à l'écriture . . .', _Le Monde,_ 26 January 1990, 27 (also in _Globe,_ 44, February 1990, 72).
'Grâce (soit rendue) à Jacques Derrida', _Revue philosophique,_ 2, April–June 1990, 167–73 (BR, 317–23).
'Oh tout finir', _Critique,_ 519–20, August–September 1990, 635–7 ( _BR,_ 298–300).
'Textes préparatoires de "La Revue internationale"', _Lignes,_ 11, September 1990, 179–91.
'Correspondances', exchange of letters between Blanchot, Hans-Magnus Enzensberger, Louis-René Des Forêts, Uwe Johnson, Francesco Leonetti, Dionys Mascolo, Iris Murdoch, Richard Seaver, and Elio Vittorini (1960–5), _Lignes,_ 11, September 1990, 218–301.
'Lettre à Diane Rubenstein [20 August 1983]', fragment of a letter reproduced in Diane Rubenstein, _What's Left: the École Normale Supérieure and the Right,_ Madison, University of Wisconsin Press, 1990, 187.
'Sur le nationalisme', response to a questionnaire, _La Règle du jeu,_ 3, January 1991, 221–2.
'Lettre à Bernard-Henri Lévy [15 September 1989]', fragment of a letter reproduced in Bernard-Henri Lévy, _Les Aventures de la liberté,_ Paris, Grasset, 1991, 311.
'Le Blanc Le Noir', _Le Temps qu'il fait,_ Cahier 6/7, special issue on Louis-René Des Forêts, edited by Jean-Benoît Puech and Dominique Rabaté, 1991, 231–2 (UV, 21–4).
'Énigme', _Yale French Studies,_ 79, 1991, 5–7.
'L'Existence posthume', response to a questionnaire, _La Règle du jeu,_ 6, January 1992, 181. Author's blurb, _Thomas l'Obscur,_ nouvelle version, Paris, Gallimard: L'Imaginaire, 1992, 5–6.
'Adresses', response to a questionnaire on Salman Rushdie, _La Règle du jeu,_ 10, May 1993, 206.
'Appel à la fondation d'un Parlement international des écrivains', signed by Maurice Blanchot and more than 300 others, Strasbourg, 31 July 1993.
'Pour l'amitié', in Dionys Mascolo, _A la recherche d'un communisme de pensée, entêtements,_ Paris, Éditions fourbis, 1993, 5–16.
'"Dans la nuit surveillée"', with two extracts from _L'Espèce humaine by Robert Antelme, Lignes,_ 21, January 1994, 127–31.
'Appel à la vigilance', petition against the rise of the far right in Europe, signed by Maurice Blanchot and some 2,000 others, _Le Monde,_ 13 July 1945, 15 (originally published, without Blanchot's signature, in _Le Monde,_ 13 July 1993, 8).
'Pour une reconnaissance légale du couple homosexuel', petition in favour of legal recognition of lesbian and gay couples, signed by Maurice Blanchot and 232 others, _Le Nouvel Observateur,_ 9–15 May 1996, 44–5.
'Extracts from a letter to Roger Laporte (9 December 1984)', in Leslie Hill,
'Introduction', _Blanchot and the Demand of Writing,_ Carolyn Bailey Gill (ed.) ( London, Routledge, 1996), 9–10.
'A letter [to Roger Laporte, 24 December 1992]', _Blanchot and the Demand of Writing,_ 209–11.
'Une lettre de Maurice Blanchot [2 September 1996]', letter to Bruno Roy, _La Quinzainel ittéraire,_ 1–15 November 1996, 5.
'Appel à la désobéissance civile contre les lois sur l' immigration', petition signed by sixty-six film-makers and theatre directors and fifty-five writers, including Maurice Blanchot, _Le Monde,_ 14 February 1997, 8.
## Works in English translation
_The Blanchot Reader,_ edited with an Introduction by Michael Holland, with translations by Susan Hanson, Leslie Hill, Michael Holland, Roland-François Lack, Ian Maclachlan, Ann Smock, Chris Stevens, and Michael Syrotinski, Oxford, Blackwell, 1995.
_Death Sentence,_ translated by Lydia Davis, New York, Station Hill Press, 1978.
_The Gaze of Orpheus and Other Literary Essays,_ translated by Lydia Davis, edited with an Afterword by P. Adams Sitney, New York, Station Hill Press, 1981.
_The Infinite Conversation,_ translated by Susan Hanson, Minneapolis, University of Minnesota Press, 1993.
_The Last Man,_ translated by Lydia Davis, New York, Columbia University Press, 1987.
'The Last One to Speak', translated by Joseph Simas, _ACTS: A Journal of New Writing,_ 8/9, 1988, special issue on 'Translating Tradition: Paul Celan in France', edited by Benjamin Hollander, 228–39.
_The Madness of the Day,_ translated by Lydia Davis, New York, Station Hill Press, 1981.
_Michel Foucault as I Imagine Him,_ translated by Jeffrey Mehlman, in _Foucault/ Blanchot,_ New York, Zone Books, 1987.
_The Most High,_ translated by Allan Stoekl, Lincoln, Neb., University of Nebraska Press, 1995.
_The One Who Was Standing Apart From Me,_ translated by Lydia Davis, New York, Station Hill Press, 1993.
'Our Clandestine Companion', translated by David B. Allison, in _Face to Face with Levinas,_ Ralph A. Cohen (ed.), Albany, NY, State University of New York Press, 1986, 41–50.
'Our Responsibility', in _Texts for Nelson Mandela,_ New York, Seaver, 1987. _The Sirens' Song,_ translated by Sacha Rabinovitch, Gabriel Josipovici (ed.), Brighton, Harvester, 1982.
_The Space of Literature,_ translated by Ann Smock, Lincoln, Neb. and London, University of Nebraska Press, 1982.
_The Step Not Beyond,_ translated by Lycette Nelson, Albany, NY, State University of New York Press, 1992.
_Thomas the Obscure,_ translated by Robert Lamberton, New York, Station Hill Press, [1973] 1988.
_The Unavowable Community,_ translated by Pierre Joris, New York, Station Hill Press, 1988. Vicious Circles, followed by _'After the Fact',_ translated by Paul Auster, New York, Station Hill Press, 1985.
_When the Time Comes,_ translated by Lydia Davis, New York, Station Hill Press, 1985.
'Who?', in _Who Comes After the Subject?,_ Eduardo Cadava, Peter Connor and Jean-Luc Nancy (eds), New York, Routledge, 1991, 58–60.
_The Work of Fire,_ translated by Charlotte Mandell, Stanford, Stanford University Press, 1995.
_The Writing of the Disaster,_ translated by Ann Smock, Lincoln, Neb. and London, University of Nebraska Press, 1986.
_A_ | _L'Amitié_
---|---
_AB_ | _Aminadab_
_AC_ | _Après coup, précédé par Le Ressassement éternel_ ( _Vicious Circles, followed by 'After the Fact'_ )
_AM_ | _L'Arrêt de mort_ ( _Death Sentence_ )
_AO_ | _L'Attente L'Oubli_
_BR_ | _The Blanchot Reader_ (contains a number of uncollected articles and a selection of essays from _FP, PF, LS, LV, EI, A,_ and _PA_ )
_CI_ | _La Communauté inavouable_ ( _The Unavowable Community_ )
_CLP_ | _Comment la littérature est-elle possible?_ ( _BR,_ 49–60)
_CQ_ | _Celui qui ne m'accompagnait pas_ ( _The One Who Was Standing Apart From Me_ )
_DH_ | _Le Dernier Homme_ ( _The Last Man_ )
_ED_ | _L'Ecriture du désastre_ ( _The Writing of the Disaster_ )
_EI_ | _L'Entretien infini_ ( _The Infinite Conversation_ )
_EL_ | _L'Espace littéraire_ ( _The Space of Literature_ )
_FP_ | _Faux Pas_ (some of the essays collected in _Faux Pas_ may be found in _BR_ and _GO_ )
_FJ_ | _La Folie du jour_ ( _The Madness of the Day_ )
_GO_ | _The Gaze of Orpheus and Other Literary Essays_ (contains a selection of essays from _FP, PF, EL, LV,_ and EI)
_IM_ | _L'Instant de ma mort_
_K_ | _De Kafka à Kafka_ (reprints a series of essays already collected in _PF, EL, EI, A_ )
_LS_ | _Lautréamont et Sade_
_LV_ | _Le Livre à venir_ (a selection of essays from this volume may be found in _BR, GO,_ and _SS_ )
_MF_ | _Michel Foucault tel que je l'imagine_ ( _Michel Foucault as I Imagine Him_ )
_MV_ | _Au moment voulu_ ( _When the Time Comes_ )
_PA_ | _Le Pas au-delà_ ( _The Step Not Beyond_ )
_PF_ | _La Part du feu_ ( _The Work of Fire_ )
_RE_ | _Le Ressassement éternel_ ( _Vicious Circles_ )
_SS_ | _The Sirens' Song_ (contains a selection of essays from _PF, EL, LV, EI_ )
_TH_ | _Le Très-Haut_ ( _The Most High_ )
_TO1_ | _Thomas l'Obscur_ 1941
_TO2_ | _Thomas l'Obscur_ (nouvelle version) 1950 ( _Thomas the Obscure_ )
_UV_ | _Une voix venue d'ailleurs_
|
{
"redpajama_set_name": "RedPajamaBook"
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| 3,078
|
The Spirit of '43 (br: O espírito de 1943) é um desenho animado estadunidense de 1943, produzido pelos Estúdios Disney como parte do "esforço de guerra" (war effort) daquela país durante a II Guerra Mundial. A duração é de curta-metragem e é protagonizado pelo Pato Donald. Embora o desenho tenha sido concebido como uma "propaganda de guerra", dentre os diversos desse tipo que foram lançados pelos diferentes estúdios, ele é notável por ter apresentado (pela primeira vez) a figura de um velho pato escocês (característica que mais tarde seria aproveitada na criação do Tio Patinhas) e também a de uma versão esbanjadora do Donald, que se parece muito com o futuro personagem Gastão.É uma sequência de "The New Spirit" (1942).
Sinopse
O Pato Donald é um operário que, ao receber seu salário no final da semana de trabalho, de imediato corre para gastá-lo em divertimentos. Mas ao meio caminho ele sofre um dilema de consciência, personificado pela aparição de duas versões de si próprio: o pato poupador, representado pela figura de um escocês, que o aconselha a guardar o dinheiro para pagar os impostos de guerra; e a do esbanjador, que o incentiva a gastar tudo rapidamente.
Curiosidades
Outro desenho de propaganda de guerra com o Pato Donald é Der Fuehrer's Face, no qual ele é um trabalhador na Alemanha Nazista.
A intenção do filme é clara, ao tentar apelar para o patriotismo da classe operária do país que sofre com a cobrança dos pesados impostos destinados a custear a guerra.
Ele foi relançado pela Disney em DVD em 2004, juntamente com outros desenhos.
Ligações externas
Curtas-metragens da Disney
Filmes dos Estados Unidos de 1943
Curtas-metragens de 1943
Filmes dos Estados Unidos
Filmes sobre a Segunda Guerra Mundial
Filmes em língua inglesa
Curtas-metragens de animação
|
{
"redpajama_set_name": "RedPajamaWikipedia"
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| 6,832
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Erich Brauer ist der Name folgender Personen:
* Erich Brauer (Ethnologe) (1895–1942), deutscher Illustrator und Ethnologe
Erich Brauer (Schauspieler) (1914–1989), deutscher Schauspieler und Theaterintendant
Erich Brauer (1929–2021), österreichischer Maler, Grafiker, Bühnenbildner, Sänger und Dichter, siehe Arik Brauer
|
{
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| 7,150
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<div id="copy"><a href="https://opengeo.tech/">Website</a> • <a rel="author" href="https://opengeo.tech/stefano-cudini/">Stefano Cudini</a></div>
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Q: Can we propogate signal from android application to know another user having the same application in android phone I am building an android app as my project. I need to implement a functionality which can let me know the same types of users (having my application in their phone) around in 1-5 meter distance. Is there anything that i can use or can i propogates any signals to do so. I dont want to connect those users but I am just interested in knowing the presence of my application in the vicinity/area.
The only thing I am aiming is mapping the users who has same applications. You can consider the below given scenario.
Example: suppose My application is 'TESTAPP'. There are 100 students on a floor, out of those 10 have my application. If one of 10 student turn on the application, then
1. I want to know that, there are more 9 students present at the same floor.
2. if possible i wan to map them by propogating a signal through 1st user.
Note: I am Creating a cloud based webservice for android and using geolocation.
A: You could use Bluetooth or WiFi P2P. See http://developer.android.com/training/connect-devices-wirelessly/nsd-wifi-direct.html
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\section{Introduction}
\IEEEPARstart{I}{n} machine learning, support vector machine (SVM) can solve binary classification problems with a well geometric interpretation \cite{Vapnik:94_svm}. An SVM classifier distinguishes two classes of data by finding a decision boundary from the training samples in the feature space. Among all possible decision boundaries, the SVM seeks the one with the minimum classification error and the maximum decision margin, as shown in Fig. \ref{linear-svm}. Linear SVM was originally devised for binary classification problems \cite{Vapnik:94_svm}. Later on, non-linear SVM \cite{Boser:92_kernelsvm} was proposed to achieve a higher classification accuracy with non-linear decision boundary by applying a kernel transformation. In addition to binary classification, SVM is also utilized to handle multi-class classification problems \cite{Hsu:02_multisvm}. Although neural networks with deep structure have demonstrated excellent classification accuracy in many applications, the SVM is believed to be more effective for classification problems with limited training data \cite{Matykiewicz:12}. Moreover, SVM classifiers feature low computational complexity in inference, which is well suited for edge devices with stringent energy budget. Examples include epileptic seizure detection \cite{Altaf:13_ISSCC}, motion tracking for autopilot \cite{Lee:17_vlsi}, and preliminary object classification and recognition \cite{Jeong:17}.
\begin{figure}[t]
\centering
\includegraphics[width=0.41\textwidth]{figures/linear-svm.pdf}
\caption{A linear SVM classifier finds the decision boundary with the largest margin.}
\label{linear-svm}
\end{figure}
For inference, the SVM model is pre-trained and loaded in the devices. However, a well-trained model may not be possible when input features vary over time. For instance, seizure patterns change over time, a pre-trained model \cite{Altaf:13_ISSCC} could become obsolete for accurate long-term detection. Under this situation, model adaptation can be applied for the classification model. In \cite{Lee:13_JSSC}, the recorded data are transmitted to an external device wirelessly and the updated model was sent back for model adaptation. However, the off-line model adaptation poses several issues, including potential data loss and privacy leakage \cite{Xu:14_iot}. Therefore, SVM model adaptation through on-line training is essential \cite{Huang:18_VLSI}.
However, on-line SVM training is challenging for hardware realization. It can be formulated as solving an optimization problem to minimize the cost function associated with the classification error and the magnitude of the margin. Several decomposition techniques, such as sequential minimum optimization (SMO) \cite{Platt:99_SMO}, LIBSVM \cite{Chang:11_libsvm} and cutting-plane methods \cite{Joachims:09_cutplane}, have been developed to solve the SVM training problem. Among these techniques, the SMO algorithm is widely used because it iteratively solves a two-variable quadratic programming (QP) problem, rather than directly solving a large-scale QP problem. However, the nested structure of the SMO limits potential acceleration through hardware parallelism, which leads to a long computation latency.
A hardware-efficient alternative direction method of multipliers (ADMM) \cite{Boyd:11_admm} algorithm is exploited in this work to achieve a short training latency. The ADMM-based SVM training algorithm features a less-loop structure, which can be processed in parallel. ADMM has been utilized to solve linear SVM with feature selection \cite{Ye:11_evsvm}. However, applying the ADMM algorithm to non-linear SVM training for hardware implementation is difficult because large-scale matrices are involved for optimization. Low-rank approximation by employing the Nystr\"{o}m method \cite{Drineas:05_nystrom} is adopted in this work to greatly reduce the matrix dimension, yielding feasible hardware mapping. Several datasets are used to verify the functionality of the proposed training algorithm. The contributions of this work include
\begin{itemize}
\item Applying low-rank approximation to the Nystr\"{o}m method reduces the overall computational complexity and the memory storage reduction by 97\% and 87\%, respectively for the training algorithm.
\item Hardware for eigenvalue decomposition is shared for matrix inversion in Nystr\"{o}m method and solving the linear system in iterative update phase, which achieves 37\% of silicon area reduction.
\item {Pre-computation for matrix matrix achieves computational complexity reduction by 22\% for hardware mapping. }
\item {Combining the common terms by employing intermediate variables reduces the computational complexity by 51\%. }
\item Memory sharing for intermediate variables that reduces the memory usage by 60\% for storing training samples.
\item {A seizure detection chip with on-line model adaptation demonstrates the effectiveness of the proposed SVM training algorithm and the hardware-efficient mapping methodology.}
\end{itemize}
The remainder of this paper is organized as follows. Section II briefly reviews inference and training for SVM. Section III describes the ADMM-based training algorithm and rank approximation. Section IV demonstrates the performance of the ADMM-based training algorithm and shows the superiority of the ADMM-based algorithm. Section V shows the proposed design techniques for efficient hardware mapping. An design example for epileptic seizure detection is shown in Section VI. Finally, Section VII concludes this paper.
\section{Preliminaries of SVM Classifier}
An SVM classifier can be linear or non-linear based on the type of the decision boundary. Fig. \ref{linear_vs_nonlinear} shows both linear and non-linear SVMs. The decision boundary of the linear SVM classifier is an affine hyperplane that separates the data linearly. However, a linear SVM classifier fails to well classify two classes of data when the data are not linearly separable. In contrast, a non-linear SVM classifier improves the classification performance by applying a kernel that transforms the original features into a higher-dimensional feature space \cite{Hofmann:08_kernel}. Generally, the non-linear SVM classifier achieves a better classification performance than the linear one.
\begin{figure}[t]
\centering
\includegraphics[width=0.5\textwidth]{figures/linear-vs-nonlinear.pdf}
\caption{Binary classification using (a) linear SVM classifier and (b) non-linear SVM classifier.}
\label{linear_vs_nonlinear}
\end{figure}
\subsection{SVM Inference}
Inference is the prediction process for the data that are not in the training dataset. The SVM inference features low computational complexity. Thus, it is suitable for low-power applications.
\subsubsection{Linear SVM Inference}
Considering the original $m$-dimensional data $\bf{x}$ and a mapping function $\phi$ that extracts $n$ features from $\bf{x}$ (i.e., $\phi :{\mathbb{R}^m} \to {\mathbb{R}^n}$), the corresponding classification function $\widetilde {{y}}$ can be calculated by
\begin{align} \label{E:1}
\widetilde {{y}} = {\rm{sign}}\left( { \phi {{\left( {\bf{x}} \right)}^T}{\boldsymbol{\beta }}} + {\beta _0} \right),
\end{align}
where $\boldsymbol{\beta} \in \mathbb{R}^{n}$ is denoted as the model weight and ${\beta _0} \in \mathbb{R}$ is the bias.
\subsubsection{Non-linear SVM Inference}
A kernel is applied to transform the data from the feature space to a higher-dimensional one where the data can be classified more easily \cite{Hofmann:08_kernel}. Considering $N$ training samples $\left\{ {\left( {{{\bf{x}}_i},{y_i}} \right)} \right\}_{i = 1}^N$, where ${{\bf{x}}_i} \in \mathbb{R}^{m}$ with the associated labels, ${y_i} \in \{ - 1, + 1\} $, and the kernel function ${{k}}\left( {{{\bf{x}}_i},{{\bf{x}}_j}} \right)$, the corresponding classification function $\widetilde {{y}}$ for the unlabelled data $\bf{x}$ can be determined by
\begin{align} \label{E:2}
\widetilde {{y}} = {\rm{sign}}\left( {\mathop \sum \limits_{i = 1}^N {\alpha _i}{y_i}k\left( {{{\bf{x}}_i},{{\bf{x}}}} \right) + b} \right),
\end{align}
where $\boldsymbol{\alpha} \in \mathbb{R}^{N}$ and $b \in \mathbb{R}$ are the model weights and the bias, respectively. A commonly used kernel function is the radial basis function (RBF) (the so-called Gaussian kernel), given by
\begin{align} \label{E:3}
k\left( {{{\bf{x}}_i},{{\bf{x}}_j}} \right) = {\rm{exp}}\left( {\gamma \left\| {{{\bf{x}}_i} - {{\bf{x}}_j}} \right\|_2^2} \right),
\end{align}
where ${\gamma}$ is a negative scaling factor that controls the decay rate of the Gaussian kernel. The RBF demonstrates a non-linear mapping in the Euclidean distance between each training sample ${{\bf{x}}_i}$ and unlabelled data $\bf{x}$.
Inference for both linear and non-linear SVM classifiers only require multiplications and additions, except that the exponentiation is needed for non-linear SVM inference. The hardware for the exponentiation can be efficiently realized by Coordinate Rotation DIgital Computer (CORDIC) \cite{{Volder:59_CORDIC1}, Meher:09_CORDIC2}. Additionally, only the non-zero model weights and the corresponding features need to be stored, thus SVM classifiers are suitable for edge devices.
\subsection{SVM Training}
The objective of the SVM training is to learn a decision boundary that can separates two classes of data. The SVM model weights associate with a minimized cost that is related to the misclassified error and the model regularization cost.
\subsubsection{Linear SVM training}
Given the training samples ${{\bf{x}}_i}$ and their corresponding labels $y_i$, SVM training can be formulated as a minimization problem given by
\begin{equation} \label{E:4}
\mathop {\min }\limits_{{\boldsymbol{\beta}} \in {\mathbb{R}^n},{\rm{}}{\beta _0} \in \mathbb{R}} \mathop \sum \limits_{i = 1}^N {\left( {1 - {y_i}\left( {{\beta _0} + \phi {{\left( {{{\bf{x}}_i}} \right)}^T}{\boldsymbol{\beta} }} \right)} \right)_ + } + \frac{\lambda }{2}{||\boldsymbol{\beta}||}_2^2,
\end{equation}
where { ${\left( f(\bf{x}) \right)_ + } = {\rm{max}}\left( {1 - f(\bf{x}) ,0} \right)$} is a Hinge loss function which quantifies the error of mis-classified data \cite{Bishop:06_book}. The regularization term is weighted by a parameter $\lambda$ to control the sparsity of the SVM model weights. The solutions $\boldsymbol{\beta}$ and ${\beta _0}$ in (\ref{E:4}) are the model weights and the bias, respectively, for the linear SVM classifier.
\subsubsection{Non-linear SVM training}
Based on the Karush-Kuhn-Tucker (KKT) condition \cite{KKT}, the model weights $\boldsymbol{\beta}$ of linear SVM classifier can be expressed as a linear combination of the features and their corresponding weights $\boldsymbol{\alpha}$, as represented by
\begin{align} \label{E:5}
{\boldsymbol{\beta }} = \mathop \sum \limits_{i = 1}^N {\alpha _i}{y_i}\phi \left( {{{\bf{x}}_i}} \right),
\end{align}
which builds the connection to transform linear SVM training to non-linear SVM training. Substituting $\boldsymbol{\beta}$ in (\ref{E:5}) for (\ref{E:4}) and defining the kernel function $k\left( {{{\bf{x}}_i},{{\bf{x}}_j}} \right) = \phi {\left( {{{\bf{x}}_i}} \right)^T}\phi \left( {{{\bf{x}}_j}} \right)$, non-linear SVM training can be reformulated by
\begin{align} \label{E:6}
\mathop {\min }\limits_{{\boldsymbol{\alpha }} \in {\mathbb{R}^N},b \in \mathbb{R}} \mathop \sum \limits_{i = 1}^N {\left( {1 - \left( {{{\boldsymbol{\Psi }}_{i \cdot }}{\boldsymbol{\alpha }} + {y_i}b} \right)} \right)_ + } + \frac{\lambda }{2}{{\boldsymbol{\alpha }}^T}{\boldsymbol{\Psi \alpha }},
\end{align}
where ${\boldsymbol{\Psi }} \in {\mathbb{R}^{N \times N}}$ is the kernel matrix which measures the similarities among all the training samples, and each element in ${\boldsymbol{\Psi }}$ is defined by
\begin{align} \label{E:7}
{{{\Psi }}_{ij}}: = {y_i}{y_j}{{k}}\left( {{{\bf{x}}_i},{{\bf{x}}_j}} \right), i, j = 1, 2, 3, ..., N.
\end{align}
For both linear and non-linear SVM classifiers, the computational complexity for the training is much higher than that for inference. A training algorithm that can be efficiently mapped on hardware is required for edge devices.
\section{ADMM-based Training Algorithm}
A hardware-friendly ADMM algorithm is considered in this work. It partitions a large optimization problem into several smaller sub-problems that are easier to solve \cite{Boyd:11_admm}. The ADMM-based SVM training algorithm allows for parallel processing in each iteration, reducing the processing latency.
\subsection{ADMM-Based Training Algorithm for Linear SVM}
\begin{algorithm*}[t]
\caption{ADMM-based Linear SVM Training Algorithm} \label{algorithm:1}
\begin{algorithmic}[1]
\Require
${\bf{X}} \in \mathbb{R}^{N \times p}$: training samples matrix;
${\bf{Y}} \in \mathbb{R}^{N \times N}$: diagonal matrix with labels;
\Ensure
${\boldsymbol{\beta}} \in \mathbb{R}^{p \times 1}$: model weight;
${\beta _0} \in \mathbb{R}$: bias;
\State {\bf{initialize}}: ${{\boldsymbol{\beta }}^{\left( 0 \right)}},\beta _0^{\left( 0 \right)}$
\Repeat
\State Solve linear equation system \hfill \break
$\left( {\begin{array}{*{20}{c}}
{\lambda {\bf{I}} + {\mu _1}{{\bf{X}}^T}{\bf{X}}}&{{\mu _1}{{\bf{X}}^T}{\bf{1}}}\\
{{\mu _1}{{\bf{1}}^T}{\bf{X}}}&{{\mu _1}n}
\end{array}} \right)\left( {\begin{array}{*{20}{c}}
{{{\boldsymbol{\beta }}^{(k + 1)}}}\\
{{\beta _0}^{(k + 1)}}
\end{array}} \right) = $
$\left( {\begin{array}{*{20}{c}}
{{{\bf{X}}^T}{\bf{Y}}{{\bf{u}}^{(k)}} - {\mu _1}{{\bf{X}}^T}{\bf{Y}}\left( {{{\bf{a}}^{(k)}} - {\bf{1}}} \right)}\\
{{{\bf{1}}^T}{\bf{Y}}{{\bf{u}}^{(k)}} - {\mu _1}{{\bf{1}}^T}{\bf{Y}}\left( {{{\bf{a}}^{(k)}} - {\bf{1}}} \right)}
\end{array}} \right)$
\State
${{\bf{a}}^{(k + 1)}} = {S_{\frac{1}{{{\mu _1}}}}}\left( {{\bf{1}} + \frac{{{{\bf{u}}^{(k)}}}}{{{\mu _1}}} - {\bf{Y}}\left( {{\bf{X}}{{\boldsymbol{\beta }}^{(k + 1)}} + {\beta _0}^{(k + 1)}{\bf{1}}} \right)} \right)$
\State
${{\bf{u}}^{\left( {k + 1} \right)}} = {{\bf{u}}^{\left( k \right)}} + {\mu _1}\left( {{\bf{1}} - {\bf{Y}}\left( {{\bf{X}}{{\boldsymbol{\beta }}^{\left( {k + 1} \right)}} + {\beta _0}^{\left( {k + 1} \right)}{\bf{1}}} \right) - {{\bf{a}}^{(k + 1)}}} \right)$
\Until
$\left\| {\left( {\begin{array}{*{20}{c}}
{{{\boldsymbol{\beta }}^{(k + 1)}}}\\
{{\beta _0}^{(k + 1)}}
\end{array}} \right) - \left( {\begin{array}{*{20}{c}}
{{{\boldsymbol{\beta }}^{(k)}}}\\
{{\beta _0}^{(k)}}
\end{array}} \right)} \right\|_2^2 \le \varepsilon $
\end{algorithmic}
\end{algorithm*}
ADMM is exploited to perform variable selection efficiently for SVM training \cite{Ye:11_evsvm}. A similar formulation and optimization procedure are adopted in this work. SVM training in (\ref{E:4}) can be re-written in an equivalent constrained minimization problem in vector form by introducing an auxiliary variable ${\bf{a}} \in {\mathbb{R}^N}$:
\begin{align} \label{E:8}
\mathop {\min }\limits_{{\boldsymbol{\beta }},{\rm{}}{\beta _0}} \mathop \sum \limits_{i = 1}^N {\left( {a_{i}} \right)_ + } + \frac{\lambda }{2}{||\boldsymbol{\beta }||}_2^2, {\quad}s.t.{\quad}{\bf{a}} = {\bf{1}} - {\bf{Y}}\left( {{\bf{X} \boldsymbol{\beta} } + {\beta _0} \bf{1} } \right),
\end{align}
where $\bf{Y}$ is the diagonal matrix with training labels, and $\bf{X}$ is the matrix with training samples, each row of which is a feature vector of a training sample. The augmented Lagrangian function transforms the original problem into an unconstrained convex optimization problem with additional $\ell_{2}$-norm penalty \cite{Ye:11_evsvm}:
\begin{align} \label{E:9}
\begin{split}
L\left( {{\boldsymbol{\beta }},{\beta _0},{\bf{a}},{\bf{u}}} \right) = \mathop {}& \sum \limits_{i = 1}^N {\left( {a_{i}} \right)_ + } + \frac{\lambda }{2}{||\boldsymbol{\beta }||}_2^2 \\
&+ \left\langle {{\bf{u}},{\bf{1}} - {\bf{Y}}\left( {{\bf{X} \boldsymbol{\beta} } + {\beta _0}{\bf{1}}} \right) - {\bf{a}}} \right\rangle \\
{}&+ \frac{{{u_1}}}{2}\left\| {{\bf{1}} - {\bf{Y}}\left( {{\bf{X} \boldsymbol{\beta} } + {\beta _0}{\bf{1}}} \right) - {\bf{a}}} \right\|_2^2,
\end{split}
\end{align}
where $\bf{u}$ is the Lagrange multiplier and ${u_1}$ is the parameter for the penalty term. The ADMM optimizer solves the convex problem in (\ref{E:9}) by splitting it into three sub-problems with respect to variables $\boldsymbol{\beta}$, $\bf{a}$, and $\bf{u}$:
\begin{align} \label{E:10}
\begin{split}
\left( {{{\boldsymbol{\beta }}^{(k + 1)}},{\beta _0}^{(k + 1)}} \right) = {\rm{arg}}\mathop {\min }\limits_{{\boldsymbol{\beta }},{\beta _0}} L\left( {{{\boldsymbol{\beta }}^{(k)}},{\beta _0}^{(k)},{{\bf{a}}^{(k)}},{{\bf{u}}^{(k)}}} \right), \\
{{\bf{a}}^{(k + 1)}} = {\rm{arg}}\mathop {\min }\limits_{\bf{a}} L\left( {{{\boldsymbol{\beta }}^{(k + 1)}},{\beta _0}^{(k + 1)},{{\bf{a}}^{(k)}},{{\bf{u}}^{(k)}}} \right), \\
{{\bf{u}}^{(k + 1)}} = {\rm{arg}}\mathop {\min }\limits_{\bf{u}} L\left( {{{\boldsymbol{\beta }}^{(k + 1)}},{\beta _0}^{(k + 1)},{{\bf{a}}^{(k + 1)}},{{\bf{u}}^{(k)}}} \right),
\end{split}
\end{align}
where $k$ denotes the iteration. In each variable update step, the Lagrangian function $L$ is minimized with respect to a single variable while the other variables remain fixed. Fig. \ref{admm} illustrates the conceptualization of optimization process for a two-dimensional convex problem. The $\boldsymbol{\beta}^{(k)}$ is used to update $\boldsymbol{\beta}^{(k+1)}$, and $\boldsymbol{\beta}^{(k+1)}$ is then utilized to update $\boldsymbol{\alpha}^{(k)}$, as shown in (\ref{E:10}). Similarly, the new updated variables $\boldsymbol{\beta}^{(k+1)}$ and $\boldsymbol{\alpha}^{(k+1)}$ are used for the update of $\bold{u}^{(k)}$. A variable is updated and the updated variable is then used in the next iteration with respect to another variable.
\begin{figure}[t]
\centering
\includegraphics[width=0.3\textwidth]{figures/admm.pdf}
\caption{The optimization process of ADMM for a two-dimensional convex optimization problem.}
\label{admm}
\end{figure}
{\bf\textsl{Algorithm 1}} shows the ADMM-based linear SVM training algorithm. The derivative of $L$ with respect to each target variable is set to $0$ to find the closed-form solution for each sub-problem in (\ref{E:10}). A gradient descent method is exploited to minimize the Lagrange function along the negative-gradient direction. The variables are iteratively updated in Steps 3 to 5. The algorithm terminates when the model weight $\boldsymbol{\beta}$ and the bias $\beta _0$ both converge.
Updating ${{\bf{a}}^{(k)}}$, which involves the Hinge loss function, can be solved efficiently. By manipulating the terms in (\ref{E:9}), updating ${{\bf{a}}^{(k)}}$ is reformulated as
\begin{align} \label{E:10-a}
\begin{split}
{{\bf{a}}^{(k + 1)}} &= {\rm{arg}}\mathop {\min }\limits_{\bf{a}} \frac{1}{u_1}\sum \limits_{i = 1}^N {\left( {a^{(k)}_{i}} \right)_ + } \\
&+ \frac{1}{2} \left\| {{\bf{1}} + \frac{{{{\bf{u}}^{(k)}}}}{{{\mu _1}}} - {\bf{Y}}\left( {{\bf{X}}{{\boldsymbol{\beta }}^{(k + 1)}} + {\beta _0}^{(k + 1)}{\bf{1}}} \right) - {{\bf{a}}^{(k)}} } \right\|_2^2.
\end{split}
\end{align}
The scalar form, which can be applied to vector form element-wisely, is given by
\begin{align} \label{E:10-b}
\begin{split}
{}&{\rm{Let}} \quad {S_\delta }\left( \theta \right) = {\rm{arg}}\mathop {\min }\limits_{x } \delta \cdot {x_ + } + \frac{1}{2}\left\| {x - \theta } \right\|_2^2, \\
{}&{\rm{Then}} \quad {S_\delta }\left( \theta \right) = \left\{ {\begin{array}{*{20}{c}}
{\theta - \delta ,}\\
{0,}\\
{\theta ,}
\end{array}} \right.\begin{array}{*{20}{c}}
{\theta > \delta }\\
{0 \le \theta \le \delta }\\
{\theta < 0}
\end{array}.
\end{split}
\end{align}
where $\delta$ is a scalar and $\theta$ is a term that excludes the target variable $x$ (see \cite{Ye:10_proposition} for details). The variable ${{\bf{a}}^{(k + 1)}}$ can be computed efficiently, as described in Step 4 in {\bf\textsl{Algorithm 1}}.
In Step 3, solving the linear equation dominates the overall computational complexity. To solve a linear system $A\bf{x} = \bf{b}$, a conjugate gradient method \cite{Hestenes:52_conjugate} or a matrix inversion method (i.e., $ {\bf{x}} = A^{-1}\bf{b}$) can be applied. Although the conjugate gradient method avoids matrix inversion and guarantees to solve the linear system within finite iterations, a long latency is usually needed for convergence. By contrary, the matrix inversion method achieves a shorter latency, which is more suitable for real-time applications. Additionally, the system matrix in Step 3 is fixed since the training sample matrix $\bold{X}$ is fixed during the entire training process. Given a fixed system matrix, the inverse of the system matrix can be computed only once and reused for each iteration. This significantly reduces the computational complexity for SVM training.
\subsection{ADMM-Based Training Algorithm for Non-Linear SVM}
The original formulation for non-linear SVM training in (\ref{E:6}) can be solved by the ADMM algorithm and the kernel matrix ${\boldsymbol{\Psi}}$ can be decomposed into several matrices. This transforms a non-linear SVM training problem into a linear SVM training problem with a similar structure \cite{Lee:10_stochastic}. The approximate kernel matrix ${\boldsymbol{\tilde \Psi }}$ can be decomposed by
\begin{align} \label{E:11}
{\boldsymbol{\Psi }} \approx {\boldsymbol{\tilde \Psi }} = {\bf{V}} {{\bf{V}}^T},
\end{align}
where $\bf{V} \in {\mathbb{R}^{N \times r}}$ is a rank-$r$ matrix. By substituting ${\bf{V}} {{\bf{V}}^T}$ for ${\boldsymbol{\Psi}}$, (\ref{E:6}) can be reformulated as
\begin{align} \label{E:12}
\mathop {\min }\limits_{{\boldsymbol{\alpha }} \in {\mathbb{R}^N},b \in \mathbb{R}} \mathop \sum \limits_{i = 1}^N {\left( {1 - \left( {{\bf{v}}_i^T{{\bf{V}}^T}{\boldsymbol{\alpha }} + {y_i}b} \right)} \right)_ + } + \frac{\lambda }{2}{{\boldsymbol{\alpha }}^T}{\bf{V}}{{\bf{V}}^T}{\boldsymbol{\alpha }},
\end{align}
where ${\bf{v}_i}$ denotes ${\bf{V}}_{\rm{i}}^{\rm{T}}$, which is defined as the $i$-th row of $\bf{V}$. By replacing ${{\bf{V}}^T}{\boldsymbol{\alpha }}$ with ${\boldsymbol{\eta }}$ and ${\bf{YV}}$ with $\widetilde {\bf{X}}$, (\ref{E:12}) can be rewritten as
\begin{align} \label{E:13}
\mathop {\min }\limits_{{\boldsymbol{\eta }} \in {\mathbb{R}^r},b \in \mathbb{R}} \mathop \sum \limits_{i = 1}^N {\left( {1 - {y_i}\left( {{\bf{\widetilde x}}_i^T{\boldsymbol{\eta }} + b} \right)} \right)_ + } + \frac{\lambda }{2}{{\boldsymbol{\eta }}^T}{\boldsymbol{\eta }}.
\end{align}
Compared with (\ref{E:4}), (\ref{E:13}) is exactly the form of the original problem for a linear SVM training problem. Thus, (\ref{E:13}) can be solved by {\bf\textsl{Algorithm 1}} and the temporary weight coefficient $\boldsymbol{\eta}$ and bias $b$ can be obtained. The original weight coefficient $\boldsymbol{\alpha}$ for the non-linear SVM classifier in (\ref{E:6}) can be recovered from an overdetermined linear system with $N$ unknown variables and $r$ linear equations, given by
\begin{align} \label{E:14}
{\boldsymbol{\eta }} = {{\bf{V}}^T}{\boldsymbol{\alpha }}.
\end{align}
The overdetermined linear system requires at most $r$ non-zero elements for valid solutions. That is, $\boldsymbol{\alpha}$ is a sparse solution, which is an $N \times 1$ vector with $r$ non-zero elements.
\begin{algorithm}[t]
\caption{ADMM-based non-linear SVM Training Algorithm} \label{algorithm:2}
\begin{algorithmic}[1]
\Require
${\bf{X}} \in \mathbb{R}^{N \times p}$: training samples matrix;
${\bf{Y}} \in \mathbb{R}^{N \times N}$: diagonal matrix with labels;
\Ensure
${\boldsymbol{\alpha}} \in \mathbb{R}^{N \times 1}$: weight vector;
$b \in \mathbb{R}$: bias;
\State
${{\boldsymbol{\Psi }}_G} = {\left[ {\left. {{y_i}{y_j}k({x_i},{x_j}} \right)} \right]_{i,j = 1, \cdots N}}$
\State Perform EVD
${\left[ {{{\boldsymbol{\Psi }}_G}} \right]_{MM}} = {\bf{QD}}{{\bf{Q}}^T}$
\State
${\bf{V}} = {\left[ {{{\boldsymbol{\Psi }}_G}} \right]_{ \cdot ,M}}{{\bf{Q}}_{ \cdot ,r}}{\bf{D}}_{r,r}^{ - 1/2}$
\State
${\bf{X}}^{'} = {\bf{YV}}$
\State
$\left[ {{\bf{w}},b} \right] = {\rm{LinearADMMSolver}}\left( {{{\bf{X}}^{'}},{\bf{Y}}} \right)$
\State
${\alpha _i} = \left\{ {\begin{array}{*{20}{c}}
{{{\left[ {{{\bf{Q}}_{ \cdot ,r}}{\bf{D}}_{r,r}^{ - 1/2}{\bf{w}}} \right]}_i}}\\
0
\end{array}\begin{array}{*{20}{c}}
{, \quad i \in M}\\
{, \quad i \notin M}
\end{array}} \right.$
\end{algorithmic}
\end{algorithm}
\subsection{Low-rank Approximation}
Apparently, the sparsity of the ADMM-based SVM model is related to the rank $r$ in (\ref{E:11}), and a dense solution occurs when $\boldsymbol{\Psi}$ is a rank-$N$ matrix. However, a rank-$N$ matrix $\bf{V}$ associated with inversion of an $N \times N$ matrix is infeasible for dedicated hardware mapping when hundreds to thousands of training samples are considered. To reduce the matrix dimension, Nystr\"{o}m method \cite{Drineas:05_nystrom} is exploited in this work to reduce the rank of $\bf{V}$ for approximating the matrix $\boldsymbol{\Psi}$ \cite{Lee:10_stochastic}. Such approximation reduces the dimension for matrix inversion from $ N \times N$ to $r \times r$.
The Nystr\"{o}m method finds the rank-$r$ matrix $\bold{V}$ for approximating an $N \times N$ symmetric matrix, as given in (\ref{E:11}). The procedure is illustrated as follows. \\
\begin{enumerate}[Step 1:]
\item Specify an integer $c$ with $r \le c \le N$.
\item Randomly choose $c$ elements in set $\{1,2,3,...,N\}$ to generate a subset $M$.
\item Perform eigenvalue decomposition (EVD) on matrix ${{\boldsymbol{\Psi }}_{MM}} = {\bf{QD}}{{\bf{Q}}^T}$, where ${{\boldsymbol{\Psi }}_{MM}}$ is a sub-matrix of $\boldsymbol{\Psi }$ with corresponding row and column indices in subset $M$, ${\bf{Q}} \in \mathbb{R}^{c \times c}$ is an orthogonal matrix, and ${\bf{D}} \in {\mathbb{R}^{c \times c}}$ is a diagonal matrix with decreasing non-negative eigenvalues.
\item Retain the first $r$ entries of $\bf{D}$ as an approximation to produce ${{\bf{W}}_{c,r}} = {{\bf{Q}}_{ \cdot ,r}}{{\bf{D}}_{r,r}}{{\bf{Q}}_{ \cdot ,r}}^T \in {\mathbb{R}^{c \times c}}$.
\item Perform matrix inversion on ${{\bf{W}}_{c,r}}$ to produce ${\bf{W}}_{c,r}^ + = {{\bf{Q}}_{ \cdot ,r}}{\bf{D}}_{r,r}^{ - 1}{{\bf{Q}}_{ \cdot ,r}}^T$.
\item Approximated matrix $\widetilde {\boldsymbol{\Psi }} = {{\boldsymbol{\Psi }}_{ \cdot ,M}}{\bf{W}}_{c,r}^ + {{\boldsymbol{\Psi }}_{ \cdot ,M}}^T$. \\
\end{enumerate}
From (\ref{E:11}), Step 4, and Step 6, the matrix $\bf{V}$ can be generated as
\begin{align} \label{E:15}
{\bf{V}} = {{\boldsymbol{\Psi }}_{ \cdot ,M}}{{\bf{Q}}_{ \cdot ,r}}{\bf{D}}_{r,r}^{ - \frac{1}{2}},\quad {\bf{V}} \in {\mathbb{R}^{N \times r}}.
\end{align}
In the Nystr\"{o}m method, the parameters $c$ and $r$ determine the approximation performance and the succeeding classification accuracy. The overall ADMM-based algorithm with rank approximation for non-linear SVM training is summarized in {\bf\textsl{Algorithm 2}}.
Low-rank approximation reduces the computational complexity for both matrix inversion and variable update. Assume that the rank of the kernel matrix is reduced from $N$ to $r$ by applying the low-rank approximation, the dimension of the input matrix ${\bf{X}}$ in {\bf\textsl{Algorithm 1}} reduces from $N \times N$ to $N \times r$. Compared to the design without low-rank approximation (i.e., ${\bf{X}}$ is full-rank matrix), the computational complexity is approximately reduced by a factor of $N/r$ for each iteration.
\section{Performance Evaluation}
Four datasets: CHB-MIT Scalp Epilepsy \cite{Shoeb:09_MIT, Goldberger:00_chbmit}, Wisconsin Breast Cancer \cite{UCI}, Pima Indians Diabetes \cite{UCI}, and hand-written digit database MNIST \cite{MNIST} are used to verify the effectiveness of the ADMM-based algorithm with rank-approximation. Table \ref{dataset-table} shows the dataset information, including the number of features and the number of data samples for training and inference. Since this work focuses on the binary classification, two classes (digits 4 and 5) in the MNIST database are selected for performance evaluation. In the experiments, the standard deviation for the RBF kernel function $\gamma$ is set to be in the range of [-10, -0.1]. The regularization parameter $\lambda$ and penalty coefficient $u_{1}$ are set to 10 and 1, respectively.
\begin{table}[t]
\centering
\caption{Verification Dataset}
\includegraphics[width=0.5\textwidth]{figures/dataset-table.pdf}
\label{dataset-table}
\end{table}
\subsection{Settings for Approximated Kernel Matrix}
\begin{figure}[t]
\centering
\includegraphics[width=0.43\textwidth]{figures/rank-error.pdf}
\caption{MSE analysis for the approximated matrix under various configurations of $(c,r)$. $R_c = c/N$ and $R_r=r/c$, where $N$ is the number of total columns.}
\label{rank-error}
\end{figure}
In this work, a design methodology is proposed to decide parameters $c$ and $r$ for the Nystr\"{o}m method. The value of $c$ affects both memory storage and computational complexity for implementing the Nystr\"{o}m method and the overall rank reduction is determined by the value of $r$. As an example, the kernel matrix for the MNIST dataset which selects 2048 training samples with 1024 training samples per class is considered. Fig. \ref{rank-error} shows the mean square error (MSE) between the original kernel matrix ${\boldsymbol{\Psi }}$ and the approximated matrix ${\boldsymbol{\tilde \Psi }}$ with respect to parameter pairs $(c, r)$. As can be seen, the approximation error is small for a large value of $c$ because more matrix information is preserved. Additionally, the error can be minimized by setting $c$ to be identical to $r$. Therefore, the design strategy is to choose a small $r$ and to set $c = r$.
\subsection{Classification Accuracy}
\begin{figure}[t]
\centering
\includegraphics[width=0.48\textwidth]{figures/accuracy.pdf}
\caption{The classification accuracy for (a) training and (b) inference, where the rank reduction ratio $R=r/N$.}
\label{accuracy}
\end{figure}
Fig. \ref{accuracy} shows the classification accuracy for four datasets for training and inference with respect to the rank reduction ratio $R$. For the MNIST dataset, the training accuracy is higher than 95\% even when the rank is reduced to only $2^{-5}$ of the original rank. A 32$\times$ rank reduction is achieved with only a 2\% accuracy drop in inference, which demonstrates the effectiveness of the proposed algorithm. Given the same rank reduction ratio of 2$^{-5}$, only 2.3\% training accuracy and 1.2\% inference accuracy are dropped for the Epilepsy dataset. For the Breast Cancer dataset, a 10.1\% training accuracy drop with 3.2\% inference accuracy drop is achieved. For the Diabetes dataset, the training accuracy decreases significantly, but the inference accuracy remains the same without significant degradation. It is noted that the best classification accuracy for this Diabetes dataset is in the range of 70\% to 76\% \cite{diabetes_result}. In our experiments, the inference accuracy without rank reduction is 70.1\%. By applying the rank approximation, the inference accuracy for Diabetes fluctuates is nearly 70\% and even higher in some cases.
\subsection{Performance Comparison for Rank Approximation}
\begin{figure}[t]
\centering
\includegraphics[width=0.47\textwidth]{figures/convergent-trend.pdf}
\caption{The training accuracy of the ADMM-based algorithm versus the iterations with respect to rank reduction ratios $R$.}
\label{convergent-trend}
\end{figure}
\begin{figure}[t]
\centering
\includegraphics[width=0.45\textwidth]{figures/convergent-time.pdf}
\caption{The convergence time for the ADMM and SMO algorithms.}
\label{convergent-time}
\end{figure}
Fig. \ref{convergent-trend} shows the training accuracy versus the iterations with respect to the rank reduction ratio $R$. The training procedure is terminated when a training accuracy of 99\% is achieved. As the number of rank increases, the SVM model initially achieves a better training accuracy and improve its accuracy with less iterations. However, its computational complexity in each iteration also becomes larger than the one of the model with a lower rank. The computational complexity of matrix inversion also increases with number of training samples. Although the model with a smaller rank has a lower initial training accuracy, the training accuracy is improved rapidly to roughly 98\% after similar iterations, and reaches to 99\% gradually. The computational complexity of matrix inversion in each iteration also becomes smaller.
\subsection{Superiority over the SMO Algorithm}
Fig. \ref{convergent-time} shows the time for convergence to reach a training accuracy of 95\% with respect to the number of training samples in the MNIST dataset \cite{MNIST}. The rank reduction ratio $R$ is associated with the model complexity and determines the reduced rank $r$ for rank approximation. The ADMM-based algorithm achieves a 17.3$\times$ faster convergent speed than the SMO algorithm under the case of 512 training samples when simulated on an Intel i7-6700 CPU. The speed improvement becomes larger when the number of training samples increases. The ADMM-based algorithm even overwhelms the SMO algorithm in speed by 9.8$\times$10$^7$ times for the case of 2048 training samples. This is because the SMO algorithm only optimizes two scalar weight coefficients in each iteration. The SMO algorithm fails to update the critical coefficients associated with support vectors since its nested-loop structure. The convergence speed of the ADMM-based algorithm can be further improved through dedicated hardware design.
\section{Efficient Hardware Realization}
Hardware complexity of the ADMM-based algorithm can be further reduced since many operations in {\bf\textsl{Algorithm 1}} can be shared and pre-computed. In this work, several design techniques are proposed for efficient hardware realization.
\subsection{EVD Hardware Sharing}
Since eigenvalue decomposition (EVD) is required in the Nystr\"{o}m method, the matrix inversion in the iterative update step in {\bf\textsl{Algorithm 1}} can be computed by the same EVD hardware. The EVD can be efficiently realized by QR decomposition \cite{Niu:13_QRD} or cyclic Jacobi method \cite{Yang:15_EVD}. The cyclic Jacobi method can be realized by regular rotation operations, which is preferable in hardware implementation. The cyclic Jacobi method performs the EVD by nullifying the off-diagonal elements of the target matrix through a sequence of Givens rotations. It can be greatly accelerated by processing the disjointed rows and columns in parallel.
\begin{algorithm}[t]
\caption{Hardware-efficient ADMM-based Linear SVM Training Algorithm} \label{algorithm:3}
\begin{algorithmic}[1]
\Require
${\bf{X}} \in \mathbb{R}^{N \times p}$: training samples matrix;
${\bf{Y}} \in \mathbb{R}^{N \times N}$: diagonal matrix with training labels;
\Ensure
${\boldsymbol{\beta}} \in \mathbb{R}^{p \times 1}$: weight vector;
${\beta _0} \in \mathbb{R}$: bias;
\State Compute
${\bf{A}} = \left( {\begin{array}{*{20}{c}}
{\lambda {\bf{I}} + {\mu _1}{{\bf{X}}^T}{\bf{X}}}&{{\mu _1}{{\bf{X}}^T}{\bf{1}}}\\
{{\mu _1}{{\bf{1}}^T}{\bf{X}}}&{{\mu _1}n}
\end{array}} \right)$
\State Perform EVD ${\bf{A}} = {\bf{QD}}{{\bf{Q}}^T}$
\State
${\bf{Z}} = {{\bf{Y}}}\widetilde {\bf{X}}{\bf{Q}}{{\bf{D}}^{ - \frac{1}{2}}}$, where $\widetilde {\bf{X}} = \left[ {\begin{array}{*{20}{c}}
{\bf{X}}&{\bf{1}}
\end{array}} \right]$
\State {\bf{initialize}}: ${{\bf{a}}^{\left( 0 \right)}},{{\bf{u}}^{\left( 0 \right)}}$
\Repeat
\State
${\bf{B}} = {{\bf{u}}^{\left( k \right)}} + {u_1}{\bf{1}} - {{\bf{\hat a}}^{\left( k \right)}}$
\State
${\bf{S}} = {{\bf{Z}}^T}{\bf{B}}$
\State
${\boldsymbol{\theta }} = {u_1}{\bf{1}} + {{\bf{u}}^{\left( k \right)}} - {u_1}{\bf{ZS}}$
\State
${{\bf{\hat a}}^{\left( {k + 1} \right)}} = {S_1}\left( {\boldsymbol{\theta }} \right)$
\State
${{\bf{u}}^{\left( {k + 1} \right)}} = {\boldsymbol{\theta }} - {{\bf{\hat a}}^{\left( {k + 1} \right)}}$
\Until $\left\| {{{\bf{u}}^{\left( {k + 1} \right)}} - {{\bf{u}}^{\left( k \right)}}} \right\| \le \varepsilon $
\State
${\left[ {\begin{array}{*{20}{c}}
{\boldsymbol{\beta }}&{{\beta _0}}
\end{array}} \right]^T} = {\bf{Q}}{{\bf{D}}^{ - \frac{1}{2}}}{\bf{S}}$
\end{algorithmic}
\end{algorithm}
\subsection{Pre-computed Inverse System Matrix}
In Step 3 of {\bf\textsl{Algorithm 1}}, since the training samples are fixed in a single training process, the system matrix ${\bf{A}}$ is also fixed. The inverse of the system matrix can therefore be pre-computed and the precomputed inverse system matrix ${\bf{A}}^{-1}$ can be reused in each iteration. The matrix inversion is computed only once, which greatly reduces the hardware complexity. The matrix inversion, as shown in {\bf\textsl{Algorithm 3}}, is realized by EVD, which can be realized by a shared EVD hardware for the Nystr\"{o}m method, as shown in Step 2 of {\bf\textsl{Algorithm 2}}.
\begin{figure}[t]
\centering
\includegraphics[width=0.5\textwidth]{figures/reuse-schedule.pdf}
\caption{Scheduling of memory storage for intermediate variables.}
\label{reuse-schedule}
\end{figure}
\subsection{Common Term Combination}
Redundant computations in {\bf\textsl{Algorithm 1}} should be avoided to enhance the hardware utilization. The update equations in Step 3 to Step 5 of {\bf\textsl{Algorithm 1}} can be re-written as
\begin{align}
\label{E:16}
\widetilde{\boldsymbol{\beta }}^{(k+1)} &= {\bf{A}}^{-1} \widetilde {\bf{X}}^{T}{\bf{Y}} \left( {\bf{u}}^{(k)} + {u_1}{\bf{1}} - {u_1}{\bf{a}}^{(k)} \right), \\
\label{E:17}
{\bf{a}}^{(k+1)} &= {S_{\frac{1}{{{\mu _1}}}}} \left( \frac{1}{{u_1}} \left( {u_1}{\bf{1}} + {\bf{u}}^{(k)} - {u_1}{\bf{Y}} \widetilde{\bf{X}} \widetilde{\boldsymbol{\beta }}^{(k+1)} \right) \right), \\
\label{E:18}
{\bf{u}}^{(k+1)} &= \left( {u_1}{\bf{1}} + {\bf{u}}^{(k)} - {u_1}{\bf{Y}} \widetilde{\bf{X}} \widetilde{\boldsymbol{\beta }}^{(k+1)} \right) - {u_1}{\bf{a}}^{(k+1)},
\end{align}
where $\widetilde{\boldsymbol{\beta }}$ denotes $\left[ {\begin{array}{*{20}{c}} {\boldsymbol{\beta}}&{\beta_0} \end{array}} \right]^{T}$. By defining a new variable ${\boldsymbol{\theta}}$, given by
\begin{align} \label{E:19}
{\boldsymbol{\theta}} = {u_1}{\bf{1}} + {\bf{u}}^{(k)} - {u_1}{\bf{Y}} \widetilde{\bf{X}} \widetilde{\boldsymbol{\beta }}^{(k+1)},
\end{align}
(\ref{E:17}) and (\ref{E:18}) can be further simplified as:
\begin{align}
\label{E:20}
{\bf{a}}^{(k+1)} &= {S_{\frac{1}{{{\mu _1}}}}} \left( \frac{1}{{u_1}} {\boldsymbol{\theta}} \right) = \frac{1}{{u_1}}{S_1 \left( \boldsymbol{\theta} \right) = \frac{1}{{u_1}}} {{\bf{\hat a}}^{\left( {k + 1} \right)}}, \\
\label{E:21}
{\bf{u}}^{(k+1)} &= {\boldsymbol{\theta}} - {u_1}{\bf{a}}^{(k+1)} = {\boldsymbol{\theta}} - {{\bf{\hat a}}^{\left( {k + 1} \right)}},
\end{align}
as shown in Step 9 and Step 10 of {\bf\textsl{Algorithm 3}}. Note that ${\boldsymbol{\theta}}$ is computed in Step 8 and can be reused in Step 9 and Step 10, which reduces the computational complexity. The minimization solution $S$, which involves Hinge loss function, is reformulated through variable transformation so that the multiplications of $1/u_1$ in (\ref{E:20}) and $u_1$ in (\ref{E:21}) are mutually cancelled out, greatly reducing the hardware complexity.
\begin{figure*}[t]
\centering
\includegraphics[width=0.7\textwidth]{figures/system-architecture.pdf}
\caption{Architecture of the ADMM-based SVM processor with on-chip learning for epileptic seizure detection \cite{Huang:18_VLSI}. }
\label{system-architecture}
\end{figure*}
\subsection{Memory Sharing for Intermediate Variables}
The matrix ${\bf{A}}^{-1}$ remains constant throughout the entire training process. It requires an extra memory with a size of $(p+1) \times (p+1)$ to store ${\bf{A}}^{-1}$. Instead of introducing extra memory, a memory reuse strategy is proposed to achieve efficient hardware mapping. By substituting $\widetilde{\boldsymbol{\beta }}$ in (\ref{E:16}) for (\ref{E:19}), ${\boldsymbol{\theta}}$ can be computed as:
\begin{align}
\label{E:22}
{\boldsymbol{\theta}} &= {u_1}{\bf{1}} + {\bf{u}}^{(k)} - {u_1}{\bf{Y}} \widetilde{\bf{X}} {\bf{Q}}{\bf{D}}^{-1}{\bf{Q}}^{T} {\widetilde{\bf{X}}}^{T}{\bf{Y}}^{T}{\bf{B}} \\
\label{E:23}
&= {u_1}{\bf{1}} + {\bf{u}}^{(k)} - {u_1}{\bf{M}}{\bf{B}},
\end{align}
where ${\bf{B}}$ is defined in Step 6 of {\bf\textsl{Algorithm 3}}, and ${\bf{Q}}{\bf{D}}{\bf{Q}}^{T}$ is the result of the eigenvalue decomposition of the matrix ${\bf{A}}$. The matrix ${\bf{M}}$, which combines all the constant terms, is an $N \times N$ matrix. It is impractical to store such a large matrix. By leveraging pre-computation and the memory reuse, a new variable ${\bf{Z}}$, which is an $N \times (p+1)$ matrix, is introduced to store the pre-computed matrix and to reuse the memory space occupied by training sample matrix $\widetilde{\bf{X}}$, which is also an $N \times (p+1)$ matrix, as shown in Step 3 in {\bf\textsl{Algorithm 3}}. Therefore, $\widetilde{\bf{X}}$, which has already merged into $\bf{Z}$, is no longer needed in the subsequent steps. The proposed technique results in a computational complexity reduction without extra memory overhead.
Scheduling for memory storage is further improved for intermediate variables to achieve high memory utilization. The ${\boldsymbol{\theta}}$ can be computed by
\begin{align}
\label{E:24}
{\boldsymbol{\theta}} &= {u_1}{\bf{1}} + {\bf{u}}^{(k)} - {u_1}{\bf{Z}}{\bf{Z}}^{T}{\bf{B}} \\
&= {u_1}{\bf{1}} + {\bf{u}}^{(k)} - {u_1}{\bf{Z}}{\bf{S}},
\end{align}
where variable ${\bf{S}}$ is introduced to split the computation into two steps, as shown in Step 7 and Step 8 of the {\bf\textsl{Algorithm 3}}. Fig. \ref{reuse-schedule} shows the memory allocation for intermediate variables over time. The intermediate variables $\bf{B}$, $\bf{S}$, ${\boldsymbol{\theta }}$, $\bf{\hat{a}}$, and $\bf{u}$ share two caches and have minimum dependency for few steps. For example, the variable $\bf{B}$ is generated in Step 6, and only be used in Step 7 to compute the variable $\bf{S}$. After Step 7, the variable $\bf{B}$ is no longer required and therefore the occupied memory can be released for other variables, such as variable $\bf{S}$. Only two ($N \times 1$) memory banks are needed for these intermediate variables.
\section{Design Example}
An SVM-based seizure detector with on-chip learning for epilepsy neuro-modulation is demonstrated in \cite{Huang:18_VLSI}. It adopts the proposed ADMM-based training algorithm to achieve high efficiency in both energy and area metrics. Fig. \ref{system-architecture} shows the system architecture of the ADMM-based seizure detector. The architecture includes a FFT-based feature extractor to extract spectral-energy features, SRAM memory banks to store training samples and intermediate data, and an SVM processing engine to handle all the computations that requires in both SVM training and inference phases.
Low-rank approximation enables a lower matrix dimension for iterative update and the matrix inversion. This greatly reduces the complexity for the hardware mapping. The rank is reduced from 256 to 16, which results in a 97\% computational complexity reduction and an 87\% memory storage reduction compared to the direct-mapped hardware mapping.
The SVM processing engine includes a CORDIC-based processing element (PE) array, register banks, and an index generator for Jacobi rotation indices. The CORDIC-based PE is exploited to support all the linear and non-linear operations. The PE array can be configured as a multiply-and-accumulate (MAC) or an adder tree to support different vector manipulations in the iterative update step. Additionally, a systolic PE array is designed to perform high-speed EVD. The hardware for EVD is shared for Nystr\"{o}m method in rank approximation and matrix inversion for solving the linear system in the variable update phase, which reduces the silicon area by 37\%. The approximate Jacobi method \cite{Gotze:93_TC} is adopted to reduce the processing latency. Unlike the cyclic Jacobi method, which requires several cycles for vector rotation and angle generation, the approximate Jacobi method only needs one cycle. The approximate Jacobi method achieves an 8.9$\times$ convergence speed than the conventional cyclic Jacobi method. By rotating the disjointed row and column pairs in parallel, the processing latency is reduced by 8 times. The overall latency reduction for EVD is reduced by 98.6\% compared to the direct-mapped design that realizes the cyclic Jacobi method.
By decomposing the matrix $\bf{A}$ into $\bf{Q}\bf{D}{\bf{Q}}^{T}$, $\bf{Q}$ and $\bf{D}$ can be combined with $\bf{Y}$ and $\widetilde{\bf{X}}$ to form the matrix $\bf{Z}$. The EVD of $\bf{A}$ dominates the computational complexity of a single update iteration. By this pre-computation scheme, the EVD of $\bf{A}$ is computed only once, and 22\% of the computational complexity is saved for an iteration compared to the design without pre-computation. The computational complexity can be further reduced by 51\% by reusing ${\boldsymbol{\theta }}$ and $\bf{Z}$. The overall computational complexity is reduced by 62\% by exploiting pre-computation and common term combination. Two caches are exploited to store intermediate variables in the iterative update step. Since the variables are alternately used and only active for a short period, the memory banks that store these variables can be shared. The register memory is reduced by 60\% through proper scheduling. Furthermore, since the training sample memory $\bf{X}$ is dispensable in iterative update step, the SRAMs used for storing $\bf{X}$ are fully reused for storing $\bf{Z}$. No extra memory is needed to store the pre-computed $\bf{Z}$.
The seizure detector chip supports both inference and training. It is the first silicon proof of SVM training on chip. The chip can perform SVM training in 0.78 second and only dissipates 2.9 mW. Compared to an Intel i7-2600 CPU, the chip achieves a 153,310$\times$ higher energy efficiency and a 364$\times$ higher throughput-to-area ratio for SVM training.
\section{Conclusions}
SVM is a powerful classifier for applications that lack sufficient training samples and ask for low power dissipation. It features simple computation in inference but the computational complexity for training is still very high. On-line training has drawn attention for model adaptation at the edge devices. Compared to the conventional SMO algorithm, ADMM-based algorithm features a short convergence latency and sparse weights. Low rank approximation via Nystr\"{o}m method is applied for feasible hardware mapping, which greatly reduces the overall computational complexity for SVM training. Four open datasets are used to demonstrate the effectiveness of the ADMM-based algorithm with low rank approximation. To achieve efficient hardware mapping, several hardware techniques are proposed to further reduce the hardware complexity. Pre-computation for inverse matrix and combining common terms jointly reduce the computational complexity by 62\%. Memory sharing for intermediate variables reduces 60\% of memory usage. As a proof of concept, a seizure detector chip showcases the high efficiency of the proposed architecture for SVM training. The chip dissipates less than 3 mW and achieves orders of magnitude improvement in energy efficiency when compared to a high-end CPU. The proposed algorithm allows low-power, real-time SVM training on chip, providing a promising solution for smart edge devices.
\section*{Acknowledgment}
The authors thank the Taiwan Semiconductor Research Institute (TSRI) for technical support on chip design.
\bibliographystyle{IEEEtran}
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{
"redpajama_set_name": "RedPajamaArXiv"
}
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Q: Using prevAll to select all sections of a table containing a keyword Here's the jsFiddle with the relevant table HTML.
I want to use jQuery to highlight or remove sections of a table if they contain certain keywords.
<table>
<tr class="mark"></tr>
<tr><td>apple</td></tr>
<tr><td>1</td></tr>
<tr><td>2</td></tr>
<tr><td>3</td></tr>
<tr><td>4</td></tr>
<tr><td>5</td></tr>
<tr class="mark"></tr>
<tr><td>pear</td></tr>
<tr><td>1</td></tr>
<tr><td>2</td></tr>
<tr><td>3</td></tr>
<tr><td>4</td></tr>
<tr><td>5</td></tr>
<tr class="mark"></tr>
<tr><td>apple</td></tr>
<tr><td>1</td></tr>
<tr><td>2</td></tr>
<tr><td>3</td></tr>
<tr><td>4</td></tr>
<tr><td>5</td></tr>
<tr class="mark"></tr>
<tr><td>pear</td></tr>
<tr><td>1</td></tr>
<tr><td>2</td></tr>
<tr><td>3</td></tr>
<tr><td>4</td></tr>
<tr><td>5</td></tr>
<tr class="mark"></tr>
<tr><td>apple</td></tr>
<tr><td>1</td></tr>
<tr><td>2</td></tr>
<tr><td>3</td></tr>
<tr><td>4</td></tr>
<tr><td>5</td></tr>
<tr class="mark"></tr>
<tr><td>pear</td></tr>
<tr><td>1</td></tr>
<tr><td>2</td></tr>
<tr><td>3</td></tr>
<tr><td>4</td></tr>
<tr><td>5</td></tr>
<tr class="mark"></tr>
<tr><td>apple</td></tr>
<tr><td>1</td></tr>
<tr><td>2</td></tr>
<tr><td>3</td></tr>
<tr><td>4</td></tr>
<tr><td>5</td></tr>
</table>
Right now, I can use
$("td:contains('pear')").parent().prev().nextUntil(".mark").css("background-color", "red");
to accomplish what I want to do (highlight all sections, delimited by ===, that contain "pear"). But how can I do this with prevAll for the cases where the delimiter row isn't directly previous to the text I'm searching for? I tried $("td:contains('pear')").parent().prevAll(".mark:first").nextUntil(".mark").css("background-color", "red"); which I think should work, but only one section is highlighted. I think I'm missing a key piece of understanding about how jQuery deals with situations where I want to map actions like prevAll() to a list of elements.
Please ignore the bad table layout; this is simply what I have to work with T_T
A: This should work:
$("td:contains('pear')").parent().each(function() {
$(this).prevAll(".mark:first").nextUntil(".mark").css("background-color", "red");
});
See: https://jsfiddle.net/aszepeshazi/9czm7g3y/2/
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{
"redpajama_set_name": "RedPajamaStackExchange"
}
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I can't take seriously any blog that is meant to tell people 'how to write', yet is riddled with spelling mistakes and grammatical errors. Often, it looks like a first draft.
Sorry, you find the typos so disturbing.
But like Mark Twain said, "I have no use for a man that can only spell a word one way".
You realize, of course, that a "standard" way to spell words, and punctuate a narrative was only made-up virtually yesterday. But alas, I fear you've made up your mind.
Editing and proofreading is what God made editors and proofreaders for. It's not that I don't try, or believe it's unimportant; I'm just not very good at it. Hence my call for proofreaders.
What I am good at is telling stories and understanding technologies.
And after 5 decades wrestling with the Muse, I've made some astonishing discoveries and I've codified the subject of writing fiction in a very accessible way.
Perhaps you'll run across the book someday (when it's all cleaned up) so that you can see past the typos and gain the tools you need.
I wish you nothing but the best, my friend.
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{
"redpajama_set_name": "RedPajamaC4"
}
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We provide oral health services at more than 60 school sites in Pierce County through our School-Based Oral Health Program. We provide oral health screenings, to identify decay and other dental needs. We provide sealant applications, to reduce the chances of children developing decay on their molars. We also provide fluoride varnish, to strengthen and protect teeth as they develop. Our goal is to ensure children in our community are starting their lives off with healthy mouths.
We coordinate with the Tacoma-Pierce County Health Department (TPCHD) who acts as the "clearinghouse" for providers in the counties' School-Based Oral Health program. At the bottom of this page are the districts and schools for whom we provide service. We only provide services through this program at the sites below at this time.
If your child attends one of the schools below and you would like to sign them up for our program, please complete our Parent Permission Form that will be sent home with your child. Please submit the form to your child's school office. Please note that if our date of service for your child's school has passed, we may not return within the school year.
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{
"redpajama_set_name": "RedPajamaC4"
}
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Home / News / GDPR – and what it means for the UAE
I'm sure you've seen it all over the web - Linkedin, Twitter, news, blogs – people talking about something called GDPR (or sometimes GRDP or GDRP), companies are all of a sudden spamming you with notices of changes in their data privacy policies and requesting consent for marketing. And you are left wondering: what exactly is GDPR and how does it affect me?
GDPR – What is it?
GDPR is simply the abbreviation of a new set of EU rules on personal data protection known as the General Data Protection Regulation. And when we say personal data protection, we do not refer to cyber security, but to the processes, policies, internal controls and measures an organization needs to put in place to ensure that it processes data legally, fairly, transparently and securely. If cyber security is an important milestone for data protection, it is by far insufficient to enable privacy compliance. The GDPR contains 99 articles and will come into effect on the 25 May 2018 replacing the 20+ year-old EU Directive on personal data protection. While the GDPR is an EU legislation, it will affect companies targeting EU markets – whether these companies are based in Europe or not. The GDPR seeks to address directly the challenges of the new era of big data, artificial intelligence (AI) and complex data ecosystems. One of its main goals is to sufficiently empower individuals with control over their data and secure an appropriate standard of enforcement.
Why care if you are not based in the EU?
With its extra territorial scope, the GDPR will apply to companies based in the UAE that collect, store, access, use or manipulate in any other way data of people in the EU. It will apply as well to companies based in UAE that have establishment within the EU. This means that any company offering goods or services to the EU market would potentially be affected by the GDPR. The GDPR is not industry specific. Any e-commerce, hospitality service, airline company, social media, big retail, but not limited in any way to these indicative examples, can be affected by the GDPR.
Putting GDPR aside, privacy and data governance should be high on UAE's agenda given that the country strives to position itself as a hub for innovation with specific focus on digital and AI. For instance, the UAE has appointed a dedicated Minister of State for Artificial Intelligence and has developed a UAE Centennial 2071 Strategy aiming to transform the UAE into the best country in the world. One of the pillars for achieving this is to promote advanced technology and engineering. High-budget projects on digital transformation have been multiplying, along with initiatives around cryptocurrencies and blockchain. For example, the UAE has launched an end-to-end virtual courtroom (ADGM eCourts).
While there is no general data protection law in the UAE which applies uniformly across the country, there are provisions in relation to privacy and personal data protection contained in the UAE Constitution, the Cyber Crime Law (Federal Decree Law No. 5 of 2012), and the Telecommunications Law (Federal Law by Decree No. (3) of 2003). In addition, Dubai International Financial Center (DIFC), a free zone, has its own data protection regime, which is comparable to the EU legislative framework. Generally, the UAE has acknowledged the importance of privacy and personal data protection. At the Data Privacy Day this year, the Ministry of Finance stated that privacy and data protection is more important than ever before and that implementing best international practices on data protection is crucial for establishing a sound economic environment that gains the trust of investors and financial institutions.
GDPR being considered as the highest standard of privacy protection world-wide, UAE companies willing to embed privacy within their culture can only benefit by using GDPR as a reference regardless whether it applies to them directly or not.
Why is GDPR a big deal?
First, because of the fines. The GDPR foresees fines that can reach up to 4% of the group/global turnover of companies or EUR 20 million (or about AED 85 million), whichever is higher. These are maximum fines, so don't panic!
Second, because of its reach. With its extended territorial scope globally, the GDPR will affect a large number of companies overseas.
Third, reputational impact. Data is the oil of our era and its protection is key. Privacy has become a trend that is not going away anytime soon. This is evident from the high-profile cases on data and privacy breaches making the headlines daily. Facebook, Equifax, Yahoo, Uber, to name just a few, have seen the value of their shares dropping drastically after their data cases, followed by a decreased trust amongst customers.
Dealing with big data without well-embedded privacy and data governance, as a foundation, is not sustainable anymore, so organizations better adapt. GDPR is perceived as the new "gold standard" for privacy and many countries intending to reform their existing privacy regulations or introduce a privacy regulation will most probably look to the GDPR. So, whether GDPR directly applies to your organization or not, do you really want to be the one that will remain in silo and choose to be outdated and vulnerable to reputational damage? With individuals becoming increasingly aware of their rights, such as the "right to be forgotten", this trend is only likely to increase.
So, what's next? Don't panic but start doing the right thing!
Yes, there are potentially big fines associated with GDPR, but don't panic. It would be surprising if as of 26 May (a Saturday), regulators open a witch-hunt for companies and start fining them millions. We all know that 100% compliance with any new legislation is difficult. Therefore, UAE companies should take a risk-based approach and start doing their homework to limit the risk of a fine and/or reputational damage and demonstrate accountability. In case of a breach leading to a potential fine, regulators will likely take into account multiple factors while determining the amount of the fine. These could be the severity and impact of the breach on individuals' rights and the ability of the organization to demonstrate that they have not been neglecting privacy and have done their due-diligence.
What does this mean for companies?
Take it step-by-step but don't wait too long to start your privacy improvement exercise as it is a journey rather than a tick-box exercise. You don't want to wait too long and take the risk of being under the spotlight of a data or privacy breach. As a start you could engage into a privacy health check, if you haven't done one yet, and determine your key risk areas and how important privacy is to your organization. Then, embark onto a privacy remediation program. Depending on the size of your organization and how data-driven you are, this could mean six months, a year or even two. One should not underestimate the time and resources necessary for a successful privacy improvement program. Most of the time, privacy is something quite new to companies, which means that designing the necessary privacy controls and processes is one thing, but effectively implementing these is another. Privacy improvement programs are change management programs – they take time as they often require an important shift in organizational culture. Therefore, it is important that you start as early as you can, as neither compliance with 99 articles nor successful implementation of privacy within an organization can be achieved in a day.
What others do
More and more companies in the UAE are becoming increasingly aware that GDPR is likely to apply to them. But the fact that companies world-wide are taking action on privacy and GDPR is creating a general sense of urgency. Some companies in the region have already embarked onto fully-fledged privacy improvement programs, privacy audits and started appointing Data Protection Officers or similar specialized functions with privacy expertise to help them embed privacy compliance effectively. Companies have started looking at how they manage consent collection and withdrawal, revisiting their privacy notices, and conducting privacy trainings and awareness campaigns.
Generally, organizations have started realizing that privacy is a core component of data governance and that with GDPR or not, privacy has become a competitive necessity. The trend of privacy as a pure compliance matter is fading away and some companies have built strategies around privacy and integrated it in their Corporate Responsibility agenda.
The bottom-line is that establishing privacy as a core value within your organization can be an opportunity to reposition your brand on the market and build a competitive advantage, because nowadays privacy equals consumer trust.
Written by Dessi Vitcheva
Claes Mårtensson joins Transcendent Group
We are very excited to announce that Claes Mårtensson, from April 1st, is a member of the Audit & Governance...
Phil Knight joins Transcendent Group Norway.
Welcome Phil!
Transcendent Group Belgium celebrates 1 year
Transcendent Group Belgium had the great joy to celebrate our one-year anniversary in January. Transcendent Group's office in Brussels is...
Documentation is key to IDD-compliant inducements
Insurance Distribution Directive, having taken effect on the 1st of October, brings about a yet new regulative instrument for insurance-based...
Who is responsible for compliance with GDPR, PSD2 and AML?
New regulations bring new opportunities and new risks to businesses.
GRC Conference
Independence of the Data Protection Officer
How do we ensure the independence of the Data Protection Officer?
|
{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
| 7,748
|
The Hong Kong Film Award for Best New Director is an annual Hong Kong industry award presented to a director who is considered the best of the year.
History
In 2002 the new category of Outstanding Young Director was created as a way to recognise up-and-coming directors, the first award was presented to Stephen Chow for his 2001 film Shaolin Soccer. Two years later at the 23rd Hong Kong Film Awards the award was renamed Best New Director and continues to be presented.
Winners and nominees
References
External links
Awards for best director
Awards established in 2002
Hong Kong Film Awards
|
{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 5,638
|
{"url":"https:\/\/socratic.org\/questions\/how-do-you-find-the-critical-numbers-of-f-x-x-2-3-x-1-3","text":"# How do you find the critical numbers of f(x)=x^(2\/3)+x^(-1\/3)?\n\nSep 17, 2017\n\nThis function has one critical value of $x$, at $x = \\frac{1}{2}$, and gives a local minimum value there of $f \\left(\\frac{1}{2}\\right) = \\frac{3}{{2}^{\\frac{2}{3}}} \\approx 1.89$\n\n#### Explanation:\n\nThe derivative of this function is $f ' \\left(x\\right) = \\frac{2}{3} {x}^{- \\frac{1}{3}} - \\frac{1}{3} {x}^{- \\frac{4}{3}}$. Getting a common denominator of $3 {x}^{\\frac{4}{3}}$ allows us to write $f ' \\left(x\\right) = \\frac{2 x - 1}{3 {x}^{\\frac{4}{3}}}$. This is equal to zero when $2 x - 1 = 0$, which means $x = \\frac{1}{2}$. It's also undefined at $x = 0$, though the original function is undefined there as well.\n\nHence, the value $x = \\frac{1}{2}$ is the only critical value of $f$. The sign of $f '$ changes from negative to positive as $x$ increases through $\\frac{1}{2}$, implying that there is a local (relative) minimum point at $x = \\frac{1}{2}$.\n\nThe local minimum value is f(1\/2)=1\/2^(2\/3)+2^(1\/3)=(1+2)\/(2^(2\/3)) =3\/(2^(2\/3)) approx 1.89.\n\nHere's the graph of this function:\n\ngraph{x^(2\/3)+x^(-1\/3) [-10, 10, -5, 5]}","date":"2021-09-18 04:35:09","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 16, \"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\": 0.9345055818557739, \"perplexity\": 213.34192685574487}, \"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\/1631780056297.61\/warc\/CC-MAIN-20210918032926-20210918062926-00183.warc.gz\"}"}
| null | null |
A pütchipü'ü, or pütche'ejachi (in Wayuu, "messenger of the word"; ), is the central element in the traditional administration justice system of the Wayuu people. The role of a pütchipü'ü is to solve conflicts through mediation and negotiation. This can include conflicts between different members or clans of the Wayuu community or with people or organizations outside the Wayuu people. This negotiation includes material compensation to be paid by the aggravating family to the aggrieved family group.
In 2010, this mediation system was recognized by UNESCO as part of the Intangible Cultural Heritage of Humanity. This indigenous law system has been recognized by governments of Venezuela and Colombia. The pütchipü'ui are organized by the Major Autonomous Board of Palabreros.
See also
Masterpieces of the Oral and Intangible Heritage of Humanity
UNESCO Intangible Cultural Heritage Lists
References
Wayuu
Masterpieces of the Oral and Intangible Heritage of Humanity
Traditional knowledge
|
{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 1,807
|
<?php
/**
* COciSchema is the class for retrieving metadata information from an Oracle database.
*
* @property string $defaultSchema Default schema.
*
* @author Ricardo Grana <rickgrana@yahoo.com.br>
* @version $Id: COciSchema.php 3515 2011-12-28 12:29:24Z mdomba $
* @package system.db.schema.oci
*/
class COciSchema extends CDbSchema
{
private $_defaultSchema = '';
/**
* @var array the abstract column types mapped to physical column types.
* @since 1.1.6
*/
public $columnTypes=array(
'pk' => 'NUMBER(10) NOT NULL PRIMARY KEY',
'string' => 'VARCHAR2(255)',
'text' => 'CLOB',
'integer' => 'NUMBER(10)',
'float' => 'NUMBER',
'decimal' => 'NUMBER',
'datetime' => 'TIMESTAMP',
'timestamp' => 'TIMESTAMP',
'time' => 'TIMESTAMP',
'date' => 'DATE',
'binary' => 'BLOB',
'boolean' => 'NUMBER(1)',
'money' => 'NUMBER(19,4)',
);
/**
* Resets the sequence value of a table's primary key.
* The sequence will be reset such that the primary key of the next new row inserted
* will have the specified value or 1.
* @param CDbTableSchema $table the table schema whose primary key sequence will be reset
* @param mixed $value the value for the primary key of the next new row inserted. If this is not set,
* the next new row's primary key will have a value 1.
* @since 1.1
*/
public function resetSequence($table,$value=null)
{
if($table->sequenceName!==null)
{
$this->getDbConnection()->createCommand("DROP SEQUENCE ".$table->name."_SEQ")->execute();
$createSequenceSql = <<< SQL
create sequence {$table->name}_SEQ
start with 1
increment by 1
nomaxvalue
nocache
SQL;
$this->getDbConnection()->createCommand($createSequenceSql)->execute();
}
}
/**
* Quotes a table name for use in a query.
* A simple table name does not schema prefix.
* @param string $name table name
* @return string the properly quoted table name
* @since 1.1.6
*/
public function quoteSimpleTableName($name)
{
return '"'.$name.'"';
}
/**
* Quotes a column name for use in a query.
* A simple column name does not contain prefix.
* @param string $name column name
* @return string the properly quoted column name
* @since 1.1.6
*/
public function quoteSimpleColumnName($name)
{
return '"'.$name.'"';
}
/**
* Creates a command builder for the database.
* This method may be overridden by child classes to create a DBMS-specific command builder.
* @return CDbCommandBuilder command builder instance
*/
protected function createCommandBuilder()
{
return new COciCommandBuilder($this);
}
/**
* @param string $schema default schema.
*/
public function setDefaultSchema($schema)
{
$this->_defaultSchema=$schema;
}
/**
* @return string default schema.
*/
public function getDefaultSchema()
{
if (!strlen($this->_defaultSchema))
{
$this->setDefaultSchema(strtoupper($this->getDbConnection()->username));
}
return $this->_defaultSchema;
}
/**
* @param string $table table name with optional schema name prefix, uses default schema name prefix is not provided.
* @return array tuple as ($schemaName,$tableName)
*/
protected function getSchemaTableName($table)
{
$table = strtoupper($table);
if(count($parts= explode('.', str_replace('"','',$table))) > 1)
return array($parts[0], $parts[1]);
else
return array($this->getDefaultSchema(),$parts[0]);
}
/**
* Loads the metadata for the specified table.
* @param string $name table name
* @return CDbTableSchema driver dependent table metadata.
*/
protected function loadTable($name)
{
$table=new COciTableSchema;
$this->resolveTableNames($table,$name);
if(!$this->findColumns($table))
return null;
$this->findConstraints($table);
return $table;
}
/**
* Generates various kinds of table names.
* @param COciTableSchema $table the table instance
* @param string $name the unquoted table name
*/
protected function resolveTableNames($table,$name)
{
$parts=explode('.',str_replace('"','',$name));
if(isset($parts[1]))
{
$schemaName=$parts[0];
$tableName=$parts[1];
}
else
{
$schemaName=$this->getDefaultSchema();
$tableName=$parts[0];
}
$table->name=$tableName;
$table->schemaName=$schemaName;
if($schemaName===$this->getDefaultSchema())
$table->rawName=$this->quoteTableName($tableName);
else
$table->rawName=$this->quoteTableName($schemaName).'.'.$this->quoteTableName($tableName);
}
/**
* Collects the table column metadata.
* @param COciTableSchema $table the table metadata
* @return boolean whether the table exists in the database
*/
protected function findColumns($table)
{
$schemaName=$table->schemaName;
$tableName=$table->name;
$sql=<<<EOD
SELECT a.column_name, a.data_type ||
case
when data_precision is not null
then '(' || a.data_precision ||
case when a.data_scale > 0 then ',' || a.data_scale else '' end
|| ')'
when data_type = 'DATE' then ''
when data_type = 'NUMBER' then ''
else '(' || to_char(a.data_length) || ')'
end as data_type,
a.nullable, a.data_default,
( SELECT D.constraint_type
FROM ALL_CONS_COLUMNS C
inner join ALL_constraints D on D.OWNER = C.OWNER and D.constraint_name = C.constraint_name
WHERE C.OWNER = B.OWNER
and C.table_name = B.object_name
and C.column_name = A.column_name
and D.constraint_type = 'P') as Key
FROM ALL_TAB_COLUMNS A
inner join ALL_OBJECTS B ON b.owner = a.owner and ltrim(B.OBJECT_NAME) = ltrim(A.TABLE_NAME)
WHERE
a.owner = '{$schemaName}'
and (b.object_type = 'TABLE' or b.object_type = 'VIEW')
and b.object_name = '{$tableName}'
ORDER by a.column_id
EOD;
$command=$this->getDbConnection()->createCommand($sql);
if(($columns=$command->queryAll())===array()){
return false;
}
foreach($columns as $column)
{
$c=$this->createColumn($column);
$table->columns[$c->name]=$c;
if($c->isPrimaryKey)
{
if($table->primaryKey===null)
$table->primaryKey=$c->name;
else if(is_string($table->primaryKey))
$table->primaryKey=array($table->primaryKey,$c->name);
else
$table->primaryKey[]=$c->name;
$table->sequenceName='';
$c->autoIncrement=true;
}
}
return true;
}
/**
* Creates a table column.
* @param array $column column metadata
* @return CDbColumnSchema normalized column metadata
*/
protected function createColumn($column)
{
$c=new COciColumnSchema;
$c->name=$column['COLUMN_NAME'];
$c->rawName=$this->quoteColumnName($c->name);
$c->allowNull=$column['NULLABLE']==='Y';
$c->isPrimaryKey=strpos($column['KEY'],'P')!==false;
$c->isForeignKey=false;
$c->init($column['DATA_TYPE'],$column['DATA_DEFAULT']);
return $c;
}
/**
* Collects the primary and foreign key column details for the given table.
* @param COciTableSchema $table the table metadata
*/
protected function findConstraints($table)
{
$sql=<<<EOD
SELECT D.constraint_type as CONSTRAINT_TYPE, C.COLUMN_NAME, C.position, D.r_constraint_name,
E.table_name as table_ref, f.column_name as column_ref,
C.table_name
FROM ALL_CONS_COLUMNS C
inner join ALL_constraints D on D.OWNER = C.OWNER and D.constraint_name = C.constraint_name
left join ALL_constraints E on E.OWNER = D.r_OWNER and E.constraint_name = D.r_constraint_name
left join ALL_cons_columns F on F.OWNER = E.OWNER and F.constraint_name = E.constraint_name and F.position = c.position
WHERE C.OWNER = '{$table->schemaName}'
and C.table_name = '{$table->name}'
and D.constraint_type <> 'P'
order by d.constraint_name, c.position
EOD;
$command=$this->getDbConnection()->createCommand($sql);
foreach($command->queryAll() as $row)
{
if($row['CONSTRAINT_TYPE']==='R') // foreign key
{
$name = $row["COLUMN_NAME"];
$table->foreignKeys[$name]=array($row["TABLE_REF"], $row["COLUMN_REF"]);
if(isset($table->columns[$name]))
$table->columns[$name]->isForeignKey=true;
}
}
}
/**
* Returns all table names in the database.
* @param string $schema the schema of the tables. Defaults to empty string, meaning the current or default schema.
* If not empty, the returned table names will be prefixed with the schema name.
* @return array all table names in the database.
*/
protected function findTableNames($schema='')
{
if($schema==='')
{
$sql=<<<EOD
SELECT table_name, '{$schema}' as table_schema FROM user_tables
EOD;
$command=$this->getDbConnection()->createCommand($sql);
}
else
{
$sql=<<<EOD
SELECT object_name as table_name, owner as table_schema FROM all_objects
WHERE object_type = 'TABLE' AND owner=:schema
EOD;
$command=$this->getDbConnection()->createCommand($sql);
$command->bindParam(':schema',$schema);
}
$rows=$command->queryAll();
$names=array();
foreach($rows as $row)
{
if($schema===$this->getDefaultSchema() || $schema==='')
$names[]=$row['TABLE_NAME'];
else
$names[]=$row['TABLE_SCHEMA'].'.'.$row['TABLE_NAME'];
}
return $names;
}
/**
* Builds a SQL statement for renaming a DB table.
* @param string $table the table to be renamed. The name will be properly quoted by the method.
* @param string $newName the new table name. The name will be properly quoted by the method.
* @return string the SQL statement for renaming a DB table.
* @since 1.1.6
*/
public function renameTable($table, $newName)
{
return 'ALTER TABLE ' . $this->quoteTableName($table) . ' RENAME TO ' . $this->quoteTableName($newName);
}
/**
* Builds a SQL statement for changing the definition of a column.
* @param string $table the table whose column is to be changed. The table name will be properly quoted by the method.
* @param string $column the name of the column to be changed. The name will be properly quoted by the method.
* @param string $type the new column type. The {@link getColumnType} method will be invoked to convert abstract column type (if any)
* into the physical one. Anything that is not recognized as abstract type will be kept in the generated SQL.
* For example, 'string' will be turned into 'varchar(255)', while 'string not null' will become 'varchar(255) not null'.
* @return string the SQL statement for changing the definition of a column.
* @since 1.1.6
*/
public function alterColumn($table, $column, $type)
{
$type=$this->getColumnType($type);
$sql='ALTER TABLE ' . $this->quoteTableName($table) . ' MODIFY '
. $this->quoteColumnName($column) . ' '
. $this->getColumnType($type);
return $sql;
}
/**
* Builds a SQL statement for dropping an index.
* @param string $name the name of the index to be dropped. The name will be properly quoted by the method.
* @param string $table the table whose index is to be dropped. The name will be properly quoted by the method.
* @return string the SQL statement for dropping an index.
* @since 1.1.6
*/
public function dropIndex($name, $table)
{
return 'DROP INDEX '.$this->quoteTableName($name);
}
}
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 4,900
|
**C ONTENTS**
_Title Page_
_Dedication_
_Author's Note_
_She yearns for things that..._
**B OOK 1: THE CHIMERA PROJECT**
1. The End
2. Fatboy and His _Moko_
3. The Burst and Transient Source Experiment
4. Save the Whales
5. 111000111
6. Mr. Dawson's Tree Museum
7. Saturday Night
8. Evensong
9. Trust
10. The Man from Subeo
11. Motukiekie
12. Water Works
13. Sea of Green
14. The _Möbius_ Trip
15. Butt Mop
16. The Rainbow Warrior
17. Laundry Piles
18. Waewaetoroa Passage
**B OOK 2: THE LONG WHITE CLOUD**
1. Bambi
2. New Zealand's Most Wanted
3. 'Tis the Season
4. Silent Night
5. White Christmas
6. On Christmas Day
7. Sanctuary
8. An Unnatural Disaster
9. FTBY DNT GO
10. Candid Camera
11. Zeta
12. Xena
13. Shapes in the Mist
14. Epiphany
15. Immunity
16. Before the Storm
17. Kaitiakitanga
18. New Year's Eve
19. The Battle for Auckland
20. Line of Fire
21. Fateful Lightning
22. Silence in the Mist
23. The God from the Machine
24. Hobson Street
25. The Dream
26. Te Kenehi Tuarua
27. The Beginning
_Acknowledgments_
_About the Author_
_Copyright_
_For Rachel, Nicki, Frances, Ray, and Nancy,
my team of secret agents,
without whom nothing would have happened._
_And for Claire B.,_
_who was right._
**A UTHOR'S NOTE**
Each year, I visit a lot of schools and talk to them about my writing. As part of my talk, I often run a competition and the children who achieve the highest levels in that competition win the right to have their name used as the name of a character in one of my books.
Congratulations to Rebecca Richards and Gemma Shaw, whose names appear as the names of characters in this book. Congratulations also to Lucy Southwell, whose name appears in this book by virtue of her father, Graham Southwell, being the highest bidder at a charity auction to raise money for Hospice New Zealand.
_She yearns for things that once were._
_He dreams of things that might be._
_They must learn from the past to change the future._
_But they are running out of time._
**T HE END**
_They took all the trees_
_Put 'em in a tree museum_
_And they charged the people_
_A dollar and a half just to see 'em._
_Don't it always seem to go_
_That you don't know what you've got_
_Till it's gone_
_They paved paradise_
_And put up a parking lot._
_—Joni Mitchell, "Big Yellow Taxi"_
**The end of the world** started quietly enough for Tane Williams and Rebecca Richards, lying on their backs on a wooden platform on Lake Sunnyvale. Which wasn't really a lake at all.
Sunnyvale School was set in a small valley. A nice little suburban valley. A hundred years ago, it had been a nice little swamp where Pukeko and Black Stilts had competed for the best nesting positions, and croakless native frogs had snared insects with their flicking tongues. But now it was a nice little suburban valley, surrounded by nice little homes belonging to nice little homeowners who painted their fences and paid their taxes and never gave any thought to the fact that when it rained, all the water that ran through their properties also ran through the properties below, and the properties below those, and so on until it reached the lowest point of the valley floor. Which happened to be Sunnyvale School.
As a consequence, Sunnyvale School had to have very good drainage. When it rained hard, as it often did in Auckland in the spring, an awful lot of that rain made its way down from the hillsides and ended up on the playing fields and courts of the small but cheerful school.
And sometimes the water, sauntering its way down the slopes with a mind and a mischievous personality of its own, would playfully pick up odds and ends along the way with a view to blocking those very good drains that the council had put in many years ago after the first and second (and possibly the third) time the school had flooded.
Sometimes it worked, and sometimes it didn't. It depended on what the water happened to find in its path. Little sticks and paper food wrappings washed right through the big metal grills of the drains. Small branches, stones, and other large objects generally just ended up at the bottom of the homeowners' nice little properties.
But light twigs and pieces of plastic sailed merrily down the surface of the water and blocked the drains beautifully.
That was what had happened this particular time, and the sports fields of Sunnyvale School were covered in at least four inches of water, high enough to lap at the doorsteps of the cheerful little classrooms across the way, but fortunately not quite high enough to get inside.
Tane and Rebecca lay on their backs on the small wooden viewing platform in the center of the two main playing fields and looked up at the stars, for the rain had stopped many hours ago, and the night was clear and beautiful.
Neither of them were pupils of Sunnyvale School; in fact, both of them were far too old to attend the school, and for another fact, both of them were in their second year at West Auckland High School.
However, when they were younger, they had both gone to Sunnyvale School, which was why they knew that when it rained really hard during the day and stopped at night, it became a magical, wonderful place to be.
The stars above shone down with a piercing intensity that penetrated the haze of lights from the suburban homes around the valley. The moon, too, was lurking about, turning the weathered wood of the small platform to silver. All around them, the lights from the sky above reflected in the inky blackness that was Lake Sunnyvale. The lake that sometimes appeared on the playing fields after a particularly heavy rainstorm.
There were stars above and stars below, rippling slowly in the light breeze, and it was like being out in the center of the universe, floating through space on your back.
Tane and Rebecca thought it was the coolest place to be. On Lake Sunnyvale. After the rain.
Tane tossed a pebble into the air, and there was a satisfying _plop_ a few seconds later as it landed. They both raised their heads to see the widening circles of ripples, shaking the foundations of the stars around them. Then, as if controlled by the same puppeteer, they put their heads back down together.
Tane's feet were pointing one way, and Rebecca's were pointing the other, so the tops of their heads were just about touching. If they had been boyfriend and girlfriend, they might have lain down side by side, but they weren't, so they didn't.
From an open window in a house somewhere on the surrounding slopes, an old Joni Mitchell folk song reached out plaintively across the water to them.
Rebecca said again, "Time travel is impossible." She said it more firmly this time, as if that were simply the end of the discussion, and the judge's decision was final, and no correspondence would be entered into.
Now, ordinarily Tane would have given up at that point, because Rebecca was almost certainly right. After all, it was Rebecca, and not Tane, who had aced her Level One Physics exams the previous year, the top student in the entire country, at the age of just thirteen! Which had been no real surprise to Tane, who had been in the same classes as his friend as she had confounded science teacher after science teacher and math teacher after math teacher by somehow, instinctively, knowing as much about the subject they were teaching as they did.
Some teachers enjoyed having Rebecca in their class because she was very, very clever, if a little rebellious and uncontrollable at times. But other teachers found it stressful to have a girl amongst their students who took great delight in correcting them whenever they made mistakes.
So if Rebecca said that time travel was impossible, then time travel was impossible. But there was something about the stars that night. Something about their slow drift through the heavens above and below them, something about the beautifully random and randomly beautiful patterns they made.
Or then again, it might just have been that Tane liked to argue, and he especially liked to argue with Rebecca.
"I read a book once," Tane said. "I can't remember what it was called, but it was about these grad students who go back in time to medieval days to rescue a missing historian, and they fight—"
_"Timeline,"_ interrupted Rebecca, who also loved a good argument and especially enjoyed showing that she knew more than Tane. "Michael Crichton, 1999."
"Yeah, that's it. But anyway, they manage to create this...like...pinprick in the fabric of time somehow, and then they kind of transmit themselves through it."
"I know. I read it," said Rebecca. And then, perhaps because she realized that she was sounding a bit negative, she said, "I mean, the science was quite good in it, about the fabric of space-time, and the quantum foam, all the way up to the part where they transmit themselves through this tiny hole into the past."
Tane twisted his head around to look at her, but it hurt, and all he could see were her shoes, so he twisted back again. He thought for a moment. True, he wasn't as good at math and science as she was. Tane's strengths were in art and music, and he was a school legend on the harmonica, but even so, the time-travel thing sounded at least feasible to him.
"Why?" he asked eventually. "Why couldn't they transmit themselves?"
"Try to think logically," Rebecca said firmly but not unkindly. "How could you transport a live human being through a pinhole of any kind?"
"What about a fax machine!" Tane said suddenly. "You put a piece of paper in at one place, and it gets sent along a telephone wire and comes out in another place."
"No, it doesn't."
"Yes, it does," said Tane, starting to get into the argument, even though he knew she was going to turn out to be right.
"No, it doesn't," repeated Rebecca. "A copy of the piece of paper comes out. The actual piece of paper you sent stays right where it was. All you are sending is an electronic image of the paper, just like a digital photograph of it. _Fax_ is short for _facsimile,_ which means 'copy.'"
"So...um..." Tane was losing and he knew it.
"We can transmit pictures, sound, even movies, through wires, or through the air in radio waves. But we can't transmit a solid object. Not even a piece of paper."
And that was pretty much the end of the conversation for the moment. They stayed on the platform for a while longer. Neither of them really wanted to go home, for reasons of their own. They talked about school a bit and made some jokes about some of the people in their classes. It was about ten o'clock, after they had sloshed their way through Lake Sunnyvale back to the road, when Tane resumed the argument, as if they had never left off. Which just showed that he had been quietly thinking about it the whole time.
"Well, if we can't transmit people through time, what about sounds, pictures, and movies, like you said?"
Rebecca had to actually think about that for a moment, which was a small victory for Tane. He pulled out his harmonica and played a slow blues riff as they walked. "Nope," she said at last. "If I understand the science right"—and Tane thought she probably did—"then you could only send stuff backward. You couldn't transmit to the future because that hasn't happened yet."
"But you could send it to the past?!"
"Well...theoretically. But let's say we invented some kind of radio transmitter that could broadcast through time. Something that could transmit messages through the quantum foam. Nobody could listen to the messages we were sending, because in the past they wouldn't have invented a radio receiver that could pick up the transmission."
"Oh," said Tane, thinking that Rebecca, as usual, made perfect sense.
They reached Rebecca's house and stopped.
All the lights were off, but one of the windows flickered with the blue glow of a television. Her mum was watching TV, which was no great surprise, because that was pretty much all her mum did all day and all night. At least since Rebecca's dad had died.
"Oh," said Tane again pointlessly, and glanced up at the sky just in time to catch the brief flash of a shooting star.
That was when the inspiration struck him. That was the moment when it all seemed so clear.
"So what if someone in the future had already invented a time radio transmitter and was sending messages back to the past, waiting for someone to invent a receiver?"
He wasn't sure if that sounded silly or not, so he just waited for the usual rebuff from his friend.
It didn't come.
"What's that again?"
"Well, let's just say that some time in the future, someone invents one of those transmitters you were talking about. And just say they are sending out messages, through that foamy stuff, just waiting for someone in the past to invent a receiver."
"Well, I...um..."
"What if we built a receiver and just listened. Just waited for a signal from the future."
"Well, the whole concept of quantum foam is not even proven. And I wouldn't have the slightest idea how to build a receiver," Rebecca mused. "But it's an interesting idea."
That may not have sounded like much, but it wasn't very often that Rebecca thought that Tane had an interesting idea, so it was kind of an important day, if only for that reason.
Although, in hindsight, it was actually an important day for much bigger reasons than that.
"You want to come on another march with me?" Rebecca asked, walking slowly backward up the driveway toward her darkened house.
"Of course," Tane said automatically. "What are we protesting about this time?"
"Whales," Rebecca said.
Tane shook his head. "I've got no problems with whales. They've never bothered me."
Rebecca laughed. "It's a couple of weeks away. I'll remind you."
She turned and disappeared into the carport and inside her home.
**F ATBOY AND HIS MOKO**
**It was a couple of** weeks before the subject of the quantum foam came up again, and in all that time, Rebecca never mentioned it once, which made Tane think she had forgotten about it.
However, a lot of other things happened during those weeks, some of which were to have far-reaching and long-lasting effects, such as Tane's older brother asking Rebecca out to the movies or Rebecca getting arrested. Another thing that happened was that Rebecca and her mother were kicked out of their home.
Tane's older brother's name was Harley. Harley had been a chubby kid, and with a name like that, he quickly picked up the nickname Fatboy, as in the classic Harley-Davidson motorcycle. He hated the name when he was ten, but by the age of fifteen, thanks to years of rugby, the weight had turned into muscle; yet the name had stuck, and Fatboy—Fats to his friends—kinda liked it. It was no coincidence that when he got his full motorcycle license, he came home the same day with a genuine Harley-Davidson Fat Boy.
Fatboy was a musician. The kind with dreadlocks and a leather jacket, not the kind who played in a symphony orchestra. And he wasn't a rock star about to be discovered and have number-one hit singles around the world. At least Tane didn't think so.
Fatboy was a session musician. He played guitar and he was pretty good at it. Good enough that he had dropped out of school before his sixteenth birthday to pursue it. ("Over my dead body," their mother had said, but somehow she had survived.) Fatboy had a kind of natural affinity for music, which must have run in the family because the same aptitude flowed through Tane's veins.
Fatboy was seventeen, and although he wasn't particularly good-looking, he always seemed to have at least two girlfriends on the go at any one time, which seemed a bit excessive to Tane, who (in his opinion) was much better-looking but didn't have a girlfriend at all.
Rebecca, on the other hand, had had a couple of boyfriends, but they hadn't lasted long. She was quite nice to look at, quite pretty in fact, in her rebellious, punky way. But some boys were afraid of her intelligence and thought she was a bit of a know-it-all, which in fact she was. Other boys thought she was going out with Tane, which in fact she wasn't.
The day that Fatboy asked Rebecca out, she was at Tane's house playing chess. It was Sunday. The day after Lake Sunnyvale.
The lounge of Tane's house was huge and split over three levels, so it was almost three lounges connected into one. Up through the center of the three levels grew the massive trunk of a hundred-and-fifty-year-old tree. It wasn't technically inside the house. It was more like there was a hole up through the center of the house in which the tree grew, with huge plate-glass windows on all four sides.
They were on the lower of the three lounges, nestled into the embrace of the native bush that surrounded the home. From the middle lounge, the house had million-dollar views out across the valley to the lights of the city, and from the uppermost lounge, a long balcony led off to a series of treetop-level rope walkways. This high in the mountains, you could walk out amongst the trees and surround yourself with birdsong on a warm summer's evening.
It was a very expensive house that Tane's father was able to afford because of his very successful career as an artist. Rangi Williams painted animals, usually New Zealand native animals, in their own environment. He often disappeared into the bush for days at a time, armed only with his sketch pads. His paintings sold all around the world. There was something especially appealing, apparently, about paintings of New Zealand's native creatures painted by a descendant of the Maori, New Zealand's native people.
The architecturally designed house was a stark contrast to Rebecca's home, which was old brick and tile and was nestled on the hillside that encircled Sunnyvale School.
Tane advanced his knight to attack Rebecca's rook. He liked to attack his opponents' rooks early in the game, as it weakened their attack later on.
"Are you sure you want to do that?" Rebecca asked.
Tane considered the move. And he considered Rebecca. Sometimes she would say that because he had just made a really dumb move. But other times she would say that to make him think that he had just made a really dumb move when really he hadn't.
He examined the board once again and removed his hand from his piece.
"I'm sure."
Rebecca shrugged and moved her rook. On her next turn, she captured his queen.
"Bum," Tane said quietly. It _had_ been a really dumb move.
The throaty roar of an engine was followed by a spray of gravel from the driveway as Fatboy spun his motorcycle around and flicked down the stand. He always parked like that, roaring up at high speed and spinning the back wheel around on the gravel. He said he liked to leave the bike facing the way he was going, not where he had been, but Tane thought he did it just to show off. It annoyed their mother to distraction, as she had to spend hours picking pieces of gravel out of her prizewinning flowerbeds.
Still, he knew she was always secretly pleased when Fatboy came to visit, which wasn't all that often now that he had his own flat in the city.
He left his motorcycle boots at the door. Not even Fatboy was brave enough to tramp all over their still-new-looking carpets wearing outside footwear. He left his jacket on, though, as he swaggered inside and was just taking off his helmet when he noticed Tane and Rebecca lying on the floor with the chessboard.
" _Kia Ora,_ Rebecca. _Kia Ora,_ little bro," Fatboy greeted them in Maori.
" _Kia Ora,_ Fats," Rebecca replied.
Tane just said, "Whatever."
Fatboy changed his culture like other people changed their clothes. A couple of years ago, he had been totally into his "homeboy" phase, walking around with an outsized baseball cap and calling everyone "m'homey." Now he was into this Maori heritage phase. Tane couldn't help wondering what was next.
Rebecca glanced up as Fatboy took off his helmet. "Cool!" she said.
Tane looked to see what she was talking about. Fatboy was grinning at them through a brand-new _moko,_ a Maori face tattoo. The left half of his face was inscribed with a fern-like design that seemed to swirl and dance with a life of its own.
Tane turned back to the game, shaking his head. "Mum is going to kill you."
Fatboy laughed. "No, she won't. It's cultural."
Tane ignored him and launched a vengeful, ill-considered attack on Rebecca's queen.
"What are you doing?" Rebecca asked slowly.
"Revenge," Tane said with a mock sneer. "Teach you to take my queen!"
Rebecca shook her head and studied the board. "You can't play chess like that."
Tane looked up. Fatboy was watching the game with interest.
"You look like a gang member," Tane muttered.
Rebecca said, "I think it looks great. Really suits you."
She smiled at Fatboy, which annoyed Tane more than a little. She had known Fatboy almost as long as he had, and she was the last person he would have expected to be taken in by the whole dreadlocks, leather jacket, rock-star-in-the-making persona. And now that Fatboy had a face tattoo, he was "cultural" as well. Tane thought things couldn't get a whole lot worse.
Things got much worse, very quickly.
Fatboy stopped at Rebecca's smile. Whatever he had come home for, he seemed to forget about it. It might have been to get their mum to do his washing or ask to borrow the Jeep (because as cool as it was, sometimes a motorcycle could be a little inconvenient); but whatever it was, he forgot it and stopped, looking at Rebecca as if seeing her for the very first time.
Tane looked at Rebecca, who had already turned her attention back to the board, and he thought he understood why. The short, spiky haircut with the blond tips. The pierced nose and the determinedly unfashionable clothes. Her eyes were too big for her face, but that with her fine cheekbones gave her a slightly elfish appearance (in a rebellious, punky kind of way!). The slightly awkward girl he had grown up with was blossoming into the kind of lovely young lady who would attract the attention of people like his older brother.
_Get lost,_ he willed Fatboy. _Go and show Mum your_ moko, _or whatever you came here for._
"Are you winning?" Fatboy asked, turning on his false rock-star charm for the first time with Rebecca. _Surely_ Rebecca would see through that in a second.
"I'm doing all right," Rebecca said, glancing up. Tane gave Fatboy a dirty look.
"What are you doing on Saturday night?" Fatboy asked.
Tane glared at Fatboy and bit his lip. It started bleeding.
"Nothing," Rebecca said cautiously.
"I'm recording with Blind Dog Biscuit all this week," Fatboy said, name-dropping with relish, "but I'm free on Saturday. How would you like to go to the movies? On the Harley," he added for good measure.
"I, um..." She looked at Tane for inspiration, but he just shrugged.
"It'd be fun," Fatboy said.
"Okay. That'd be okay," Rebecca said after a moment, trying unsuccessfully to act cool.
"I'll pick you up at seven," Fatboy said. "Blind Dog Biscuit might be having a wrap party later on for their new album. We can go to that too, if you want."
Rebecca shrugged. "Okay."
Fatboy disappeared into the depths of the house, leaving Tane staring after him, trying not to show his annoyance.
Rebecca moved her queen to the end of the board. "Checkmate," she said with a smile, and looked up at him. "Your lip is bleeding," she noted with some concern.
"It's nothing," Tane said.
She looked at him closely. "Tane, you don't mind if I go to the movies with Fatboy, do you?"
He shook his head quickly and said with just a trace of irritation, "No, why would I?"
Just then, there was a scream from the kitchen.
The week passed surprisingly quickly. In fact, when he looked back on it the next weekend, Tane felt as if time had somehow compressed itself, and they had somehow time-slipped from one weekend straight to the next and missed the five days of school in between. Which made him think idly about his time-radio-transmitter idea.
On the Saturday night when Rebecca was out with Fatboy, he spent a lot of time thinking about his time-radio-transmitter idea. But she hadn't mentioned it at all, so he felt he was probably just wasting his time and brain cells.
On Sunday morning, Tane arrived at Rebecca's house just after eight, the panicky phone call from Rebecca ringing in his ears the whole way. He'd cycled the few miles between their houses at a breakneck pace.
It was drizzling, which made cycling dangerous and not much fun, but he made it without incident. He left his bike lying in the carport next to the old two-door sedan with the missing front bumper and raced up the cold concrete steps.
Rebecca met him at the front door, crying. She was trying to hold it back but was failing, so her words came out in muffled lumps between the sobs. She put her arms around Tane's neck, and he hugged her.
"Is this something to do with Fatboy?" he asked. "Last night..."
She shook her head. "No. No. He was nice. It's this."
She uncurled from around his neck and showed him a piece of paper she was clutching. He read it quickly. It was a notice from their bank of the mortgagee sale of their house.
"I got home really late after the party last night. I had such a good time, and Fatboy was such a gentleman. And I got to meet the band, and they weren't like I thought they'd be; they were just regular guys. But the party went on all night, and I didn't get home till four this morning."
Tane thought that was far too late for a fourteen-year-old girl to be out at night. Especially with _his_ older brother.
Rebecca continued, "I was tired, but really happy. I'd had such a great time. Then a man came to the door this morning, at seven-thirty; he got me out of bed. I was barely awake. He wouldn't talk to me and insisted that I wake Mum up. When she came down, in her dressing gown, he asked her for her name, and when she told him, he just handed her this envelope, and said, 'You have been served,' and walked away. Just like that. He didn't say anything else."
"Oh, Rebecca," Tane said, feeling useless.
"Mum just threw the envelope in a drawer and went to watch TV. She didn't even open it."
Judging by the sounds from the other room, it sounded like Mum was still watching the show.
"So I opened the drawer, and look!"
Rebecca took Tane into the small dining room of the house. On the old glass and metal dining table were piles of envelopes and pieces of paper.
"They're bills and mortgage demands; they go back months."
Tane picked up an electricity account. "This one's up-to-date," he noted.
Rebecca nodded. "Dad had set up most of them to be paid by automatic payment. Things like the power and the phone. But others have just been accumulating. And it looks like the mortgage hasn't been paid for months!"
It took Rebecca three or four days after school to straighten out the accounts.
Tane helped as much as he could, which mostly consisted of opening the envelopes for her and sorting out bills into piles from the same company, and also by date.
It wasn't good news at the end, though. The house was definitely going to have to be sold, and with all the unpaid bills, Rebecca and her mum would have nothing. No house and the only money they'd have to live on was the small allowance her mum got from the government as a widow with a dependent.
"What will you do?" Tane asked when they finally worked out all the numbers and realized just how dire the situation was.
Rebecca shrugged. "I tried to talk to Mum, but she was watching _The Beautiful Years,_ and Dr. Messenger was just about to poison his wife's sister, so she wouldn't listen to me." There was the faintest trace of bitterness in her voice.
"So what will you do?"
Rebecca sighed. "I think we're going to have to move to Masterton, to live with Grandma."
Tane caught his breath and held it for a moment before releasing it. Masterton was a six-hour drive away. That meant he would hardly ever see her. If ever.
"When?" he asked.
"The lady at the bank said we have about a month, until the house sells."
Tane nodded. He looked at Rebecca, head in her hands, leaning forward over the lined pad she'd used for her final calculations. The light from the single bulb above the table made little shadows around her eyes. He thought she suddenly looked a lot older.
He wished there was something he could do. But there wasn't.
"Just a month," she said slowly, shaking her head. "It's not much time."
"No," Tane agreed, trying to think of a way to take her mind off her problems.
To his surprise, though, her mind was somewhere else entirely.
She said absently, "So we'd better go and see Professor Barnes as soon as possible."
Tane raised an eyebrow. "Professor Barnes?"
"As soon as possible."
**T HE BURST AND TRANSIENT SOURCE EXPERIMENT**
**The university corridors were long** and winding. The high arched walls of the outside quickly disappeared to high, polished wood panels, which in turn gave way to white-paneled walls, white-paneled floors, and white-paneled ceilings.
This deep inside the university, the corridors branched frequently and unexpectedly.
The maze of corridors would have stymied Tane, but Rebecca had been there a few times to visit her dad, when he'd worked there, before the accident.
Rebecca's father had been one of the brightest minds in the Earth Science department at the university. Some said the only brighter mind was that of his wife. Rebecca's mother.
His car had been crushed by a furniture truck that had failed to take a bend. The truck driver was a sensible, experienced, and sober driver. He just took a bend a fraction too fast one day and killed Rebecca's father.
Rebecca was familiar enough with the layout to understand the rather confusing array of small blue signs posted on the walls at corners as they hurried down the long passageways to a pair of white-paneled double doors marked GEOPHYSICS LAB.
Rebecca caught Tane by the arm. "Let me do the talking, and agree with everything I say," she said.
"Are you sure you should be doing this?" Tane asked. He'd asked the same question a couple of times at school that day and had gotten pretty much the same answer, but he asked it again anyway. "I mean, with the house and everything."
It was Friday, and after spending most of the week straightening out her family's bills, it did seem a little odd that she was now spending energy on what was really nothing more than a whimsical conversation they'd had while lying on a platform at night on Lake Sunnyvale.
Then again, he thought, maybe it took her mind off her worries.
"I already told you," Rebecca said. "After we move to Masterton, I won't have the opportunity to do this. This is the only lab in the entire country that has access to live feeds from BATSE."
Tane still had no idea why they needed live feeds from BATSE, or even what on earth BATSE was, and he wasn't even really sure he wanted to know; but Rebecca had promised to explain it all in very simple terms, "terms that even an artist can understand," she'd said, after they had the data. Which was what they were here for.
She pushed open the doors, and they stepped into the lab, which didn't look like a lab at all. Whatever Tane had been expecting, it was not simply a row of desks with a computer on each one.
A balding man with gray hair and beard and narrow, wire-framed glasses looked up from his desk and rose to greet them. He looked very much the image of a university scientist, which of course he was.
"Rebecca," he said warmly. "You've grown." He looked as though he didn't know whether to hug her or shake her hand, so ended up doing neither.
"This is my good friend Tane," Rebecca said, and the man shook Tane's hand; then, as if that had made his mind up, he shook Rebecca's as well. She continued, "Tane, this is Professor Barnes. He used to work with my dad."
"How's your mum?" Barnes asked.
"She's...okay. She's fine."
"So, Rebecca, how can I help you?" the professor asked after a moment or two longer of meaningless small talk. "You said something about a piece of art? But I couldn't work out what that has to do with geophysics."
Rebecca smiled. "Tane is an artist, and a very good one, and he has this great idea of creating a piece of art from transmissions from the depths of space. Sort of an 'art of the universe' thing. It's kind of hard to explain." She gave Barnes a look as if to say, "Artists, you know."
"I see. I think," Barnes mused, and turned to Tane. "Any particular kind of transmissions you had in mind? There are a lot of them, you know."
Rebecca cut in before Tane would feel that he had to answer. "I was telling him about BATSE and the gamma-ray bursts. Tane felt that we could use the raw data from one of those bursts and model it digitally to create visual patterns in the computer."
Barnes looked confused. "Well, I suppose that could be considered as art. Actually, I paint a little myself. Landscapes, still lifes, that sort of thing. I have quite an original technique using flat knives. Maybe that would be of use to you, too. I could take some photographs and e-mail them to you. There is one painting in particular—"
"That would be great." Rebecca cut him off with such a wide smile that it didn't seem at all rude. "And the BATSE data?"
Barnes considered for a moment, then said, "Well, I don't suppose it would do any harm. As long as you're only using it for artistic purposes. We're the only lab in the country that has access to live feeds of this data, you know."
"Really?" Rebecca said, wide-eyed. "I didn't know that."
"There was a burst this morning. I downloaded the data a few moments ago. It's all just bits and bytes, though. Raw numbers, if you know what I mean. Are you sure that will be of any use to you?"
"Oh yes. We'll put it into some imaging software that will create representations of it, no problem," Rebecca said.
"Very well, then, I'll write it onto a disk for you." He disappeared through an unmarked door at the back of the lab and reappeared a few moments later with a CD.
"You promise you won't let anyone else have a copy of this? It's not top secret, but the data is copyrighted; it belongs to NASA."
Rebecca said sweetly, "Who on earth would I give it to?" She took the CD from Barnes's outstretched hand and passed it straight to Tane.
The data made no sense to Tane at all. It was as Barnes had described, just long strings of ones and zeros. Raw computer data. But he also didn't understand Rebecca's explanation of what this was all about.
He lay on his bed, watching her tap carefully at the keyboard on his study desk. They were at Tane's house because he had a new and powerful computer. Tane knew Rebecca envied his computer and resented a bit the fact that he mainly used it to play games. Rebecca didn't have a computer at all.
"Okay, try and stay with me on this," she said, adding a line of code to the computer program she was working on. Tane concentrated furiously on her face, in the hope that by doing so, he might somehow "stay with her on this."
"I did some reading about quantum foam, and I found out that scientists think it may be possible to detect the stuff, if it exists, by looking for fluctuations in these gamma-ray bursts."
She stopped, staring at her code for a moment, then added a pair of brackets around part of an equation.
"Go back a bit," Tane said. "What are gamma-ray bursts, and where do they come from?
"Okay. They are explosions of a type of radiation called gamma rays, which are a bit like X-rays or radio waves but with an extremely short wavelength."
That didn't help Tane a bit, but he nodded as if it did.
Rebecca continued, "Nobody knows what causes them, but they seem to come from all over the galaxy. In 1991, NASA sent up a satellite called the Compton Gamma Ray Observatory. It has an instrument on it called the Burst and Transient Source Experiment."
"BATSE!" Tane exclaimed.
"Right, BATSE. It's the most sensitive detector of gamma rays in the world. What we have on this disk is data from a burst of gamma rays that was received by BATSE this morning. It's probably just random data from random gamma waves. But if the quantum foam scientists are correct, then the gamma rays could be affected by the quantum foam, and therefore, theoretically, and don't ask me how, it could be possible for someone in the future to transmit a message using these fluctuations in the gamma rays."
"Nope," Tane said. "You lost me at BATSE."
"Doesn't matter. We'll analyze the data and look for patterns. That's what this program is for. If we find them, great. If not, well, it was a wild idea to start with."
She pressed a couple of keys on the keyboard and said, "Hold your breath."
Tane looked at the screen and waited, but apart from a single, flashing cursor, nothing was happening.
"What's it doing?" he asked.
"My program looks for patterns. It could take a while, though; there is a lot of data to look through, and the patterns could be quite complex. But if it finds a series of numbers that form a definable pattern, it will stop and we'll be able to look at it."
"Okay, cool," Tane said, although he had really been hoping that it would have been flashing colors and weird patterns on the screen as it worked, a bit like the ambience patterns you got on Windows Media Player.
They watched the flashing cursor for a bit, but it got very boring very quickly.
"Why wouldn't NASA have found these patterns?" he asked after a while.
"They're not looking for them," she said. "They're trying to find out what causes the bursts. They think it might be two neutron stars colliding, or maybe a neutron star being swallowed by a black hole."
"Really," Tane said, nodding his head and wondering what on earth she was talking about.
"There are some scientists from Texas who are looking for fluctuations, but they're trying to prove the existence of quantum foam, and they're still trying to work out what a quantum gravity signature would look like. Nobody is looking for messages in the rays."
"Just us."
"As far as I know."
Tane watched the cursor for a while longer, then turned and looked at his friend, looking at the cursor. Her eyes never left the computer screen.
He had known Rebecca all of his life. A lot of people said things like that, but in their case it was true. They shared the same birthday, and their mothers had shared a maternity ward at the hospital. They had even lived close to each other in those very early years, before Tane's father's career had taken off and he and Tane's mum had built the big house amongst the trees up in the mountains. Tane and Rebecca had played together in kindergarten and now went to the same high school. For nearly fifteen years, there had never been a time when he had not known the spiky-haired girl sitting next to him.
"I'm going to miss you when you go to Masterton," he said.
Rebecca said quickly, "Don't forget the march tomorrow. We need to get down to the station before nine to catch the bus into the city."
"I'll set my alarm clock," Tane promised.
**S AVE THE WHALES**
**On the day of the** big march, Rebecca got arrested. The Japanese prime minister was visiting Auckland. What he was doing there, Tane wasn't sure. Some trade summit or international congress of some kind; he didn't really know or care.
The protest march was set to start at the same time as the prime minister's flight arrived at the airport, which was mid-morning on Saturday. It would wend its way up through the central city, arriving at the prime minister's hotel just about the same time he would be.
Tane gripped his placard with both hands and held it high. Just to show his support for the cause. The cause was something to do with whales and saving them from Japanese whaling boats, but it was actually more complicated than that and had something to do with fishing quotas and scientific whaling and all sorts of other things that he was sure were worth fighting for, or against. Whichever it was.
Rebecca certainly believed so, and Tane couldn't let her march by herself. He hadn't let her march by herself on the anti-nuclear march or the anti-GE (genetic engineering) march, although he'd had the flu on the day of the climate-change march, so she had done that one by herself.
There had probably been a time when Rebecca didn't feel strongly enough about some issue to want to march in protest about it, but that was probably when she was in preschool, and Tane couldn't remember it.
He thought most people in the difficult situation that Rebecca now found herself in would have lost interest in protest marches. It was she who needed saving at the moment, not the whales. It could have been another excuse to take her mind off her problems, but Tane thought it was rather more than that. He'd known Rebecca a long time, and he knew that when she committed to something, she always saw it through.
Which was why they were both there at the very front of the marchers. On the front line of the battle, as it were.
A long swelling chant began at the back of the line of marchers: " _Ichi, ni, san, shi_...don't kill whales, leave them be. _Ichi, ni, san, shi_..."
"What's this itchy knee business?" he asked Rebecca in a quiet moment between chants.
She rolled her eyes. "It's 'one, two, three, four' in Japanese. It's because the—"
"Yeah, yeah, I get it now," Tane said as the chant began again.
" _Ichi, ni, san, shi_...don't kill whales, leave them be!"
"Thanks for coming, Tane," Rebecca said after a while.
"Gotta save those whales!" Tane said enthusiastically, waving the banner around and accidentally clouting a large man with a shaved head who was wearing a leather jacket and marching next to them.
"Sorry," Tane said.
The man grinned and nodded to show that no harm was done.
It was an officially sanctioned, organized march, which meant that the road was blocked off by police cars with flashing lights at each intersection along the route.
Another police car preceded them, rolling slowly forward just a few yards in front of Tane and Rebecca.
Along the way, early morning shoppers either raised their arms in the air and shouted in solidarity or just stared curiously at the throng, which was wide enough to completely cover the roadway and stretched away behind them. There had to be a thousand marchers, Tane thought, although he wasn't much good at estimating the size of crowds.
The march started down on the Auckland waterfront and proceeded straight up Albert Street to the huge Sky City casino complex, with its massive three-hundred-sixty-yard-high Skytower.
They turned right just before the casino onto Victoria Street, then stopped at the entrance to Federal Street, where the Japanese delegation's hotel was.
Wooden barricades prevented the marchers from entering the street, so they had to wait, milling and chanting, completely blocking the road.
It didn't take long for the prime minister's motorcade to arrive. First came a police car, then a large black van that had to be filled with security guards. Then a long black Mercedes limousine.
The chanting rose to a crescendo as a line of police officers formed a human barricade behind the wooden one. Behind the blue line of police, Tane could see the slender figure of the Japanese prime minister emerge as a large man in a dark suit opened the door of the limousine and stood to attention.
Several New Zealand dignitaries that Tane didn't recognize stood in front of the hotel, waiting to greet the man.
And that might have been the relatively peaceful end of it, if it hadn't been for the prime minister stopping as he got out of the car, turning to the protestors, and waving cheerily.
Maybe he was just being friendly. Maybe he was waving to someone he knew. But it was the worst thing to do to a crowd that had been winding itself up, chanting and shouting over the past twenty minutes while marching. It was like pouring gasoline onto a barbecue.
There was an angry roar from the crowd, like that of a wounded animal; then suddenly the wooden barricades were down, toppling under an onrush of protestors. The police linked arms and stepped forward to meet the onslaught. Behind them, more police officers drew batons and waited.
The Japanese prime minister and the other dignitaries scurried toward the hotel, all thoughts of ceremony vanishing in the face of the wild beast that lunged toward them.
Tane tried to push himself backward, but it was impossible with the press of the crowd behind him, and he found himself crushed up against a huge policeman with a beard and bad breath. The air squeezed out of his lungs with the pressure from behind, and an overwhelming feeling of claustrophobia enveloped him.
The thin blue line held, though, the storm of protestors safely contained on the outside. All except one, Tane saw through a thin gap in the blue uniforms. A small, quick shape, a blur of movement, and Rebecca was halfway toward the Japanese delegation, dodging around the larger, slower policemen like a rugby player evading tacklers.
She almost made it, shouting and screaming something about whales and murder, when one of the large, dark-suited men grabbed her by the arms, pinning her and forcing her to the ground.
At that point, the line fractured and split apart in a dozen places, the fury of the crowd intensifying as one of their own was attacked. Suddenly there were protestors running everywhere, some battling police batons with their makeshift placards.
The bearded policeman whirled away from Tane, and he managed to fight his way sideways, unable to see Rebecca, unable to do anything but try to claw breath back into his lungs and get out of the running, crushing crowd.
He found a small oasis amongst the huge concrete pillars at the base of the Skytower and slumped to the ground, exhausted.
In the end, they had to call in the riot police to clear Federal Street. Over a hundred people were arrested, but most were released without charge after being processed at the Auckland Central police station, just a few blocks away.
Tane waited outside for four hours until Rebecca finally emerged, bruised and disheveled but defiant.
"That was awful," she said. "They photographed us, took our fingerprints, and jammed us all into these tiny cells while someone decided what to do with us."
"I tried to get to you," Tane said, which wasn't really true but seemed like the right thing to say.
"You couldn't have done anything," Rebecca said. "They had me into the police van in three seconds flat. In handcuffs!"
She rubbed her wrists, and Tane could see red marks where the cuffs had been.
"It's so unfair," she raged quietly. "They're the criminals, killing whales and calling it research, but we're the ones who end up with a criminal record!"
"Don't worry about it," Tane said. "You're still a kid. They have to erase all record of the arrest the day you turn eighteen. I read that somewhere."
She was silent.
"Really," he insisted, trying to make her feel better. "It's nothing. It won't matter at all."
He was wrong, though—as it turned out, Rebecca getting arrested mattered quite a lot.
**111000111**
**Tane's computer was very new** and very powerful. It had a shiny silver case and a flat nineteen-inch screen, with a cordless mouse and keyboard. It was very expensive. He'd gotten it for his fourteenth birthday, so it was pretty much the latest processor, and it had a lot of memory, a fast hard drive, and a high-powered graphics processor, and it was generally really quick at anything, especially games.
So it was a little bit hard for Tane to understand why it was taking so long for it to run Rebecca's program. She started to explain it to him once or twice, but her explanation about the program code made no more sense to him than the code itself.
The only good thing was that once it was running, it kept running all by itself, without needing assistance from anyone. Not that Tane felt he would be able to give it any kind of assistance anyway.
They had started it running on Friday night, the day before the protest march, and it was still running the next Sunday. And Monday.
More days passed. A week. Then another. Sometimes Tane would wake up during the night and would feel that the insistent flashing of the cursor was reaching out to him. But mostly he just wondered if it was really doing anything at all, or if it was just stuck in some mindless loop because of some bug in Rebecca's program.
A third week went by. That week, Rebecca had another date with Fatboy, although she said she didn't feel very much like dating, with all that was happening. But she went anyway. That night, the flashing cursor seemed like a warning light.
The day of the auction, November 14, was the day that the program finally paused and displayed some data on the screen, but Tane wasn't there to see it. Neither was Rebecca. Painful as it was, they were both at the auction.
It was a bright, chirpy day and for that reason the auctioneer, who looked just like a dapper little English gentleman but who spoke with an Australian accent, held the auction outside in the small backyard. Tane had mown the lawn, and they had both spent an entire weekend weeding to make the place presentable. He thought it looked as good as it could look, as the potential buyers, the tire-kickers, and the nosy neighbors gathered around.
The auction turned out to be a disaster. It was over in less than ten minutes, and the house, Rebecca's family home, sold at a bargain-basement price to some flamboyantly dressed young entrepreneur.
Rebecca shuddered once or twice as the hammer fell, and Tane put his arm around her shoulders.
At that price, Rebecca and her mother wouldn't even be able to fully repay the bank. They'd have to keep making mortgage payments, plus pay the creditors, and they'd have no money for the move to Masterton. It was an all-around disaster.
Rebecca was crying silently as Tane led her inside. He offered to make her a cup of cocoa, but she shook her head and said she wanted to go to bed for a while.
He rang his mum and told her he'd be home a bit later than expected. She didn't mind and asked if there was anything she could do to help, but there wasn't really.
When dinnertime came, he found a few bits and pieces in the cupboards and the freezer and made some savory pancakes, but neither Rebecca nor her mother would eat them.
He ate them himself at the dining table, which was still covered in mounds of paper, and a bit later, he went to check that Rebecca was okay.
He pushed the door open noiselessly. The light from the hallway spilled inside. It was not a little girl's room, and it was certainly not a typical teenage girl's room. In place of the posters of boy bands, there were posters for Greenpeace and Amnesty International. The books on her bookshelf were by Stephen Hawking and Salman Rushdie, rather than Meg Cabot or Jacqueline Wilson.
She was lying on top of the bedcovers, still fully clothed but sound asleep. In sleep, the heaviness and the tiredness lifted from her face, and there was a stillness and a calm about her that wrenched Tane's heart.
He pulled a blanket over her and gently brushed his hand against her cheek by way of a goodbye.
It was still light, but only just, when he stowed his cycle in the garage and wound his way up through the many levels of his parents' house to his room, and there, waiting for him on the screen, was a long line of ones and zeros.
Rebecca's software had found a pattern.
"Thanks, Tane, thanks for everything." Rebecca smiled at him tiredly over the Sunday-morning cup of cocoa he'd made her. He'd quietly let himself back into the house at about eight, using Rebecca's key, which he had borrowed on his way home the night before. "You're a good friend. We'll keep in touch, after I move down to Masterton, won't we?"
"Of course we will," Tane said. "We'll probably be on the phone every night, and I can come and visit you on long weekends, stuff like that."
She smiled and nodded her agreement, although they both knew that it would probably never happen.
She said, "Yesterday was such a nightmare. I don't even want to think about what we're going to do now."
"When do you think you'll leave?" Tane asked.
"I dunno. I think the settlement date is in about three weeks, and we'll probably have to be out before then. I expect I'm going to have to organize it. Mum doesn't seem to want to have anything to do with it. I don't know how we're going to pay for it, though."
Tane had a thought and said, "Maybe my mum and dad can help out. A loan of some kind."
Rebecca looked down at her mug. "I'd say no and tell you that we're too proud and all that. But I guess we're not. We're just desperate." She laughed. "I just hope my BATSE analyzer finishes running before we leave."
"It already has."
"What?!!"
"I was going to tell you, but it didn't seem like it was important."
Rebecca put down her cocoa slowly, phrasing her words carefully. "Not important? Not important! It's probably nothing, just a random series of numbers that happens to look like a pattern, but if it is really a pattern, then we could be talking about the scientific discovery of the century!"
"I was going to tell you—"
"Bigger than the splitting of the atom. Bigger than the invention of the airplane. Bigger than...than...big. But don't get excited, because it is probably nothing."
It was a little hard not to be just a bit excited, Tane thought, considering that Rebecca was just about jumping out of her skin. "Do you want to come and have a look?" he asked.
"We'll take Mum's car. It'll be faster," was her answer.
Tane hesitated. "I don't have my license."
"Me neither," Rebecca said as she picked the car keys off the hook by the door.
**0001100110010000100000011001100011**
**0000011001110011000001100111000000**
**1001100111100000000110011100101010**
**1100000000110001010101100000111101**
**0101100010000011111010101000000001**
**1111100110000111011110001100010100**
**0001001000001100001010101110001111**
**110001111010100101**
Rebecca was still wearing the same clothes as the previous day, Tane noticed. But then the previous day had been a day from hell, so he supposed it didn't matter.
She had been poring over a printout of the numbers for nearly an hour, trying to make sense of it, but as far as Tane could see, it made no sense at all, not even to her.
"Look," she said, running her hand along a line of ones and zeroes. "It looks like a pattern all right, and parts of it repeat, which is not the sort of thing you'd expect from a random explosion of gamma rays."
Tane looked, and it appeared pretty random to him. Just a long series of ones and zeroes printed in fine black ink on a plain white piece of paper. "Really?" he asked.
"Look, see this section?" She drew some quick lines in pencil on the printout, marking out a series of six digits. _001100._ "Look how many times that sequence repeats throughout the data. Nine times. Too often to be a coincidence. Here's another sequence: 0101100."
She pored over it some more, making notes on another piece of paper and occasionally adding up some figures on a pocket calculator. "The next step is trying to decipher it. Like cracking a code."
Tane peered at it, squinting, turning his head slowly to each side, trying to see some kind of picture in the numbers, like one of those hidden picture paintings.
Something stirred a faint breath of recognition, as though he had seen patterns like this before, a long time ago. He breathed out slowly, which was his usual way of trying to relax into a memory that would not quite come. The memory tickled and teased agonizingly at the far corners of his brain, but the picture would not come into focus. The harder he tried, the more it kept darting just out of reach.
After a while, Rebecca shook her head. "Maybe it is just random after all. Depending on how you group the digits, there could be any different number of recurring sequences, or none at all. They say that if you had enough monkeys and enough typewriters, one of them would eventually type _Hamlet._ "
"What are you talking about?" Tane snorted. "Monkeys and typewriters? Hamlet?"
"Not literally," Rebecca tried to explain. "It just means that if you have enough random..." She saw the grin on his face and threw her pen at him.
"But wouldn't they be better off using word processors?" Tane asked.
"Who?"
"The monkeys."
"Whatever."
The excitement of the morning had worn off by lunchtime (cheese on toast, provided by Tane's mother). If there was any kind of a pattern there, it was proving to be elusive, and the dark circles under Rebecca's eyes had returned. Even her hair, normally spiky and sharp, was limp and lifeless.
"I thought I was on to something for a while there," Rebecca sighed. "I really thought I had it."
"What?"
"No. It's wrong," she said sadly. "Binary code. That's the language that computers talk in—I mean, way down in the depths of their operating systems. Information is stored in bits, which can be on or off—"
"Yeah, I get it," said Tane, even though he didn't. "Move along."
"Well, the BATSE data is actually in binary code. My program just displays it in ones and zeroes to make it easier for us to read. So I had this bright idea of translating the binary code into ASCII characters."
"Um, ooohkaaay."
"God, it's like talking to a monkey some days," Rebecca stormed.
"Maybe, but I've written this really cool play about this guy named Hamlet—"
"ASCII characters," Rebecca cut him off with just a brief smile, "are eight-character binary codes for the letters of the alphabet. For example, 01000001 is _A,_ 01000010 is _B,_ and so on. So I translated the whole string of characters using ASCII."
"And?"
"Oh." She seemed distracted. "Nothing, just gibberish. So then I tried it in BASE 64, which is another kind of computer code, but no luck; then I realized there were two hundred fifty-eight characters, which is two too many for ASCII or BASE 64, which must be in groups of eight, so I took off the first two characters and tried that in BASE 64."
"And?"
She handed over a printout.
**ZkIGYwZzBnAmeAZyrj8czAMVDDVz9UA/MO8YycAkxQ==**
"'Zuhkiggy wazzabeen ameezy,'" Tane read out importantly. "You know this is a really significant moment, like when Alexander Graham Bell invented the telephone and said, 'Mr. Watson, come here.' Except our first message says, 'Zuhkiggy wazzabeen ameezy.'"
"Bell didn't invent the telephone," Rebecca said sourly.
"Maybe that's the language they speak in the future," Tane said. "Maybe 'Zuhkiggy wazzabeen ameezy' is future-speak for 'Hi, what's up?' or..." He trailed off, seeing that she was not laughing. "Sorry."
A tear appeared at the corner of Rebecca's eye and found its way slowly down her cheek.
"Let's go for a walk," said Tane, feeling stupid and useless.
From the back of the house, a narrow, little-used track wound through the trees and eventually joined up with one of the main hiking trails through the bush of the mountains. Not even forty-five minutes of steady walking along that brought them to the top of one of the dams that were scattered through the mountains to provide water for the city below.
This was a small, earthen dam that almost seemed a part of the hillside. They leaned on a solid wooden handrail, hewn from a half-round log, and looked down at the lush bush-covered hillside below.
The memory that had been tickling at Tane's mind was still there. He tried not to think about it. That was what his dad always said. Stop trying so hard. Let your subconscious sort it out.
"It's beautiful here," Rebecca said. "I love it. All you can hear is the water and the wind in the trees and the birds—" She stopped, interrupted by a loud bell-like sequence from just above their heads. "There, what's that one?"
"Korimako," Tane said. "Bellbird. And that's a saddle-back over there."
"How do you know all these?" Rebecca asked with a grin. "Are you a secret bird-watcher?"
Tane smiled and shrugged. "Nah, just Boy Scouts. Had to learn them all in Scouts. If you passed a test, you got a badge that...you...could..."
The delicate tendril of a memory tugged again at the back of his mind, tantalizingly close. He shut his eyes for a moment, and then, after what seemed like an age but was no more than a second or two, he said, "We've got to get back to the house."
"What is it?" Rebecca asked, but Tane was silent.
He flattened the sheet of paper out on the coffee table and smoothed it with his hand.
"You said it would be like cracking a code," he said, and pointed to the short pattern of numbers that Rebecca had marked: _001100._ "When I was a scout, I got a lot of badges. I got my New Zealand Birds badge, and I also got my Morse code badge. Now, if this was Morse code, then that would be a comma."
Rebecca froze and turned to him. Her eyes were wide. She didn't seem to be breathing. "Morse code?" she said in a dry voice.
"It's a method of signaling that ships—"
"I know what Morse code is," she rasped.
"Well"—he hesitated—"I can't remember much of it, but I can remember a few bits. If the ones were dots and the zeroes were dashes, that would be a comma."
"Okay. So what would this be, before the comma?"
"Ummmmm..."
"Google," said Rebecca determinedly.
A few moments later, they had a printout of all the Morse code characters.
Half an hour later, they were no closer to solving the puzzle.
"It might be Morse code," Rebecca said. "But Morse code characters aren't all the same length, and that makes it really hard to decipher. Take this first bit, for example: 00011001100. That could be _O, I, M, I, M,_ or it could just as easily be _M, N, P, W_ or any of a hundred other combinations, none of which make sense!"
"Yeah," Tane said slowly, poring over the printout. "But if I was right about the 001100 being a comma, then we have 00011 comma 1000010000 comma 1100011000 comma, and so on."
"Well, 00011 is eight," Rebecca said, checking the chart, "and 10000 is one, 11000 is two, so that would give us eight, eleven, twenty-two, thirty-two, thirty-nine...." She paused and looked at Tane. "Holy crap!" she said. "I think it _is_ Morse code."
They looked at each other for a long time, desperately eager to carry on and translate the code but strangely afraid to. Slowly their eyes dropped back to the paper.
"That's it," Tane said after a moment. "See, 101010; that's a period. That must be the end of it."
"No, there's still a lot to work out. But let's start with this bit," Rebecca said.
They looked intently at the sheet.
**8,11,22,32,39,40,3.**
Tane said, "And..."
There was silence for a few moments.
"I don't know," Rebecca admitted. "They're just numbers. The fact that we picked them out of a gamma-ray burst using Morse code is pretty significant, but as for what they mean..."
"Are you sure this isn't just more monkeys typing _Hamlet_?" Tane asked after a while. "More random noise that just coincidentally happens to make Morse code characters? I mean, why use Morse at all? Why not use that ASCII stuff, or Moonbase 64 or whatever it was?"
"Base 64," corrected Rebecca. "Anyway, that's easy. Binary takes eight bits, eight ones or zeroes, to make a single letter like _A._ Morse code does it in less than half that. _E_ and _T,_ for example, take just one digit. _E_ is dot, and _T_ is dash. So they can fit many more letters into a single message. Maybe there's some kind of a limit, like on a text message."
"They need to change their phone company," Tane muttered.
"Eight, eleven, twenty-two...," Rebecca read out loud.
"We're going to have to think creatively on this," Tane said. "Think outside the box. How many numbers are there?"
"Seven."
"The lowest number is three, and the highest is forty."
"Yes. The numbers start at eight and go upward, until they get to forty; then it suddenly drops back down to three."
Rebecca said, "Maybe it's a series, and we have to work out the next few numbers in the series."
Tane said nothing and closed his eyes.
Rebecca said, "It can't be letters of the alphabet, because there are only twenty-six of those. Maybe if I calculated the differences between each pair of numbers or..."
"No," said Tane suddenly, "you're thinking too logically. Try to think creatively."
"What do you mean? How can you think 'creatively'? We have to approach this logically."
Tane considered that for a moment, then said, "Okay, here's an example. It's a puzzle that my dad once gave me."
He took the pen and wrote a series of letters on a clean piece of paper: O, T, T, F, F, S, S, E.
"What's the next letter in that series?" he asked.
"Do we have time for this?" Rebecca shook her head.
"Try it."
"Okay, then." She took the pen from him and made a few notes on the paper. " _E_ would be the obvious answer, because the other ones are doubled, but the _O_ is not, so that can't be right. _O_ is the fifteenth letter of the alphabet, _T_ is the twentieth, _F_ is the sixth—"
"See, you're already being too logical," Tane said. "It's much simpler than that."
"Simpler?" Rebecca looked confused. "How can it be? There's no numerical consistency. Ah, but working backward, _S_ comes after _T,_ and _E_ comes after _F,_ so maybe—"
_"N,"_ Tane said. "The answer is _N._ "
"No, it's not. That's not logical—"
"One, two, three, four, five," Tane read out, pointing at the letters. "It is the first letter of each word. The next one is _N_ for _nine._ "
"That's stupid," Rebecca said. "Anyway, you're the creative one. You think creatively."
"Okay, then," Tane said. "Well, first of all, we need to consider that there are all kinds of numbers. Phone numbers. PIN numbers. Combination-lock numbers."
Rebecca agreed. "Dates are usually written as numbers."
"Room numbers, house numbers, decks of cards have numbers."
"Then there are serial numbers, like on money." Rebecca stopped, seeing the look on Tane's face.
"Money...," he said very slowly.
Rebecca waited, watching him closely. Tane tried to keep his face steady, but it kept wanting to break out into a huge goofy grin.
"Well, share!" she said impatiently.
"What...if..."
"What?!"
"Six seemingly random numbers from one to forty..."
"Seven numbers, and from three to forty," corrected Rebecca.
"No, six, and from one to forty. The first six numbers fall between one and forty; it just happens that the first number is an eight. Then after those six numbers there's another number between one and eight."
Rebecca looked at him blankly.
"Don't you see?!" Tane shouted. He dived off the bed and ran out of the room, leaving Rebecca sitting there stunned and not quite sure why.
He was back in a few seconds clutching a newspaper, which he flipped quickly through, then thrust in front of her face. "Don't you get it?" he cried. "The six numbers are the Lotto numbers! The last number is the Powerball number!"
Rebecca, who had stood up when Tane had started rushing like a mad thing around the room, suddenly found herself sitting back down again without intending to.
"Holy crap!" was all she could say, looking at the winning numbers from the previous week's draw, crumpled in the paper in front of her nose: 3, 22, 27, 30, 39, 40, and 7 for the Powerball.
"Somebody sent us Lotto numbers!" Tane said. "From the future!" And then repeated himself two or three times.
"But who would..." Rebecca trailed off.
"I don't know!" Tane shouted.
"But why would—"
"I don't care!" he hooted.
"But for which draw?"
There was a long pause, and Rebecca slowly took the newspaper from Tane's hand and lowered it to the desk beside the computer, mesmerized by the short sequence of digits.
"I don't know," Tane said finally.
"If we are right about this, it'll be the scientific discovery of the century," Rebecca whispered.
"If we are right about this, we're going to be rich!" Tane cried.
It took at least an hour before Tane could settle himself down enough to even look at the piece of paper again, but the numbers were still there, and they still looked a lot like Lotto numbers.
Rebecca, when she had calmed down, turned back to the computer printout with all the ones and zeroes on it.
"What about the rest of it?" she murmured thoughtfully. "We've only worked out the first bit."
"I know," Tane agreed. "Let's get started."
**1100000000110001010101100000111101**
**0101100010000011111010101000000001**
**1111100110000111011110001100010100**
**0001001000001100001010101110001111**
**110001111010100101**
Rebecca nodded. "The periods are easy to spot. They are the 101010 sequences."
Tane said, "It looks like more numbers. That first 11000 is a two."
"I think you're right," Rebecca said, "Two, zero, two, period."
She quickly translated the next few characters and printed it out in neat block letters.
**202.27.216.195,**
"What does it mean?" Tane asked.
"I don't know." Rebecca frowned and tapped the pencil on the paper. "And this next bit."
**001110111100011000101000001001000**
**00110000101010111000111111000111101**
**0100101**
"I don't think it's numbers. There's 00111011110, then a comma."
It took them about half an hour to come up with a combination that worked, but when it did, it was obviously correct.
" _G U E S T_ —guest!" Rebecca said with relish.
Tane was still puzzling over the next section. "Does 'Compton1' mean anything to you? That's the only combination that seems like a word."
"Yes!" Rebecca exclaimed. "The Compton Gamma Ray Observatory, remember? That's where they pick up the BATSE data!"
"Of course." He rubbed at his temples. "My head hurts."
"Not much to go now," Rebecca said brightly. "I'll take the next bit, and you do the last sequence." She smiled sympathetically. "My bit is longer."
Tane looked at his segment. "0101." It made just a few combinations: _NN, KE, TR, TETE, TEN,_ or simply just a _C._ He discounted the _TEN._ All the other numbers had been represented in digits, not spelled out. For reasons that he couldn't quite fathom, he decided that _TR_ was the most likely.
He looked over at the section Rebecca was working on.
**111000111111000111**
She already had half a page worth of combinations scribbled out.
"I'm getting there," she said. "I think _I, A, M, S, I,_ maybe a _J_..."
"I am Sij," Tane laughed, but the laugh suddenly died on his lips.
Rebecca noticed Tane's expression. "Why do you look so worried all of a sudden?"
"It's much simpler than that," Tane said strangely. "Every Boy Scout in the world knows that one. Dot, dot, dot, dash, dash, dash, dot, dot, dot. Then it repeats."
"Yes?" asked Rebecca, who had never been a Boy Scout.
"SOS!"
**M R. DAWSON'S TREE MUSEUM**
**The four science buildings at** West Auckland High School formed a huge cross. Physics, Chemistry, Biology, and Geography, two-story gray concrete structures with a classroom on each level. The blocks were surrounded by the concrete of the playing courts and the asphalt of the teachers' parking lot.
In the center of the cross was an open courtyard where the corners of the four buildings touched, and in the 1990s, Mr. Dawson, a forward-thinking biology teacher, had claimed the area for his department. He had ripped up the cobblestone paving, thrown out the wooden bench seats, and wheelbarrowed in, with the help of a tenth-grade biology class, a load of fertile earth.
Then, with the blessing of the principal, he had planted the courtyard with New Zealand native plants. His aim was to re-create the wilderness of New Zealand as it had been two hundred years ago. Before Western civilization with its six-lane motorways and oxidization ponds and two-story gray concrete classrooms. He installed a small pump and piped in water from his biology classroom—carefully filtered to remove the chlorine and fluoride—and irrigated the plants with a small stony-bottomed brook that babbled happily across the courtyard and into a drain on the other side.
Mr. Dawson was only forty-two when he died of a massive stroke caused by a heart attack. There was neither rhyme nor reason for it. He wasn't overweight or unfit.
Some people just die young.
But Mr. Dawson's death, just a few months after completing his garden, ensured that the project, which had never been intended to last more than a couple of years, became a permanent fixture at West Auckland High School.
Students were not usually allowed in the Fred Dawson Memorial Garden, except when they were working on biology or environmental projects. The last thing the school wanted were chewing gum and chip packets destroying the carefully maintained environment.
Rebecca Richards was one of the few exceptions to this rule, but then, she was an exceptional student in many ways. The courtyard got a willing caregiver, and Rebecca got a quiet place to get away from the other fourteen-year-old girls discussing makeup and hair and fashion and bra sizes and the current romances of their favorite singers and TV stars.
Rebecca loved the peace and the seclusion of the garden. Tane, however, thought it was somehow sad in a small way that he did not quite understand.
Tane sat on a punga log, strategically placed as a seat, and stared quietly at the clear water of the brook, chewing slowly on a peanut butter sandwich. Rebecca, on the other hand, could not stay still for a second. She sat down; she jumped up. She found a weed and pulled it out; she fidgeted. She picked her nails; she scratched her head. She didn't touch her lunch at all. Tane had never seen her like this.
Oddly, Tane wasn't thinking about the Lotto numbers, and he wasn't thinking about the SOS. Both of them had talked and thought about little else over the past week, and Tane had found it almost impossible to concentrate on his studies, which was becoming a serious problem with final exams now so close. It was November 20, and the exams started on the 30th.
But just at that moment, he was thinking about chess. Not the game in general, but one chess set in particular. The chess set in question was extraordinarily beautiful. It was made of marble and all the pieces were famous sculptures. The king was Michelangelo's _David,_ and the rook was Rodin's _Thinker._ Tane had been saving for most of the year to buy the chess set and had only one payment left to make. It wasn't for himself. It was for Rebecca. For Christmas. And he knew she'd love it, and he knew that she would know that he had saved for most of the year for it.
But strangely the Lotto numbers changed all that. If they really were Lotto numbers, then he and Rebecca were about to be rich beyond anything they could have imagined. And the fact that he had saved all year for the chess set would be meaningless. Rebecca would be able to buy twenty marble chess sets if she wanted.
He supposed it was a small price to pay. The sudden wealth would mean they could buy anything they wanted and, more importantly, that Rebecca and her mother could afford to buy a new house in Auckland and not move to Masterton.
But all that depended on whether they were right about the Lotto numbers. And then there was the question of which draw. Tane had had a moment of panic, worrying that the draw with their numbers might have already come and gone, but they had found an Internet site with all the Lotto results on it and had checked back six months. To their great relief, their numbers hadn't yet come up, so they resolved to buy a ticket every week until they did.
There was also the problem of the section of the cryptic message they had not yet solved. And the worrying SOS in the tail of it.
"The next Lotto draw is tomorrow night," Rebecca said. "What are we going to do about the problem?"
Tane looked up from the brook. Rebecca was pacing around the moss-covered dirt in small, darting movements, like a bird. "Sit down," he said, patting the log next to him.
She sat but kept twitching as if she wanted to keep pacing.
_The problem._ They had been discussing _the problem_ almost since they had realized what the numbers meant. There was a Lotto draw every Saturday night. The _problem_ was that to buy a Lotto ticket, or claim a prize, you had to be sixteen years or older. And Tane and Rebecca weren't.
It was a real problem. Tane's parents were out of the question. They didn't believe in gambling. Tane's mum believed that it was introduced by the pakeha, the European settlers, to keep the Maori people debt-ridden and downtrodden. There was no chance there.
Rebecca's mum was another possibility, but she hadn't left the house in over a year and wasn't likely to anytime soon.
Tane continued, "We have to do something. What if the numbers come up tomorrow night and we miss it! The Powerball Jackpot is up to six million dollars. How many adults would you trust with that kind of money? They could just claim the prize and keep it for themselves. We would never be able to prove that we gave them the numbers."
Rebecca stopped fidgeting for a while and said, "Then it has to be Fats."
Tane turned away from her and said softly, so that his words were almost swallowed by the muttering of the small brook and swept away downstream, "Fatboy is the last person I would trust with six million dollars of my money."
"But at least he's family!"
"Only by blood. Not by choice."
Rebecca put her hand on his arm. "How can you say that? He's your big brother."
"I don't trust him," Tane said.
"I really don't understand why you're so down on him all of a sudden. Is it because he took me out a couple of times?"
The recorded sounds of a native parrot reverberated suddenly off the painted concrete walls around them. Tane looked at his friend. She was a different person than the one of just a few days ago. The light inside her was back, even brighter than before, despite the fact they had not yet proved, or won, anything. The light was excitement, he thought, but much more than that, it was hope. Her hope was a brilliant beacon in the darkness that had descended on her over recent weeks.
"Of course not. It has nothing to do with that."
"Well, I say we have no other choice. We'll sign him to a contract if that makes you feel any happier. Give him ten percent of the winnings—"
"Five."
"Okay, five percent. Now where is the message?"
Tane pulled a grimy, much folded, much looked at piece of paper from a back pocket, and they looked at it together, despite having both committed it to memory many times over.
**8,11,22,32,39,40,3.202.27.216.195,GUEST,COMPTON1.SOSSOS.TR**
Rebecca said, "I'll ask Fatboy tonight, and if he agrees, we can write up some sort of a contract tomorrow."
Tane looked up in surprise. "Are you seeing him tonight?"
"Yeah, he's picking me up after school. We're going to the movies."
"I thought we were walking home together, to do some math study."
"Oh yeah." Rebecca was silent for a moment. "I forgot to tell you. Sorry."
"Final exams are only a few weeks away."
"I know. But I'm pretty comfortable with my math."
She was, and Tane knew it. She would probably top the country, even without studying. He said a little mean-spiritedly, "Well, if you're so clever, how come you haven't solved the rest of the message yet?"
**202.27.216.195,GUEST,COMPTON1.SOSSOS.TR**
Rebecca ignored the barb. "How far in the future do you think the message has come from?"
"Who knows. Maybe thousands of years. What I'd like to know is who sent it."
"Me too."
The bell sounded distantly, beyond the concrete confines of the Fred Dawson Memorial Garden. The end of lunchtime.
That night, Friday night, while Rebecca was at the movies with Tane's big brother, Tane figured out the rest of the message.
Fatboy laughed. It was a friendly chuckle, but it sounded like an evil cackle to Tane, like the villain in a pantomime: "Moo-ha-ha-ha."
Fatboy said, "Five percent? Listen, little bro, I'm happy to go shares in a Lotto ticket with you; just don't tell Mum. But I'll chip in a third of the price, and we'll go thirds on any prize. You won't win the big one; the odds against that are enormous."
Rebecca and Tane shared a secret look. Saturday morning had dawned clear and calm in the forests surrounding Tane's house. It was the day of the big weekly draw, and that meant time was desperately short. Fatboy had picked up Rebecca from her place and brought her around on his Harley, because she'd told him that she and Tane wanted to discuss something important with him.
When he had found out that it concerned a Lotto ticket, he had roared with laughter. _Moo-ha-ha-ha!_
Tane said stubbornly, "All right, ten percent." He crossed out the figure on the contract that he had prepared, using his best legalese, and wrote in "ten percent."
He handed the paper to Fatboy, who promptly tore it across the middle and laughed again. "Forget it. It's a silly bit of fun. You don't need a contract, and if I buy you the ticket, then we go thirds. Take it or leave it."
Tane growled under his breath, and Fatboy asked cheerily, "What, have you got it rigged or something?" and laughed again.
"Okay," Rebecca said, a little too loudly. "We go thirds. You buy the ticket, you claim the prize—"
"If we win," interjected Tane quickly.
"If we win. And you guarantee that you will give two-thirds of the money to me and Tane."
"Boy Scout's honor," said Fatboy, who, unlike Tane, had never been a Boy Scout.
Tane glared at Rebecca for a while before bringing out an envelope. The six Lotto numbers and the Powerball Jackpot number were clearly listed on the side. Inside were some crisp new five-dollar bills.
"Just these?"
"That's all," Tane responded. "For tonight's draw, and each week from now on."
"That's just one line. The more lines you buy, the better your chances."
"We don't want any other numbers, thanks all the same."
Fatboy persisted. "Look, you can't buy less than four lines, but if you're going to buy a ticket, you might as well give yourself a decent chance. Get a lucky dip with ten lines on it."
"No." Rebecca transfixed Fatboy with her gaze and spoke in a quiet, steady voice that eliminated all room for doubt. "No. Get four lines, then; you can choose the other numbers. But whatever you do, make sure you get the numbers on the side of the envelope."
Fatboy stopped laughing and looked at them both a little strangely for a moment, then shook his head as if dismissing a fanciful idea. "Cool, kids." He took the envelope with a sweep of his hand and stood up. "See you, Becks. See you, little bro."
"If we win the six million, you promise you'll share it?" Tane asked one last time.
"If we win the six million, I'll eat my motorcycle helmet!" Fatboy was out the door, his final laugh echoing back in through the open window. _Moo-ha-ha-ha._ Then the throaty roar of his bike and he was gone.
"He'll be honest," Rebecca said in a conciliatory way to Tane, who felt anything but conciliatory.
He nodded. "He'd better. But you're right. What else could we do? Imagine if the numbers come up and we haven't got a ticket!" He looked at her for a moment and she looked at him. Then, at exactly the same time, they both burst out laughing. It had nothing to do with Fatboy, Tane thought. It was just the sheer excitement of it all, and maybe a bit of relief at having finally solved _the problem._
Now the new problem was how to fill in the time until the live televised Lotto draw at eight o'clock that evening. Tane looked at his watch. Only nine hours and forty-seven minutes to go.
He said, "Our share would be four million dollars. What could we buy with that?"
"A new house," Rebecca murmured, almost to herself. "But don't forget the SOS. There's more to this whole thing than getting rich. Whoever sent those numbers did so for a reason."
Tane stood up with a mischievous glint in his eye. "You'd better come see this."
Tane's computer was on in his bedroom. Rebecca sat on the bed while Tane took the chair and opened an Internet browser. She watched with a fascinated frown as he typed some numbers into the address bar.
**202.27.216.195.**
"Those are the numbers from the message...," she realized.
Tane nodded. "It's an IP address. We learned about those last year."
"An Internet address. Of course!" Rebecca actually smacked herself on the forehead like they do in cartoons. "I should have recognized the pattern. What Web site does it take us to?"
"Have a look."
She stood behind him as the page slowly loaded onto the screen. The first thing that came up was the bright blue NASA logo. The next was a series of letters that they both recognized: BATSE. Below that were Username and Password boxes. Tane carefully typed "guest" into the Username, and "Compton1" into the Password.
He said, "It took me a couple of goes to get it right, because it was case sensitive."
"You clever bunny," Rebecca breathed. "Is this what I think it is?"
Tane nodded. "All the BATSE data." He pointed at a list. "That's the one we got from Professor Barnes. And that one arrived yesterday. These are all the ones in between. Want to analyze them?"
"Wanna try and stop me!" She almost kicked him out of the chair.
A moment later, her program was whirring away.
"It'll be quicker this time," she said. "I reprogrammed it to look for the Morse code patterns. It's quite clever but also quite complex. Do you want to know..."
"I—"
She didn't give him time to answer. "I went back and examined the raw data, and I found out something interesting. You see, the bursts are radiation waves, like an AM radio signal using amplitude modification to convey the ones and zeroes—"
"I'll take your word for it." Tane grinned.
"No, it's simple!" Rebecca said. "Imagine waves on a beach. There are big waves; those are ones. And there are little waves; those are zeroes."
"Okay." Tane nodded. "That much I understand."
"But sometimes there are gaps between the waves. And those are the gaps between the Morse code characters!"
"That is clever! And it'll make it much easier," Tane said. They had spent ages trying to work out all the possible combinations of the first message.
"Yeah, and faster..." She paused, noticing Tane's sly smile. "What are you smiling at?"
"I think I know who sent the message."
"You do?! Who?"
"Well, my guess is that was the final part of the message. Kind of like a signing off."
"TR?"
"Who do you know with the initials TR?"
Rebecca looked blank, and Tane's grin grew bigger.
"Isn't it obvious? TR. Tane and Rebecca. It's us. We sent the message to ourselves!"
Rebecca blinked rapidly a few times but said nothing as the enormity of that sank in.
Eventually she said, "Makes you wonder about the SOS, then, doesn't it."
Tane looked at his watch again. Nine hours and forty-three minutes to go.
**S ATURDAY NIGHT**
**Tane's watch said twenty minutes** to go, and he was sure the hands were standing still. But as he watched, the second hand inexorably flicked over. From where he was sitting, on the soft leather sofa in the middle lounge of his parents' house, the lights of the city blazed up into the clear night sky. The flashing lights of an airplane made a staccato string of pearls through the sky above the city. He barely noticed it. The second hand ticked over again.
Rebecca's program had already decoded a second message from the future and was busy working on the third. The second message was just as cryptic as the first.
**PROFVICGRNCHMRAPRJCTSTOPIT.**
**BUYSUBEONTLS.DNTGOMST.DNTTLNE1.**
Rebecca was clicking her fingers in front of Tane's face to get his attention.
"Concentrate," Rebecca said. "This is important."
Tane didn't think it was all that important, but it was taking Rebecca's mind off the last twenty minutes, so he tried to concentrate, for her sake.
It was hard. His hands were shaking and he wanted to vomit. If the numbers were right, then Rebecca would be able to pay all their bills, and they wouldn't have to move to Masterton. Everything would be all right.
"It's called the grandfather paradox, and it goes like this. What if you went back in time and killed your own grandfather?"
"Why?"
"Why what?"
"Why would I go back in time and kill Grandad? He's nice. I've got nothing against Grandad."
"Tane! Focus! That doesn't matter. We're just saying _if._ Okay. _If_ you went back in time and killed your grandfather when he was just a boy, then you would have never been born. Therefore, you could not have gone back in time to kill your grandfather. And so you would have been born, and so you could go back and kill him, but then you wouldn't have been born...and around and around it goes."
"Grandad takes me fishing," Tane said, but quickly added, when he saw that he was about to get thumped, "But I get it, I get it!"
"Some people say that time is like a Möbius strip. An endless loop with no start, no end, and a single surface, called the present."
Tane just shook his head. This science stuff was elementary to Rebecca, but it was way beyond him. His watch said nineteen minutes to go. _What if the numbers were wrong! What if they weren't Lotto numbers at all?_
"What's a Möbius strip?" he asked.
"Oh, come on! Do you ever stay awake in math?" Rebecca cried, and jumped up. She disappeared into Tane's room for a second and reemerged with paper, scissors, tape, and a pen. Tane watched intently as she cut a long strip from the paper and held the ends together in a loop.
"A Möbius strip is a piece of paper with only one side and one edge."
Tane tried to imagine that. "No way. If a piece of paper has a top, then it has to have a bottom. How can a piece of paper have only one side?"
"Watch."
Rebecca took one end of the strip of paper and twisted it over, just once. She taped it to the other end. "There you go. A piece of paper with only one side."
Tane took it and examined it. "Nope. Look, a top and a bottom. Or I suppose you'd say an inside and an outside." He knew he wasn't going to win this argument, but it was always fun to try.
Rebecca offered him the pen. "Draw a line longways, around the strip. Don't lift your pen off the surface. Stay on one side of the paper only."
Tane shook his head but took the pen and started drawing.
_What if there was no Lotto win? No great scientific discovery? And Rebecca would still go to Masterton._
After a few seconds of drawing, he found himself right back where he started, joining up with the start of his line.
"So?"
"So you drew on only one side of the paper, right?"
"Yeah?" He looked at the Möbius strip. He had drawn around both the inside and the outside of the loop.
"See, it has only one side."
Tane frowned and forgot about his watch for the first time that day.
"But what has this got to do with us?"
"It's like we're on that loop. And when someone in the future sends a message to the past—"
"When _we_ send a message to ourselves..."
"Whoever. But it's like they have made a hole in the paper and passed the message through to where we are in the past. But instead of paper, it's quantum foam, and the message is the gamma-ray burst."
"And what has my grandfather got to do with all this?"
"Well, they sent the Lotto numbers, right? But when we win—"
"If."
"Okay. Just suppose for a moment that it is us sending the messages."
"It is!" Tane insisted. "Think about it. Who else would know that we had thought of analyzing the BATSE data just at that precise time. Only us!"
"All right, us. If we win the Lotto but then forget to send the numbers to ourselves, then we won't win the Lotto, and around and around it goes!"
"Wow." Tane could think of nothing else to say, really.
Rebecca held up a notebook. "So I've got this notebook, and I have recorded the exact dates and times of the gamma-ray messages. Along with what the messages said, of course. Sometime in the future we have to send the messages, and any others that arrive, exactly as we received them. Otherwise, _kaboom,_ the grandfather paradox."
"Leave my grandad out of this," Tane muttered. "And where are we even going to get the gamma-ray time-messaging-machine thingy from anyway?"
"That part I'm not sure of. In the meantime, let's see that new message again."
Rebecca opened the notebook and they pored over it together. Rebecca lightly drew some lines in pencil to separate what she thought were the words.
**PROF VIC GRN CHMRA PRJCT STOP IT. BUY SUB EON TLS. DNT GO MST. DNT TL NE1**
She said, "I think it's like text-messaging. That kind of truncated English."
Tane picked up the rest of the paper that Rebecca had made the Möbius strip from. "So what have we got?"
"I think the first part is easy. _PROF VIC GRN_ has to be Professor Vic Green. I don't know who he is, but we can Google him or check with the universities later."
"What about _CHMRA PRJCT_?"
"Something project. Chim, cham, chem, chom, chum. Where's your dictionary?"
It was five to eight by the time they found the word.
" _Chimera!_ That's the only word that fits." Rebecca pronounced it slowly. "Ky–mere–rah."
"What does it mean?"
Rebecca looked, and frowned. "In Greek mythology, it's a monster with a lion's head, a goat's body, and a serpent's tail."
Tane blanched, remembering the SOS. "I don't like the sound of that!"
"Wait, in biology, it means an organism formed by grafting tissues or splicing genes from two or more different organisms."
"Chimera Project. Stop it. We are supposed to stop the Chimera Project. That's what this whole thing is about." Tane frowned. "I'm starting to wish they'd sent the message to someone else."
"We sent the message, according to you. Who else were we going to send it to?"
Tane's watch said seven fifty-seven. "Better turn the TV on," he said, and did so.
Rebecca was still examining the message. "We have to buy a 'SUB EON TLS,' whatever that is, and don't go to the 'MST.' Mast, mest, mist, most, must. Don't go in the..."
"Masterton," said Tane brightly. "Don't go to Masterton!"
"Okay," Rebecca said, "and the last bit is easy. 'Don't tell anyone.'"
The live, televised Lotto draw came on the TV, and Tane turned the sound up. He could hardly breathe. If the numbers were the same... _What if they weren't?_ Then again, from the cryptic clues in the last message, maybe they'd be better off if they weren't!
Rebecca and Tane sat on the couch to watch the short program, the original crumpled piece of paper on the coffee table in front of them. The numbers stared back at them: 8, 11, 22, 32, 39, 40, 3.
"What time is your mum coming home tonight?" Rebecca asked idly during the theme tune and preamble.
"Not till after eleven. Why?"
His dad was away in the bush, and his mum was out at some community council meeting.
"No reason," Rebecca said quietly.
Tane dragged his attention away from the screen and looked at her. He asked thoughtfully, "How's your mum? Will she be okay on her own tonight?"
"She's fine. Stop worrying."
Tane stopped worrying, but only because the Lotto hostess, elegant and sophisticated in a long blue gown, came on and started talking. Her blond hair was pulled back in a tight bun, and her smile was wide and toothy.
After an interminable introduction, the overly effusive hostess started the barrel rolling, and the Lotto balls began to tumble.
He barely felt Rebecca's hand slip into his. She was scarcely breathing.
The first number out was thirty-two.
"We've got that! We've got that!" Tane yelled.
Rebecca still wasn't breathing.
The second number was eleven.
"We've got that too! It is this week's draw! It is this week's draw!"
The next number rolled down the slope from the barrel and Tane froze.
"Thirty-six? Thirty-six?" He screamed, "It can't be thirty-six!"
The ball stopped rolling. The Lotto hostess announced, "Thirty-nine."
Rebecca collapsed against Tane.
He said, "Thirty-nine. It looked funny when it was rolling. It was a thirty-nine."
Rebecca didn't reply. She hadn't breathed since the start of the show. She finally took a breath after the next number, though. Eight. By that stage it was just a formality.
Forty. Twenty-two.
He didn't even bother watching the bonus numbers and found to his great surprise that he was hugging Rebecca, and she was hugging him.
The Powerball Jackpot had been sitting at $6,325,450 by the time the Lotto booths had closed at seven o'clock that evening.
It was almost anticlimactic seeing the three ball come wobbling up the little tube.
"We proved it. Messages through time. It's the scientific discovery of the century!" Rebecca breathed out slowly, and added almost as an afterthought, "And we're rich!"
"No." Tane shook his head. "Not yet. At the moment, we've got nothing. Fatboy is the rich one. Let's see if he does the right thing."
Rebecca nodded. "He will. But now that we know it really works, we've got the important stuff to discuss."
Tane knew what she meant. That was one thing his mind kept coming back to again and again. This wasn't a get-rich-quick scheme.
It was a cry for help.
**E VENSONG**
**Sunday was a good day.** A day of celebration. Rebecca stayed over in the guest room, and Tane's mum made French toast for breakfast.
They didn't mention the Lotto win to her, even though to keep it inside when it kept trying to burst out was like trying to hold in an enormous belch after drinking a whole can of Coca-Cola. Tane wasn't quite sure why they kept it a secret. "Don't tell anyone," the message had said. But did that include his own mum?
Tane had tossed and turned during the night, dreaming of Greek monsters with teeth of fire, but even so, he was up at six, before the sun, and Rebecca was already awake when he got up. It felt a bit like the day after Christmas when all the excitement is over and done with, but the real fun of playing with all your presents is about to begin.
They just talked until his mum got up and made the French toast. They talked about the money mainly. What to do with it. What to spend it on.
There were some dark thoughts lurking, but those didn't get a mention in the early light of Sunday.
By nine o'clock, they had solved most of the second message. Professor Vic Green turned out to be a woman, Professor Victoria Green, a highly respected geneticist. According to the Auckland University Web site, she was currently heading a private research laboratory on Motukiekie Island in the Bay of Islands.
But _SUB EON TLS_ turned out to be the biggest surprise: _SUBEO NTLS._
"It's a submarine!" Tane's eyes were wide. "We're supposed to buy a submarine! Cool!"
"I wonder why?" Rebecca said curiously.
Subeo was a British company. They had achieved international fame a few years earlier when they had produced what they called the world's first underwater sports car: the Subeo Gemini, a two-man submarine, designed mainly for recreational purposes.
The Subeo Aquarius had followed, a three-man version that also proved quite popular for commercial operators. But the latest model, not yet released, according to their Web site, was the Subeo Nautilus. If the Gemini was a sports car and the Aquarius a sedan, then the Nautilus was an underwater motor home. It was large enough for six people, an entire family, and could stay underwater for months on end. It was the first of the Subeo products to incorporate a diesel engine as a generator, to charge the banks of sealed-cell batteries that were the power source for the submarine.
Tane ran through three sheets of specifications that he downloaded as a PDF file before he came to the price.
"Holy cow!" he said.
"Where? Let me see." Rebecca grabbed the piece of paper off him.
"One and a half million pounds! What's that in New Zealand dollars?" She found a currency converter on the Internet. "That's over four million dollars!"
"We'll only just have enough, after Fatboy takes his share," Tane said grimly, the thoughts of riches vanishing before his eyes. "Are you sure we have to buy this thing?"
Rebecca nodded. "I wish I wasn't. There is a reason they, we, went to so much trouble to send this message back through time. I don't know what it is, but I know that we have to buy a Subeo Nautilus, and if they're in England, then we'd better get started."
Tane looked carefully at her. If they spent all their money on a submarine, then how would they afford to buy a new house for Rebecca? He opened his mouth to say so, but decided against it. Instead he just sighed and composed a quick inquiry e-mail to Subeo. He hesitated before clicking the SEND button but eventually sent the message on its way through cyberspace.
"Where the heck is Fatboy?" he wondered aloud.
That evening, after Rebecca had gone home and there was still no sign of his brother, Tane went for a walk out amongst the treetops. His dad had built the ropewalk himself, off the end of the deck, but it was sure and safe, if a little wobbly in places. There was a heavy rope for a base, like that of an anchor rope from a sailing ship. On each side, at about shoulder height and just in arm's reach, two more slender ropes gave you something to hold on to.
Tane didn't bother to open the safety gate; he just swung his legs over it and walked the first few steps without even holding on to the handrails, balancing like a tightrope walker.
He came out here often. Some days because he wanted to, and other days the evensong of the native birds seemed to call to him.
The breeze had come up with the closing of the day, but the sun had yet to disappear behind the mountainside, so it was pleasantly warm. The leaves on the trees that surrounded him ruffled softly, but the branches and the rope were still. All around him, birds sang joyfully in an enveloping chorus.
His dad came out here a lot also. He said it beat the hell out of watching television in the evening, and Tane supposed he was right.
The last time he had seen his dad was a couple of weeks ago, just before he had gone bush on another painting project. Fatboy had come around that day to show off his _moko,_ which he had somehow managed to persuade their mum to keep a secret.
Fatboy had walked in and taken off his helmet, and after an initial look of surprise, their father's face had cracked slowly into a smile, and his eyes had sparkled with pride. His dad had embraced Fatboy and pressed their foreheads and noses together in a _hongi,_ the traditional Maori greeting. Then he had hugged his eldest son, and Fatboy, the cool, leather-clad, rock-star-in-the-making that he was, had hugged him back without embarrassment or backslapping.
Tane thought back on that now and shook his head. He and his brother couldn't be more different. He was getting messages from the future, but Fatboy was still stuck in the past.
Where the heck was Fatboy anyway? He had not called, and when they tried his mobile phone, it went straight to voice mail. Had he even watched the Lotto draw? Did he know? Maybe he did, _and that was why he hadn't called._
A tui landed on the rail just in front of Tane's left hand. The distinctive white feather under the bird's chin looked like a miniature clerical collar. The parson bird, the early European settlers had called it, because it looked like a churchman. Tane didn't move. The tui looked at him suspiciously for a moment, then fluffed up its feathers and began to sing. The call of the tui was legendary, and they seemed to sing a different song every time you heard them. This bird, this day, had a slow, sad, rhythmical pattern that sounded like a lullaby.
After a while, the bird stopped and looked back at Tane, turning its head from side to side in small darting movements.
Tane raised a hand slowly toward the tui, inviting it to perch on his finger. The tui took a step backward on the rope. _I'm not scared,_ it seemed to be saying, _but I'm not stupid._ Then it was gone in a whirling dash around and through the branches of a nearby macrocarpa.
Tane stayed put for a while, looking out across the valley toward the concrete spires of the city.
It would be a shame when this was all gone. He knew it was coming. The subdevelopers with their tractors and bulldozers would be here one day. Already, across the ridge in the next valley, he could see the brown scar where a construction crew had felled the trees and cleared the bush, preparing the foundations for a new lodge and conference center.
One day, this ropewalk would be something he would tell his grandkids about, and they'd laugh, he thought, unsure whether to believe him.
He walked on to where he knew there was a new nest of young fantails. The mother was busy feeding them worms and didn't notice him.
Where the heck was Fatboy?
**T RUST**
**Fatboy rang when Tane was** getting ready for school, bleary-eyed and headachy. He'd been unable to sleep the night before.
His mum answered the phone, and from the tone of her voice, he knew instantly who it was.
His schoolbag dropped, spilling his pencil case and English study notes across the kitchen floor. He barely noticed.
"Harley wants to speak to you," his mum said, holding up the phone.
Tane had to restrain himself to avoid snatching the phone away from her. "Where are you?"
Fatboy's voice sounded cheerily in his ear. "We're millionaires, little bro. All three of us. We need to get together."
"We have school today." Tane tried to sound cool, as if he had never even remotely considered the idea that Fatboy might have absconded with the money.
"Can't you take the day off?" Tane could hear the excitement beneath the words. Fatboy, too, was trying hard to maintain the ice-cool rock-star persona, but the excitement was seeping out into his inflection like a little kid with a new toy.
"No way. Exams start next Monday," Tane said steadily.
"Lunchtime, then. I'll meet you both at McDonald's."
Fatboy rang off, and Tane realized that he had forgotten to ask why he'd had his mobile phone turned off.
He speed-dialed Rebecca's number.
Lunchtime was at twelve-thirty, but by twelve thirty-seven, Fatboy still hadn't showed. Tane could barely restrain his excitement, and a Quarter Pounder and fries didn't help.
Rebecca, by contrast, was oddly silent and ate nothing.
"What's troubling you?" Tane eventually asked. "Is it the SOS?"
"No, no. Yes, I'm excited. It's just..." A small tear squeezed itself out of her left eye, and she wiped it away quickly. "I know you should never say 'What if.' What if this had happened, what if that had happened. But I just can't help thinking, what if we had thought of the whole thing a year earlier? Fourteen months earlier."
Tane reached across the small Formica-topped table and put his hand on her arm. He knew where she was heading.
Rebecca said, her voice choked, "They could have warned us. We could have told Dad to stay at home that day. Everything would be different." Rebecca struggled to contain the sobs. "Mum..." She couldn't continue.
"You're right," Tane said. "You should never say 'what if.'"
He wanted to say more, something to ease Rebecca's pain, but he couldn't find quite the right words, and then it was too late because Fatboy pulled up, right outside their window in a brand-new, metallic green, soft-top Jeep Wrangler. There was a nervous-looking man sitting beside him in the passenger seat.
Rebecca grabbed Tane's napkin and patted at her eyes. By the time Fatboy and the stranger sat down, her smile was forced but believable. Just.
Fatboy slid across the bench seat with a flourish and put his arm around Rebecca's shoulders. He grabbed Tane's Coke. " _Kia Ora,_ kids."
"Hey!" Tane protested.
"Buy yourself another," Fatboy laughed. "Hell, buy yourself the whole goddamn factory if you want to."
The stranger sat timidly on the end of the bench seat. He was tall and balding in a flat line across the top of his head. He had a black mustache. He said a little unsteadily, "Actually, the Coca-Cola Amatil factory would be worth considerably more than the six million you have available to invest."
"Don't you just love lawyers!" Fatboy and his _moko_ grinned at them. "They're so literal."
"Nice wheels," Rebecca said cautiously.
"Yeah. Goes like stink too," Fatboy agreed. "I almost got it up on two wheels coming around Seymour Road."
Tane thought that might go some ways to explaining the rather nervous-looking lawyer.
"And we need a lawyer, do we?" he asked pointedly.
Fatboy reached inside his jacket and pulled out a thick, folded orange-colored booklet. He tossed it in front of Tane, who picked it up and examined it. _Realize Your Dreams. Winner's Information from the NZ Lotteries Commission._
"Page twelve," said Fatboy. "Seek professional advice. Tane, Rebecca, this is Anson Strange; Anson, my brother Tane and my um...Rebecca."
They both shook the man's hand.
Fatboy continued, "I didn't have a spare helmet for the Harley with me. But it was no problem. I just ran next door to the Chrysler yard and picked up the Wrangler."
"I hope you got a discount," Tane muttered. "Why couldn't we get hold of you?"
Fatboy looked aggrieved. "I was on my way to Wellington to claim our winnings. You can't just run into the nearest Lotto shop and ask them to cash out a six-million-dollar ticket!"
"We tried your mobile phone."
"Can't use them on a plane."
Rebecca asked, "Are you sure we need a lawyer? I thought you said that was all unnecessary."
Fatboy answered, "Things are different now. With six million dollars to play with, we've got to do things properly. Otherwise we might go and do something stupid like spend it all on a flight to the moon or something."
"Or a submarine," said Tane under his breath.
Rebecca said, "Does anyone else know about our win? Like your mum and dad?"
"Or the press?" Tane added.
Fatboy shook his head. "Nobody. I requested that our details remain anonymous." He turned to the lawyer. "Anson, would you give us a few minutes?"
Anson rose dutifully and went to stand in a queue behind the counter.
"What are you guys into?" Fatboy asked as soon as Anson was out of earshot.
"What do you mean?" Rebecca asked innocently.
"You knew those numbers were going to come up. I'm not entirely stupid. Something's going on."
"Nothing's going on," said Tane quickly.
"Can you do it again? Pick the numbers?" Fatboy stared directly at Tane.
"No."
"Maybe."
Tane and Rebecca said it simultaneously. Tane was louder.
Tane could just about see the dollars ticking over behind Fatboy's eyes.
"I want in," Fatboy said. "I want a part of it."
_You want a part of everything,_ Tane thought, and said sullenly, "You don't even know what _it_ is."
"That's true." Fatboy grinned. "But I want in anyway. How's that? Is it illegal?"
Tane didn't think it would worry Fatboy if it was. "No."
"There are some big up-front expenses," Rebecca said cautiously.
"Bigger than six million dollars?"
Rebecca nodded slowly. "Maybe."
Fatboy whistled. "But you could do it again, right? The Lotto thing?"
"Maybe," Rebecca repeated, but it was clear that Fatboy interpreted that as a definite yes.
"So if I understand what you're saying...We all pool our money together now, and we all get an equal third of anything later."
"You don't know what you're getting into," Tane insisted. "Stay out of it."
The more Tane insisted, the more interested Fatboy seemed to get.
Fatboy said slowly, "Okay, I'm in. My two million for a one-third share."
"We need to talk about this," Tane said.
"I'll give you a minute," Fatboy said, and went to chat with his nervous-looking lawyer.
"No way." Tane was adamant, although he kept his voice low.
"We don't know what we're getting into," Rebecca reasoned. "He might be a good person to have around."
"I'd rather sell my soul to the—"
"Look, he came through all right with the Lotto ticket, didn't he?"
"I still don't trust him. And anyway, he thinks this is all some kind of moneymaking scheme."
"Well, trust him or not, and whatever he thinks, we need his share of the money."
"No, I figure we'll just scrape through without him."
Rebecca dropped her eyes and there was a sudden catch in her voice. "Sure. And where are Mum and I going to live? On the submarine?"
Tane's next sentence froze on his tongue.
"Come on, Tane."
"I think we'll regret it."
Rebecca reached over and kissed him on the forehead.
Tane realized that he already regretted it.
They told Fatboy everything. Tane hadn't intended to tell him any more than they had to, but once you showed a tiny corner of the picture, it just kind of led to more questions and they led to more questions, and before long, Fatboy knew as much as they did. He didn't quite believe them at first, but the Lotto ticket was a quite convincing argument.
Tane thought Fatboy would change his mind when he found out that it wasn't just about picking the Lotto and was quite surprised when he didn't.
After that, Fatboy waved over their patiently waiting lawyer, who recommended a trust to cope with the financial arrangements.
After school, Fatboy picked them both up in the new Jeep, and they went to the bank to sign some documents and pick up cash-cards for their new bank account, which had almost as many zeroes in it as the messages from the future. Then Fatboy took them around to Tane's house to check their e-mail.
There had been no response from Subeo, but Fatboy, ever practical, suggested a phone call to the UK. He felt they could probably afford the cost of the call.
It turned out that the Subeo sales representative was in Sydney on a visit to the Australian Navy, and Fatboy didn't have to tell him too many lies to persuade him to add Auckland to his itinerary.
It was Rebecca who finally asked the question that Tane had been dreading. "What are we going to do about Professor Green?" she asked.
That brought reality crashing back into the excitement of the money and the trust and the submarine purchase. There was a reason for all this, they suddenly remembered. Some time in the future, they were going to be in big trouble.
Tane said, "If we believe the instructions in the message and understand them correctly, then we are supposed to somehow stop the Chimera Project."
"But what _is_ the Chimera Project?" Rebecca wondered.
"Look it up on Google?" Fatboy suggested.
"I did," Rebecca said. "Nothing. So I looked up Professor Green and the research facility. There's a lot of stuff about their research into rhinoviruses—"
"Rhino viruses?" Fatboy looked up incredulously.
Rebecca smiled. "Nothing to do with rhinos. It's the virus that causes the common cold."
"Oh." Fatboy looked quite disappointed.
Rebecca continued, "But researching the common cold isn't going to cause a worldwide disaster, so that can't be the reason for the SOS. But then again, Prof Green is a geneticist, and playing with genes can be playing with fire. Maybe there's something else going on at the lab that they don't want the rest of the world to know about."
"The Chimera Project," breathed Tane.
"And we're supposed to stop it," Fatboy said.
"I think we should go and see the professor," Rebecca said, "ask her about the project, and perhaps ask her to stop it."
"And if she says no?" Tane asked.
"We'll worry about that when the time comes."
"Why don't we just tell the police?" Fatboy asked. "Or the government? Or someone official. I mean, what are the three of us supposed to do?"
Tane looked at Rebecca, who shook her head. "Not yet. I doubt they'd believe us, but more importantly, the message told us not to tell anyone. Until we know the reason why, we should follow the instructions."
Tane looked at her closely. That made sense, but he couldn't help wondering if there was a bit more to it than that. Was she afraid that someone else would whisk this discovery away from them and claim credit for their ideas and hard work?
The next day, school or no school, exams or no exams, SOS or no SOS, Tane and Rebecca took their new cash-cards and went shopping. It wasn't every day that you became a millionaire, after all. They didn't go wild, though. Nothing extravagant.
Rebecca bought a new pair of jeans that she'd been admiring for a while, then a couple of pairs of shoes to go with them. Tane bought a new leather jacket that he thought looked really cool.
Then Rebecca bought a few CDs—music she couldn't afford to buy before, just twenty or so of her favorite groups—and Tane bought a new joystick for his computer and a couple of new computer games. There were a few other odds and ends, too.
At lunchtime, Tane said he was going to take Rebecca to the best restaurant in the whole of Auckland, and according to the driver of the limousine they hired, the best restaurant in town was Number Five, which had just reopened up in City Road, near the Sheraton Hotel.
It was very nice. So nice, in fact, that it wouldn't let them in. Not in the clothes they were wearing. But half an hour in Smith & Caughey's fixed that, and Tane toasted Rebecca across a very fine cut of eye-fillet steak wearing the first suit he had ever owned and the second he had ever worn. The first being at Rebecca's father's funeral.
By the end of the day, the score was fifty-seven CDs, eleven DVDs, one top-end laptop computer for Rebecca, twenty-two articles of clothing, one computer joystick, four computer games, seven pairs of shoes, two books, four items of jewelry (of which Tane's new necklace was by far the most expensive), a bicycle, two pairs of sunglasses, a life-sized stuffed toy baboon that sat in a spare seat in the limo and laughed at them the rest of the day, three mobile phones, and a two-storied cliff-top house for Rebecca and her mother to live in that happened to have a boatshed, looking out over the water at West Harbor.
Nothing extravagant.
**T HE MAN FROM SUBEO**
**The man from Subeo was** Arthur Fong, which sounded Chinese, although he wasn't. He said he'd be there on Thursday evening, November 26, at seven-thirty on the dot and rang the doorbell as the clock in the hallway just ticked over.
Tane, Rebecca, and Fatboy had gathered at Rebecca's new house for the meeting. Fatboy had picked up Tane after school, and they had had a quick dinner of fish and chips while waiting for Arthur Fong to arrive.
It was Tane who answered the door. He'd jumped up like a shot and practically sprinted for the door while the others were still registering the sound of the bell. Then, not wanting to appear too eager, he had sedately strolled down the polished wooden floors of the hallway to the front door.
The door was solid kauri inlaid with panels of stained glass. It was a nice door. It was a nice home. It wasn't new; in fact, it must have been fifty years old, but it was elegant, and a lot of money had been spent restoring it. None of which had really mattered to Tane, Rebecca, and Fatboy when they had found the place. What had sold them were two things. Firstly, it was vacant and available for immediate possession. Secondly, the back lawn led straight down to the edge of a high cliff above a secluded inlet of the upper harbor. At the bottom of the cliff, down a series of wooden staircases, there was a large, brown, slightly ramshackle boatshed.
From an upstairs room, the sound of a television washed faintly through the floorboards. Rebecca insisted that her mum was only grieving, that her mind was all right, but she had not questioned her daughter when she told her that they had bought a new house. Had not asked where the money had come from. Had just moved in, quietly accepted the room that Rebecca pointed out to her, and turned on the television.
As sad as it was, it was also convenient. It was good that she didn't ask too many questions.
Arthur Fong was tall. Thin of face but wide of bottom, he was rather like a pyramid in shape and when he realized that his appointment was with three teenagers, suddenly found several pressing reasons to leave.
"Sit down," Fatboy said, and added, "please," for good measure.
Mr. Fong sat down. People had a habit of doing things when Fatboy told them to.
"Listen," Mr. Fong said, "I admire your initiative. If this is for a school project, then I'd be happy to send over some brochures, even some of our technical drawings, which we don't normally release. But I am on a very tight schedule."
Tane said, "Mr. Fong—" But Fong held up his hand to interrupt him. "I have spent time—and money—flying over here because I thought I was going to be meeting with a company who was genuinely interested in purchasing one of our products." He rubbed vigorously at his face with both hands, a gesture of tiredness and frustration.
"Can I get you a cup of tea?" Rebecca said demurely, and Tane glanced at her. That was not really like her.
That seemed to soften his attitude slightly, although he declined.
"Not had a good week?" Rebecca asked.
Mr. Fong smiled tightly. "You could say that. I've had flights delayed, lost luggage, canceled orders, and now a wasted trip to New Zealand, so excuse me if I seem a little brusque. You do realize, don't you, that the price of the Nautilus is over a million pounds. It is not a toy!"
"Canceled orders?" Rebecca asked casually.
Fong said nothing.
"In Australia?" she coaxed.
Fong sighed. "Yes. Six months of negotiations all down the drain. And now this." He made moves to get up again.
"Why did they cancel?" Rebecca asked quickly but still with a casual tone. "Is there something wrong with the submarine?"
"Of course not," Fong said indignantly.
"Because if there are problems with it, then—"
"The sub passed every test they gave it with flying colors. The cancellation was all to do with bureaucracy and politics in upper management. The sub is fine. It's brilliant, in fact."
"So where is the sub now?" Rebecca coaxed.
Fong looked at her and smiled, realizing where she was heading.
"It's still in Sydney," he said. "But please be serious. It costs a million pounds. I don't know what that would be in New Zealand dollars—"
"Four million, one hundred twelve thousand, two hundred and twenty-nine dollars," Rebecca said from memory. "And ten cents. At today's rate."
Fong rose to his feet.
"It was nice to meet you. But right now, I am going to leave. I don't like my time being wasted."
"Your time is not being wasted," Fatboy said. "We represent a trust that has substantial funding. The Nautilus you have in Sydney. We'll buy it."
"A trust," Fong said skeptically.
"I said we'll buy it."
Mr. Fong looked at Fatboy with a kind of exasperation, as if he were speaking to an idiot who wouldn't see sense. "Sure. It's yours," he said. "Just write me out a check for, hell, round it off to four million New Zealand dollars. It's yours."
The doorbell sounded and Rebecca went to answer it.
Fatboy stretched out a hand and said, "Mr. Fong, you have a deal."
Fong ignored the hand.
Fatboy continued, "There are two conditions. You ask no questions, and you don't inform the press. This deal is just between you and us."
Mr. Fong looked at him cynically for a moment, but then laughed and shook Fatboy's hand. "Absolutely. Anything you say. No questions asked. And the check?"
Fatboy shook his head. "We don't have a checking account yet, but—"
"What a surprise." Fong didn't sound surprised at all. "Then I'm afraid the deal is off."
Rebecca's voice came from the doorway. "Mr. Fong, I'd like you to meet our lawyer, Anson Strange."
"Just in bloody time," Tane said out loud, without intending to.
**M OTUKIEKIE**
**The engine of the small** plane roared and spray flew past the window. Tane peered out at the water slipping away faster and faster underneath the hull of the small seaplane. The harbor was smooth, but even so, small waves drummed faster and faster at the hull of the craft.
"I thought it would be bigger than this!" Tane shouted over the noise of the engine, gripping the back of the seat in front of him.
Rebecca seemed unconcerned by the fact that they were about to take off from the surface of the sea and head thousands of feet up into the sky in a piece of motorized tin that was probably built by the Wright Brothers. Fatboy was in the copilot seat, joking with the pilot.
The pilot, a blond-haired man who seemed far too young to be in charge of an airplane, heard and called back, without taking his eyes off the controls, "It's a Grumman Super Widgeon. It's quite large compared to most seaplanes you see nowadays."
"I'd hate to see an ordinary Widgeon," Tane shouted.
He'd been on planes before, plenty of times. But they had video games in the backs of the seats and sixteen music channels and a cabin crew who brought you cold drinks and cookies.
And they took off and landed on land.
This was the fastest way to get to Motukiekie, though. Professor Green had recommended the company herself. Tane was a little surprised that she had agreed to see them, but it seemed that having a famous scientist for a father opened quite a few doors for Rebecca, even if (or perhaps because) he was no longer around.
It was a Friday, but there was no school, as it was the last Friday before exams, and it was officially a study day. Which is exactly what Tane's parents thought he was doing. Studying. At Rebecca's house. Not flying to the Bay of Islands in a prehistoric seaplane.
The rippled surface of the ocean outside turned to a mosaic of blue tiles, painted in flowing brushstrokes, then to a continuous blur, and then it disappeared, and all Tane could see was the sky as the plane banked around in a tight circle, back over the city.
They climbed as they turned, but they were still so low as they passed back over the wharves and the harborside apartments and office blocks that Tane could see people eating lunch.
They kept climbing and by the time they passed the enormous Skytower, the largest building in the Southern Hemisphere, it was well below the large floats hanging outside Tane's window. From this angle, it looked surprisingly small.
The city slipped away beneath them, and the harbor bridge beckoned, a gray coat hanger joining the central city to the North Shore.
"You should fly under it!" Fatboy grinned at the young pilot.
Tane gripped the back of the seat.
"That's illegal." The pilot smiled back. "I'd lose my license. It's been done, though."
"Really!" Rebecca exclaimed.
"Captain Fred Ladd, back in the sixties or seventies. He was a bit of a legend apparently."
"Did he lose his license?" Tane wanted to know.
"Yeah, but they gave it back to him. He was a bit of a legend, after all."
It took little more than an hour to reach the Bay of Islands, even in this old museum piece. The pilot seemed to know the area well.
"That's Cape Brett," he said. "My great-grandfather used to be the lighthouse keeper there. Of course, it's all automated nowadays. Down to your right, that's the Hole in the Rock."
It was a tiny island with a hole punched right through one end. As they passed over it, a launch packed with tourists cruised right through the gap, seemingly oblivious to the danger from the rock walls that surrounded them.
"If you like big-game fishing, we can organize a tour for you," the pilot said. "There's great marlin fishing up here, but if you prefer kingfish, tuna, shark—"
"Sportfishing is murder," Rebecca said quietly, but loud enough to be heard.
The pilot took no offense. "Then I guess you'd be more interested in the bird sanctuary on Roberton Island."
"Another time," Fatboy said. "We're on a tight schedule, this trip."
"Well, that's where we're headed, right there." The pilot pointed. "That's Motukiekie Island."
There were islands everywhere, but the one he was pointing at was easily recognizable thanks to the small complex of buildings surrounded by a wire fence, not far from a small cove at the end of the island. It seemed a stark contrast to the lush verdure of the surrounding islands.
The seaplane began to descend, and they flew low over a small hill at the opposite end of the island.
Fatboy drew in his breath suddenly. "Look at that. A Pa."
Tane heard the undertone and looked, but he saw only a lumpy hillside with curious circular ridges. Fatboy was right, he realized; it was indeed the remains of an ancient Maori fort, a Pa. But it still seemed like just a lumpy hillside to him.
Then, as the plane passed alongside the hill, preparing to land, it all changed. For a moment, the hillside seemed to come alive in his mind. He saw women in flax skirts weaving baskets, children playing. Strong-shouldered men preparing a feast in a dug-out earth pit. Suddenly, an attack by warriors of a neighboring tribe, screaming and charging up the hill at the wooden battlements with their _wahaika_ and _taiaha_ —their weapons—held high.
Just for a moment.
Then the floats of the plane hit the water, and a cloud of spray obscured the hillside, and when it cleared, the Pa was gone and he was looking at nothing more than a lumpy hillside.
Professor Green met them at the jetty herself, which seemed a little unusual, Tane thought. Surely the head of the organization didn't meet all her visitors personally at the wharf?
"Call me Vicky," she said brightly, her emerald-green eyes matching her name. Her hair was red and constantly trying to escape from the loose bun she had it pulled back into. Tane figured that loose hair was probably not a good idea in a science laboratory, but Vicky's hair seemed to have a mind of its own.
"We don't normally allow visits, for security reasons," she said, leading them up a concrete path through dense native bush. "But I thought I could make an exception." She looked sadly at Rebecca. "I knew your father. By reputation anyway. And I did meet him once at a conference in Dunedin."
Rebecca nodded silently.
Vicky continued, "And is your mum still conducting research into climate change? I seem to remember that her work was quite radical. Groundbreaking. But I never saw anything published in the scientific journals."
"She's taking a sabbatical this year," Rebecca said.
Tane thought of the blue flickering windows of Rebecca's mother's room and said nothing.
They reached a high wire fence, topped with vicious-looking barbed wire. There was a gate set into the fence and beside it an electronic keypad.
Tane glanced over as the professor tapped in a four-digit pattern. He caught the last number. Three. Like the Powerball number. Like the three of them.
The path continued inside the fence, and they made their way through some brightly colored flowerbeds to another door, another keypad, and through some polished corridors to her office.
Vicky fussed around them, getting them each a glass of cool water, despite the fact that they hadn't asked for one.
There was a painting on the wall of her office that Tane recognized immediately and knew that Fatboy would, too. It was one of their father's works, entitled _Tuatara Dawn._ It was worth a lot of money, Tane remembered.
"So how can I help?" Vicky's emerald-green eyes flashed brightly once again.
Rebecca began, "Well—"
"What kind of security reasons?" Tane interrupted, looking up at the CCT security camera mounted in the corner of Vicky's office. Another had stared at them in the main entrance area.
"I'm sorry?"
"Earlier you said you don't usually allow visitors, for security reasons. What kind of security reasons? Do you work with dangerous viruses here?"
Vicky laughed, a soft, bell-like trill. "Good heavens, no. Nothing like that. All our work is with rhinoviruses, nothing dangerous. The security, and the reason we operate all the way out here on an island, is to get away from protestors, who have no idea what we are doing but object to it anyway. Because we're a genetics lab, they assume we are creating genetically modified tomatoes or cloning sheep or something like that. Do you know what rhinoviruses are?"
Fatboy answered smugly, "Of course. It's the common cold."
Tane caught a brief smile from Rebecca.
"Come and have a look," said Vicky.
She talked as they walked along a short corridor with large glass windows looking in on a laboratory where technicians and scientists in white lab coats and plastic hair caps were doing unguessable scientific things with microscopes and test tubes.
"This is our level-one lab. That's what we call a biocontainment level. The lab is sealed while people are working in there, but there's no real danger to anyone."
"And if there was an accident?" Fatboy asked.
"Well, you might catch a cold, I suppose." Vicky laughed again, pleasantly.
Tane had been expecting to meet some evil scientist with devious plans, thick glasses, and maybe a Persian cat on her lap. Vicky Green didn't fit the bill at all.
Vicky continued, "We made provisions for a level-two lab when we built the complex, but we haven't used it yet."
Tane noticed that her eyes involuntarily flicked toward a solid-looking door at the end of the corridor as she spoke.
She continued, "That would be for any dangerous pathogens, like influenza or hepatitis C. Labs go all the way up to level four, you know, but that's only for people who are working with the really deadly viruses like Ebola. The United States has one at the CDC, their Centers for Disease Control, and I think the Russians have a couple."
They stood and watched the lab staff at work for a while.
"What area of rhinovirus research are you conducting?" Rebecca asked.
"Well, our main area is conserved antigens. Are you familiar with that field?"
"Slightly," said Rebecca.
"I'm not," Tane said quickly.
"Okay, do you know how your body's immune system works?"
"Antibodies?" Fatboy queried.
"Well, that's a part of it. _Antibodies_ are your body's watchdogs against viruses and bacteria. _Macrophages_ are your body's soldiers. When the antibodies recognize something dangerous—a _pathogen_ —they latch on to it, smother it, and send out a call for the macrophages to come along and swallow it up. But viruses like the rhinovirus keep changing. Mutating. Your body learns to recognize one rhinovirus but next winter along comes a new one with a different shape, and your antibodies don't recognize it."
She stopped and looked around them, to make sure they were following her, which they were. "We are looking for conserved antigens, which means looking for common characteristics."
She drew a felt pen out of her pocket and, seemingly absentmindedly, drew a diagram on the glass window looking into the lab. Tane suspected she did things like that a lot.
Her diagram looked a bit like a flower, a central circle, with smaller circles surrounding it, joined to the center by stalks.
"Suppose this is a rhinovirus. Our antibodies recognize these shapes here"—she pointed to the smaller circles—"but then along comes a new virus." She rubbed out the small circles with her thumb and drew in small triangles. "With a different shape, which they don't recognize."
"Then it wouldn't be able to smother it," Rebecca said.
"Right, but look, the stalks are the same on both viruses. That is what we call a _conserved antigen._ What if you had antibodies that could recognize the stalks, instead of just looking at the overall shape?"
"Wow," said Tane.
"You'd cure the common cold," said Rebecca.
"What about the Chimera Project?" asked Fatboy, and there was a sudden silence.
Tane winced. They had to be subtle, he thought. Fatboy was as subtle as a bull in a china shop.
"What is the Chimera Project?" Vicky asked after a while.
"We were rather hoping you could tell us," Rebecca said.
Vicky thought for a minute, then shook her head. "Never heard of it. It could have something to do with genetics, though. A chimera is what we create if we splice together genes from two different organisms. The University of California created a 'geep' a few years ago, part goat, part sheep, but there was a lot of hoo-ha about that, and you don't hear about that sort of thing very much anymore."
Tane looked carefully at her. Was she telling the truth? If there was no Chimera Project, then maybe they could just get back to Auckland, cancel the order for the submarine, and get on with spending the six million dollars. That sounded like a good plan. However, he couldn't get three small letters out of his mind: _S, O,_ and _S._
Rebecca said, "What if you were to genetically splice together two, or more, different cold viruses? To help find your conserved antigen. What if you did that?"
Vicky laughed, a little too quickly this time. "A chimera rhinovirus. I'm afraid that's just science fiction, young lady."
The trip back was silent. Even the pilot sensed the mood and cut his usual cheery chatter. It wasn't until they were almost about to land that Rebecca said what they were all thinking.
"Professor Green was lying through her back teeth."
**W ATER WORKS**
**WTRWKSBTMP1000:2.80,24,341,55,500. 80,24,342,54,499,1.80,24**
**Rebecca's software, trawling through weeks** of gamma-ray bursts, had found the next pattern, but this one made no sense at all.
All three of them stared at the characters dotted across the computer screen, trying to see order in the chaos. The early Saturday sun cast long streaks of light across the carpets of the lounge of the West Harbor house but did nothing to illuminate the puzzle.
"So you've checked earlier messages too?" Fatboy asked.
"Weeks of them," Rebecca replied. "The messages start on the day we visited Dr. Barnes."
"As if they knew you would visit him that day."
"Exactly. That can't be just a coincidence."
"I still don't understand where we're going to get this time transmitter from." Fatboy frowned.
"Me neither," Rebecca said with a smile. "I'd invent one, if I had the slightest idea of where or how to start."
"SOS means an emergency," said Tane, whose mind was somewhere else entirely. "It means 'help, save us,' but from what?"
"Water Works," said Rebecca, looking at the printout. "Like on Monopoly. You know, the Electric Company and the Water Works."
The other two looked at her and she shrugged. "Still doesn't make much sense, though, does it."
"Maybe it's a plague," Tane said. "Maybe Dr. Green is going to accidentally create some horrible disease and wipe out half of mankind!"
Fatboy asked, "What if we just went and saw her again? Maybe she'd listen to us if we told her about the message."
"Maybe she'd deny everything again and have us arrested," Tane said.
"What are you up to?" It was Rebecca's mother, drifting through the room. They hadn't heard her come in.
"Runescape," Tane lied quickly.
"What's that?" she asked vaguely.
"It's an online game where you get to be a kind of a character, called an..." He trailed off as she drifted out of the room, not listening to his answer. Tane stared at the computer screen, careful not to look at Rebecca.
"I think we have to involve the authorities," Fatboy said. "If it is the end of the world that we are talking about, then that's too big a problem for the three of us to deal with."
"You're right," Rebecca agreed. "But first we have to prove it. So far it's all just guesswork, and as Tane says, maybe we have misunderstood the message."
"And what if we tell the authorities, and they don't believe us, and they alert Vicky, and we lose our chance to do something about it?" Tane said. "Then the end of the world will be our fault!"
"Maybe that's why the message said, 'Don't tell anyone,'" Fatboy considered.
Rebecca said, "We need to know more about the project. I mean, what's it about? What are they really trying to do? I think we need to go back to the island."
Fatboy shook his head. "She's not going to admit anything."
"I know," Rebecca agreed. "That's why we have to go when she's not around. When nobody's around."
"Ooooh kaaay..." Tane drew out the words.
Rebecca continued, "We get in there at night when nobody is working, and go through her files. Find out what she's up to, then figure out what to do about it."
"That sounds pretty reasonable," Tane said, "but what about her security? The barbed-wire fences and alarms."
"Worry about that next," Rebecca said. "Are we agreed on the basic idea?"
"Sounds pretty good to me," Fatboy said. "And I might have an idea about the security."
Tane just nodded. "It's okay with me."
Fatboy said, "Okay. We definitely do it at night, when nobody's around. Wear masks so the security cameras can't identify us later." Fatboy idly doodled while he talked. Tane looked at the paper. It was a rough map of Motukiekie.
Rebecca was busy on her new computer and soon had a satellite image up on the screen, courtesy of Google Earth. The lab was clearly visible, a large cleared area amidst the dense bush of the island. Northwest of the lab, along a narrow path, was another complex of buildings, consisting of one large block and a number of smaller ones on a hill nearby.
"What are those buildings?" Tane wondered. "More labs?"
Rebecca shook her head. "Accommodation. The scientists need somewhere to live, don't they?"
Fatboy studied the picture and added a few notes to his map.
"Okay, but how do we get to Motukiekie Island without being seen? I don't want to get arrested." Rebecca was suddenly quiet and Tane knew what she meant. She'd already been arrested once. Who would look after her mum if she was in prison, or reform school, or wherever it was they sent teenage criminals?
"Underwater," he said with dawning awareness.
"The submarine!" Fatboy snapped his fingers.
"The Subeo Nautilus," breathed Rebecca. "So that's why we had to buy the submarine."
"This just might work," Fatboy said, "but there is still the problem of their security system. I think I caught the first few numbers when Professor Green pressed them on the keypad."
"Me too!" Rebecca said cautiously.
Fatboy closed his eyes, remembering, and said, "Five, one, something, maybe four...I couldn't see the last numbers because of the way she was holding her hands."
"That's right." Rebecca nodded. "Five, one, and four. I saw them too, but they're no good. We might as well have nothing without the last number."
"Three," Tane said, "like the Powerball number."
"Are you sure?" Fatboy asked.
Tane nodded. "I'm not totally useless."
Fatboy ignored him. "Okay, then. When? Any ideas when we should do this?"
Rebecca said quickly, "Soon. As soon as possible. Before it's too late."
Tane asked, "But how do we know when 'too late' is?"
"That's easy," Rebecca said, staring Tane straight in the eye. "The day after Professor Green releases the virus, or whatever it is that she is going to do...that's a day too late."
Fatboy said solemnly, "The sooner the better, then."
"Okay. Moving on. We still have more message to decode," Rebecca said slowly. "Bottom P. Bat Map. Butt Mop."
"Butt Mop?!" Tane burst out laughing, a little too loudly. Then Rebecca got the giggles, which set Tane off again, and the combination was too much even for Fatboy, who laughed so hard his hat fell off.
Exam week came and went, and Tane was convinced that he failed every darn thing. He couldn't concentrate, he couldn't focus, he couldn't even remember. Things that he had known all year suddenly seemed like distant memories. Subjects he was strong in were all of a sudden his weak points.
It was unfair, he thought. How many of the other kids sitting the exams were trying to solve cryptograms from the future and planning to break into a genetics laboratory? Well, just one other that he knew of, and she was going to breeze through her exams and probably still top the country despite everything that was going on.
**S EA OF GREEN**
_As we live a life of ease_
_Every one of us has all we need_
_Sky of blue and sea of green_
_In our yellow submarine_
_—The Beatles, "Yellow Submarine"_
**Imagine the biggest, best, most** exciting toy you ever unwrapped on Christmas morning. Then times that feeling by ten. It still doesn't even come close.
Technically, this wasn't a Christmas present, but with only a couple of weeks to go until the twenty-fifth, it certainly felt like one. The drudge and slog of exam week was over, and the freedom of the holidays was here. Tane hadn't felt he had done very well, but that seemed of little importance now.
He had felt like celebrating on the weekend and had been going to suggest something to Rebecca, but she was going out with Fatboy, so he had sat home by himself and watched TV.
Tane stood at the entrance to the boatshed and stared at their brand-new, bright yellow, six-person submarine. Rebecca stood beside him and Fatboy beside her.
The single one-hundred-fifty-watt bulb hanging on a piece of wire from the ceiling reflected off the rounded sides of the Subeo Nautilus, giving the whitewashed walls of the boatshed a warm yellow glow, which complemented the streaky orange hues of the sunset across the harbor behind.
Arthur Fong and Wee Doddie, the mad Scottish engineer who had arrived with him, were still on board going through some final delivery checks, although that seemed a little bit late in the day considering that the sub had sailed, if that was what you called it, under its own steam from the delivery ship anchored out in the gulf, all the way up through the harbor, under the harbor bridge, and out through the upper harbor to Beachhaven. It had traveled underwater, as a bright yellow submarine sailing through the harbor might have attracted the kind of attention that Mr. Fong had promised not to attract.
"What are we going to call her?" Rebecca asked.
Tane looked at her and noticed that Fatboy's arm was casually draped around her shoulders.
"Tane, you're the creative one. You think of a name," Fatboy said.
He already had. "Möbius Dick." He thought it was a clever play on the name of the infamous whale.
"You gotta be kidding," Fatboy laughed.
Tane bit back his annoyance. "How about just Möbius, then?" he suggested.
"I like Möbius," Rebecca offered.
Fatboy shrugged. "I could go with Möbius."
Wee Doddie, who insisted that was his real name, climbed out of the top hatch grinning cheerfully. He was in his fifties or sixties, tough and wiry, completely bald, utterly mad, and had a gold ring through each eyebrow. His forearm was tattooed with a picture of two dolphins above the word _Dreadnought,_ which Tane thought might be the name of a submarine he had once served on.
He was their training instructor, according to Arthur, although Tane wondered if that was such a good idea. Not because of his knowledge or experience, which were undoubtedly excellent, but because of his thick accent.
He was from "Gluzgi," which eventually turned out to be Glasgow, and when he spoke, it was just a jumble of syllables that their brains instinctively tried to turn into English words. Only none of it made any sense.
"Rate kids way rid utter gore. Giddy arses blown will tea coffee a waste-bin."
Tane looked at Fatboy who looked at Rebecca.
Rebecca struggled to suppress a smile. "I think he wants us on board."
"You sure?" Tane asked doubtfully.
"My uncle Iain talks like that," Rebecca whispered. "What he said was, 'Right, kids, we're ready to go; get your asses below and we'll take her for a wee spin.'"
Doddie jabbed a thumb toward the open hatch and said emphatically, "Wedgie wheaten fur? An invite o'da queen? Giddy arses blown will tea coffee a waste-bin."
"Come on, then," Fatboy said.
Doddie shook his head and folded his arms as they climbed the short ladder onto the top hatch. "Do they nay spike English reindeer."
"I think I understood that one," Tane whispered to the others as they climbed inside.
Doddie's voice came clearly in through the open hatch. "Aye, an than yer arse fell orf."
It was dark by the time they took the _Möbius_ for her first dive, which suited all of them just fine, as the last thing they wanted to do was draw attention to themselves.
Arthur and Wee seemed to accept, if not to understand, the need for discretion.
It turned out the bright yellow color had nothing to do with the old Beatles song and everything to do with safety. It made the craft more visible underwater, and most commercial submarines were painted yellow for that reason.
Everything inside was space-conscious. Not an inch of room seemed to be wasted, and every surface seemed to have at least two or three uses. The main control room of the sub was separated from the central cabin by a thick pressure door, operated by a wheel lock. In the event of a rupture or leak in the rear of the sub, they could seal the control room and steer the boat to the surface.
The central cabin had three round portholes on each side, made of an impossibly thick glass that still somehow gave a good view of the surrounding ocean. There were floodlights built onto the hull of the sub outside the portholes on each side. In the central cabin were three bunks on either side.
At the back of the main cabin was a small galley and at the rear of the sub, through another watertight door, was the battery room with its rows of sealed-cell batteries, tiny diesel-powered generator, and storage cupboards in every conceivable nook and cranny. There was a marine toilet in one corner of the battery room, which had been modified in some way for use in the pressurized hull of a submarine, but Tane could not understand most of what Doddie had to say on that subject.
Everything was computerized and automated. They didn't have to blow ballast tanks or adjust diving planes or any of the stuff Tane had seen in the old submarine movies, which was a relief. There was no _Ah-OO-gah_ diving horn either, though, which was a shame.
The Subeo Nautilus was controlled with a two-handled joystick, like that of a small airplane. Forward to go down, backward to go up, turn to go left or right. The top hatch was actually a pressure chamber, like an air lock, which meant that it could be opened underwater. The hatches were synchronized so that they couldn't both be open at the same time while the boat was submerged. Of course, it paid to be wearing scuba gear when you did this, unless you were particularly good at holding your breath.
The sub could go forward, or backward, although the steering was "krarp" in reverse according to Wee Doddie.
A detachable buoy on the rear of the hull provided a number of important links with the outside world. Once released, it floated to the surface. It had an air intake, to provide air for the diesel generator as well as for the passengers; a radio antenna; and a solar panel that could recharge the sub's batteries while it was underwater.
But the bit that Tane liked best about the buoy was the gimbal-mounted video camera, linked to an LCD screen, which could be controlled with a small joystick on the console.
"A periscope!" Tane exclaimed.
Arthur Fong showed them how to change the Sofnolime scrubber cartridges in the carbon-dioxide filter. If they were staying for a long period below the surface, without sucking in fresh air through the buoy, then they would have to change the cartridges regularly or the carbon dioxide would build up in the craft's atmosphere and kill them.
They all paid particular attention to that part.
Wee Doddie showed each of them how to run the sub and made each of them do it over and over until he was "suss fee" that they could "date purply": _satisfied_ that they could _do it properly,_ Rebecca translated.
They stayed well clear of the harbor bridge for fear that someone on the bridge or in one of the high-rise towers in the central city might notice the bright underwater lights of the _Möbius,_ and they kept pretty much to the center of the harbor to keep away from any anchor chains of boats moored near the shores.
It was black underwater at night, darker than Tane had expected. The lights penetrated a certain distance into the water, turning the black into a dirty, murky green, and there were surprisingly few fish. Tane had expected to see schools of them, but there were only a few, here and there, caught in the lights of the submarine.
He was about to ask Doddie about it when a dead fish floated into the lights of the _Möbius,_ belly-up, bloated, its head stuck inside one of the plastic rings of a beer six-pack holder. Tane held his breath and watched it drift slowly, diagonally, across the bow and into the darkness beyond the lights. Rebecca and Fatboy didn't see it. They were busy playing with the periscope camera.
It was only one fish, but it took some of the fun out of the test dive for Tane.
Arthur left the next morning for Sydney, so they spent two full days, or rather nights, training with Wee Doddie and one day out amongst the islands of the Hauraki Gulf, well away from the eyes of any spectators on the myriad of yachts that painted the surface of the gulf in long loving brushstrokes.
When Arthur returned on Thursday, there was a small ceremony that included the handing over of the keys to the boat (Doddie had been insistent that it was a "boot" not a "shape") and the signing of all sorts of papers that basically said that if they drowned themselves, then it was their own silly fault and not the silly fault of Subeo UK Ltd., its shareholders, directors, or subsidiaries.
"Are we old enough to sign these?" Tane whispered to Fatboy, hesitating over the dotted line.
Fatboy shushed him. "If it makes Fong feel better..."
Rebecca impulsively hugged Wee Doddie when the taxi arrived to take him and Arthur to the airport.
She said, "Spooner greet cup laddies. Text February thing!"
_It's been a great couple of days. Thanks for everything!_
Tane and Fatboy looked at each other with raised eyebrows, but Wee Doddie gave her a big mad smile and tousled her hair affectionately.
"Suez can spike English off troll," he said.
**T HE MÖBIUS TRIP**
**The next day, they left** before dawn to be clear of the crowded harbor before daylight. This time of the year, the harbor and the inner gulf were awash with boats of all kinds enjoying the summer sun. Sailing boats, for yachting was a fire in the blood of many New Zealand people; motor boats, because some people just like to go fast; and a multitude of smaller craft, from Windsurfers to Jet Skis to kayaks.
There were families and fishermen and foreign freighters. Tane hoped they wouldn't run into any scuba divers, and luckily they didn't.
The plan was simple. Too simple, Tane thought, but they just didn't have any more information to make more detailed plans with. It would take them two days to cruise up to the Bay of Islands in the _Möbius._ That second night, they would sneak onto the island, break into the lab using the passcode, and look around for...for what?
He was glad that Rebecca was going to be there. He'd have no idea what to look for and wouldn't even know it if he found it. But no doubt she would make sense of what was in the filing cabinets, or the petri dishes, or whatever it was that they found.
And how to stop the Chimera Project. That bothered him the most. But he guessed that would depend on what they found.
They stocked the compact refrigerator of the _Möbius_ with five days' supply of perishables and packed the small larder with bread and tins.
There was no bath or shower on board, for space reasons, but Tane cycled down to the local boat shop and picked up a few bars of sailor soap, the kind that lathers in salt water.
"We can take a bath every day," he said. "Just open the hatch."
"Yes, Mother," Fatboy said with a grin.
Because there were only three of them, there was plenty of room on the three spare bunks for extra clothes, towels, and assorted gear that you would usually take on a camping trip. The three black wet suits, masks, fins, and assorted other gear took up one of the bunks.
Tane brought a pack of cards, and Fatboy even squeezed his guitar on board. Rebecca brought her computer, which by now had a wireless connection that allowed her to access the Internet from anywhere. Even, they discovered, using the aerial on the buoy, from underneath the surface of the ocean.
Exams were over, and whatever the results might be, there was nothing Tane could do about that until the next year. They were now on vacation, and when the weather shone out clear, blue, and calm on the first morning, it seemed that all was right with the world.
Tane's mum and dad had been given a story about a camping trip, which they had no reason to disbelieve, and Rebecca's mum came down to the kitchen the night before they left while they were packing up the last few odds and ends.
Tane could never quite understand, although he would never say this to Rebecca, how normal her mother looked. How _sane._
She wore jeans and a casual T-shirt, just like any other mum, around the house. Her hair was always brushed in a neat bob, and she always wore makeup. Not over the top, just a little, like his mum wore to go to the shopping mall.
She came down just like any other mother and wished them well on their boat trip.
"Thanks, Mum," Rebecca said. "I've made you a casserole. Enough for two or three days. It's in the fridge, and there's leftover pizza in the freezer if you get hungry."
"Oh, you didn't need to fuss around like that," her mum said. "I'm quite capable of looking after myself."
Tane caught a small glance from Rebecca and knew that being capable of looking after yourself and actually doing it were two different things.
Rebecca gave her mum a kiss and said, "See you in a couple of days, Mum. Love you."
"Love you too," her mum said absently. "Take care, now." Then she disappeared back into her room and they heard the theme tune start from _Survivor: New York._
The _Möbius_ made better speed underwater, but the air quickly got dry and tinny, so when they were well clear of any other boats, and far enough away from land not to be seen, they surfaced and opened the twin hatches to let the fresh cool sea air wash through the boat.
It was a sunny day, and there was a small flat area in between the two glass domes of the cockpit where you could just about lie down comfortably. Rebecca changed into her bikini and did some sunbathing, while Fatboy steered and Tane lounged half in and half out of the hatch, playing a happy bluesy tune on his harmonica.
The _Möbius_ might have been cool, and she was very discreet, but she wasn't fast, and it seemed to take an eternity even to get out to the tip of the Whangaparaoa Peninsula, at the northern end of the Hauraki Gulf.
"Don't spend too long out there; you'll burn," Tane said. He had put sunblock on his face and arms half an hour ago and had put on a hat. Rebecca was still soaking up the sunshine, swaying gently with the movement of the boat in the water and squealing occasionally when a rough wave splashed spray over the front fins.
"Yes, Mother," Rebecca laughed, and it would have been funny, except that Fatboy had said the same thing the day before.
"Seriously. They reckon the ozone hole is extra large this year. You'll burn fast."
She craned her neck up past the right-hand dome and squinted at him. "I know, you're right," she said. "But it's luvverly! Nice to know you care, though."
Tane shrugged. The sun was nice and warm on his skin; that was true. And the occasional splash of seawater provided a pleasant, refreshing spray.
He felt, rather than saw, Rebecca looking at him out of the corner of her eye and knew she had something on her mind. He shut his eyes and let the sun warm his eyelids while he waited for her to come out with it.
Rebecca said after a while, "Why do you care? Why are we such good friends? Why do you like me? Some of the kids at school won't even talk to me."
Tane thought about it for a while and played a few more bars before answering. "I guess they haven't known you as long as I have."
"What's that got to do with anything?" she asked, then said, "Chuck us the sunblock."
Tane carefully slid the plastic bottle down the rounded yellow side of the _Möbius_ into her outstretched hand. He said, "Think of it like this. Other people see you and they only see you as you are right at that moment. And they make up their minds about you based on that one tiny instant in time."
"Yeah?"
"But I've known you all my life. I look at you and I see the little girl who wet her pants in the sandbox at kindy and blamed Mary Mackey. I see you at your tenth birthday and at your dad's funeral. I look at you and I see all the different yous, and..." He paused. "I wonder about the ones still to come."
Rebecca laughed, a slightly embarrassed sound. "Jeez, you talk a load of crap. I'm not sure if you're being nice or being rude!"
"So why do you like me?" Tane asked.
Rebecca said immediately, "What on earth makes you think I like you?"
She kept a straight face long enough to make Tane think she was serious, before dissolving into another fit of laughter.
Tane thought that, despite what she said, she was secretly quite pleased with his answer. It felt good. Like the old days, before Fatboy stuck his nose in.
"Of course, some days," he said, "you can be a real pain in the ass."
"What do you mean..." She looked around and saw his smile and threw the bottle of sunblock at him. It bounced off the hatch and disappeared into the ocean.
"Oops," she said.
They stopped for a while in the evening, and Rebecca, who was the most experienced cook, made a quick fry-up on the gimbal-mounted electric stove. Electric, not gas, Arthur had pointed out, because it was easier to replace electricity than oxygen.
They ate on board and continued for another hour after dinner into a sheltered cove on the lee side of Little Barrier Island. There was no point in continuing farther, as the light was fading, and running with the lights on used up the batteries twice as fast.
"It's deserted," Tane said, scanning the cove with binoculars. "Let's park the sub up on the beach for a while and go for a swim."
"Hauturu." Fatboy shook his head, giving the island its Maori name. "We're not allowed to land there. It's a wildlife sanctuary."
"We're not allowed to break into science labs either," Tane pointed out, "but that's not stopping us. And what do the wild lives need sanctuary from anyway?"
Rebecca smiled but said softly, "Us."
They did go for a swim eventually, after Fatboy jury-rigged an anchor by tying a nylon rope to the bow and diving down to secure it around a heavy rock in shallow water near the northern tip of the small cove. It seemed strange that the _Möbius_ did not have an anchor of her own, but maybe, as Rebecca surmised, nobody ever thought she would need one.
Once the boat was secure against the gradual draw of the tide, and with the sun a dusky memory on the horizon, they fooled around for a while, dive-bombing each other off the end of the boat, dunking each other, and generally acting like a bunch of lunatics until the light faded a little too far, and the water darkened a little too much, and they climbed back on board for the night.
They sealed the hatches, released the buoy, and with a whisper of bubbles, slowly sank to the bottom of the cove, landing gently in soft sand between a small underwater ridge and a few scattered boulders.
And there, in the tranquility of Little Barrier Island, they slept, serenaded only by the gentle sounds of the sea and the hum of the air hose from the buoy.
**B UTT MOP**
**Rebecca downloaded the next batch** of BATSE data the next afternoon while Tane had his turn steering the sub. A few boats had been cruising the area, and the fine weather of the previous day had disappeared, replaced by squally showers, so they had taken the precaution of submerging and continuing the trip in the relative calm of the ocean depths.
"More of the same," Rebecca said, peering at the characters on the screen of her laptop. "Just more of the same."
Since the "Water Works" and the as-yet-indecipherable "Butt Mop," there had been nothing but numbers. Always a series of numbers, separated by commas, then a full stop, then another series. Not Lotto numbers; they didn't fit the pattern. Something else. Something with a _lot_ of numbers!
Every day it seemed there were more transmissions captured by the satellite and uploaded to the BATSE Web site. They had set both Tane's computer and Rebecca's shiny new laptop running Rebecca's program, day and night, and were working through the backlog of transmissions that had been coming in ever since they had visited Dr. Barnes at the university.
Rebecca had tried every combination and calculation she could think of to make some sense out of the numbers, but the answer still eluded her. She was convinced, and had said so many times, that the solution would be something not logical. Something lateral. Something that Tane's creative imagination would be needed to solve.
"Come on, Tane," she said. "We need you to think outside the box."
Tane stared out at the monotonous sameness of the ocean. It was hard to think creatively when you had a splitting headache, and he had one now. He hadn't slept much the previous night but had lain awake worried about breaking into the lab. That was against the law. It was a criminal act. He had never broken the law before (unless you counted that packet of gum he had "borrowed" from the corner store when he was seven). He had rolled over and over in his tiny bunk, and now his head throbbed with the pulse of the engines. _Think outside the box._
He wasn't the only one feeling the stress of the mission. He could see it on the faces of the others, especially Rebecca. She could not afford to go to jail. Could any of them? No. But especially not Rebecca. Yet it had to be done. What were the consequences if they did it? What would be the consequences if they didn't?
In the clear open water here, there was nothing much to look at. Just the occasional school of fish or curious shark. Ahead at Poor Knights Islands there was a world-famous diving spot, renowned for its clear waters, colorful marine life, and the wreck of the _Rainbow Warrior._ But that was a bit off course for them, so he had to be content with the blue-green infinity of the open water.
_Think outside the box._ It was a well-worn phrase from an old puzzle that he had once seen. Nine dots that formed a three-by-three grid. Join all the dots using just four lines. It seemed impossible to most logical people, but creative thinkers quickly realized that it was easy, if you allowed your four lines to extend beyond the confines of the grid. Outside the box.
He sketched the puzzle on a notepad, then completed the lines to make the answer.
Something struck him about that phrase. _Think outside the box._ He stopped trying to solve it and let his mind drift idly, like the ocean around them. The dead fish in the beer rings came to mind, but he shook that away quickly. Quick flashes of many things, superimposed on each other—Fatboy's hand on Rebecca's shoulder, the nervous lawyer absentmindedly pulling at his mustache, the fantail family naively fearless in their flimsy nest, the chessboard he had bought Rebecca for Christmas. And as it usually did, the answer just floated into his mind and was there for quite a while before he realized it.
The chessboard. _Think outside the box._ Black and white chess pieces, black and white boxes on the chessboard. Boxes. _Think outside the box._ The chessboard itself was a box, made up of smaller black and white boxes. Think _inside_ the box!
"The chessboard," he finally said out loud.
Rebecca was in the main cabin, sitting on one of the bunks, working on her computer. She looked through the pressure door at him. "What chessboard?"
"Any chessboard," Tane said quickly.
Fatboy had been lying on one of the other bunks, but now he walked over and sat in the codriver's seat. "Go on," he said purposefully. He was wearing his cowboy hat, which seemed a little silly to Tane.
"A chessboard is made up of eight boxes by eight boxes, right? Some black, some white."
"Yes."
Rebecca was starting to get it, Tane thought.
"Suppose you had a chessboard that was made up of one thousand boxes by one thousand boxes. That would be one thousand squared." He picked up the notepad and wrote it: 10002. "In the first row, instead of alternating black and white, suppose the first, say, eighty were white, and the next—what was it..." He checked the printout.
**BTMP1000:2.80,24,341,55,500.80,24,342,54,499,1.80,24**
"...twenty-four were black. And so on and so on for all the rows. What would you have?"
"A bloody great, stupid-looking chessboard," said Fatboy, who was cool and popular but wasn't really all that bright.
"A photograph," said Rebecca, who was.
"Or a fax," said Tane who, just at that moment, thought he was the brightest of them all. "A fax is just a lot of rows of black and white dots, arranged in a particular pattern to form a picture."
"Brilliant!" Rebecca cried. She put down her computer and rushed into the control room. She threw her arms around Fatboy's neck and gave him a huge hug from behind.
_Hey,_ thought Tane, _it was me who solved it!_
"What should we do?" asked Fatboy. "Get a big piece of paper and start to draw it up?"
Rebecca shook her head. "It'd be easier to do on the computer. I'll do it in Photoshop. I'll just create an image one thousand dots by one thousand dots and save it as a..." She stopped, and then strangely, and rather tiredly, began to laugh again.
"What?" asked Tane, a little defensively, thinking she was laughing at him.
"Save it as a _bitmap._ "
It took Rebecca nearly two hours to take the data they already had and convert it into dots on the bitmap. All the weeks of data they had received and it made up less than a third of the image. It was clear enough from that what the image was going to be, though.
"It's a diagram," Rebecca announced from the main cabin, studying the image on the screen of her laptop. "A schematic."
"Of what?" Fatboy asked.
"Of what the hell do you think?" Tane answered a little testily, his head still throbbing.
Rebecca said, "Easy, Tane. Of the gamma-ray time-message-sender device, Fats. I have a feeling that we are going to need this, sooner than we thought."
"How soon?" Fatboy asked, but nobody answered.
Rebecca came back into the control room and, there being no spare seat, sat on Fatboy's knee. Tane stared out of the viewing window of the sub. He said, "I think I have a name for it."
Rebecca looked up. "Yes?"
"Well, it's like a telephone that sends text messages through time, right?"
"Well, sort of."
The name had floated into Tane's mind in between throbs of pain. _Tele_ was ancient Greek for "distance," and _phone_ was ancient Greek for "speech." So a "telephone" was a machine that let you speak across a distance. The ancient Greek word for "time" was _chronos._
"It's a Chronophone."
Rebecca said, "I like it."
"Wouldn't it be a Chronograph?" Fatboy asked. "I mean, it's more like a telegraph, sending Morse code, than a telephone."
"Maybe, but a Chronograph is a kind of watch, so I think Chronophone is best," Tane insisted.
"Actually, Fatboy is right," Rebecca said.
"Okay, whatever." Tane shrugged and turned away.
There was an awkward pause. Fatboy coughed.
After a moment, Rebecca said brightly, "Chronophone will do fine. But what a paradox."
Tane groaned, "Oh, here we go. I suppose you want me to kill Grandad again."
"You killed Grandad?!" Fatboy asked.
"No, Fatboy, don't worry about it. What paradox?"
"Think about it." Rebecca's eyes were wide. "We're in the middle of sending ourselves plans for the Chrono...phone, right, from the future."
"This much we know already."
"But where did we get the plans from?"
"Which 'we' are you talking about?" Tane asked.
"Okay," Rebecca said. "Call the future Tane and Rebecca 'them.' Now, where did they get the plans from? From us, right?"
"From us?" Fatboy queried. "How did we send the plans to the future?"
Rebecca sighed with exasperation. "We didn't send them. We just hung on to them. Think about it. Tane and I are the future Tane and Rebecca, just not yet. So future Tane and Rebecca get the plans from us, but where did we get them from? _From them!_ "
"So who had the plans in the first place?" Fatboy asked.
"Exactly!" Rebecca thundered, leaving Fatboy no more the wiser.
"And this has something to do with Grandad?" Fatboy asked, and couldn't understand why Tane and Rebecca got the giggles.
"The plans must have come from somewhere," Fatboy insisted.
"Maybe this is one of those kinds of questions that our brains just can't comprehend," Rebecca said, wiping laughter tears from her eye. "Like the infinite size of space. Or what existed before the universe."
"Or why in automatic cars you pull the gear lever backward to go forward, and forward to go backward," Tane contributed, and even Fatboy started laughing, in a confused kind of way.
Eventually Fatboy said, "Okay. So I don't really get all this paradox stuff, and I don't understand where the plans for the Chronophone came from, but did you guys ever think about what a weird coincidence this whole thing is?"
Rebecca, still sitting on his knee, twisted her head around to look at him. "Yeah, I know."
She idly lifted off his cowboy hat and put it on.
Tane asked, "What do you mean?
Fatboy hesitated. "Well..."
Rebecca said, "He means that us, well, you really, Tane, thinking up the idea of receiving messages from the future, just at the right time to stop Dr. Green and her Chimera Project, is a very unlikely coincidence. Right, Fats?"
Fatboy nodded. "Did anything else unusual happen, when you thought up the idea?"
Tane asked, "Like what?"
"Like, did you see anything, hear anything..."
Tane shut his eyes, remembering. "There was a shooting star."
Rebecca looked up sharply. "What?"
"The moment that I thought of the idea of messages from the future, I saw a shooting star. That's it. Nothing else."
Fatboy shook his head. "Maybe it wasn't an ordinary shooting star. Maybe the shooting star was actually a thought from the future hurtling through the atmosphere in some cosmic ray from the depths of the universe just before embedding itself in your brain."
Tane said, "That's nuts."
Rebecca was already adding it to the notebook where she kept the message dates and times. She spoke as she wrote. "Please remember to send a shooting star above West Auckland on the twenty-seventh of October at exactly nine-fifty-three p.m. That was the right time, wasn't it?"
"Close enough."
Fatboy grinned. "So maybe the whole thing wasn't Tane's idea at all."
"Bite me," Tane laughed. "I thought of it by myself."
Fatboy said, "Sure, with a little help from the future and their intergalactic brainwashing machine!"
He put his arms around Rebecca's waist and she leaned back into him.
Tane gritted his teeth. "I thought of the basic idea," he said.
His annoyance must have shown because Rebecca said, "He was only joking, Tane."
Tane snapped back, "Well, I'm sick of it."
He regretted saying it immediately. The confines of a submarine were no place for bickering and fighting, but the words were in the open now and there was nothing he could do about that.
"Easy, Tane," Fatboy said calmly.
Once started, it was difficult to stop, and his head was pounding.
"Easy! You blackmail us out of two million dollars, worm your way into our project, and start barking orders around like you're running the show. And you tell me to take it easy."
"Hey, Tane," Rebecca said gently.
"No, stuff it!" Tane shouted. "And stuff you! None of this, the submarine, the money, none of this would be here if not for me, and you just weasel your way into everything and take it all!" Even Tane knew that it wasn't the money that he was talking about.
"Chill out," Fatboy said, his jaw set.
"I'm sick of it," Tane cried, his voice suddenly hoarse. "I'm sick of it and I'm sick of you."
"Enough!" Rebecca said suddenly and firmly. "That's enough, Tane." She stood and took Fatboy firmly by the hand and led him back into the sleeping cabin. She pulled the watertight door shut between the two compartments, pausing only to say, "We'll keep out of your way until you've calmed down."
The clang of the metal door and the whir of the wheel-lock turning were like a knife in Tane's chest. For some reason, he had thought she would agree with him. That she would tell Fatboy to back off and remember that he was just a one-third partner and there only by the grace of Tane and Rebecca. Friends since forever.
But it hadn't turned out like that at all.
"Crap!" he yelled suddenly at the top of his voice into the void of the water all around. "Crap!"
Then, after a quick consultation with the map, he changed course, just slightly.
It was a childish display of emotion. An ill-considered, impulsive thing to do.
And it probably saved their lives.
**T HE RAINBOW WARRIOR**
**"Holy mother...," Fatboy said,** his body still halfway through the doorway from the bunk compartment. "Where the heck are we?"
Tane stirred uncomfortably. He had slept in the driver's seat, and his back and neck now ached. He stretched, trying to work out the knots.
"Sorry about yesterday," he said.
"We all have days like that." Fatboy flashed a quick smile. "Don't worry about it."
Tane smiled. Fatboy picked his guitar out of a narrow locker, sat on one of the fold-down seats, and began to play. _Music to soothe the savage beast,_ thought Tane.
Rebecca emerged silently from the compartment behind Fatboy and said nothing.
Fatboy asked, "Was it something to do with Rebecca? Do you not like that we—"
Tane interrupted, to avoid more pain, "I was just tired and grumpy, I'm okay now."
He avoided Rebecca's eyes.
"Where the heck are we?" she said at last, startling Fatboy, who dropped a chord but picked up the tune and continued.
Through the large twin glass domes of the _Möbius,_ the rusted bow of a ship reached plaintively out toward them. The water was clear and still, and the early light of the morning washed the bow in cobalt hues. The seabed around the ship merged with the blue of the sea behind, in a soft vignette. The railings of the bow were intact but encrusted with green and brown marine life. In death, the ship was the foundation of life.
"The _Rainbow Warrior,_ " Tane said.
"Wow," Fatboy said, "it's amazing."
Beyond the prong of the bow, the rest of the ship lay on its side on the sandy bottom of the ocean, an artificial reef. The back of the ship was broken, and she lay twisted in her final resting place.
Tane wasn't quite sure why he'd diverted the sub. It just seemed like the right thing to do. For Rebecca.
"We are a long way off course," Rebecca said curtly.
"No," Tane said, "only a couple of hours. I...just...felt like a diversion."
"It'll cost us a day."
"But it's only a couple of hours out of our way," Tane protested.
"It means we won't make Motukiekie today. We'll have to stop over for an extra night, and do"—she hesitated—"the thing, tomorrow night instead. It's cost us a day."
"It's just one day," Tane mumbled. "I'm sorry."
"Tane, something bad is going to happen. We don't know what and we don't know when. One day extra might mean one day too late. We already talked about this."
_Crap!_ Tane thought. _Crap!_
Rebecca's eyes were frosty, but she shut them for a while, and when they finally opened, they were clear.
"What a tragedy," she murmured, watching a school of colorful fish flit through the encrusted handrail on the point of the bow.
"The guy who died?" Fatboy asked.
Fernando Pereira, a Dutch photographer, had drowned when French secret agents bombed the _Rainbow Warrior_ in Auckland harbor.
"That too," she said softly.
They spent the third night at sea on the wave-washed bottom of a bay just south of Cape Brett, which they had flown over a few weeks earlier in the Grumman Super Widgeon with its underage-looking pilot.
The island was closer now, and in Tane's mind it loomed large and frightening. Breaking and entering was against the law. And what if there was some virus loose on the island? The uncertainty and fear grew as a storm blew up overnight, and even on the seafloor, the ferocity of the crashing waves above thudded against the little yellow boat, seesawing them from side to side and shunting them small amounts across the sand of the bay.
By ten-thirty, when they were just bunking down for the night, the nerves and the uncertainty became overwhelming. _I can't do this,_ Tane realized. _I can't go onto an island in the middle of the night and break into a science laboratory that might be infected with a superplague._
He would be letting them down. He knew that. But what choice was there? _I can't go through with this,_ he thought again as another violent wave rocked the sub from side to side. He took a deep breath and held it for a moment before opening his mouth to speak. They'd think him a coward, but he had to let the others know.
"I'm really not sure if I can go through with this," Fatboy said while Tane's mouth was still open like a fish. He shut it with a snap.
"What do you mean?" Rebecca asked, emotion rising in her voice.
"I've done a few dodgy things in my life," Fatboy said, "but I've never broken the law. Well, not a big law. Not the kind of law they put you in jail for. Maybe we should just wait and try to talk to Vicky again."
Rebecca said in a small but determined voice, "That won't help. And if we don't do this, who else will?"
"I'm just not sure if I can go through with it," Fatboy said again quietly.
"It's gotta be done," Tane said firmly, as if he had never considered any alternative. "And we've got to do it."
There was a long pause.
"I know," Fatboy said at last, reluctantly.
The next day they rounded the tip of Cape Brett, passing Motukokako Island and its famous Hole in the Rock, then into the Bay of Islands itself.
"Waewaetoroa Passage," Fatboy said, looking at the chart. "That's Waewaetoroa Island on the right and Urupukapuka Island on the left. We could go all the way around, but it would be quicker to go through the passage between the islands."
"Any reason why not?" Rebecca asked.
"Not really." Fatboy frowned. "There are a lot of big rocks and shoals, and it gets quite shallow at the other end, but boats navigate through there all the time."
"On the surface of the water," Tane noted.
"Yeah, but I think we'll be okay."
Fatboy drove the sub, as he had proved the most adept at handling its idiosyncrasies, and in the confined space and currents of the passage, that might prove to be vital.
The mouth of the passage slipped past them before they realized it. Just two gradual upward slopes on the seabed that slowly resolved into the underwater sides of the two islands.
Beneath the surface of the water, the Waewaetoroa passage was far from a straight and clear path between the islands.
Rocks jutted out at odd angles from the sides of the islands, which were at times shoals and other times sheer vertical cliffs. Huge boulders on the seabed rose up to meet them, well beneath the keels of the pleasure boats that frequented the passage but big enough to stop larger ships from traveling this way. And certainly big enough to keep Fatboy on his toes.
He had to slam the boat into reverse at one point as they rounded a curve on the side of one of the islands and were confronted by an unexpected underwater ridge, almost a reef, teeming with fish of many kinds and colors.
Rebecca sat in the codriver's seat and pored over the chart holder, marking rocks and sketching in ridges and rises with a pencil.
Huge fronds of seaweed reached out toward them from the sides of the islands and from many of the scattered boulders. At times the weed blocked visibility, and at other times it threatened to wrap itself around the main propeller and draw them into a watery grave.
It was a scary, underwater obstacle course.
"I'm glad we're taking this slowly," Fatboy said at one point.
"And in daylight," Tane added.
Finally the massive shapes to each side began to fall away, and deeper water beckoned. They continued on for a while as Tane raised the buoy and opened the iris on the video camera.
A fuzzy green shape bounced around on the screen as the light chop above rocked the buoy about.
"That's Motukiekie," Tane breathed.
Motukiekie Island. Their second visit, but a vastly different trip from the first one in every possible sense.
Motukiekie. Professor Green. The Chimera Project. It was suddenly very real. Far too real. The dangers that lay ahead of them were exposed on the video screen in front of them, a fuzzy green shape on a blue ocean.
Rebecca heard it first. "What's that?" she said.
It started off as a kind of far-off rumble, but quickly became a throbbing, whooshing sound that reverberated through the hull of the sub.
"I don't know," murmured Tane, twisting the joystick to spin the small camera around on the flat gray buoy above.
The answer was suddenly there on the screen before them.
"Dive, dive, dive!" Tane yelled. Fatboy had already rammed the controls forward and hit the manual override for the trim tanks, flooding them all at once to make the _Möbius_ sink like a stone.
"What is that?" screamed Rebecca.
Tane grabbed at the switch for the winch that lowered the buoy, but his fingers slipped and it took him two goes to get the motor winding and the buoy lowering.
The bow of a Navy frigate looks large from any angle, but bearing down on you in the water, it looks like the end of the world.
"I hope it's deep enough," Fatboy muttered as the hull hit the sand and the ocean floor with a thud that jarred the vehicle.
The rumble of the engines in the ship grew louder as it closed in on them, and they could hear each individual rotation of the huge propellers.
The sound passed right overhead, shaking the hull and its terrified occupants, but then the ship, and the stark terror, passed them by.
"What the hell is a frigate doing here?" Tane wanted to know, raising the buoy again once he was convinced there were no more surprises like the first one.
The ship had slowed once it had passed them and was rapidly disappearing around the end of the island.
"What is it?" Rebecca asked. "One of ours?"
Tane nodded. The New Zealand flag was clearly visible on the short pole at the stern of the ship. "Must be the _Te Mana_ or the _Te Kaha._ "
"Probably on exercises," Fatboy said.
"Will that stop us?" Rebecca asked.
Tane and Fatboy looked at each other.
"I don't see why," Tane said after some thought. "They won't be interested in Motukiekie. Just cruising past, I think."
Even so, they stayed on the ocean floor for more than an hour, to make sure the frigate would not return, before resuming their journey to the island.
When the last rays of the sun were a long-ago memory, they took the wet suits off the top bunk.
"Who's coming with me?" Rebecca asked, pulling the wet-suit trousers up over her bikini.
"I am," Tane said.
Fatboy said, "I can go if you like, Tane. I'd be happy to do it."
"No," Tane said firmly, "I started this, and I'm going to help finish it."
He put on his weighted belt, heavy enough to compensate for the buoyancy of the wet suit.
Fatboy helped Rebecca with hers, then handed them the oxygen bottles. Not full scuba gear, just small pressurized metal bottles, about the size of a small Coke bottle. There were no straps; they simply gripped the mouthpieces tightly with their teeth. Waterproof flashlights dangled from their wrists on flexible cords.
It was three a.m. when Tane and Rebecca slipped out through the open top hatch of the _Möbius_ and slowly swam toward the island.
**L AUNDRY PILES**
**They came ashore on the** beach at the base of the wharf. Moonlight silvered the waves in the channel, but here in the shadow of the island, it was dark. A large boat was moored back along the wharf, but there were no lights and it looked deserted, so they decided to ignore it and carry on. If anyone was on board, they were well asleep at this time of the morning.
They hid the fins, weight belts, and oxygen bottles behind a pillar beneath the end of the wharf but left on their masks to hide their faces from the security cameras.
Rebecca started off down the wharf but stopped after just a few yards and touched her hand silently to Tane's arm to stop him also.
She squatted and cupped her hand around the end of her flashlight so it couldn't be seen from a distance. Before turning it on, she instinctively also put her back to the island, to shield the light from any watching eyes. In the dim light, Tane could see something on the ground.
Tane squatted with her and examined the find. It was a cloth of some kind. Made of a soft pink material. He picked up one end of it. It immediately resolved into a distinct shape. It was a woman's nightie. A smaller, separate shape slipped out of the end as he raised it, and Rebecca caught it with the light. It was a sensible, white pair of ladies' panties.
"Seems a strange place to leave your laundry." Rebecca's voice, low and soft in his ear.
Tane nodded his head, unsure of what to say. It was a disturbing thing to find.
It took them a while to find the entrance to the path, even though they had been here once before. Tane risked a quick flash of his flashlight.
"Over there," he whispered, and followed Rebecca as she trod lightly and carefully up the concrete pathway.
He looked back at the water just once, before it was obscured by the trees surrounding the track. It was dark and deep and peaceful. It had been that way for millions of years and would still be the same in another million. Just looking at the expanse of water gave him a steadiness and a resolve to complete this mission, no matter what.
He caught up to Rebecca and whispered in her ear, "Kids will read about this in history books one day."
He felt her smile rather than saw it in the darkness.
After only a few paces, Rebecca stopped again. Another pile of clothing, this one a small mound. She separated it with her flashlight. A white lab coat. Inside of that a bright red T-shirt and a pair of jeans. Inside the jeans, which were still done up, a pair of blue boxer shorts. Underneath the clothes was a pair of Nike's, complete with socks. A watch fell out of the sleeve of the lab coat when she moved it.
"This might explain the nightie," Tane said with a grin. "Maybe a couple of scientists out skinny-dipping for a bit of a lark."
Rebecca nodded, as if she agreed, but said, "In other circumstances maybe. But not here. Not now. And why are the jeans still fastened?"
The pathway leveled out as they approached the top of the rise and emerged from the native bush into the cleared area that housed the laboratory complex. Here they were no longer in the shadow of the island, and the flat area in front of them was bathed in moonlight.
They found no more piles of "laundry" as they silently approached the building complex.
They had both memorized the security code for the gate, but the moonlight brushed it with silver, and Tane could see that there was no need. It was not locked. It was not even shut.
He looked at Rebecca, and the moonlight shining into her diving mask showed a furrowed brow. Why was the gate unlocked?
They moved through the gate and crept forward along the short concrete path that led to the lab.
Back at home, and on the _Möbius,_ the plan had seemed simple. Open the gate with the keycode. Open the door to the lab the same way. Snoop around inside and just like in the movies, surely they would instantly find all the incriminating evidence they would need to shut down the project and save the world from extinction.
But no one could have planned or prepared for the reality. The door to the lab was open, swinging idly in the light breeze that blew in from the east, here on the more exposed top of the island.
Not that long ago, a red-haired professor with a big bright smile had opened this door for them and welcomed them into the lab. But nobody was here to welcome them this time. The light from inside spilled outward from the doorway, clearly illuminating the smashed doorjamb and ruined hinges.
"Something's really wrong here," Tane whispered. "That door's been smashed open. Someone's broken in before us!"
Rebecca grabbed his arm and pulled him down into a crouch beside her. Enough light spilled out from the broken doorway for him to see her eyes, stark, wide, and terrified inside the scuba mask.
"No," she said. "It's worse than that. Much worse. We've got to get off this island now!"
"What is it?" Tane asked, wondering what had her so spooked.
"Look at the door, Tane," she said. "Look at the hinges. Someone—or some _thing_ —has smashed that door open from the _inside!_ "
All the breath in his body suddenly froze. _What the heck..._
He rose slowly and took a step backward. Rebecca's hand slipped into his. Another step, then another. Walking backward, unable to take their eyes off the smashed door and the carelessly spilling light.
They were just back past the gate when the bright spotlights crashed on all around them, twenty or more, brilliant white lights that hurt his eyes, which were still adjusted to the darkness. He spun around and instantly recoiled, shielding his face with his arm, desperately frightened, unsure whether to turn and run or to stand still.
Rebecca screamed and clutched at him. In that single instant, he felt like a possum on a highway at night, transfixed by the headlights of an onrushing car, knowing that this was certain death but paralyzed beyond saving himself.
A voice—deep, authoritative, American—shouted at them from behind the screen of lights, "Drop your weapons. I say again, drop your weapons and lie on the ground, facedown, toward my voice. Drop your weapons, or we will open fire."
_Weapons?!_ Tane tried to scream, "It's just a flashlight!" but his throat wasn't functioning.
"We are unarmed!" Rebecca shouted. "We are unarmed."
Tane found his voice and joined in. "We are unarmed," he called, lying facedown as he did so. "We're just kids!"
He saw only dark dirt as strong boots thundered around him and strong arms twisted his arms up behind his back before some tight plastic wire was pulled harshly around his wrists, and he was hauled to his feet.
His mask was wrenched down and left hanging around his neck. One of the bright lights was shone in his face. He winced from the pain of the glare. Even with his eyes tightly shut, it was bright enough to hurt. It felt like a dream, like he was in some strange bright fantasyland where nothing made any sense and nothing was expected to.
"Where are the others?" the same voice shouted. "Where are the rest of you?"
If he could have, he would have helped them willingly. "On the submarine," he would have said. But his throat had closed up again and he couldn't speak.
Beside him he heard Rebecca's voice. "There are no others! We are just kids!"
The man's voice again, talking, not shouting this time. "Crawford, this is Crowe. Any more warm spots showing on the scope?"
As he spoke, the soft chop of a helicopter—a very quiet helicopter—slid smoothly through the dark sky overhead.
Then the man was back a few inches from Tane's face. It was strange how muffled his voice was, in the midst of all the fear, darkness, bright lights, and confusion. As if the man were talking from behind a glass window. Or a mask of some kind.
"Where are the others? Where are the hostages?"
_What others? What hostages?_
When there was no answer from either of them, the man's voice softened a little. "Get them back to the boats. We'll interrogate them properly on board the ship. Crawford, remain in position. I want to know if a mouse farts on this rock. Red and Blue Teams exfil now."
Strong hands gripped their arms, at least two fully grown men to each child, wrenching their arms up behind their backs so high that it brought tears to Tane's eyes. He heard Rebecca cry out in pain, and a fierce anger flared inside him. How dare they do that to a girl. To Rebecca. But there was nothing he could do.
The frigate was the _Te Mana._ Her name was stenciled in huge letters on her side, near the stern. She looked very different from the side than she had from beneath the bows. They boarded her via thick rope netting hung down the side of the ship, a quick clip of some cutters behind them freeing their hands for the climb.
Compared to the darkness of the island, the deck of the frigate was a city of light. Men and women in naval uniforms kept their distance from the black-clad soldiers who had arrested them.
"Take them below?" one of the soldiers wanted to know.
"Not till we're a hundred percent that they're clean." The soldiers all looked the same, but the voice was the first one they had heard on the island, obviously the leader of the team. "What does the RPAD say?"
A large man was scanning them with a handheld device connected to something on his back. All of the men were armed with an unusual, rounded-looking rifle, with some sort of sprayer attached to the front of it. They wore black, armored space suits, with oxygen masks covering their faces and a small flat TV screen pulled back over their heads, some kind of night-vision system, Tane thought.
"Nothing," the man said. "No pathogens."
There was a hiss and a click and the man with a red "1" on his shoulders pulled his mask away from his face. He was thin, with a face that looked to have been chiseled out of granite. Even when he spoke, there was no expression, no movement on his face except for his mouth.
"Where are the rest of your people? And where are the hostages?"
Tane looked at Rebecca. None of it made any sense. No sense at all. What people? What hostages? How had the soldiers known they were coming? Rebecca's jaw was clenching and unclenching, and veins stood out in her neck. _As long as we don't get caught._ That was what they had said from the very beginning. _As long as we don't get caught. I can't afford to go to jail._
The man continued, "Where are you holding the scientists? Have they been harmed in any way? And what about the fog? How did you create that?" He glared at Tane for a moment, then switched his gaze to Rebecca. "Cooperate with us now, or things are going to get real ugly, real fast."
_Fog? What fog?_ It wasn't making a whole lot of sense, Tane thought. Perhaps if they told them the whole truth, they would understand. They wouldn't believe them at first, but they could show them the Lotto numbers and the other messages, and they would have to believe them, wouldn't they? And they hadn't actually committed any crime yet. They were about to, but they hadn't had the chance to go through with it.
"We got a message from the future," he said quietly.
"What?!" the leader asked.
Tane looked him in the eye and continued a little more strongly, "We found a way to detect—"
"Oh my GOD!" Rebecca screamed, staring wild-eyed at the island and holding up her hands to shield herself.
Tane jumped out of his skin and stared past the soldiers to the black shape of the island behind. He strained his eyes, desperately seeking the cause of the alarm but could not see what Rebecca was seeing.
He wasn't the only one. There was not a man on the deck of the ship who did not instinctively turn to face the island. Those with weapons already had them raised, seeking targets, in an awesome display of instant reactions.
It all happened at once—the scream, the men turning to face the island, the slap on his arm—that he hardly noticed that Rebecca was running. Running away from the soldiers, away from the island, straight toward the side of the ship.
The leader shouted, "Hey!" But Rebecca was already three or four yards away from them and moving fast. A split second later, Tane was moving, too. One of the soldiers made a grab for him, but Tane ducked out of his reach and sprinted after Rebecca. The tall man with the scanner was there, but he had to drop the device to grab at Tane and that slowed him down, just enough. There were running boot-steps behind him but Tane didn't look back.
Rebecca reached the railing at the edge of the ship and leaped up lightly, one-footed, onto it before diving down into the darkness.
Tane didn't risk anything so fancy. He just grabbed the railing with two hands and hurdled it. Then he was falling, and falling, and falling.
The deck of the frigate was a long way from the surface of the ocean, and in the dark, it seemed to take forever to reach the water.
In that fraction of an instant before he splashed deep beneath the waves, he saw what Rebecca must have already seen, or perhaps had just guessed would be there. The dim, underwater glow of the _Möbius,_ and the raised flat platform of the periscope buoy, just visible in the moonlight.
**W AEWAETOROA PASSAGE**
**The shock of landing felt** like a car crash; then dark water enveloped him.
His nose filled with salty water. It seemed to take an age to kick back to the surface, but then he shot up above the waves like a cork, the wet suit giving him buoyancy. He gasped in a huge lungful of air and coughed and spluttered the water out of his sinuses.
Then there was a loud splash nearby. Someone had followed him over the side of the ship!
He jackknifed in the water and began to swim down. It was hard. Harder than he had realized. The wet suit kept trying to drag him back up to the surface.
He opened his eyes despite the sting of the salt and saw the bright lights of the _Möbius,_ barely a few meters away, the submarine itself just a dark smudge in the water behind the glare of the lights. A couple more strokes and he was nearly at the sub. The top hatch was open, he saw, and Rebecca was already inside, silhouetted against the hatch lights.
She was waving frantically at him, and he suddenly realized that she had been down there, holding her breath, waiting for him to come so she could close the hatch.
Suddenly a strong hand closed around his ankle pulling him backward in the water. He kicked frantically but the grip was viselike. Desperately, he kicked out with his free foot and felt it connect solidly with something soft, like a person's face. The steel ring on his ankle loosened, and he kicked away down toward safety.
The hatch neared, and then Rebecca's hand reached up and pulled him closer.
He dived down into the hatch. Rebecca was already spinning the wheel to close it, but she seemed tired, listless. He swiveled around inside the narrow compartment and grabbed the wheel off her, twirling it around. The hatch snicked shut and locked, and in the same instant, Tane's hand found the flush lever.
Compressed air roared and the water level dropped, an inch, two inches, then enough for him to get his mouth and nose above water. He sucked in the tinny air and thought it tasted wonderful. Rebecca, where was Rebecca? She was just floating in the chamber and had not raised her head.
The water level continued to drop, and Tane grabbed Rebecca by the hair, wrenching her head above the water. She didn't breathe in.
"Oh crap!" Tane muttered. There was nowhere to lie her down in the narrow chamber, so he pressed her body against the rounded wall with his, the black neoprene suits clinging to each other. He pressed into her stomach with his fists and a torrent of seawater poured out of her mouth. She still wasn't breathing.
They had done this stuff over and over in first-aid classes at school, but that all seemed like a million years ago. _Finger in the mouth, check for obstructions._ He remembered that much at least. There were none. _Seal the nostrils._ Her mask, still hanging around her neck, was getting in the way, so he tore that off over her head. He gripped her nose tightly between two fingers. The rest of the training was a blank, so he just sealed his mouth on hers and blew. He counted to three— _was that in the training?_ —and blew again. The water was around his waist by then. He breathed into her lungs again, and by the time the chamber was empty, she was breathing on her own.
The only thought in Tane's mind was, _It works. That stuff they taught us. It really does work._
He spun the wheel for the side hatch and dragged Rebecca's semiconscious body out onto the floor of the main cabin. He could hear the engines of the _Möbius_ running at full speed.
Fatboy looked back through the open door from the control room. "What's wrong?" he shouted.
"She's okay," Tane yelled back.
Rebecca coughed and spluttered and her eyes began to open. He was wedged against the side of the sub. Her head was in his lap.
"Tane," she said weakly.
"Are you okay?" he asked.
She just nodded and closed her eyes again. A moment later, she said, with her eyes still shut, "I couldn't hold my breath any longer."
"You did great; you did great," Tane reassured her. She had been holding her breath _for him._
She tried to get up, and Tane put an arm underneath her shoulders to assist. He helped her lie down on one of the bunks and fastened the Velcro safety strap around her waist. He unstrapped the torch from her wrist and loosened the wet-suit jacket to help her breathe.
"I need some help up here!" Fatboy yelled from the cockpit, and Tane scurried forward.
Water rushed past the steering dome as he strapped himself into the codriver's seat.
Fatboy said, "Grab Rebecca's chart, the one she marked the rocks on. We're going back through the Waewaetoroa Passage."
"At night?" Tane asked incredulously. "At this speed?"
"Got no choice," Fatboy said, and as if to prove him right, there was suddenly a loud pinging sound against the metal of the hull.
"What the hell was that?"
"Sonar," Fatboy said. "I knew that was coming. They're chasing us. Now they know exactly where we are. They're much faster than we are, but they're big and heavy and it'll take them a while to get moving."
"We can't go through the passage at night." Tane was horrified.
"They can't go through it at all. They draw too much water. They'll have to go all the way around Okahu Island, by Whale Rock. And they won't be able to ping us through the passage—too much rock for the sound to bounce off. I'm thinking that we run the passage and try to make it to the Hole in the Rock before they round the island. We can hide from the sonar behind Motukokako Island, and they won't know which way we've gone."
The pinging sounded again and again, vibrating against the little hull.
"Get that chart ready," Fatboy said. "Here we go."
The rocky mouth of the passage loomed, and then they were between the two rock walls. It was much faster this time, and not just because Fatboy was driving the boat hard. The current must flow in this direction, Tane realized. The last time they had been fighting against it. This time it was dragging them along.
"What happened up there?" Fatboy asked, not taking his eyes off the rushing rock walls ahead of them. "I was watching the ship on the periscope when it came back into the bay, and I saw them take you on board."
"I don't know," Tane replied. "Ridge coming up on your right. Better come up a bit and steer a bit left."
The ridge, a gnash of jagged, hull-piercing teeth, growled away rapidly to their right, a school of dark fish scattering in all directions as they burst through the middle of them.
_Had they known that they were coming? If not, why were the soldiers there? Had something bad happened on the island?_ There was no time to talk or even think about that now.
The pinging reverberated off the walls of the underwater canyon, bouncing around and around in echoes of echoes of echoes, creating its own wall of sound and shielding them from the ship. In here they were invisible to the sonar, just as Fatboy had said.
"Big rock, center of the passage," Tane said calmly, and Fatboy pulled the steering back, swerving up over the seaweed-covered behemoth as it loomed in their lights. Both of them scanned the water ahead intently. Their lights did not reach far enough ahead to give them enough warning of obstacles, and if not for Rebecca's map, they would have smashed into the rock of the seabed many times already.
"Ridge to your right, no, your left, your left!" Tane called as Fatboy almost steered them into a pancaked rock formation.
"Get it right, Tane." Fatboy gritted his teeth.
A large shark, apparently dozing near the exit of the passageway, twirled its tail and spun out of their way as they shot out the end of the canyon, and the seabed began to drop away beneath them.
"Now the race is on," Fatboy murmured. "If we can't make Motukokako in time, they'll pick us up on the sonar and haul us in like a fish on a line."
The pinger behind them fell silent. They were shielded from the _Te Mana_ by the islands of Waewaetoroa and Okahu.
They said nothing, lost in their own thoughts. Tane checked on Rebecca once, but she was resting and just smiled up at him.
Only when they had passed Motutara Rock did Fatboy relax a little. "I think we've got a good enough head start," he said.
Almost immediately, the pinging started again, but this was long off and distant: just a faint echo against the hull.
Tane looked at his brother in concern, but Fatboy shook his head. "I think we're okay. The sound has to reach us and then travel all the way back to the ship like an echo. That pinging is too weak. I reckon we're out of sonar range for them at the moment."
The pinging, faint as it was, grew steadily louder as they continued on toward Motukokako and its famous Hole in the Rock.
"They're chasing us," Tane said worriedly.
"No. They don't know where we are. They're just guessing. If we can make Motukokako, we can lose them." He patted the control console. "Come on, little submarine, you can do it."
Tane didn't smile. He felt sick.
Motukokako rose suddenly out of the seabed before them. A huge rock wall in the lights of the _Möbius._ The pinging was getting louder now.
"Here we go," he said, slowing the submarine down and crawling along the face of an underwater cliff. The currents were strong and volatile, throwing the small craft from side to side, but Fatboy kept the _Möbius_ as steady as he could.
Another cliff rose on the starboard side, and Tane realized that they were in the "hole," the short passageway that cut right through the rock of the island.
When the rock walls disappeared, Fatboy steered to the port side and held the craft steady in one spot, using the motor to adjust for the currents that threatened to drive them back into the cliff. The pinging grew in intensity, but Fatboy shook his head at Tane's querying look.
"We are behind the island. They can't 'see' us here. But better get that periscope up and let me know when they come around the point."
Tane raised the buoy and tried to look around. It was difficult in the swelling waves, even with the gimbals that kept the camera steady.
The moonlight illuminated the edge of the island and sea beyond, but the water kept spinning the buoy around, and he had to constantly maneuver it.
The pinging surrounded them by the time Tane caught the first glimpse of the bow of the frigate, a dark silhouette against the moonlight.
"There she is!"
Fatboy was already turning the submarine, and rock surrounded them again for a moment as they ducked back through the hole in the rock.
Tane aimed his camera at the edge of the island just in time to see the stern of the frigate disappear behind it.
"What now?" he asked.
"Wait," Fatboy replied.
Less than thirty minutes later, the ship returned, but they were through the hole to the other side of the island long before it would have been able to ping them.
"It's like playing hide-and-seek," Fatboy laughed.
It sort of was, but Tane still felt sick.
They stayed in the shadow of Motukokako for another hour as the pinging receded into the distance. Only then did they start the journey back to Auckland.
Tane spent most of the trip sitting on the floor by Rebecca's bed, watching her. She woke up at one point, looked at him, and said, "Nothing makes any sense." But most of the way she slept. She vomited once or twice and he cleaned that up and reassured her gently that she was okay.
_She had held her breath for him._
**B AMBI**
**Gazza Henderson poured the remains** of his dinner onto the last embers of his fire with a hiss and a small puff of steam.
The dregs of his coffee went the same way, but there was barely enough heat left in the fire to raise a sizzle.
Technically, the campfire was illegal here deep in the bush at this time of year, but rules like that were for hikers and tourists. He had been hunting in the forests of Northland for more than twenty years and felt as at home in the bush as he did in his own living room. He certainly knew how to put out a campfire properly.
He heaped earth on top of the fire's remains and stamped on that carefully, making sure there was no chance of residual warmth flaring up again later and causing a forest fire. He had no wish to do that. It would be like burning down his own house.
Bambi's cold dead eyes stared up at him from beside his pack. Bambi wasn't a fawn, like in the movie. It wasn't even a doe. He was a buck, a hoary old stag, but Gazza always called the deer he shot "Bambi." It was a kind of twisted joke that had started when he used to go hunting with his mate Trevor in their teens.
This Bambi was a beauty. His antlers were worth at least two hundred on the Douglas points scoring system. Plus he had the Kaipara split in the antlers, which traced his ancestry directly back to the original fallow deer released in New Zealand in 1864.
The Kaipara split wasn't worth any extra points, but it was certainly worth a few pints in the clubrooms when he got back. Besides, he was sure that Bambi would take out the New Zealand record this year.
"Won't you, old boy," he said out loud. Bambi just stared at him coldly.
The sun was setting through the tall straight trunks of the kauri of the forest, and a light haze was rising as the day cooled. It softened the shadows of the trees, already elongated by the low angle of the sun, and gave the forest a quite otherworldly feeling.
Gazza debated with the dead stag for a while whether to de-head him and leave the body in the bush. He was a trophy hunter, a sportsman, he told all his mates, not a butcher, and the whole stag was a lot of weight to lug all the way back to Russell.
As he watched, the haze thickened into a light mist, swelling up around the tree ferns and rata vines. It seemed a bit unusual for this time of day. Mists were common in the early mornings, and many times he had awoken to find himself in a whitened-out world. But not usually in the evening when the land was still warm.
In fact, he couldn't ever remember a mist through the forest at sunset like this.
The tall kauri made ghostly spires as the delicate tendrils of the fog—for it really was turning into a fog—wrapped their way around trunks and through the fanned-out leaves of the pungas. Already those trees, more than twenty or thirty meters away, were dissolving into the mist, leaving a wall of white behind the trees closer to him.
There was a vague movement in the mist, more of a stirring than anything else.
"Anybody there?" he called out. He didn't expect there to be, but the alternatives were a deer or a Captain Cooker, one of the ferocious wild boars with razor-sharp tusks that populated the Northland forests.
He reached for his rifle but left the safety on. Hunting was a dangerous enough sport in the full light of day. But in the twilight, in a fog, it could be a death sentence. The last thing he wanted to do was shoot some innocent tramper, or another hunter, thinking it was a deer or a Cooker.
But at the same time, he didn't want to leave himself defenseless in case a wild boar did suddenly materialize out of the mist and charge toward his little campsite.
There were more movements then, more stirrings in the mist, around and in between the trees.
"Anybody there?" he called out again curiously, but not nervously.
The mist was thickening all the time, and he could see barely a few meters now. He had a strong sense that someone or _something_ was out there. He slid a round into the chamber of his rifle and flicked off the safety.
"I am a hunter," he called out. This time there was a slight nervous catch in his voice. "I have a rifle. Please identify yourself."
There was no reply, and all around him now, he had the sensation that the woods, the mist, were alive. Indefinable shapes swirled around him in the fog, ghosts amongst the trees.
A short while later, two rifle shots echoed through the tall kauri of the forest.
When Gazza hadn't returned by Thursday, December 17, his wife, Lorna Henderson, reported him overdue. Gazza was never back late from a hunting trip.
**N EW ZEALAND'S MOST WANTED**
**"I can't believe," said Tane** with a serious expression, "that they fell for that corny old look-out-behind-you routine."
"Actually it was 'Oh my God,'" said Rebecca with a grin. She had woken up after that first miserable night feeling one hundred percent fine and full of vim and vigor, if with a slight raspy cough to remind her of her brush with death.
Tane wasn't sure if she knew just how close she had come to dying, or what he had had to do to save her life, and he didn't enlighten her. It was something that was probably best forgotten.
"Seriously, though," he said with a smile, "whoever those guys were, they were tough, they were hard, they were professional combat soldiers. And yet they fell for the oldest trick in the book."
"Who were they?" Fatboy asked, without any trace of humor. "And is there any way they can identify us?"
They were seated around the wooden dining room table of the West Harbor house. The journey back had been a slow one, creeping from bay to bay, point to point, watching, always watching for the sharp bow of a naval frigate. It had taken them four days, but the _Möbius_ was now safely tucked away in the boatshed, away from prying eyes.
"No, I don't think so," Rebecca said.
"What about fingerprints?" Fatboy asked.
"I've never been fingerprinted by the police," Tane said. "So even if they got my fingerprints on the island, they wouldn't help."
"I have," Rebecca said carefully. "After the Save the Whales march. And that was going through my mind the whole time after they captured us. On the rubber boat, we had our hands tied behind our backs, and when we got on board the ship, I was careful not to touch anything. Not even the guardrail."
Tane remembered how she had skipped lightly over it.
"What about the netting?" he asked. "When we climbed up the side of the ship."
Fatboy shook his head. "They won't be able to lift fingerprints off wet rope," he said. "I am almost certain of it."
"Which just leaves the air bottles and weight belts," Rebecca said. "Should we go back to the island and get them?"
"Too risky," Fatboy said. "They'll be watching the island from now on. Better just pray they don't find them."
"What I think we need to work out most urgently," Tane said, "is what they were doing there. There was minimal security when we visited the island a few weeks earlier, and then when we turn up, suddenly the entire fifth cavalry is waiting."
Rebecca nodded. "I think something bad has happened on that island."
"Something to do with those piles of clothes you found?" Fatboy asked.
"Maybe," she replied, "and the broken door."
"The soldiers might have broken that door down," Tane said hopefully.
"From the inside?" Rebecca reminded him.
"Well, at least it's off our shoulders now," Fatboy said. "Now that the army and the navy are involved, it's not our problem anymore."
"Thank God for that," Tane agreed.
"Maybe," said Rebecca without conviction.
Tane looked closely at her. "Maybe?"
"It's not up to us anymore," Fatboy said emphatically. "We tried. We failed. Let the authorities sort it out."
"They won't," Rebecca said. "Or they can't. Or they just don't."
"You've got to be kidding," Fatboy said. "They've got soldiers and scientists and...and...other stuff. Of course they can sort it out."
Rebecca looked down at the tabletop, which was covered with copies of the almost-complete diagram, neatly printed out on crisp new sheets from their new inkjet printer.
"Tane, Fats, if the authorities are going to sort it out, then why have we been getting SOS messages from the future? They are going to fail. Just like we failed. The only people who can sort this out is us—that is, if we can figure out these messages and do the right things. In time."
Tane caught the sting in the tail of that and glanced away.
Rebecca asked, "What about the Chronophone?"
With two computers working almost nonstop, they were quickly chewing through the backlog of BATSE data. There was now less than a third of the image to go. The plans lay tantalizingly close to completion in front of them.
"We can't build it yet," Fatboy said. "Not until we have the missing bit. But we could start buying some of the components."
"Where from?" Tane asked. "The local Chronophone shop?"
Rebecca laughed, and Fatboy said, "They're all just standard components, looking at the diagram. Resistors, transistors, diodes. We can pick them up at any RadioShack. But I'll get a mate of mine to look over it in any case. Goony. He's the electronics technician at the recording studio. What he doesn't know about circuit boards is not worth knowing."
"Will he help us build it?" Rebecca wondered out loud.
"Probably," Fatboy replied.
"Can we trust him?" Tane asked.
"I don't think we have any choice," Fatboy responded. "But anyway, he doesn't have to know what it does to assemble the components."
"Hi, Rebecca. Hello, boys," a new voice said, and they all looked up with somewhat guilty expressions. They hadn't heard Rebecca's mum come in.
"Hi, Mrs. Richards," Tane said. "How are you feeling?"
He saw Rebecca's cross look out of the corner of his eye.
"I'm fine, thank you, Tane. How is your mum? I haven't seen her in such a long time."
"Oh, she's fine, thanks," Tane replied.
"What's happening on your program?" Rebecca asked conversationally, casually turning the Chronophone plans upside down on the tabletop as her mother poured herself a glass of orange juice from the fridge.
"Oh, I don't know." Her mum looked quite annoyed. "It's getting hard to follow. They keep interrupting it with those blasted news reports."
"News reports?" It was Tane who asked.
"Oh, all those cordons up past Whangarei. Mad cow disease, something like that."
The three of them sat openmouthed as Rebecca's mum disappeared back upstairs.
"There's a TV in the lounge," Rebecca said unnecessarily.
The newsreader was as smooth and polished as a greenstone tiki. They watched with increasing shock as the details of a quarantine zone were announced. A police spokesperson was interviewed and urged residents to stay calm but to avoid any unnecessary travel. Mad cow disease had been discovered on a Northland farm, a cattle disease potentially damaging to New Zealand's dairy exports.
The main highway had been cordoned off at Kawakawa to the south of Russell and at Kerikeri to the north. All ancillary roads had also been blocked. There was no movement in or out of the quarantine zone without special permission of the police and military who were policing it.
In the same bulletin, but declared as an unrelated story, the police were searching for two teenagers and an unknown number of accomplices in connection with the disappearance of a prominent scientist from a research station in the Bay of Islands.
A photo of Professor Green was displayed, along with a potted summary of her career, followed by a police sketch of Rebecca and Tane. It wasn't a good likeness.
"What the hell is going on?" Fatboy wanted to know when the bulletin had finished.
"Disappearance!" Rebecca's mouth just about hit the floor. "So that's what the soldiers meant by 'Where are the hostages?' They think we kidnapped Vicky!"
"And the rest of her team!" Fatboy realized.
Tane said, "But we didn't. So who did? And what's this about mad cow disease?"
"It's not mad cow," Rebecca said. "That's just a story, to avoid panic."
"So what is it?" Tane asked.
Rebecca said chillingly, "Isn't it obvious? Something bad _did_ happen on the island. The Chimera Project."
Tane got up and turned the sound off on the TV. The sudden silence in the room seemed to take all the warmth out of the day, despite the unfettered sunshine outside.
Rebecca said, "I think we were too late. That whatever it was that we were supposed to stop has already happened."
**'T IS THE SEASON**
_He sees you when you're sleeping,_
_He knows when you're awake._
_He knows if you've been bad or good,_
_So be good for goodness sake!_
_—J. Fred Coots and Haven Gillespie,_
_"Santa Claus Is Coming to Town"_
**The week leading up to** Christmas was full of the noise and fun and shopping that is like no other time of the year.
Carolers sang joyously on street corners, whole streets of houses were festooned with thousands of tiny lightbulbs, and children sat fearfully on Santa's knee in the shopping malls and asked for outrageous gifts like ponies and spaceships and magic rings that could make them fly.
Shops promised pre-Christmas sales, large corporations sponsored nighttime light shows and drew Santa's sleigh on the clouds in colored laser light.
Drivers and shoppers alike grew increasingly irritable as stressed-out motorists and frazzled parents gesticulated at each other over places in motorway queues, or parking spaces, or the last stock of a special-priced toy.
In other words, the year progressed just like those that preceded it, and apart from the occasional slightly disturbing news report from the north, life was pretty good.
To Tane, Rebecca, and Fatboy, it all seemed like sheer insanity.
The man in the Santa suit glanced casually over at Tane again. Too casually. For the third time.
"Let's get out of here," he said to Rebecca, who was comparing the component numbers on some electrical parts with the list that Fatboy had given them.
Fatboy was still circling the block. He had given up looking for a space after half an hour of trying. Kmart had a sale on and there wasn't a free parking space this side of Western Springs.
"But we haven't got the capacitors yet." She frowned, running her finger along a row of small, labeled plastic bins.
"Santa Claus has been watching us for the last five minutes."
Rebecca looked around at the man, who quickly looked away. "Might be a store detective. The way you're acting, I'd be watching you too."
"A store detective in a Santa suit?" Tane took a deep breath and tried to calm down. It didn't help. A police car went past outside with its siren crying, and he accidentally knocked over an entire stand of _Dummies_ books. He picked them up quickly, before the sales assistant could come and help.
Santa was watching them again, Tane realized, half hidden on the other side of a circular stand of electronics magazines. He was curiously thin-faced for a Santa, who was usually a chubby, jovial fellow. This Santa had a scar above one eye, not quite masked by the stuck-on bushy eyebrows. He looked like a spy in a Santa costume. Or a hired killer. Or a soldier.
"He's still got his eye on us," Tane whispered, covering his mouth with his hand in case Santa could read lips.
"You'd better watch out," Rebecca said quietly.
"What? Why?" Tane hissed urgently.
"You'd better not cry..."
"Cry?" Tane stared at his friend in confusion.
"Got it," said Rebecca, picking up a couple of small metal objects out of a plastic tray. "Let's get out of here."
"Cry?" Tane asked again.
Rebecca laughed gaily and sang in Tane's ear, "He's making a list and checking it twice; he's gonna find out who's naughty and nice..."
Tane groaned and punched Rebecca lightly on the arm.
For all her frivolity, though, he noticed that her hands were shaking as she handed over the money at the counter.
Fatboy swept into the curb in the Wrangler as they emerged from the electronics shop. They swung on board, Rebecca in the front, Tane in the back, and Fatboy pulled out again in a seamless maneuver.
He handed a newspaper to Rebecca as he steered carefully around a family of four lugging two long plastic kayaks across the parking area.
"Made the front page!"
Rebecca looked at it and wordlessly handed it back to Tane.
HUNT CONTINUES FOR TEENAGERS, the headline blared over the police drawings of him and Rebecca. Rebecca's was a reasonable resemblance, he thought, but his picture looked like an axe-murderer.
Rebecca had dyed her hair jet black since the escape from the soldiers and had started wearing a cap. It was summer, so they all wore sunglasses. All in all, Tane didn't think they were in any danger of being recognized by a passing stranger. But what if one of their school friends recognized the pictures?
"What do we do?" he asked.
"Nothing," Fatboy replied immediately. "It doesn't change a thing. They'll never recognize Rebecca from a picture like that, although yours is a pretty good likeness."
"Bite me," Tane said.
The Wrangler was an open-top jeep, which would have seemed like a lot of fun on such a lovely, sunstruck day. But it made Tane feel elevated and conspicuous. There was nothing to hide behind.
They sat silently in a queue to get out of the car park. Tane pretended to rest his face on his hand to shield himself from an elderly couple in an old Volvo in the next lane.
"The worst thing is feeling like a criminal all the time," he said, almost to himself, "when we haven't done anything wrong."
"I know what you mean," Rebecca said in a small voice, and Tane thought, not for the first time, that she was a lot more fragile than she was making out. She had that tired, heavy look about her again. Perhaps she felt that she, of all of them, had the most to lose.
The car radio was tuned to a news talk station.
"Any news on the plague?" Tane asked.
"Nothing," Fatboy replied. "Quarantine zone is still in place, apparently, but there's been no other reports."
"Let's hope it stays that way," Tane said.
**S ILENT NIGHT**
**A strange fog rolled through** the streets of New Zealand's northernmost city—Whangarei—on the night before Christmas, blanketing the city. Streets that were usually full of festivity and Christmas Eve revelers became unusually, deathly silent.
By morning, not a creature was stirring in the city of nearly fifty thousand people, except for the birds in the trees and the small animals of the undergrowth; the pets, now howling and yowling for their owners; a few farm animals; and, of course, the occasional mouse.
**W HITE CHRISTMAS**
_Sleep in heavenly peace_
_Sleep in heavenly peace._
_—Josef Mohr, "Silent Night"_
**Crawford landed the Sikorsky MH-60S** Knighthawk helicopter in the middle of the main road, touching down with a gentle sigh.
Crowe was out before the engine had even been switched off, ducking his head against the wash from the rotors.
He stopped next to his second in command, Big "Mandy" Manderson, and stared to the north. His shoulder itched and he absently reached up to scratch it, realizing the futility only as his finger connected with the solid Kevlar of his armored combat biosuit. The suit was black. It was a positive-pressure suit, so even if it was damaged, the higher air pressure inside would keep pathogens from getting in. Scientists in biohazard labs wore positive-pressure suits but the air pressure was about the only similarity. Crowe's suit, and those of his team, were made of bulletproof Kevlar with ceramic chest and stomach plates. They were not only protection from germs, but also protection against human germs with guns. For all that, it was lightweight and about as comfortable as a black combat biosuit could be.
"Still think it's terrorists?" Mandy asked in his slow Southern drawl.
Crowe didn't answer.
It was dark. The sun was still hiding a couple of hours away behind the forests and the ocean to the east, but even so, all over New Zealand, excited children would be waking up—with that natural, internal alarm clock that kids have on Christmas morning—and eagerly unwrapping presents.
Except in Whangarei.
Crowe stood in the middle of the road and looked at the fog. He had never seen a fog quite like it before. The edge of it was so well defined that it looked as if you could walk up to it and touch it. It was as if a puffy cumulus cloud had dropped from the sky and settled on the road in front of him.
A road sign just in front of the fog was perfectly clear, but everything behind it was blanketed in white. WHANGAREI 2 KM the sign announced.
The calls to emergency services had started at eleven the previous night. The local police station had gone off the air, so police officers from surrounding areas had rushed to the scene. They had reported seeing the fog; then their radios had fallen silent.
"What did you find out?" Mandy asked.
Specialist First Class Evans, the unit's newest and youngest recruit, appeared at his side, holding a bunch of large-scale photographs. "The satellite images you asked for," he said, handing them over to Crowe.
Crowe flicked through them quickly. They were in series, with a date/time stamp at the top of each photograph in black computer lettering. The photos themselves were grainy black-and-white weather satellite images showing the top of the North Island of New Zealand.
"There it is," Crowe said, pointing to a small fuzzy spot on one of the photos. "That's the fog over Motukiekie."
He flicked forward a couple of photos. "Here it is moving south, across the water."
He looked up at Manderson. "It comes inland here, to the east of Russell, but misses Russell, Paihia, Kawakawa, and travels through the Russell forest, then down the coastline through this area here. What is this area?"
"Mainly forests and farmland," Mandy said, consulting a map.
"Okay, it rolls over the top of this small town here, Hikurangi, and ends up in Whangarei."
Crawford joined them, the rotors of the Knighthawk slowly winding down behind them.
Crowe pursed his lips and nodded, then looked up. "Any luck finding those kids?"
Mandy shook his head. "No leads as yet. The local guys are currently reviewing the security camera footage from the lab. Going back a couple of weeks. See if the cameras noticed anything suspicious in the days leading up to the...accident."
"What about the days when the scientists disappeared? What do the cameras show on those days?"
"Nothing. Not a thing. The image is completely fogged for a couple of days. Some kind of jamming equipment perhaps."
Crowe looked at the high bank of fog a short way up the highway and said nothing.
Manderson straightened up to his full height and stood next to Crowe, facing the mist. He said, "Something's been worrying me, Stony."
Crowe said nothing but looked at the big man.
Manderson continued, "How did those kids get from their submarine onto the island without us noticing?"
"They swam." Crowe frowned.
Manderson shook his head. "Crawford was overhead in the helicopter the whole time. He would have seen them on the heat-scope."
Crawford said, "I saw nothing until they appeared at the end of the wharf."
"Underwater!" Crowe realized.
"But they had no air tanks when we picked them up," Manderson noted.
Crowe said, "Get some men back to the island. Search around the wharf."
Evans said suddenly, "Skipper, the mist!"
Crowe swung around back to the wall of fog in front of them. It was glowing.
"What the hell...?" Manderson drawled.
Crowe wasted no time. "All teams, seal masks and get up here now. Set your fields of fire and kill zones. Be ready for anything."
Jackboots sounded all around them as the members of Red and Blue Teams fanned out across the road, dropping to one knee or even lying on the roadway, their special-issue XM8 automatic weapons trained on the glowing mist.
The glow slowly split in two and gradually resolved itself into two distinct lights, brightly luminescent in the mist.
"It's a car!" Crawford said.
"No," Crowe said, "the lights are too large, too far apart. It's a truck."
It was crawling toward them. Just rolling slowly forward through the mist. The lights brightened, yard by yard.
The dark bulk of the truck gradually materialized through the ethereal white clouds. It was big, an eighteen-wheeler.
The long snout of the truck now poked its way out of the wall of the edge of the fog. The name SLIPSTREAM WARRIOR was painted in bold lettering across the front of the hood.
"Hold your fire," Crowe called. "We may have a survivor."
At a painfully slow crawl, the big truck rolled forward out of the fog, past the WHANGAREI sign, and gathered momentum down a short slope before a small bridge across a stream. It was no more than fifty or sixty yards in front of them now.
"They're not in a hurry, are they," Evans murmured.
"Stay alert," Crowe ordered.
As it turned out, there was no need. For alertness. The truck failed to take the small bridge on a mild bend in the highway. It never turned. It never tried to turn. It just rolled forward in a dead straight line and hit the railing of the bridge on an angle.
The concrete wall of the bridge disintegrated under the impact of the massive truck, and the juggernaut toppled slowly off the side of the bridge and crashed down, nose-first, into the small stream below.
The back wheels of the cab and the large trailer remained on the highway. The wheels of the truck continued to grind for a few seconds; then it stalled with a huge shudder that ran through the body of the truck like that of a dying animal and was still.
"Crawford, Manderson," Crowe said tersely, "check it out."
The two men ran in a crouch over to the dead body of the beast, sliding down the embankment beside the bridge and peering in through the shattered windshield of the truck.
"It's empty." Crawford's voice sounded in his earpiece. "No driver."
"Stranger and stranger," Crowe said.
"Around here, if you ask me, 'strange' is pretty normal," Manderson said.
Crawford spoke again, his voice suddenly low and serious. "Crowe, you'd better come and look at this."
"Evans, don't take your eyes off that fog," Crowe said, running across to the edge of the broken bridge. Crawford and Manderson were crouched over something on the bank of the stream. Crawford turned and looked up at him, and then he could see past the man, to the object they were crouching over. It was a body.
Crowe slithered down the embankment and splashed through the stream to where the other two crouched. His heart wrenched. It was the body of a small boy, half in the water, faceup in the mud. The body was covered in mud, and it was incredible that even the eagle-eyed Crawford had spotted it.
The boy couldn't have been more than four. It was all Crowe could do to say, "Get the body back to the lab."
Manderson said, "Let me do it," and Crowe remembered that Mandy had a five-year-old son of his own. More than any of the others, he would be feeling the anguish of the little boy's death.
Manderson stowed his weapon and carefully, respectfully, worked his hands into the mud underneath the neck and knees of the tiny body. He lifted and the body came free of the mud with a sucking sound.
The boy opened his eyes, took one look at Manderson's black suit and face mask, and screamed his little lungs out.
A police doctor went back in the ambulance with the boy. They still called him "the boy" because he had been unable to tell them his name.
He had done nothing but scream until the ambulance had arrived. The U.S. Army Bioterrorism Response Force soldiers in their black combat biosuits must have looked a fearsome sight to a half-drowned, terrified four-year-old.
So far, he was the only known survivor of the calamity that had embraced the city, but as an eyewitness he was useless. Only two coherent words had come from the terrified little boy the entire time, and they made little or no sense at all.
"Jerryfish," the boy had screamed over and over. "Jerryfish! Jerryfish!" and "Snowmen!"
The "jerryfish" was the most confusing. It seemed he was saying "jellyfish," but there was no rationale in that. Jellyfish were saltwater creatures and the boy had been found in a freshwater stream.
But the word "snowmen" had Crowe worried. A white biosuit could be mistaken for a snowman, particularly by a young boy. If he had seen terrorists in biosuits, that could well explain the "snowmen."
"It's starting to drift," Manderson noted.
Already the WHANGAREI road sign had disappeared into the maw of the mist.
"I noticed," Crowe replied. "It's coming south. Get the men ready to evac."
"A bit strange, don't you think?"
"Strange, why?"
Manderson looked at him oddly. "The breeze is nor'east, Stony."
Crowe peered up at the leaves on the branches of some nearby trees. It was true, he realized with a profound horror. The fog was coming south.
But the wind was blowing the other way.
**O N CHRISTMAS DAY**
**It was half past nine** in the morning. On Christmas Day.
The Christmas tree sat in the corner of the middle lounge, the largest of the three. It was a tall tree, perfectly shaped in the traditional Christmas tree cone, with deep green needles—or were they leaves?—and a dark wooden trunk scored with the intricate and random patterns of bark. It was a luxurious tree—vibrant, exciting, larger than life—that embodied the spirit of Christmas in its very form.
And it was made in China, according to the not-quite-covered-up sticker on its base. Aluminum, fiberglass, and plastic, if you cared to read the small print.
"Be there in a moment," Tane's mum called cheerily from the kitchen, busy with some final putting-away.
They had all helped clear away the remnants and dishes of the champagne breakfast, which was a Christmas morning tradition in the Williams household and happened before the presents were opened. They all helped; that was part of the tradition also. Even Rebecca.
It was the first time that Rebecca had spent Christmas morning at Tane's house. Her mum's cousin had come and picked her up, but Rebecca had asked to go to Tane's and nobody had minded.
Tane thought it was strange to be having a day of celebration with all that was going on—quarantine zones, kidnappings, nationwide police hunts, and so on—but it seemed even stranger _not_ to celebrate Christmas, and certainly it would have raised some tricky questions with his mum and dad if they had decided not to show up.
His mum wandered in at last from the dining room and found an empty armchair.
"Right, then," she said. "Who's the youngest?"
That was another Williams family tradition. The youngest person would play Santa Claus and hand out the presents to the others, starting from the oldest person. As there were no grandparents or young cousins this time, that made his dad the oldest and Tane the youngest. Rebecca shared his birthday, but she had been born in the morning and Tane had been born in the evening.
Tane didn't answer. He was too busy watching the spider. It was a big one. Not huge like an Avondale spider or a tarantula, but big enough to be scary. It was brown with an elongated body and thick, sectional legs. It had woven an intricate web in the corner of one of the large picture windows of the middle lounge, a tightly woven web, almost honeycomb in appearance, with many layers of strands on top of other layers.
The spider was quivering, shaking. He had never seen a spider do that before. A thin band of white crossed over the dark brown body. It struggled to move, and he suddenly realized what was happening. The spider had become trapped in its own web.
"Tane's the youngest," Rebecca said brightly, showing no sign of the worry that must surely be festering inside.
Tane hunted in the pile of brightly wrapped parcels until he found something with his dad's name on it.
"Merry Christmas," he said jovially, making a bit of a performance of the handover. His dad grinned and snatched it off him, scanning the tag to see who it was from and ripping off the paper with gusto. It was a book, the latest John Grisham thriller, from his uncle in Wellington.
His dad laughed now, for no real reason. Just the joy of the day.
_If only he knew!_
Tane was already rummaging for a present for his mum. His mind was elsewhere, though, and he skipped over one a couple of times before noticing it.
He found a present for Fatboy from Rebecca, and then one for Rebecca, from Fatboy.
Fatboy's present to Rebecca was a silver necklace. Tane didn't look at it too closely, but it looked expensive. Still, with over a million dollars still earning them interest in their trust account, what was money anyway? It was the thought that counted, and he hoped that Rebecca would realize the effort that had gone into the present Tane had chosen for her—the chess set.
She still hadn't opened it, saving it till last probably, because it was the biggest. Or maybe, he hoped, because it was from him.
He opened Rebecca's present slowly. It was delicately, femininely wrapped in layers of colored tissue, bound by ribbons. That wasn't like her at all. Maybe they had wrapped it in the shop for her, he thought. Inside the wrapping was a white cardboard box. He pulled the top off carefully and stared down at what was inside.
It was a brand-new harmonica. Engraved on the silver top of the instrument were the words FRIENDS FOREVER.
He gave Rebecca a hug, with a warm feeling that went from his toes up to the hair of his scalp.
His parents' presents for him and his brother were the same, he realized as he opened his. Fatboy had opened his a few moments earlier. It was a genuine hand-carved _patu pounamu,_ a greenstone club, almost a foot long with a leather cord through one end, carved with traditional symbols of their Tuhoe tribe.
"Goes with the _moko,_ don't you think?" Fatboy said proudly.
Tane put his carefully to one side, conscious of the close scrutiny of his parents. It was a kind of cool present, but he had been hoping for a new Xbox console. He forced a smile.
"Thanks, Mum, Dad, it's great!"
Rebecca came over and sat next to him as she unwrapped the chess set. Tane crossed his fingers behind his back.
It was a hit.
Rebecca actually squealed with delight as the paper fell away. She slid the wooden case out of its plastic covering and pulled each piece individually out of its velvet casing, noticing the fine detail of the replications.
She even held up the king, Michelangelo's _David,_ to show the rest of the room. "Look at the detail!" she exclaimed. "You can see every muscle on his tummy. And what a tiny willy."
They all roared with laughter.
"Thanks, Tane," Rebecca said, and hugged him warmly.
Tane never got to see what Rebecca had bought Fatboy, which was the only jarring note on the day. Fatboy unwrapped it, looked inside without revealing the contents, smiled at Rebecca, and wrapped it back up. He gave Rebecca a hug after that, but Tane looked the other way.
It was a lovely day, and the worries that faced them receded for a few hours at least. They ate, they drank more champagne than they were allowed by sneaking into the kitchen and topping off their glasses just a little at a time, and they listened to "Snoopy's Christmas" over and over on the stereo until his dad got sick of it and put on "The Little Drummer Boy" instead. They just spent time enjoying their presents and each other's company. Fatboy got his guitar out after lunch (leftovers from the breakfast) and Tane joined him on his new harmonica. The big windows were open to the forest, and the sounds and smells of the native bush drifted gently inside. The sun was fiery, but sheltered by the trees of the forest, the house was cool and peaceful.
It was a lovely day.
In the midafternoon, they went back to the West Harbor house to do some more work on the Chronophone. Rebecca's mother was still out at her cousin's, so they had the house totally to themselves. They had already purchased most of the parts for the machine, and Fatboy had arranged for his mate Goony to come over and start building it the moment the rest of the plans came through.
Rebecca went to sit on a large sofa in the corner of the living room and wrapped her arms around her legs, rocking back and forth slowly. Tane watched her silently. They had managed to put it all out of their minds for a few hours on Christmas morning, but now the reality was breathing down their necks.
At six o'clock, Tane turned the television on to watch the news. The queen was on, broadcasting her Christmas message, and he flicked over to TV3.
There was only one story. The news had apparently been running all day. Breathless reporters behind police roadblocks and helicopter shots from a distance told a story of the most catastrophic disaster in the history of New Zealand. Fifty thousand people, cut off, missing or worse. A strange fog. The evacuation of the small town of Maungaturoto, south of Whangarei, the next in the path of the fog as it drifted down toward Auckland, New Zealand's largest city.
Auckland residents were urged to remain calm and to not try leaving the city. Civil defense managers assured reporters that special teams of experts had already been brought in from overseas to deal with the problem.
Rebecca had been drinking some water, but her glass shattered suddenly on the floor. Her face was white.
"Rebecca?"
She turned and ran outside onto the patio. Tane ran after her. He sensed Fatboy following.
"Oh no. Oh no, oh no, oh no."
Rebecca gripped the edge of the patio table with both hands as if she would fall down without it. She was breathing strangely, Tane noticed, hyperventilating. She kept lurching forward.
"Oh my God, no," Rebecca said.
"What is it, Rebecca? Do you think we were exposed to a virus when we were on the island?" Fatboy asked urgently.
"No. Worse than that. Much worse than that."
_Worse than that!_ The world seemed to be spinning around Tane's head. "What's wrong, Rebecca?"
She shut her eyes for a second and breathed out a long, slow breath. "I've just put it all together. The island. The fog. The cryptic messages. I know why we had to buy the submarine."
"To visit the island. We already knew that, didn't we?" Tane asked with a growing apprehension.
She shook her head. "What if there were two reasons? A plan B in case our trip to the island failed."
"Which it did," Fatboy noted.
"What if we were right? What if this...virus...or whatever it is, is so dreadful, so devastating, that plan B is to give ourselves a refuge, a kind of a fallout shelter under the sea."
"A sanctuary," Fatboy said softly.
Rebecca asked, "What if we're supposed to hide underwater in our little yellow submarine to keep ourselves safe from a plague that is about to wipe out the rest of New Zealand?"
"Or the rest of the world," Tane said.
It was ten past six in the evening. On Christmas Day.
**S ANCTUARY**
**On Boxing Day, while other** kids all over the country went swimming or cycling on new Christmas bikes or were just playing with their Game Boys and other presents, Tane, Rebecca, and Fatboy planned for the apocalypse.
"How long do you think we will be underwater?" Tane asked.
"However long it takes," was Rebecca's answer. "Months, maybe years."
The Chronophone plans had finally finished with a last long list of numbers, followed by the single word _END._ They had printed out the schematic and pored over it.
"Why even bother?" Fatboy had asked at one point. "It's too late. It's no use to us now."
Rebecca had looked at him incredulously. "Are you insane? If we don't build the Chronophone, then we can't send the messages back through time. If we don't do that, then we won't know anything and we'll just be out there like everyone else, playing with our Christmas presents and lying on the beach, and not even knowing that we are about to be wiped off the face of the planet."
Fatboy had looked at Tane in alarm, but Tane had said, "She's right. No Chronophone, no Lotto, no nothing. We'd better get it built."
That had been three days ago. That day, Tane had opened the front door to possibly the thinnest person he had ever seen in his life. Goony seemed to be made of just cling-film wrapped around a skeleton. He introduced himself shyly with a big droopy grin and lugged inside a huge plastic toolbox that looked heavier than he was. He was supposed to be a genius at electronics and, according to Fatboy, had once built a guitar amplifier out of an old kitchen mixer just to prove he could do it.
The Chronophone plans had given him no trouble at all.
"It's a transmitter," he had said after a cursory look at the diagram. "What's it for?"
"It's a time transmitter," Fatboy had answered with a smile. "We're going to send the winning Lotto numbers back to ourselves in the past and win the Lotto."
"Yeah, right!" Goony had just given one of those big droopy grins and got on with his work.
It had been three days of intensive work. Rebecca helped analyze the plans, and Fatboy helped solder components into place. Tane just stayed out of the way and made lots of tea and coffee.
Now, nearly at the last day of the year, the Chronophone was a waterproof aluminum box the size of a briefcase sitting in the garage. It was more than just a transmitter, according to Rebecca. Built into it was a radio receiver, which received the signals they would transmit from the submarine and retransmitted them through the gamma-rays bursts.
"Don't we take it with us in the _Möbius_?" Tane asked.
"No. It needs a much more powerful aerial than the small one on the buoy," Rebecca explained.
The signal they transmitted, she told them, was not the one that would be received back in the past. It was a disruptor signal, which would disrupt the gamma-ray burst that was already on its way to Earth from the depths of the galaxy and imprint their message on the radiation of the burst. That burst of radiation would somehow seep through the quantum foam, through the fabric of time itself, into the past.
The signal from the Chronophone wouldn't even reach the gamma-ray burst for another two years. It was a hard concept to grasp. They sent a signal to the future, which ended up in the past.
By far, the hardest part of building the Chronophone was the location of the transmitter, when they had finally deciphered the schematics and worked out where that was. On the plan it showed just a tall, thin spike, with what looked like a satellite dish at the top, a serial number, and some coordinates.
As usual, it was Tane who had put two and two together.
"It's the Skytower. The top of that thing is stuffed with satellite dishes of all shapes and sizes. I think we're supposed to connect the Chronophone to one of those dishes, one with this serial number, and aim it at these coordinates."
"I hope the owner of the dish doesn't mind," Fatboy had said.
"Tough," had been Rebecca's answer. "We have no choice."
Tane steered the _Möbius_ through a cloud of sludge, whirled up by a passing freighter from the bottom of the harbor. There had been no danger of collision, but the murky water was unsettling all the same.
Rebecca was intently watching a readout on one of the control panels. "We're getting close," she said.
The next batch of messages had proved quite informative. It was a series of numbers, each with an exactly specified time. It was Rebecca who figured it out.
"This is the final piece of the puzzle," she said. "I was expecting this."
"What is it?" Fatboy asked.
"It's the timings. Each message has to be sent at an exact time, to hit the gamma-ray burst at the right moment and end up in the past at the right moment."
The tail end of the last message had been interesting, too.
**GPS,-36,50.999,174,49.876**
GPS map coordinates, Rebecca had recognized at once, and a detailed map of Auckland had shown the spot defined by those coordinates to be on the edge of Rangitoto, a large island volcano, long extinct, that dominated the view from the shoreline of most of Auckland.
They were on their way to that spot now, while Fatboy and Goony continued working on the Chronophone.
It was daylight, but time was considered to be more important than discretion. The unspoken question was always, Had they wasted too much time before they had made their move on Motukiekie? If they had been earlier, _could they have stopped the Chimera Project in time?_
"What's in front of us?" Rebecca asked. "We are almost right on the coordinates."
The GPS readout on the control panel flashed red numbers in the dim light of the cockpit.
"Nothing yet," Tane said. "The seabed is starting to slope up. We must be just about at Rangitoto by now. There's...oh crap!"
A cliff face reared up suddenly in front of the _Möbius._ Tane flicked the craft into reverse for a moment, slowing the craft quickly, and they coasted up to the wall of rock.
"Just nudge a bit to your right," Rebecca murmured, and Tane complied, drifting a little in that direction.
"Nothing," he said.
"Try going down," Rebecca suggested.
"We're just about on the seafloor," Tane said.
"Then try going up."
A few yards higher up the cliff face, it became immediately apparent what the GPS coordinates were for. The wide mouth of a cave, almost oval, except on the left where a section had broken away from the top and made a mound on the bottom rim of the cave. It was well big enough for the _Möbius._
"Want to take a look inside?" Tane asked, not waiting for an answer.
He touched a few buttons on the control panel, and the interior of the cave lit up like a cathedral in the glow of the underwater spotlights of the _Möbius._ The cave was huge.
"Probably a volcanic vent," Rebecca said, her voice full of wonder. "It could go on, or down, for miles."
Tane remembered stories of underground water channels that ran between Rangitoto Island and Lake Pupuke, just inland from Takapuna beach, and wondered if this was anything to do with that.
The cave was filled with marine life. A haze of tiny silver fish filtered past the glass dome of the driver's bubble.
"This is it," he said, maneuvering the craft around inside the underwater cavity in the side of the mountain. "This is our hideout. This is where we tuck ourselves away from whatever hell is coming for the rest of the human race, and wait until it is safe to emerge."
Rebecca nodded, examining the roof of the cave through the dome. "If we set her down close to the entrance, we'll be able to run the buoy up outside the cave to the surface. We can suck down as much air as we want and stay here indefinitely, if need be."
"I hope it's not that long," Tane said quietly. _Indefinitely_ sounded like too long a time to be living in a tiny tin tube on the floor of an underwater cave. "We can bring in supplies, food, diesel, whatever, in airtight drums and stack it up on the cave floor."
"I think we're going to get mightily sick of tinned ham and peaches," Rebecca said, trying to make light of it, but the darkening circles under her eyes wiped away any trace of humor.
"God, I hope we're wrong about all this," Tane said. But he had a terrible feeling that they were right.
**A N UNNATURAL DISASTER**
**The TV was on for** the news. The news was not good. In fact, the more they heard, the more frightening it all became.
Tane held the remote and unconsciously kept turning the volume up, until it became painful, then would turn it back down to a reasonable level, only to start all over again.
They were watching a press conference. A room full of eager reporters. A thick bush of microphones sprouting from a wooden podium labeled with HYATT OREWA.
A tall, gaunt American entered and stepped up to the podium, closely followed by a woman, scarcely half his size.
"That's him!" Rebecca caught her breath.
"Who?" Fatboy asked.
"The leader of the soldiers on the island."
A graphic came up at the lower left corner of the screen identifying the speaker as Dr. Anthony Crowe of USABRF.
Just the sight of him was enough to make Tane's heart race.
The woman stood next to Crowe. She would need a stool to reach the microphones, Tane thought, then wondered why he was thinking about such stupid details when the fate of the world was at stake. Her name was Dr. Lucy Southwell, according to the subtitles.
A large map of the upper north island was pinned to a board behind them.
Crowe wore a military uniform, but wore it casually, as if the uniform was not a symbol of pride for him, the way it was for many Americans. His face was as long and craggy as a cliff face and showed no expression; in fact, his face might as well be made of stone for all the emotion that showed on it.
Southwell pointed to the map. "The _Horouta_ is a delivery boat. Operates out of Russell, here. She makes a regular weekly supply drop at Motukiekie. Just a small boat, with a skipper and one crewman. Four days ago, she was discovered, beached in Kaingahoa Bay, about forty miles east of Motukiekie. The throttle was wide open, and the engine was still running. There was no sign of the crew."
"Odd," murmured Fatboy.
Southwell continued, "A coast guard vessel was sent to investigate. Six-man crew. It didn't return."
"Odder still," Tane said
Southwell drew a circle on the map. "An airforce Orion was dispatched to search for the missing coast guard cutter. It covered roughly this area here, which was as far as the cutter could have traveled in the time. It overflew Motukiekie but was unable to see anything due to a dense fog. By this time, the local police were involved and wisely decided that it was time to call in the experts.
"I work for the Biological Hazard Containment Unit, a part of our Ministry of Agriculture and Fisheries. A team of three colleagues of mine—two men, one woman—was sent in, in full biohazard suits and in constant radio contact with a command unit based in Russell. Naturally, at this stage, our fear was of some..." She clearly didn't want to say it. "...biological agent that had been released on the island. They...um..."
"They disappeared," Crowe intervened. "Their biosuits were found on the island when we went there to investigate."
"What about Whangarei?" a female reporter was asking. "Fifty thousand people. They can't just have disappeared."
"Actually, ma'am, that's exactly what seems to have happened," was Crowe's reply.
Strange how Americans called women "ma'am." It was such a British expression.
_Another stupid thought!_
"Disappeared to where?" the reporter persisted.
"We have no information on that at this time."
"Do you anticipate finding them alive?"
Crowe replied without emotion, "No, ma'am, we do not."
The room grew suddenly, unnaturally silent.
Southwell broke the silence. "Civil defense are evacuating everyone in the projected path of the fog and are preparing an evacuation plan for Auckland, should that become necessary. I must stress that there is no need for panic. Any evacuation will be completed in plenty of time; we are asking residents to remain calm and at home until we make an announcement."
"Yeah, right," Fatboy muttered. "Fifty thousand people missing, but don't panic!"
Southwell wore a rather drab, olive blouse and a staid gray skirt. She dressed older than her years. Her hair was pulled back in a loose ponytail, held together with a butterfly band.
"Is there any chance that a breeze will push this toxic fog out to sea?" a man asked.
"Toxic fog." Tane sounded the words out loud.
"The movement of the fog does not appear to be governed by the direction of the wind." Crowe answered the question, and the room grew completely silent again for a second or two as people tried to work out how that could be.
"Toxic fog," Tane said again, blankly. Rebecca was staring at him with a horrified expression, but he ignored her and concentrated on the television.
"If the toxic fog continues on its present course and at its present speed, when do you anticipate it will hit Auckland?" the same man asked.
Crowe answered, "It's hard to be accurate. The fog does not seem to move at a constant speed. It did not move from Whangarei at all for two and a half days. I would say we have about a week."
The questions were coming from all over the room now.
"Is it true that the fog is growing in size?"
"Yes. Substantially."
"By how much exactly?"
Crowe looked as if he would rather avoid answering that one, Tane thought. He turned the television louder.
"On our satellite photos, the fog was roughly circular and only a few hundred yards across when it entered Whangarei. It now measures several miles across."
Rebecca said, "Several miles! That means it has grown by a factor of twenty or thirty times since it rolled into Whangarei. Or even more!"
Someone asked, "How many survivors are there?"
"Currently three. The youngest is a boy of four."
"And what about the reports of snowmen in the fog?"
"We believe these sightings to be of people wearing biohazard suits, like this one." Crowe motioned to his side, and an assistant wheeled in an inflated silver suit on a trolley. It looked more than anything like a space suit, although the faceplate was narrower, not as spherical as a NASA space suit.
Southwell moved around the front of the suit and placed her hand on it.
She said, "This is a UN-issue biohazard suit. It is silver but reflective. Surrounded by white mist, this would appear white also."
A tall reporter whom Tane recognized from the TV3 news stood up and raised a hand, asking, "So there are men or women, inside the toxic fog, wearing protective suits. Can we assume that these people are responsible for the fog?"
"That would be a reasonable assumption," Crowe answered as coolly as before.
"Then it would be another reasonable assumption that these people are terrorists. Bioterrorists," the TV3 man said.
"Yes. Possibly. Probably. Yes."
"Has there been any kind of demand or ultimatum?" someone else asked.
"No."
There was a strange, stunned silence from the pack of reporters. Crowe stood impassively, waiting for the next question.
"So it is terrorists," Fatboy said calmly.
"Maybe," Rebecca said noncommittally.
"Why?" Fatboy asked. "Why here in New Zealand? What have we ever done to deserve this? What have we done to offend anyone?"
"What if..." Tane started, then paused, thinking for a second. "What if it's a demonstration? What if their plan is to choose a small isolated country, release their toxic fog, and wipe the country clean? Everybody, gone."
"Why would they do that?" Fatboy asked.
"Think about it. What kind of ransom could you demand then? From Australia, Britain, or the USA. They'd say, 'Remember New Zealand. Land of four million people. Now just feral sheep and possums. You're next if you don't pay up.'"
"How do you plan to stop the fog from reaching Auckland?" a reporter finally asked on the television.
Crowe answered slowly. "The answer to that is in two parts. Firstly, we have taken samples of the fog, and we are analyzing it to see what we are dealing with. We hope to find a way of neutralizing it before it gets to Auckland. Secondly is the matter of dealing with the terrorists, with the"—he almost smiled, Tane thought—"snowmen. We have set up a line of defense just north of Orewa. We have taken the high ground of the Waiwera hills and will be aiming to prevent either the fog or the terrorists from proceeding beyond that point."
"Who will be manning that defensive line?" It was an anonymous voice from somewhere in the crowd of reporters.
"My own men, from the U.S. Army Bioterrorism Response Force, along with your Special Air Service and units of your regular army. All will be outfitted with biohazard suits like this one. Eighty of the New Zealand Army Light Armored Vehicles will be deployed along the line, leaving twenty-five in reserve. As you no doubt know, these vehicles are also protected against chemical and biological agents."
"That's a lot of firepower," somebody said.
Crowe nodded. "In addition, we will have air-strike capability from FA18 Super Hornets flying off the USS _Abraham Lincoln,_ which will be within striking distance within three days. Whatever, whoever, is causing this, we will stop them at the Waiwera hills."
"What about the children?" the TV3 man asked. "The ones you have been looking for, from the island."
"We're still looking," Crowe said noncommittally. "We think they may have some information that will help us."
Someone shouted out, "What kind of information?" but Crowe ignored it.
When the press conference finished, Tane turned the sound down but left the television on, in case there were any more developments.
They all sat in silence for a while, until Tane finally spoke.
He said again, "Remember New Zealand, land of four million people...."
**FTBY DNT GO**
**Fatboy went around to Goony's** house to pick up a pair of overalls, and while he was gone, Rebecca came and sat next to Tane. She put her hand on his arm.
"What have we got ourselves into?" she asked.
Tane didn't answer. There was no answer to give.
He covered her hand with his own, and she leaned forward, touching her forehead to his.
"We'll always be mates," she said. "Whatever happens."
"I hope so," Tane said.
Rebecca leaned back a little and nodded. "I know so. I just wanted to make sure you knew too."
"I never doubted it," Tane lied, thinking about the argument on the submarine and feeling more and more guilty about it.
"Friends forever," she said, and sat with him silently for a while before moving off into the kitchen to get herself a drink, leaving Tane with such a warm feeling that it was as if she was still sitting next to him.
_Friends forever. Friends since forever._
The feeling was still there when Fatboy came back with a pair of clean, white overalls, emblazoned with _Telstra-Clear_ across the front and back.
"The genuine article," Fatboy said proudly. "Goony once worked for them."
Tane asked, "Didn't he ask any questions about why you wanted them?"
Fatboy nodded. "He did. The answer was a thousand bucks."
Tane laughed.
"When do we do it?" Rebecca asked. "When do we install the Chronophone?"
"Security is going to be a real problem," Fatboy said. "It's a casino, so they have tight security anyway. These days with terrorist alerts all the time, they are going to look pretty suspiciously at anyone wandering around the Skytower with a suitcase."
"Even in your lovely new overalls?" Tane asked.
"Even in my overalls."
The satellite dish they were going to use belonged to Telstra-Clear.
Tane had carefully stenciled the name of the company on the side of the aluminium briefcase also, so it would look like a toolbox.
"Are you sure you should be doing this alone?" Rebecca asked. "Wouldn't it be safer with two?"
Fatboy shook his head. "We talked about this already. Neither of you looks old enough to be a Telstra-Clear technician."
In some ways, Tane wished he was going. This was the climax of the creation of the Chronophone, the greatest invention since the telephone, or the airplane, or maybe just the greatest invention ever. And he, Tane Williams, had thought of the idea that had started it all. And nobody knew. Maybe nobody would ever know. It seemed wrong not to be there at the critical moment.
In other ways, though, he was glad he was not going. Fatboy would have to take an elevator over two hundred yards straight up, to the main observation deck, then another elevator up another fifty yards to the Sky Deck. Then it was a climb up the internal ladders to the crow's nest, a tiny platform on the _outside_ of the Skytower, three hundred yards high. But even that wasn't the end of it. The Telstra-Clear satellite dish, one of many atop the tower, was another fifteen yards above the crow's nest, accessible only via a ladder up the side of the topmost spike of the tower.
He would have to do all this lugging a heavy metal suitcase. It would take steady nerves and a fair bit of strength.
"Better get on with it," Fatboy said determinedly. Tane sensed that he was more nervous about the climb than he was letting on.
"Good luck," he said as Fatboy climbed into the overalls.
"Final test?" Rebecca suggested.
"Suppose we'd better," Fatboy replied.
They had been testing and testing the Chronophone. The last thing they wanted was for it to fail, once it was high above the ground.
"I'll do it." Tane disappeared into Rebecca's room, where her laptop was sitting on a small study desk.
He opened the small program that Rebecca had written and typed in "good luck Fatboy," then clicked SEND.
The small radio transmitter attached to the laptop would now be sending the message to the receiver built into the Chronophone. Inside the case, a small digital readout would be displaying the characters he had just typed, as it encoded it into the gamma-ray disruptor signal. Right now, that was as far as the signal would go. It needed the big satellite dish on the Skytower to be able to transmit the signal to the gamma-ray bursts.
There was no "Okay" from the kitchen to acknowledge the receipt of the message, so he tried again. "Don't look down," he typed, and sent it.
Still silence from the kitchen, which was a little odd. All the previous tests had worked fine.
He was just about to wander out to see for himself what the problem was, when a flashing light caught his eye on the side of the screen.
_Another message!_
Rebecca's software now checked the NASA site hourly for new BATSE messages and automatically decoded them.
He clicked on the flashing light and it opened the BATSE message window. As usual, it was a cryptic jumble of letters and numbers that they would have to try and figure out as quickly as they could.
**FTBYDNTGO.WTRBLSTMPS.DSVLETHM.**
**SLTABS.DNTABSRB.**
There were still some parts of previous messages that they hadn't fully understood.
**WTRWKS** for example.
He printed a couple of copies of the message on the inkjet, to show it to the others, and as he was doing so, the first nine characters caught his eye.
**FTBYDNTGO.**
He caught his breath and tried to make any other interpretation from it, other than the obvious. **FTBY DNT GO.** _Fatboy don't go!_
"Oh crap!" He grabbed the printout, knocking the chair over in his haste to get out to the kitchen. The second copy whirred swiftly out of the printer behind him.
He rushed down the short hallway and in through the swing door.
The Chronophone was open on the kitchen table, and even from the doorway he could read the words DON'T LOOK DOWN visible on the display.
Fatboy couldn't see them, though. Neither could Rebecca. He had his arms around her, and she had her arms around him, lost in each other's world. As Tane entered, her lips met his.
_Friends forever!_
Any remnants of the earlier warm feeling died a sudden cold, jagged death. His breath caught in his throat, and a black rage that he hadn't known existed inside him welled up from deep within his belly. He forced it back down and coughed, loudly. They both looked up, startled.
"What is it, Tane?" Rebecca asked in alarm, taking a quick step away from Fatboy.
Tane stared at them, breathing heavily through his nose.
"What is it?" Fatboy asked.
He looked at them both for a moment longer. "Nothing," he said tightly. There was a ringing in his ears and spots dancing across his vision. He folded the piece of paper discreetly behind his back and slipped it casually into a pocket. "Nothing." He laughed. "I thought the Chronophone had stopped working, because I didn't hear anything from you two, but I see that it's all okay." He gestured at the message on the readout.
Fatboy looked at the message and laughed.
Rebecca just looked at Tane, in a rather strange way, and said, "We were just saying goodbye."
"Yep," Tane said, "I could see."
**C ANDID CAMERA**
**When Fatboy left, Tane and** Rebecca busied themselves with the supply barge. They called it a barge, but it was really more of a cage. A large plastic-coated wire box with floats attached.
When loaded with supplies, it was just buoyant enough for the _Möbius_ to tow, without dragging the little submarine to the bottom or floating up to the surface.
Fully loaded, it had room for twelve crates.
The crates themselves were watertight plastic boxes, with a rubber seal around the rim, purchased from a local plastics shop. They would not be able to withstand pressure, but they were being stored in shallow water, so that wasn't a problem.
This was the last load. There were already over a hundred crates stacked neatly in rows on the bottom of Rangitoto Cave, as they had come to call it. By Rebecca's careful calculations, there was enough food and fresh water there to last four people for over a year, or six people for at least nine months. Rebecca's mum didn't know it, but she had a berth booked on the submarine. So did Tane's mum and dad, but they didn't know it either.
How do you explain to your parents that the country you live in is about to be devastated and that the only hope of survival is to live in a submarine in an underwater cave for the conceivable future?
This last load was probably the most important. Oxygen cylinders and Sofnolime cartridges. The oxygen cylinders replenished the air in the sub, and the Sofnolime cartridges removed the carbon dioxide that they breathed out.
While they had the air hose up to the surface, they wouldn't use either, but they had planned for a long period of time when they would not be able to draw in air from above the waves.
They would only do that when they were sure the air was safe and clean, and there was no real way of knowing that, so they planned for at least the first few months to be entirely sealed off from the rest of the world.
They worked as a team in the small wooden boatshed but said little. Rebecca loaded the cylinders and cartridges into the plastic boxes, and Tane stacked them onto the barge. He wanted to talk but found that there was little to say. There was a strangeness about Rebecca that hadn't been there before he had walked in on them in the kitchen. It was as if she had something to say but was afraid to.
He thought about showing Rebecca the message, but was too embarrassed. Instead, he tried to ring Fatboy after a while, feeling guilty about letting him go, when the message said not to. Maybe it wasn't too late. But Fatboy's phone rang and rang, then went to the answering service.
It took them over an hour to load up the barge. There were a few spare cylinders that would not fit in the last few plastic crates, so they just loaded them on board the _Möbius._ You never knew when you might need them.
Rebecca flopped, exhausted, into one of the lawn chairs in the backyard of the house and stared silently out over the water toward the city. Tane looked around to make sure they had everything and noticed the laptop still sitting on the outdoor table.
He packed it up and took it down to the _Möbius,_ holding it carefully as he negotiated the wooden staircases that led down to the boatshed.
Halfway back up the staircase he heard the phone ring. He hurried to the top in case it was Fatboy. Rebecca was waiting for him, the phone in her hand.
"It's for you," she said.
"Who is it?"
"I don't know." She shrugged.
So it wasn't Fatboy.
Tane put the phone to his ear but heard only the _pip, pip, pip_ of a disconnected line.
They walked back to the house together silently, uncomfortably. Rebecca's mum popped her head out of the open window of her room.
"There you are!" she called. "I've been looking everywhere for you!"
"Are you okay, Mum?" Rebecca called back, but her mother interrupted her.
"You're on the TV! You and Tane and um...Tane's brother."
Tane looked at Rebecca. Surely she meant the police sketches. But there hadn't been a picture of Fatboy.
Simultaneously, they broke into a run, hurtling through the ranch slider of the lounge to the big TV in the corner.
No more indentikits. The police now had photos. Photos of all three of them. Not sharp, but clear and easily recognizable.
"Where the hell did they get those from?" Rebecca breathed.
"I don't...wait," Tane said. Something about the background of the photos was familiar. Suddenly he got it. The painting on the wall behind them. It was _Tuatara Dawn._
"Oh crap, they're from the security cameras on Motukiekie," Tane said. "We should have thought of that. They have gone back through the old security tapes! Crap!"
"We've got to warn Fatboy!" Rebecca shrieked. "With that _moko,_ they'll recognize him in a second. We've got to stop him!"
"I think we're already too late," Tane said. "We've got to get out of here now!"
_Crap! Why had he let Fatboy go off without telling him about the message?_
"No! We've got to warn him!" Rebecca said, charging down the hallway to her room. "If he doesn't install the Chronophone, there won't even be a Chronophone. Or a Lotto ticket. Or a submarine!"
She grabbed her cell phone off the desk and started to dial, but stopped abruptly. She remained stuck in the doorway as if transfixed by the frame.
Tane caught up with her and stood beside her.
The second copy of the Chronophone message was still on the printer. In his haste and anger, he had forgotten about it.
"There's a new message," Rebecca said in bewilderment. "Why didn't you tell us?"
"I..."
She scanned quickly through the string of letters, then came back to the words at the start.
"Fatboy don't go," she read out slowly. She turned to face him. "Oh my God! You already knew."
She stared at him, her face just a few inches away from his.
"You knew!"
There was nothing to say.
"You knew and you said nothing. You've destroyed it all. You let Fatboy go, despite the warning. Tane!" She screamed his name out suddenly, from close range. Tane recoiled and tried to think of anything that would lessen what he had done.
"This was because you saw me and him together, wasn't it?" Rebecca said slowly. "You were going to tell us and then you saw us, and then you didn't. You stupid..."
Her legs suddenly seemed unsteady, and she took two short steps and collapsed onto the side of her bed.
"Tane," she said softly, "I just broke up with Fatboy. I told you, I was just saying goodbye."
"Oh, Rebecca," Tane breathed. "I'm sorry, I'm so sorry..."
She blocked her ears like a child. "I don't want to hear it!" she screamed. "I don't want to hear it!" She took a long breath and continued, more calmly, "Tane, I've known you all my life, and it turns out that I don't know you at all."
He moved toward her, his arms held out.
"Get away from me!" she screamed, and that was when the roof fell in.
There was a huge flash and an enormous crack of thunder that pulverized his brain. The windows were gone, smoke was swirling around in the draft from outside, and there were men everywhere. Men in black uniforms with black masks and black guns.
In a daze, he saw them dive on top of Rebecca, forcing her facedown onto the bed, her knees on the floor. He thought she might have screamed, but he couldn't be sure. Then the men had him, too, banging his face down onto the carpet, twisting his arms behind him.
The smoke danced around his head, and spots danced around his eyes from the pain of his arms, twisted so high he thought they must already be broken. Rebecca was screaming and someone else was screaming and it was him.
Then everything turned to black.
**Z ETA**
**Tane watched the chimpanzee, watching** him. There were two chimpanzees, actually, but only one was watching him. The other was watching Rebecca. His was called Z2. At least that was what it said on the cage. Rebecca's was Z1.
Z1 was the younger of the two, as far as Tane could tell, but he was no expert at guessing the age of chimpanzees. It was just the wise, almost serene expression in the eyes of Z2 and the way she sat, erectly, regally, with her hands clasped together in her lap. Hers was the face of a sad clown.
Z1 had a more mischievous expression. Impish, if you could call a chimpanzee "impish." Perhaps that would make it "chimpish," Tane thought.
The cages were set up near the center of what should have been the central restaurant of a fancy new hotel, the Hyatt Kingsgate in the seaside town of Orewa, except that the chairs and tables were stacked against the walls.
Not all the chairs and tables. The two chimpanzee cages rested heavily on the polished wooden surface of a couple of tables. Other tables were full of test tubes and petri dishes, being worked on by men in black space suits with open faceplates.
In the very center of the restaurant, three tables supported a long glass tank of some kind, covered with white tablecloths.
Rebecca and Tane sat quietly. The soldiers who had brought them here had not exactly encouraged talking, but Tane suspected that Rebecca would not have had a lot to say to him if it had been allowed. Nothing nice anyway.
They had been brought here half an hour ago. The troopers who had raided their house in West Harbor had brought them directly here, sat them down, and left them in the care of these other men, in the black space suits.
_Where was Rebecca's mum?_ he wondered. _Had she been arrested also?_
The troopers at the house had been New Zealanders. SAS, Tane guessed. But these were not. The murmured conversations all had an American burr to them. Anyway, he recognized the uniforms, the space suits. These were the men they had met on the island.
They waited, and watched the chimps. The chimps watched them back until eventually they tired of that and started watching each other and playing with the newspaper that lined the bottoms of their cages.
Eventually, a door opened and a tall man entered, talking into a mobile phone. Tane recognized him immediately. The leader of the soldiers: Dr. Anthony Crowe.
Two other soldiers entered behind Dr. Crowe. Between them they held the defiant shape of Harley Rawhiri Williams. Fatboy. His arms, like those of Tane and Rebecca, were fastened behind his back by a plastic tie, but his chin jutted staunchly. He walked with his usual swagger, and he still wore his hat. The scientist they had seen on the television, Dr. Lucy Southwell, followed.
Dr. Crowe pocketed his mobile phone and stopped in front of Rebecca and Tane. He opened his mouth to speak, but Fatboy spoke first.
"Sorry, guys," he said.
Tane glanced involuntarily at Rebecca. Would she tell? Let Fatboy know what he had done? She didn't get the chance.
"Found your voice at last," Crowe said amicably. "That's two more words than I've heard out of you the entire time I've known you. Sit down."
He motioned to the other two soldiers, who sat Fatboy down next to Rebecca.
Crowe pulled up a chair, turned it backward, and straddled it, facing them, resting his arms on the chair back.
He pulled Rebecca's notebook from a pocket on his leg and thumbed through it, frowning. Eventually, he closed it and put it back in his leg pocket.
He looked at them for a few moments, then said, with a glance at Southwell, "Cut them loose."
An even taller man with a mop of curly hair and a Texan accent said, "You sure, Stony?"
Crowe nodded. "These kids aren't terrorists. I'd stake my life on it. Him, I wasn't sure about"—he jerked his head at Fatboy—"but these two, definitely not."
The Texan produced a pair of cutters and motioned them to lean forward, so he could reach behind and cut the plastic ties.
"Just promise me you won't go jumping overboard again," Crowe said with something that approached a quick smile.
"Can't promise anything," Rebecca said.
"Well, if you must," he sighed, "but we're two stories up, and I think you'll find the landing a bit harder this time."
"How did you find us?" she asked.
"Fingerprints," Crowe answered.
"From the air bottles?"
Fatboy said, "There was a roadblock at the end of the street. I didn't even have time to call and warn you."
Rebecca glared at Crowe. He picked at something in a tooth with his tongue and stared back at her. Southwell pulled up a chair and sat next to him. After a while, Crowe asked, "Who are you? And what were you doing on the island? Why were you at the research lab?"
"You wouldn't believe me if I told you."
"You have no idea what I would believe. Especially today. Especially here."
Tane could see Rebecca thinking. He wondered if she would tell the full truth. They might not believe all of it. Which would make them not believe any of it.
Rebecca must have thought the same, because she just said, "We were trying to stop the Chimera Project."
"Why?"
_Why!_ He hadn't said _what,_ Tane realized. So he knew about the Chimera Project.
"Because we believed that bad things would come of it, if it was allowed to proceed. That there might be a man-made disaster."
"Do you think this is that man-made disaster?" Crowe asked, waving a hand toward the north.
"Do you?"
He pursed his lips a moment before answering. "I think we have a bunch of bioterrorists loose with some new weapon. What that has to do with Professor Green, I couldn't say. So tell me why you were so worried about the project. Tell me how you knew about the project."
"We met Professor Green on the island a few weeks earlier," Rebecca said truthfully, then lied, "She told us all about it."
Crowe considered that for a few moments.
"Do you believe them?" Southwell asked.
Crowe looked around. "It fits the facts. The fact that they arrived on the island after everyone had disappeared."
He turned back to Rebecca. "I sure would like to know how a bunch of kids could afford that dinky little submarine, though."
"We won the Lotto," Rebecca said evenly. "Where's the sub now?"
Southwell answered, "It's safe. Don't worry about it. We towed it to the naval base at Devonport."
Crowe said, "So you won the lottery and decided to spend your winnings on a submarine, then used it to sneak onto an island and try to break into a genetics lab?"
"Yes. That pretty much sums it up."
"And the cryptic messages in the notebook?"
"Our plans. We wrote them in code in case the notebook got lost."
Good call, Tane thought. The cryptic messages would look like code to an outsider.
"And I'm sure there's a very glib explanation for the strange transmitter in the aluminum briefcase."
"Sure," Rebecca said easily. "We were going to connect it to a mast at the Skytower. It uses gamma rays, which transmit through water. It lets us connect to the Internet, even when we are submerged." She smiled. "Check our e-mail, you know."
It almost seemed believable.
Crowe thought it over. "Gamma rays, huh?" He seemed uncertain. "There are some holes in what you're telling me. Some damn big holes. But overall, I'll buy it."
He took out the notebook and handed it to her. "Seems we got off on the wrong foot. I'm Stony Crowe, from the U.S. Army Bioterrorism Response Force."
"Rebecca Richards." She took the book tentatively, as if there might be a trap of some kind in it.
"Harley Williams," Fatboy said.
"Tane Williams," Tane said in turn.
Lucy Southwell introduced herself.
"Well, seeing as you know so much about the Chimera Project," Crowe said, "perhaps you could enlighten us a little. We've been through Professor Green's notes, but there's nothing to indicate any kind of a problem or how it could be used by terrorists."
Tane started to protest that they knew absolutely nothing about the Chimera Project, but Rebecca shot him a warning glance. An icy-cold warning glance.
"Happy to help," she said.
"Did Professor Green mention anything to you about macroscopic pathogens or bacterial clusters?"
She hadn't. Tane had never heard of either. Rebecca evidently had, though, as she said, "That's impossible."
"So you are familiar with the concept, the theory, of macroscopic pathogens."
"No," Rebecca said, as if it were a stupid question, "but I know what a pathogen is, and I know what _macroscopic_ means. But that's impossible. Surely!"
Tane had no idea what _macroscopic_ meant. He tried to work it out.
Crowe said, "Just because something is beyond the realm of what we already know doesn't make it impossible. You'd better come and look at this."
He stood and led them toward the long glass tank, still covered, in the center of the room.
They stood around the tank as he reached out and pulled off the cover in a single movement. The tank was filled with thick, impenetrable fog. Thick rubber gloves were embedded into the sides of the tank at intervals down its sides.
"You got a sample of the mist," Rebecca said. "Have you analyzed it?"
"Mmmm." Crowe seemed distracted. "Inconclusive results as yet. But that's not what I wanted to show you."
Tane looked deeply into the mist. Was there something moving around in there?
"What, then?" Rebecca asked.
"This."
Crowe placed one hand on the side of the glass tank, then tapped the glass with his free hand. There was a low whistling noise from within the tank, and suddenly, from out of the mist, a shape materialized, flying at high speed toward the palm of his hand. Tane, Rebecca, and Fatboy jumped, and even Crowe flinched involuntarily as the shape slammed into the side of the tank. He withdrew his hand.
"What the hell is that?" Fatboy asked. Tane just stared, openmouthed.
"We call them _jellyfish,_ " Crowe said. "They seem to be attracted by vibrations. Either movement or sound."
Tane could see where the name had come from, although this creature was far smaller than any jellyfish he had ever seen in the ocean. It was about the size of a large bumblebee, a bulbous shape made of a gelatinous, translucent material. The main body seemed to be in three parts, forming a Y shape, with a mass of thin fibrous tentacles trailing underneath.
"You caught that in the fog?" Rebecca asked.
Crowe shook his head. "Nope. We just sucked up a sample of the fog and released it into this tank, to help us study it. There were no jellyfish in it then."
"Then how...?"
"They just formed, out of the mist. Little dense patches at first that gradually got bigger."
"Holy crap," said Fatboy.
Rebecca said incredulously, "You're not trying to tell me that that is a macroscopic pathogen."
"What on earth is a macroscopic pathogen?" Tane asked.
Crowe looked appraisingly at him. "A pathogen is an organism that attacks another larger organism. Like bacteria or a virus attacks the human body. All the pathogens we know of are microscopic. Too small to be seen with the naked eye."
Southwell added, " _Macroscopic_ means large enough to be seen without a microscope."
Rebecca scoffed, "You're not trying to tell me that this creature is some kind of giant virus!"
Crowe almost smiled, just a brief twitch at the corners of his mouth. "A giant virus? No. Viruses are subcellular. Smaller than a human cell. They have to be. They crawl into cells to attack them. No, not a giant virus."
The small jellyfish-like creature drifted slowly away from the wall of the tank, losing definition gradually in the mist.
Crowe continued, "I attended a lecture a few years ago. At Oxford. A Doctor Hans Heinrich was the lecturer, a highly respected immunologist. He hypothesized the existence of macroscopic pathogens. Not viruses, but bacterial clusters."
He paused and looked around the little group. "Bacteria are single-celled organisms. But if you grow a whole lot of them together, then they form a colony, or cluster together, in what we call a _biofilm._ And a bacterial cluster can show characteristics quite different from those of a single bacteria. They exchange chemical signals between cells, and the cluster itself can grow into a quite specific shape. We see wave patterns, towers, and other structures. Dr. Heinrich suggested the existence of bacterial clusters that behaved as a single organism. Perhaps as large as a grain of salt. Thousands of individual bacteria, acting in concert. A single macroscopic pathogen. Invading the body, then overwhelming its defenses by the sheer volume of the bacterial cells released. To the best of our knowledge, that is what we have here."
Fatboy mumbled, "It's a bit bigger than a grain of salt."
Rebecca said, "You're not trying to tell us that these terrorists, these 'snowmen' in the fog, have developed bacterial clusters that are trained to attack humans."
Crowe shook his head. "They're not 'trained' to attack humans any more than a cold virus is 'trained' to attack us. It's just what they are. It's what they do. But we can't find any reference to bacterial clusters, or anything even remotely connected to it, anywhere in Green's journals. That is where I was hoping you might have a little more inside knowledge."
One of the other soldiers approached and said quietly, although for little purpose, as they could still hear every word, "We are ready for the test now, Doctor."
"Then get on with it," Crowe said.
"Z1 or Z2?"
Crowe shrugged. "Whichever."
"They are living creatures. Why do you call them by numbers?" Rebecca asked. "Why not give them names?"
"They're not pets," Crowe replied curtly. "Pets have names. These are lab animals."
The Texan opened one of the animal cages and the older-looking chimp, Z2, jumped out with a squeal of delight and began tousling the tall man's hair.
He smiled and Tane laughed.
"She's got character." Rebecca smiled. "I'll name her for you." She thought for a moment. "Z-two, zeto..."
"Zeta," supplied Tane.
Rebecca looked at him for a moment, before accepting it.
"Zeta," she said. "Hi, Zeta!"
Zeta looked over at her and held out a hand as if she would like to jump onto Rebecca, but the Texan held the animal firmly.
Crowe did not find it funny. "They're not pets," he repeated.
"Makes it harder to stick electrodes in their brains and vivisect them, doesn't it," Rebecca said with a little-girl innocence completely at odds with her words. "How about the other one, Z-one?"
"Xena," suggested Tane.
"Zeta and Xena," Rebecca declared. "And what little test have we got lined up for you today, Zeta?" She held out a hand to Zeta, who patted it and looked up at her with big, sad clown eyes.
"We're going to put her in the tank," Crowe said flatly, to Rebecca's look of horror.
The end of the tank was a separate box, sealed off from the rest of the tank by a glass door with thick rubber seals.
They let Zeta climb in the box, which she did willingly, trustingly, and then sealed the lid above her. In the main compartment of the tank, there were whistling, swishing noises as the jellyfish, agitated, swept around in circles in the mist.
Zeta jumped a little and turned around and around inside the small area, but otherwise didn't seem too concerned about being shut in a glass box.
"You can't!" Rebecca said, again and again. "You can't put her in the tank with that thing!"
Southwell, looking quite uncomfortable, tried to explain. "She'll give us very valuable data. Chimps are our closest cousins."
Manderson contributed, "Genetically, they are ninety-nine percent the same as humans."
"Don't flatter yourself," Rebecca muttered, but the insult went straight over Manderson's curly head.
Crowe said, "We need to know what these pathogens can do to us."
"And you need to sacrifice an innocent animal to find out?"
"It's just one chimp," Crowe said with a look of annoyance. "Fifty thousand human beings got 'sacrificed' in Whangarei. And there'll be more if we can't figure out what is going on."
With a nod from Crowe, Manderson released the seals and Zeta's compartment flooded with the fog.
"No!" shrieked Rebecca. She pressed her fingers against the glass by Zeta's head. Zeta looked at her and gave her a big clown smile. The other USABRF men gathered around to watch.
Zeta seemed bemused by the fog at first, as it started to fill her chamber, then a little confused as it thickened around her. The jellyfish whistled around in the thick of the fog, avoiding the thin vapor at the other end.
"They can't exist outside of the fog," Crowe murmured, watching intently. "They can't move, they can't live. It's their nutrition and their locomotion."
One jellyfish flashed down the side of the tank near them. By now the fog had spread evenly between the two partitions. The jellyfish whirled around Zeta, then disappeared back into the fog.
That was all. Nothing else happened.
After a while, Zeta went for a walk. Tane held his breath and could hear from the sharp intake of air from Rebecca that she was doing the same.
Zeta wandered through the main tank, comically trying to wave the fog away from in front of her eyes. She found one of the jellyfish, drifting at about her eye level, and Tane winced as she stretched out a hand toward it.
The jellyfish remained motionless. She even batted at it with her hand, swatting it like a fly, but without effect.
"It's not interested in her," Southwell said with a puzzled look.
"Okay, that's long enough," Crowe said.
"Go, Zeta!" Rebecca yelled in a mixture of delight and relief, hopping from one leg to the other and punching the air. "You go, girl!"
Zeta screeched and danced happily inside the cage, a little two-legged, two-armed Irish jig. Tane and Fatboy laughed, but Crowe just shook his head.
They reversed the procedure with the small compartment at the end of the tank, sealing off the main section before pumping in air and extracting the fog.
"Are you going to let her out?" Rebecca asked.
"She may be contaminated," Crowe replied, then, a little too quickly, said, "Dr. Southwell, would you show them the journals?"
Southwell led them to the far side of the room, where a series of notebooks were spread out on a table.
"Professor Green's notes," she said. "Do you know much about what they were researching?"
"Tell me," Rebecca said. "Vicky told us it was rhinoviruses."
"Actually, it was rhinoviruses they were researching. They did a small amount of work on NLVs, but only for a short time, to confirm some aspect of their main research. They were researching conserved antigens. That's common structures within the viruses that can—"
"She told us about that too," Rebecca interrupted. "How much do you know about the Chimera Project?"
Southwell said, "Conserved antigens proved to be elusive. Our immune systems just kept getting fooled by the changing shapes of the viruses. It was a dead end."
"So?"
"Professor Green recently gained health department approval to experiment on the other side of the equation."
"The human side of the equation?"
"Yes. They were playing around with bone marrow, where antibodies are produced, genetically engineering our immune systems to try to produce a generic antibody."
"An antibody that would recognize any kind of virus."
"Any kind of rhinovirus. That was the field they were concentrating on."
"And how," Rebecca asked, a little skeptically, "do you start to create a generic antibody?"
"The scope of the project approval was quite specific. They were genetically splicing together different kinds of antibodies, creating a..." She trailed off, seemingly unwilling to finish the sentence.
"A chimera." Rebecca finished it for her. "Is that it? Is there anything else you can tell me?"
Southwell shook her head. "Nothing. Professor Green had not yet submitted a report on the results of the research. All we have is her notes." She indicated the table again. "Would you mind having a look through? Tell us if anything sticks out."
"Of course," Rebecca said, and picked up the first journal.
Tane idly leafed through one. Rebecca was the only one with a hope of understanding them, but it was interesting to see the clearly handwritten notes, dates, and formulas that the late Professor Green had written. Vicky's handwriting was small, neat and verbose, flowing on, page after page. Tane idly wondered why she hadn't just typed up her notes on a computer.
Fatboy was the first to notice, glancing across at the other side of the room. "What are they doing?" he wondered.
Tane looked across, and Rebecca also. Two of the men had their hands in the gloves in the sides of the compartment. One was holding Zeta while the other ran a needle into her arm.
Zeta didn't like it; she screeched and snarled at the men.
"They'll be taking a blood sample," Rebecca said. "To see if the fog had any effect on her."
She looked back at the journal she was reading, only to look up again a second later to check on Zeta.
The man with the syringe had withdrawn it, but it was empty. With a frown, Rebecca walked across the room to the tank. Zeta looked sadly, imploringly, at her from inside the glass.
"What are you doing?" she asked. "You're taking a blood sample, right?"
Crowe was there now. "Miss Richards, this is our work. I'd like you to let us get on with it."
"That was a blood sample, right?"
Inside the tank, Zeta began to shiver. She sat down suddenly and looked up at Rebecca like a frightened child. The top lip of her clownish mouth drew up into a sneer.
Southwell put her arm around Rebecca's shoulders and tried to steer her away. She shook the arm away violently.
"What have you done to her?"
Zeta slumped against the side of the tank, her eyes open. Her breathing was ragged, heavy. She looked up at Rebecca one last time; then her eyes just glazed over and remained open. Her chest was still.
Crowe took Rebecca's arm this time, firmly, and drew her away from the tank.
Rebecca cried, "You killed her! What are you going to do, dissect her to see if the fog has affected her?"
Crowe said nothing.
"You are! You monsters!"
"Monsters?" Crowe hissed, the stony façade cracking for the first time. "Monsters!" He grabbed Rebecca by the back of the neck and pressed her face against the glass of the large tank. There was a whistle and a flash of white fog, and two of the jellyfish smashed into the glass, just a few thin fractions of an inch away from her eyes and mouth. She screamed. Tane jumped forward, and Fatboy was with him, but strong arms gripped their elbows, pinning them.
"There are your monsters! We don't have time to wait and see how the animal is feeling in a month or so. We have just a few days before the fog hits Auckland. We need answers right now!"
"Murderer!" Rebecca whispered, sobbing, her lips crushed against the glass.
Manderson reached in with the rubber gloves and laid the body of the chimpanzee flat on the floor of the tank. In death, Zeta had found peace once again. The forced sneer was gone, and her face held its natural sad-clown expression.
Crowe looked at Rebecca with stony eyes. "I told you not to give it a name," he said.
**X ENA**
**Rebecca could move pretty quickly** when she wanted to. The soldiers should have known that already. But they were all gathered around the tank.
Rebecca was at the cage, unsnicking the lock, and Z1—Xena—was in her arms before anybody realized what she was doing.
"Put the animal down," Crowe ordered, and he was clearly a man who was used to being obeyed.
Rebecca was a girl who was used to disobeying. "You've murdered one innocent creature today; you're not getting this one too."
Crowe moved toward her. Manderson and another soldier, Crawford, according to a badge on his helmet, moved slowly in behind her. She edged to her side. "Leave us alone!" she yelled.
"Leave her alone," Southwell said. "Let her calm down."
Crowe almost looked as if he were considering that for a moment, then he said, "No, we don't have the time. I had hoped that they might be useful in understanding Professor Green's work, but so far they've been just an obstruction."
He spoke to Crawford, behind Rebecca. "Take the animal off her and get all three of them back to Auckland." He turned to Southwell. "Get them charged with trespass or...something...so your police can keep them out of our hair."
Crawford nodded and circled around Rebecca to grab Xena. Manderson gripped her shoulders from behind, despite her shaking and struggling. Crawford put his hands under Xena's arms and began to pull. Rebecca held on tightly, and the chimpanzee, sensing the struggle, hugged her tightly back. Crawford was just trying to pry Rebecca's hands from around the chimp's back when Tane hit him broadside.
Fatboy had played rugby league in school and had a pretty tough reputation from years of tackling huge front-row forwards. Tane had never played rugby in his life. But he'd been to plenty of games to watch his brother play, and he hit Crawford in a textbook midriff tackle.
It cut him in half, knocking him to the floor and sliding both of them across the room toward the tables holding the tank. Crawford's back hit one of the table legs and the whole structure shuddered.
"Look out!" Crowe yelled.
Manderson, with incredible reactions and speed for a big man, was at the side of the tank before there was any danger of it falling, steadying it with two hands.
Crawford leaped up and angrily hauled Tane to his feet, his arm swinging back in a fist. Tane, facing the side of the tank, brought his hands up to protect himself and had just enough time to wonder, _Where are the jellyfish?_ when Southwell said, "Something's happened in the fog tank, Stony."
Crawford's arm remained tense but his eyes were on the tank. Everybody's eyes were on the tank.
The fog was swirling. Not just quivering a little from the knock to the table leg, but twirling and swirling in broad patterns inside the tank.
"Where are the jellyfish?" Manderson asked in his slow Texan drawl, echoing Tane's thoughts. His hands were on the glass sides of the tank, but the jellyfish had not attacked.
A young-looking soldier, whose name badge read EVANS, said from the other side of the tank, "There's something new forming over here."
The chimpanzee was forgotten, Rebecca was forgotten, and Tane was forgotten. Crawford's arm dropped as the men crowded around the tank.
Tane moved to where he could see. It was a larger object, not really a definable shape at all, just a mass of a white substance, more visceral than the jellyfish, with a moist, diaphanous surface, like that of a slug.
"The fog is thinning," Crowe noted. "It is being formed out of the fog."
Tane watched as the shape grew a fraction of an inch or two in diameter in front of his eyes.
"Why now?" Crowe asked, as if he had been expecting this to happen sooner or later.
"The knock to the tank?" Crawford asked.
"No. Look at the size of it. It's been forming for quite a while," Manderson pointed out.
They all watched in silence for a few moments, but the shape grew no farther.
"It's run out of fog," Crowe decided after a while. "It needs more fog to grow. But why suddenly now? What has happened to trigger that growth?"
He sounded concerned.
"Do you want to run the same tests?" Crawford asked, glancing at Rebecca and Xena. Rebecca recoiled, taking a few steps backward and twisting Xena around behind her as if to protect her.
"I don't know," Crowe murmured, not taking his eyes off the glutinous shape. "Start with the human cell tests."
Rebecca was ignored and drew closer to the tank again, intrigued, as two of the soldiers put their hands in the thick rubber gloves on the side of the tank. The big white slug quivered, but did not move.
A small petri dish was introduced through an air lock at the end of the tank, and one of the soldiers opened it carefully inside.
The fog was a lot thinner now, and it was easier to see what was going on. Inside the petri dish were a few small objects that took Tane a moment or two to recognize. There were a few hairs, human he realized, and some nail clippings. Evans picked up a hair out of the dish and dropped it carefully onto the white shape.
It touched the surface of the blob and just disappeared. At first Tane thought it had sunk into the material, but then he realized that it hadn't. It had just "melted" on the surface.
The fingernail clippings followed, with exactly the same effect.
"Test the pH," Crowe said.
Evans tore open the plastic seal on a small cardboard strip already placed inside the tank and touched it to the surface of the object. After a moment he said, "Neutral. Just slightly alkaline."
"So it's not an acid," Crowe said, deep in thought. "But it dissolves human cells."
"Like butter on a hot griddle," Manderson drawled.
Rebecca, Tane, and Fatboy might as well have been invisible, so little attention was paid to them.
The soldier-scientists spent the next hour running batteries of tests on the small white blob, but always, Tane thought, with inconclusive results. At least that was the impression he got from their expressions as they discussed, in highly scientific terms, the results of each test.
One thing was clear to Tane, though. This substance, whatever it was, was related to the disappearance of all those people on Motukiekie and Whangarei.
A fax machine set up on a table in a corner rang while the tests were going on, and Southwell went to check it. She came back with a concerned look on her face.
"Stony," she said, "the fog is moving much faster than we thought."
Crowe looked at the printout, a detailed weather satellite image of the area. His eyes opened wide. "Warkworth! We hadn't expected it to get that far south for another couple of days. It's accelerating! At this rate, it'll be here by tomorrow."
"Let's hope it doesn't get any faster," Manderson said slowly.
Tane, Rebecca, and Fatboy looked at each other in alarm.
"Had they completed the evacuation?" Crowe asked.
Southwell said, "Yes. Orewa also, and they're starting to evacuate Torbay, Albany, Greenhithe, and Helensville."
Those three towns represented the northernmost tip of Auckland. The fog was getting close to the homes of a million people.
"When were these taken?" Crowe asked, scanning the image for a date and time.
"This morning at first light."
"First light! Get back to the meteorological people. I need to know how far south it's come since then." He spun around to Manderson. "Where are the SAS and NZ Army units?"
"Base camp. Silverdale. Just down the road."
"Pull them back to Albany. We'll never have time to set up the defensive line at Waiwera before it gets there."
He looked back at the tank. "Damn! I was hoping for much more time than this. Get the equipment packed up, and get the men ready to evac. That fog is just up the road and heading this way. I don't want to still be standing around, running tests, when it gets here."
There was a flurry of activity as the soldiers worked to pack up their equipment, disappearing, probably down to the big black trailer units that Tane had seen parked outside.
Only the main tank remained, and some odds and ends of equipment, lined up by the door, when Crowe looked across at Xena, still perched in Rebecca's arms. Rebecca was sitting on a chair on the far side of the room having a long conversation with the chimp about the _Möbius_ submarine. Xena seemed interested, interjecting occasionally with screeches and sudden gestures of her own. At one point, she started looking through Rebecca's hair, apparently looking for nits, but (fortunately) didn't find any.
Rebecca noticed Crowe's gaze and instinctively drew away.
"What do you want to do, Skipper?" Crawford said.
"Evacuate the kids back to Auckland. We'll put the chimp back in her cage and take her with us."
"Liar," Rebecca spat, "I know what you're going to do with her, and you're not getting her. You're going to put her in the tank with that white blob to see what it does to her. I'm not going to let you."
Crowe sighed tiredly. "Sort it out, Mandy."
The soldiers were all gone now, packing their trailers, except for Manderson and the young man, Evans. The tall Texan didn't waste time with threats of physical violence. He just reached under a table and brought out his weapon, one of the long, strangely rounded guns they had seen earlier.
"Hand over the animal, ma'am."
"Shoot me!" Rebecca shot back. "You big brave American GI. Just shoot me!"
"Don' wanna," the lanky Texan drawled. "Will if I have to."
Tane stood and moved next to Rebecca. She glanced at him, appreciating his presence, he thought. He sensed a movement behind him.
"Give me the chimp," Manderson asked politely.
"Give him the chimp," Crowe roared suddenly, and Xena screamed.
"Dr. Crowe!" Southwell pleaded.
"Leave her alone," Tane shouted. "Leave us alone!"
Manderson's rifle drifted toward Tane. For the first time in his life, Tane stared straight at the small black circle that was the mouth, the end of the barrel of a gun. Just a tiny amount of pressure on the trigger at the other end of the barrel was all it would take. Such a small movement of one finger, and...He shut his eyes.
He opened them again as he felt a hand push him to one side.
Fatboy was an imposing figure for a seventeen-year-old. Tall, strong, toughened by years of rugby league, and the cowboy hat added even more height. The _moko_ seemed suddenly terrifying on the face of the warrior who now stepped in front of Tane. Fatboy's knees were bent, his back rigid, his chest puffed out. Tane had seen him act tough before, but this was something more. This was something deeper, something ancestral. Fatboy's eyes burned and his tongue stabbed at the soldiers. He smashed his hands into his chest.
_"Ka mate! Ka mate! Ka ora! Ka ora!"_
Fatboy faced the soldiers and their deadly weapons with the spirits of ancient warriors on his shoulders. Crowe and Manderson froze in the face of such ferocity. They had faced lethal viruses and terrorists with guns, they had faced hell, and they had faced death, but never before had they faced up to Fatboy and his _moko._
Xena was still screaming, lips flared, jumping up and down on Rebecca's lap as Fatboy continued the _haka._
_"A upa...ne! ka upa...ne! A upane kaupane whiti te ra!"_
"Leave us alone!" Tane yelled.
"Stony!" Southwell shouted.
Xena screamed again and broke free of Rebecca's grasp. She ran across the floor of the room on her hands and feet. Long drapes covered the windows and she lunged at them, tearing at the fabric. She screamed and screeched in alarm.
One of the drapes tore from its hooks, falling away from the window even as the three soldiers froze, listening intently to their earpiece radios.
Tane didn't need to hear the message to know what was being said. Through the torn drape, fog whirled and swirled around outside the second-story windows of the hotel.
Then came the sound of gunshots.
**S HAPES IN THE MIST**
**"Get them into biosuits," Crowe** ordered, his face tense.
Gunfire sounded again from outside the hotel. A few sporadic shots, followed by long bursts of automatic fire.
Southwell wasted no time. She opened a suitcase from a small stack near the door and motioned to Tane and Fatboy to do the same.
"It's a pressurized suit," she said. "You'll feel it inflate when you seal the faceplate."
She put on her own suit, then helped Rebecca into hers. Tane and Fatboy imitated as best as they could. She showed them the earpiece radios and how to strap the throat mike around their necks.
"You want to talk, press here," she said, indicating a small button on the outside of the suit, near the neck. "But don't use the radio unless it's important."
She connected her air hose and closed her face mask with a click, then helped Tane with his.
The moment Tane plugged the small earpiece into his ear, he was suddenly immersed in the battle that was raging outside. A cacophony of orders, shouts, and cries of alarm. The gunfire was constant now.
Crawford's voice was recognizable. He seemed to be coordinating the battle outside.
"Don't mind the jellyfish," he was shouting, the voice tinny but surprisingly real in Tane's ear. "They can't get through the suits. Watch out for the big ones!"
_The big ones! The big ones?_
"Crowe, this is Crawford, I've—"
The voice was cut off suddenly amidst a sustained burst of gunfire.
All three of them were in their suits now, and Tane realized that Crowe was shouting at them.
"Get behind us. We're going down the main stairway. We need to make it to one of the trailers!"
Xena leapt into Rebecca's arms.
"Let me," Fatboy said, and Rebecca passed the chimp over gratefully. They had to move fast, and Fatboy's strength was going to be needed.
"Leave the chimp behind," Crowe ordered, but Fatboy ignored him.
The sounds of shouts and confusion intensified on the radio.
"Fall back, fall back to the trailers!" That was Crawford's voice again. "Try your sprayers. The bullets don't bother them; just cut straight through them!"
_Cut through what?_
"Crawford, this is Crowe. What's going on out there?"
They were already moving, out through the double doors of the restaurant and toward the main staircase. The fog had poured in through the main doors to the hotel, flowing over and around the reception desks and rolling up the staircase, creeping up the stairs, one by one.
"They're all over the place, Stony."
"Crawford! What are all over the place?"
"The...Oh my God! Oh my God!" Crawford's voice again, desperate, despairing. The voice cut off suddenly.
"Repeat that, Crawford!"
The radio remained silent, but from outside they could still hear the sounds of firing.
"What the hell is going on?" Crowe raced along the short passageway to the stairs, close on the heels of Evans. Manderson and Southwell followed, and the three kids raced behind her as quickly as they could in the bulky, armored biosuits that were far too large for Rebecca and Tane and were awkward to move in.
The fog was halfway up the long curved staircase and climbing rapidly. The first yard, maybe two, was soft and transparent, but then the fog intensified into a dense cloud.
Crowe turned to face them. "The trailers are outside the door to the left."
Even as he said it, they heard one of the truck engines start.
Crowe continued, "I don't know what we're heading into, but we'll try to deal with whatever it is. You kids head straight for the trailer. It's armored. Once we reach it, we'll head south, out of the fog. Are we clear?"
"Clear!" Tane and Fatboy said, but Rebecca drew in a sharp breath.
"Don't go mist," she said with a quiver of terror in her voice. "Don't go in the mist."
"DNT GO MST," Tane remembered, and realized the message had had nothing to do with Masterton at all. _Don't go in the mist._
"It's our only way out," Crowe snapped. "We've got to go through the mist to get to the trailers."
"Don't go in the mist!" Rebecca screamed.
Crowe shook his head. "Evans, you're on point. Manderson, tail-end Charlie. Get moving, now!"
Rebecca didn't move. Tane was already three or four steps down, following Southwell who was following Crowe, when he realized that she was still on the top landing. The mist swirled around him, light at first but intensifying.
Evans, a couple of steps lower, disappeared into a cloud of the dense fog ahead.
"Stay close," Crowe ordered.
Manderson, realizing that Rebecca had not followed, retreated back up to the top landing and grabbed her by the arm. "Get moving," he shouted, pushing her down the stairs. She tripped and fell, sliding face-forward down past Tane and Fatboy, and stopping herself a step below Crowe. Her body half disappeared into the thicker fog ahead.
There was a sudden hissing sound, and Evans's voice came back to them, not in words, but in a strange strangled gurgle. Then silence.
There was a thud, and his weapon hit the stairs, the end of it just visible and protruding out from the fog.
Rebecca screamed and scrambled backward up the stairs, pushing herself up with her hands and feet.
"What is it, Rebecca?" Tane yelled. "What happened?"
"I didn't see," she screamed, turning and running back up the stairs. "He just disappeared!"
Tane turned and followed her, and realized that the others were with him. All of them.
They raced back into the restaurant. Silent now. The radio, too, was silent.
Crowe and Manderson turned in unison and slammed the double doors shut. Crowe used something on his weapon to coat a dense foam around the edges of each of the doors.
"What was that?" screamed Rebecca. "What's out there?"
"Calm down," Crowe shouted, not too calmly himself. "Calm down," he repeated with a bit more control. "Whatever it is, it can only survive in the fog. That foam will prevent the fog from getting in here."
Tane looked at the irregular pattern of the foam. It looked like gray-colored shaving cream. Or icing on some bizarre cake.
"What happened to Evans?" Manderson asked, checking the safety on his weapon.
"Don't know," Crowe said.
"Crowe, this is Miller," a voice on the radio said now.
"Go ahead, Miller. What is your status?"
"I have nine men with me, no injuries. We are in trailer two, heading south, out of the fog. What is your position?"
Crowe looked at Manderson before saying, "We are secure. We are in the forward command post. We have sealed the doors to prevent the fog from entering. We can probably hold out here for a while. What the hell is going on out there?"
"I wish I knew." The voice on the radio sounded frightened. "Some kind of...creature...I...don't know. But whatever they are, they are big, and they move fast, especially where the fog is thick."
Crowe said, "Okay, Miller. Keep moving. Get yourself clear."
"Roger that, Stony. We'll regroup, then come back for you."
The radio went silent once again.
"Nine men," Manderson drawled. "Plus Miller. Plus two of us."
He didn't need to say any more.
"Let's hope they made it to the other trailer," Crowe said without conviction. "Are there any other entrances to this room?"
"Fire escape."
Crowe and Manderson ran to the rear of the room to seal that door. They were just returning when Tane noticed a whisper of mist trickling in through a gap in the seal on the front doors.
"Dr. Crowe," he said urgently, pointing out the thin plume of steam.
Crowe nodded and moved forward to plug the gap with more foam.
He was only a few yards from the door when it exploded.
For just half a second, Tane thought he saw a vague white shape at the toughened glass of the door, as if something had charged at the door through the fog on the other side. There was a hissing noise also, and then the entire door just shattered into tiny squares like the glass from a car windshield.
Pieces of glass smashed into his biosuit, and he thought for a moment that Crowe, who had been much closer, had been cut to shreds by the flying shards that enveloped him. But the biosuits were made of some toughened material, designed to stop bullets, and the glass fragments bounced harmlessly off.
Fog poured into the room.
"Get back!" Crowe yelled, running back and pushing Rebecca by the shoulder. "To the fire escape."
"What was that?" Manderson shouted. "That thing at the door."
Crowe had no answer. He just said, "Try your sprayer on them. Bullets don't work!"
"Water works," Tane said, without fully realizing what he had said. _"Water works!"_
Crowe looked at him without stopping. "What does that mean, son?"
"I don't know, but water works."
They hit the door to the fire escape and burst through it, scattering the gray foam that Crowe and Manderson had carefully sprayed there a moment ago.
It was a narrow concrete staircase, flight after flight of featureless stairs with a red metal railing.
The fog was lapping at the boots of the biosuits and rising up around their knees.
"Up," Crowe shouted, bounding up the stairs.
They couldn't exactly go down, Tane thought.
He took two steps and then stopped dead. Manderson, behind him, collided with him but didn't stop and brushed past.
Tane turned and ran back into the room.
"Tane!" His brother's voice. "Where are you going?"
"The Chronophone!" Tane called back. It was in a line of equipment by the double doors, waiting to be packed out to the trucks.
If he had stopped to think, given himself a chance to be afraid, he never would have done it. But he hadn't. So he did.
The fog was still thickening inside the room. The walls faded before his eyes as it intensified.
The jellyfish stayed in the dense fog, he remembered, and hoped the same was true for the _Big Ones._
The Chronophone was where he remembered it, and he grabbed it with a snatch. It was heavier than he remembered and he stumbled but kept his feet.
The fog was alive, moving and scything around in broad patterns, and there were things in the center of those patterns.
From all around came strange hissing noises as whatever they were moved swiftly through the thickening fog.
The mist had risen up the stairwell also, but he cleared it in a single flight of stairs. The others were waiting for him, just out of the white clouds.
"Get moving," Crowe said in a gravelly voice, and that was all that anybody said.
The six of them, seven if you counted the chimp, flew up two flights of stairs and looked down from the next landing. Still the fog was rising up the narrow shaft.
"How high is this hotel?" Crowe asked.
Nobody knew the answer.
The next two flights of stairs were harder, but adrenaline gave them wings. Two more stories, though, and Tane could see that Rebecca was flagging. His own legs, unaccustomed to carrying the weight of the biosuit and the Chronophone, were rebelling also. Southwell seemed unaffected, and Fatboy, even with the weight of Xena, hardly seemed to have raised a sweat.
Looking down, Tane could see that the fog was swelling up the narrow concrete shaft after them. Slowly, inexorably rising up the stairs.
If they couldn't get above it, Tane realized, it was the end of everything.
Two more stories and they came to the final flight of stairs. Above them a cold, hard concrete ceiling blocked their escape route.
Crowe looked down at the rising mist. "Through here," he said, flinging open the door to a rooftop area, where a few comfortable loungers skirted the edges of a long rectangular swimming pool. The afternoon sun burned into a light haze over the area, a thin spread of fog. Around them the rest of the world was white. They were a small concrete platform adrift in a sea of cloud.
"Are we high enough?" Tane asked. No one answered.
The mist around them began to intensify, rolling over the edges of the concrete parapets of the rooftop area and falling out of the doorway behind them.
"Miller, are you still there?" Crowe called.
"Roger that."
"We've had to DD. We are now on the roof of the hotel. We are in extreme, I say extreme, danger. We need evac now."
"Cannot help. I repeat, unable to assist. We have just cleared the fog and are proceeding south to the new command center at Albany. Will contact the Kiwis for you and see if there is anything they can do."
"Roger that." Crowe looked around grimly. He was hard to see, even in the full light of day, thanks to the thickening mist.
"What do we do now?" Southwell asked.
"Sit tight," Crowe said. "Sit tight and pray."
The jellyfish came first. Flying through the thickening fog. The harsh whistling sound they made was the first indication that something else, besides the six humans and the chimpanzee, was alive in the fog.
"Try not to move," Crowe said. "They are attracted to movement and sound."
Even as he said it, it was clear that it was useless. The biosuits themselves gave an audible click and a hiss with every breath they took.
It didn't take the jellyfish long to find them.
One landed on Tane's arm, and he watched it, horrified, yet fascinated for a second as it extended its long filaments and probed the black armor of the biosuit, trying to find an opening. The suit was strong enough to withstand it, though, and he flicked it away with a yelp of disgust.
It was back a second later, though, so he squashed it, flattening it with a sharp slap. It fell away into the mist.
The next one he squashed stayed there, stuck to his suit, but when he looked back at it a moment later, it was mostly gone. Dissolving, it seemed, back into the fog.
More came, though, and more still. He slapped at them, smashed them, shudders running through his body at the thought of the fine tentacles needling their way inside the suit. He looked at Rebecca's back and realized with a shock that her black biosuit had turned white. Her back was covered with the jellyfish, the odd Y shapes fitting into each other like pieces of a jigsaw puzzle. Smothering her.
He forgot his own creatures for a moment and began hammering on Rebecca's back, screaming wildly as the creatures went flying, unable to get a grip on the smooth surface of the suit.
"The jellyfish can't penetrate the biosuits," Crowe said calmly.
"It's not them I'm worried about," Fatboy muttered.
There was a swirling in the fog near the door to the fire escape, and Tane thought he glimpsed a white shape through the mist.
"Here they come," he breathed.
"The pool," Rebecca suddenly said. "Water works. Get in the pool!"
"Everybody in the pool," Crowe ordered. "Now!"
"What about Xena?" Rebecca asked, but a pair of hands cut short the argument, shoving her violently in the back, toppling her headfirst into the water. She didn't see who did it, but Tane did. It was Lucy Southwell.
Tane let himself over the side of the pool and splashed into the deep water. He sank like a stone and fought a rising panic, until he realized that he was in a fully self-contained suit, with its own oxygen supply. He hoped the metal case of the Chronophone really was watertight.
Fatboy was talking, but Tane could not hear him. Crowe motioned them all into the center of the pool and held out a hand. Manderson laid his hand on Crowe's, and after a slightly confused moment, the rest followed.
Crowe's voice sounded suddenly in Tane's ear. "The radio signals don't travel underwater. But if you touch one of the other biosuits, the signal will travel directly from one to the other."
"The entire suit acts as an aerial," Manderson explained.
"Still think it's terrorists, Dr. Crowe?" Rebecca asked, a little cynically.
Crowe ignored her.
"How do you know we'll be safe in here?" Fatboy asked.
Crowe replied immediately, "They can only survive and move in the mist. You kids were right. The fog can't penetrate the water, so the creatures can't either."
The suits were all black again. The jellyfish released their hold the moment they hit the water and floated to the surface. They bobbed around there for a little while, hundreds of them, and gradually disappeared.
It was left to Tane to ask the next obvious question as vague shapes moved around the sides of the pool and across the surface of the water above them. Large, whitish shapes, indistinct and blurred through the water of the pool.
"How much oxygen do these tanks have?" he asked. "How long can we stay down here?"
**E PIPHANY**
**The water above Tane rippled** with the passage of one of the— _what were they?_ —above him. The sun, still high in the sky, diffused down through the opaque whiteness of the mist above, then softened and soothed further by the wash of the pool water into the dull brightness of a child's toy lamp.
When one of _them_ passed over the surface of the water—never breaching the surface—the resulting ripples created undulating patterns over the light blue walls and floor of the pool.
Tane sat with his back to the pool wall and watched the soft light play over the faceplate of Rebecca's suit. Here under the water, the tinted faceplates turned to mirrors, preventing any glimpse of the face inside.
She could be smiling at him. She could be scowling. He had no way of knowing.
What were they? The _snowmen._ The human mind always tries to rationalize things. To fit what it sees to what it already knows. To judge new experiences by previous experiences. Tane's mind wanted to believe that the creatures that now ruled the world above them were human, in some strange costume perhaps. But no matter how hard his mind tried to rationalize that, the image kept recurring of the shape at the door, just as it exploded into a million shards of glass. And human beings couldn't walk across water. Besides, he had a horrible feeling that they had seen one of these things being born. In the fog tank.
A darker shadow blocked the light for a moment by the edge of the pool. Xena. She had been wandering around the poolside since they had jumped in. Looking for them. Wondering when they would resurface. Rebecca hadn't mentioned Xena again. Fortunately, the snowmen were no more interested in Xena than the jellyfish had been.
Xena moved on, lurching around the poolside. She must be just about as confused as they were, Tane thought with an ironic inward laugh.
He looked around the bounds of their underwater prison. Rebecca sat opposite him, unmoving. Maybe even sleeping, although he doubted it. Southwell was next to her. Just to his right, Fatboy and Crowe were sitting next to each other against the end wall of the narrow pool, and Manderson had stretched out full length on the bottom of the pool, as if resting.
It was surprisingly comfortable, once you got used to the hiss and click of the oxygen valve. The water bore most of his weight, cushioning him in a soft cradle.
He checked his oxygen levels again. Half full. Crowe had said they had about four hours on a single tank, which was all they had.
Crowe touched him on the arm and his voice came over the radio.
"You went to a lot of trouble to save that suitcase, son."
Rebecca looked up, and Crowe motioned her to join in the conversation. Southwell did also, but Fatboy and Manderson remained where they were.
Crowe repeated his comment, then added, "These cryptic comments you keep making. 'Water works' and 'Don't go mist.' And the submarine. They're written down in that notebook of yours, aren't they? It seems that you know more than you are letting on."
"Tell him, Rebecca," Tane said, pressing his radio button. "It can't do any harm now."
After a moment, Rebecca's voice came through his earpiece. She said, "Do you remember telling us that we did not know what you might believe?"
"I remember saying something like that."
"Then would you believe me if I told you that we've been receiving messages from the future, warnings about what is going on now?"
Crowe said, "I remember Tane blurting something about that just before you jumped off my ship. Go on, convince me."
Rebecca said, "We discovered a way of deciphering messages embedded in bursts of gamma radiation, picked up by a NASA satellite."
"From whom, exactly?"
"From ourselves."
There was a brief silence. Then Crowe said, "Okay. It's a bit far-fetched so far, but under these circumstances...Carry on."
Rebecca explained, "Only so many characters can fit into a gamma-ray burst, so the messages have been extremely short and cryptic. But we figured out enough to buy the submarine, to try to stop the Chimera Project, and to end up in the mess we're in now."
Tane added, "We're not scoring too well at the moment."
"No," Crowe mused. "If it were true, and I'm not saying I believe you just yet, then it would raise some interesting complications. Have you heard of the grandfather paradox?"
"Oh God, don't start," Tane groaned. "You'll be building a Möbius strip soon."
"What?" Crowe asked, but got no answer.
"I've been thinking about the snowmen," Rebecca said.
So that was why she had been so still for so long.
She continued, "And I don't think they fit with your theory of bacterial clusters."
"Go on," said Crowe.
"And you surely don't still think we're dealing with terrorists?"
"Possibly not."
Rebecca lapsed back into silence.
A new voice joined the conversation, and Tane realized that Manderson had shifted one of his long legs across, touching Rebecca's and thus linking him in to the conversation.
Manderson said, "I might try sticking just my hand above water and seeing if I can pick up a signal. Let the others know where we are."
Crowe's helmet bobbed up and down in a nod. "Worth a try."
Manderson rolled himself into a sitting position, then squatted, tentatively raising a hand up into the air above the pool.
"Blue Three, this is..." He stopped talking and snatched his hand into the water again as fast ripples spread across the surface of the pool toward him. The light cascaded in waves over the sides of the pool as some kind of feeding frenzy took place above them.
The short flurry of activity died away as Manderson lay back down on the floor of the pool. "Won't be trying that again," he said.
"Any chance the fog will move on?" Tane asked.
"It's several miles wide and growing," Crowe answered. "It won't pass us by in time. We only have a couple of hours of air left."
"And then what?" Rebecca asked.
"You tell me," Crowe replied. "Ask your friends from the future."
Manderson asked, "How did they know that I was there? I'm in a biosuit; they can't smell me. They can't see me, except for my hand. They're not bothering Z1. How did they even know who or what I am?"
"Maybe they know what a human hand looks like," Crowe conjectured.
The words connected with some hidden memory in Tane's brain, and he said absently, "Shape recognition."
"What's that again?" Rebecca asked abruptly.
"Shape recognition," Tane repeated, wondering where he had heard the phrase before.
Rebecca removed her hand, cutting herself out of the conversation, and was still again, thinking.
Tane looked at his oxygen gauge. What would they do when the air ran out? Face the snowmen? Pray that the fog had moved on more quickly than they expected? The only thing to do now was wait it out. "Don't move too much," Crowe had said to them just after they had submerged. "It uses oxygen."
The tranquillity of the pool bottom was shattered suddenly with a huge splash, and Tane's heart leaped inside his chest as something plunged into the water at the shallow end of the pool. It was a snowman. It had to be a snowman. He cowered away from the shock wave that swept past him and fought the urge to surface. That would be fatal.
It wasn't a snowman. It was a rescue harness, attached to a long steel wire cable.
Crowe was at the harness in a second. He ignored it and grabbed the wire cable with his hand, using his free hand to key his radio.
It took Tane a moment to realize what he was doing. The steel cable acted as a huge aerial, taking the signal from Crowe's radio out above the water. He touched Crowe lightly on the ankle, to hear the conversation.
"Rescue helicopter, this is Dr. Crowe of the USABRF," Crowe said. "We are mighty glad to see you."
A New Zealand accent came back through the earpiece, terse and professional. "Dr. Crowe, how many in your party? Over."
"Six. How fast is your winch?"
"Two feet a second at full speed. Why do you ask? Over."
"Not fast enough. We will be attacked on the way up. I repeat, we will be under attack on the way up. You have to get us clear of the fog faster than that."
"We could climb as we winch. That would more than double the speed. Over."
"That'll have to do."
Crowe motioned Rebecca toward him and strapped her into the harness. He grabbed the wire again. "Crowe to rescue helicopter. Allow some slack in the line also. Then start climbing and winching at the same time. You'll whip us out of here like a slingshot."
"Roger that. Over."
"First person ready," Crowe said. "Take her away."
Rebecca grasped onto the harness tightly, as if she might fall out of it, although it was a secure-looking strap. Tane lifted a hand in a kind of goodbye wave, but she was already gone.
She was there one moment and not there the next as the whiplashing cable snatched her from the bottom of the pool like a tiny doll on the end of a bungee cord.
A moment or two later, the harness splashed back into the water, near Crowe. He pointed to Tane.
The harness felt snug and secure around his shoulders, but like Rebecca, he grasped it firmly. He had seen the speed of the whiplash and did not want to be jerked out of the harness by it. He clipped the handle of the Chronophone to a metal clip at his shoulder.
The cable above him tensed, and then suddenly the water was gone, fog rushing down past him. White shapes roared toward him, rising up with him, but then he was above the mist, hanging below a large black helicopter in the broad sunshine of a beautiful summer's day.
He wanted to scream with exhilaration. It had been a short but wild ride. He clambered over the side of the helicopter with a little help from a crewman as he was winched on board.
He looked down. The helicopter was hovering well clear of the fog. Being careful. Just as well, he thought. If you knew what was roaming around in there, though, you'd be a lot higher still.
Ten or twenty minutes later, they were leaving the fog-covered township of Orewa behind them, soaring high above the mist on the black blades of the chopper.
Crowe was leaning forward, busy on the radio, asking questions, and answering them as well. Their faceplates were open and the fresh air tasted great.
Crowe sat back after a few moments and his eyes were grim. Tane had heard why. Four of his men had disappeared when the mist had rolled in from the north.
"What about Xena?" he asked Fatboy.
"We'll go back for her later," he said carefully. "When the fog has cleared."
Tane wasn't sure if that was likely to happen or not, but he let it go. He didn't want to upset Rebecca any further.
She had been silent since they had been snatched off the rooftop, thinking, wordlessly working away inside her own mind. She looked up now, though, and said suddenly, "I know what they are."
All eyes were on her.
"I bought into the idea of bacterial clusters"—she was looking directly at Crowe—"of giant pathogens, because we didn't have any other ideas. But that didn't explain, that couldn't explain, the snowmen."
She paused, thinking, and Crowe took the opportunity to interject, "It's the best guess we've got. Until some more reasonable explanation is found. And I mean reasonable, not some fantastical story about—"
Rebecca was staring at him now, frowning, a look of realization slowly dawning on her face.
"You know, don't you? You don't want to admit it, but you know too."
Crowe interrupted, "I don't know what you're—"
"The moment that Tane said 'shape recognition.' That's when you realized. You couldn't _not_ have known. You're an immunologist. Heck, I'm just a fourteen-year-old kid, so it took me a little longer to work it out, but you must have known straightaway."
Southwell seemed shocked. "Rebecca, are you saying what I think you're saying? My God, you'd better be wrong."
"They are bacterial clusters," Crowe insisted.
"They're not! And you know they're not." Rebecca was thinking furiously now. "The strange Y-shaped jellyfish. Those... _things_...in the fog. It's so obvious. You do know. I know you know."
"What the hell are you talking about?" Tane shouted. "What are they? What are the jellyfish if they're not bacterial clusters?"
Rebecca spoke distinctly, as the rotor blades of the helicopter changed pitch in preparation for landing.
"Antibodies," she said.
**I MMUNITY**
**Manderson lowered his eyes and** smiled quietly to himself. Crowe just sighed tiredly. Only Lucy Southwell looked kindly at Rebecca and said, "You know that's impossible, don't you?"
Manderson looked up with a bemused expression and said, "I suppose that would make the big ones, the snowmen, phagocytes of some kind."
"Macrophages," Rebecca said firmly. "Mother Nature's immune system. Now triggered by Dr. Vicky Green. Against the human race."
Southwell put a hand on her arm. "Rebecca, it's an imaginative idea but just not very likely. Antibodies are simple proteins. They're microscopic."
"I never said they were human antibodies," Rebecca said, and wouldn't say anything else until the helicopter had landed on the lined green surface of the main playing field at the North Harbor Sports Stadium in Albany.
The Command and Control Center was set up in a sponsors' lounge on the fourth floor of the stadium. Through huge plate-glass windows, the green rectangle of the rugby ground was now home to a number of helicopters and row upon row of armored fighting vehicles, preparing for battle.
Tane, Rebecca, and Fatboy were waiting to leave. Their transportation was coming up from the central city. All vehicles here apparently were already hard at work, transporting troops and equipment to build the defensive line.
"They are antibodies," Rebecca finally spoke again, in a small but determined voice. "Antibodies and macrophages. Accept it. You have to. You can't defeat what you can't understand."
Crowe glanced momentarily up from a detailed topographical map of the surrounding area that he and a gray-haired officer from the SAS had been poring over for about fifteen minutes, discussing something called kill zones, along with fields of fire and "claymores."
Crowe said without any further trace of humor, "Rebecca, even if that were possible, think about what you're saying. That would make us—human beings—pathogens. Antibodies attack pathogens."
"I know," Rebecca said softly.
Crowe shook his head and turned back to his work. An SAS trooper entered, saluted, and passed a note to the SAS officer.
Rebecca said, "We think of the Earth as a lump of rock, floating through space. Just a big stone, conveniently placed in a nice warm spot for us to grow on, like mold on cheese. But that's just a way of thinking about it. What if we thought of this planet in a different way. As a complex web of interrelated ecosystems, host to billions upon billions of smaller organisms." She paused. "Not all that unlike the human body when you think about it."
Crowe ignored her, sketching in a line of defendable positions on the map.
Manderson just sat quietly in the corner. Of all of them, only he seemed unfazed by what they had just been through.
A young soldier in the uniform of the regular New Zealand army came in with a stack of orders, which Crowe checked and the other man signed.
Through the window, Tane saw the first line of fighting vehicles began to move out.
Rebecca stood up and crossed to the map table. She leaned over it, her hands on the table, interrupting their work.
"You know what global warming is?" Rebecca asked calmly. "I do. The world has a fever. We are pathogens. Mother Nature is sick and the sickness is us!"
Crowe looked up at her through half-closed lids. Almost a display of emotion, Tane thought.
"I lost four men today," he said slowly. "I am not in the mood, and I don't have the time for your childish environmental fantasies. Get her out of here."
This last was to Manderson, who rose without question and moved behind Rebecca.
Rebecca didn't budge. She laughed, a little hysterically, which was unusual for her, but then again, it had been a very unusual day, Tane thought.
She said, "We don't inhabit a place: We infest it. We poison rivers; we pollute the skies and chop down the trees. We drill holes deep into the earth and suck out all the goodness. We are malignant and highly infectious."
Manderson grasped Rebecca by the arms, but Lucy Southwell intervened, drawing Rebecca away from the table. "What are you saying, Rebecca? That Professor Green somehow created an antidote to the human race?"
"No. I think these things have been there all along. Locked in our genes. Some kind of safety cutout. A self-destruct mechanism for the human species. I don't think Vicky Green invented them. I don't think she even discovered them. But by playing around with the building blocks of life, I think she finally triggered them against us."
Southwell said, "That's crazy, Rebecca. Listen to what you're saying. You're wrong."
She led Rebecca across to the large window and stared out at the bush-covered ridge in the distance and the blue skies above that.
Tane and Fatboy followed. After a while, Fatboy asked, "But what if she is right?"
"She's not," Southwell said. "I've studied this field my whole life. It's just not possible."
Somehow she sounded less sure than she had a moment ago.
Fatboy repeated his question. "But if she is right?"
Southwell sighed. "An antibody exists for only one purpose—to destroy an infection. An antibody has no conscience, no morals, no power to decide. It just does what it was created for. It binds to an infectious particle and disables it, to make it easier for a macrophage to absorb it and destroy it. That's all it does. If what you are saying is true, then that's it. That's the end of the human species."
"I know," Rebecca said. "And maybe it's all we deserve."
"For Christ's sake, get that child out of here!" Crowe shouted, shaking his head erratically from side to side. Even the stoic Manderson seemed shocked at the uncharacteristic display of emotion from his commanding officer.
He motioned to Tane and Fatboy, who didn't argue but pushed open the double doors to the lounge and began to walk along the short corridor to the wide concrete staircase. Fatboy took the Chronophone. Manderson followed to make sure they did as they were told, and Southwell helped Rebecca along behind them.
Rebecca was crying now, and Tane wanted to comfort her but wasn't sure that she'd want him to; besides, Lucy seemed to be doing that job.
They exited the building and moved slowly past one of the huge black trucks and trailers of the USABRF team. The snout of the truck was tucked into the lee of the building.
An army Land Rover pulled to a halt by a row of ticket gates, and a uniformed soldier got out expectantly. A young-looking blond girl in the uniform of the transport corps.
"Why won't he listen?" Rebecca asked between sobs. "What's wrong with him?"
Manderson spoke up then, and in the Texan's slow Southern drawl, Tane heard a whisper that maybe he wasn't quite so convinced that Rebecca was wrong.
"What d'ya think is wrong with him? The skipper has spent his entire life fighting against dangerous germs and nasty bugs." Manderson turned and spat some gum into a plastic rubbish bin by the back wheels of the big black truck.
"An' you just told him he is one!"
**B EFORE THE STORM**
**Private Gemma Shaw drove quickly,** expertly, without speaking, at a regulation sixty miles per hour, heading west on the Northwestern Motorway.
Tane wondered how fast she'd drive if one of _them_ was behind her.
Convoys of trucks passed them on the other side of the motorway, great olive-green behemoths with huge jagged tires, long columns of them that stretched into the distance. But in their direction, the motorway was clear, at least until they got out of the Albany basin.
Private Shaw carefully braked and brought the army Land Rover to a halt.
Tane stared at the scene in front of him. Two hundred thousand people lived on the North Shore of Auckland City, and it seemed that all of them were jammed into little metal boxes down the four lanes ahead. There seemed to be no order to it. No careful lines of cars. It was just a jumble of multicolored pieces, as if someone had emptied a LEGO set down the motorway. The cars spilled from lane to lane, invading the shoulder and even the narrow grass of the median strip, rasping paint off their doors as they scraped along the median barriers. There were five and in some places even six cars squeezed into the narrow asphalt corridor.
There were family wagons, and sedans, and tradesmen's vans, stuffed to the gunnels with belongings and people. Every second vehicle seemed to be a big, square four-wheel drive, spewing black and brown diesel fumes from its exhaust. Motorcycles somehow found chinks in the solid metal armor of the roadway, weaving and winding their way through.
Just past Bush Road a late model Audi had been abandoned in the middle of the center lane. There was no way to get it off the road; instead, it was bulldozed along by the Toyota SUV behind. Whenever the Audi veered to the left or the right, a clip from a car in one of the side lanes steered it back into line. Already it was a wreck from the constant battering. And yet, with the relentless pressure of the traffic, it kept moving, as if it too wanted to escape the horror that was creeping across farmlands, through gullies, and down the highway, a few miles to the north.
Members of a North Shore evangelical church walked the length of the motorway, clambering over car bonnets when they had to, handing out muesli bars and bottled water and religious tracts. Voices were shouting and horns were blaring, and from at least one vehicle, now pushed to the side, a column of smoke rose out of the engine.
"How the hell are we going to get through that?" Fatboy asked in dismay.
"Won't be a problem, sir," Private Shaw said, and performed an extremely nonregulation U-turn in the middle of the motorway, driving the wrong way down the on-ramp, under the overpass, and back up the off-ramp on the other side of the motorway. There was a police roadblock on the off-ramp to prevent people from doing exactly what Shaw was doing. They needed the eastbound lanes clear for the convoys of trucks. The army vehicle and Shaw's ID got them through the roadblock without problem, though.
Shaw turned her headlights on full-beam, even though it was daylight, as a warning to the oncoming traffic.
They passed police cars at irregular intervals down the motorway, trying ineffectually to create some order out of the chaos.
They saw any number of minor nose-to-tail accidents, but the drivers did not even bother to stop. One car was scraping its front bumper along the roadway in front of it.
Not that they were moving far. A yard at a time if they were lucky.
Most of the cars were packed with belongings. Suitcases strapped onto roof racks, backseats stuffed with cardboard boxes and canvas bags. They passed one car with an elderly woman sitting in a kayak strapped onto a roof rack, wearing a bicycle helmet for protection. The car was driven by a middle-aged couple. There was no room in the backseat for the old lady because that was taken up by three ferocious-looking rottweilers.
"What are we going to do?" Tane asked. "When we get home?"
"Tell Mum and Dad what's going on," Fatboy said. "Then get over to Rebecca's house."
Rebecca's mum had been questioned and released, and was now back at the West Harbor house, according to Crowe.
They had tried phoning them from the stadium but got only a recorded voice telling them to try again later. The entire telephone system was overloaded across Auckland as a panicking population tried to contact friends or relatives.
Rebecca had stopped crying now, but there was a strange sadness about her. More than that, a sense that she didn't care anymore. That nothing mattered. It was like a wall around her, and even Fatboy didn't try to penetrate it.
Tane wondered if she was right. About the antibodies and macrophages. He had known her for the whole of his life, of her life, too, and she was seldom wrong about anything.
Yet Crowe had been so insistent.
"At the start, there would have been just one," Rebecca said, mostly to herself. "A small cloud of mist rising out of a test tube or a glass flask. Thickening. Growing. Maybe it was late at night. Maybe no one was there to see. Then, in the mist, a macrophage grew, and it waited. It waited for a pathogen. Maybe it was a night watchman or maybe a scientist working late."
She paused, and stared out the window for a while.
"And those cells became a resource. Food, if you like. And with them, the fog grew, and then maybe there were two of the macrophages. And the fog spread, creeping along corridors and under doorways, finding its next target, and then there were three or four of the creatures. By the morning, the mist covered the island and the people were gone."
"Don't think about it," Fatboy said. "It's upsetting you."
It wasn't just Rebecca it was upsetting.
Rebecca ignored him and continued, "Then the fog reached Whangarei, wiping out a few scattered farmhouses and small towns on the way. And there were fifty thousand people. Fifty thousand germs to be disinfected."
"What is she talking about?" Private Shaw was getting nervous.
Rebecca said, "It's going to reach Auckland soon, and most of a million people will still be here, and it will use their cells to grow. And it will grow. If fifty thousand people makes a fog several miles wide, then how much fog will a million human bodies make?"
"Rebecca, stop it now!" Fatboy said.
"Rebecca," Tane said.
She closed her eyes and shook her head gently. "Sorry, guys. Long day."
"A long, strange day," Fatboy agreed.
Tane said, "Crowe and the rest of those scientists know what they are doing. They don't think we are a disease, and I don't either."
Rebecca lapsed into a strange, moody silence and said nothing more.
"What about the Chronophone?" Tane asked.
They still had to install the device, currently cradled on Tane's lap.
"We'll do it later," Fatboy said. "It'll be easier to get into the city on the bike; we can just cut through all the traffic."
A radio beeped, and Private Shaw held it to her ear, giving their location and direction to someone on the other end before pulling over and stopping.
"What are you...?" Fatboy began to ask, but his question was answered as another Land Rover appeared behind them, headlights flashing.
Big "Mandy" Manderson got out of the other vehicle. He was grinning widely. They got out to meet him.
"Managed to commandeer a vehicle," he said. "Got some good news for you."
"Really?" Rebecca said sardonically, understandably, considering the circumstances.
"In times like these, any good news is a blessing," Manderson said.
There was a movement behind him, and a small brown bundle clambered out through the passenger window, chattering and waving its hairy little arms.
"Xena!" Rebecca cried as the chimp ran across and leaped up at her.
"Just wandered out of the mist about twenty minutes ago," Manderson said. "God knows how she found her way back down through the hotel in the fog. Must have followed the main road south. Just pure luck, I guess."
Maybe, Tane thought, but animals had a strange instinct sometimes.
Rebecca hugged the chimp like a long-lost daughter, and Xena hugged her back.
"Hi, Xena," Tane said. She screeched joyfully at him, and he found himself strangely pleased to see the happy little chimp again.
"Please say thanks to Dr. Crowe for us," Rebecca said as Manderson climbed back in the Land Rover.
He smiled, and said, "Stony is a bit busy at the moment."
"You didn't tell him, did you." It was a statement, not a question.
Manderson smiled again and shook his head. Rebecca walked over before he could pull away, leaned in the window, and kissed him on the forehead, just below that unruly mop of curls.
"Thanks, Mandy," she said. "Is there any way of reaching you, if we need to? All the phone systems are out."
He reached into the rear of the vehicle and handed her a portable radio. "It's already set to the right frequency," he said. "Just holler if you need anything."
"I think you'll be too busy to worry about us." Rebecca smiled.
"You take care now," he said. "Go and hide somewhere nice and deep in that little submarine of yours."
Rebecca hugged Xena tightly as Manderson headed back up the motorway. Toward Albany.
Toward the fog.
**K AITIAKITANGA**
**Private Shaw deposited them on** the red gravel driveway of the house nestled into the bush, and said goodbye with a regulation wave.
Tane wondered where she was going to and hoped it was south, away from the fog.
The house was silent. The enveloping trees blocked the dwindling light of the fading sun, spreading longer finger-like shadows across the weatherboards and giving it a forlorn, moody appearance.
A note on the front door explained the silence. _Gone to Waitakere Marae._
Like many others, his dad and mum had sought a place of refuge in uncertain times.
"I'll get the car." Tane unlocked the front door and opened the garage door for the others to enter. His mum's car, a bright red Volkswagen, occupied the left of the garage. There was a larger space for his dad's Jeep, but it was not there.
The spare keys to the VW were on the hook inside the pantry, and Tane tossed them to Fatboy.
Rebecca sat silently in the back, playing hand games with Xena.
"Do you think that Crowe and his men, and the army, will be able to hold back the fog?" Tane eventually asked the question that was on all of their minds.
"They don't know how," Rebecca answered, and there was something about the way she said it that made Tane turn around and look closely at her for a moment.
"Do you?" he asked.
Rebecca didn't answer, but she didn't deny it.
"Rebecca," Fatboy said gently, "is there a way to stop the antibodies? To defeat the macrophages?"
"I don't know," Rebecca said, shaking her head. "It's just something that was in one of the messages. I'm not sure."
"What!" Tane was flabbergasted. Xena put her hands over her eyes and peeked out from between her fingers.
Tane desperately tried to remember the content of the last message. Or was it something in an earlier message, one that they had failed to decipher?
Fatboy said, "Rebecca, try to remember. You are talking about the lives of hundreds of thousands, millions, maybe billions of people."
She was silent. Fatboy and Tane looked at each other in growing concern.
"Come on, Rebecca," Tane said lightly. "Let's be heroes and save the world!"
She just said, with a weariness that filled the small car around them like a black shroud, "I think the world is doing a pretty good job of saving itself at the moment."
She said no more after that.
They came down out of the mountains and headed south toward the Marae.
The road they were traveling had been the scene of a battle. The aftermath was everywhere. A battle of desperate people, trying to force their way along one of the main feeder roads to the Northwestern Motorway. Earlier in the day, this road must have been jammed solid with cars. Broken-down or fuelless vehicles were shunted haphazardly onto the verges. A few sat in the center of the road, and they had to drive carefully around them. Most showed signs of damage.
When Rebecca finally spoke again, it was to say, "I'm hungry."
Tane realized then that they hadn't eaten all day.
"Want a Big Mac?" he asked, seeing a McDonald's sign ahead of them and trying to be funny.
She just sighed tiredly and said, "It won't be open."
Of course it wouldn't be open. That was the point of his joke, which didn't seem at all funny now.
The light was on, the great golden _M_ glowing like an ancient tribal beacon down the road before them. The crew must have left in a hurry and forgotten to turn the sign off. There was no way it was going to be open.
It was open.
With an expression of disbelief, Fatboy pulled into the drive-through.
"Can I please take your order?" a bright young girl in a blue McDonald's uniform asked from behind the small window.
"A Big Mac," Fatboy said cautiously. "Two Big Macs. Combos. What are you having, Tane?"
"Same same."
"Make that three."
"Certainly, sir," the girl said cheerfully. Her tag said her name was Helen. "Would you like to upsize those to a super-combo?"
Fatboy stared blankly at her for a moment. "All right," he said.
She took his offered money and said, "Please drive on to the next window."
The young man at the next window handed them their food, and as they pulled out, Tane and Fatboy looked at each other in amazement.
The whole transaction had been so utterly commonplace that for a moment Tane wondered if he was dreaming the rest of it all, the fog, the antibodies, and that the normalcy, the _insanity_ of the fast-food outlet was really the reality, in an insane world.
But they passed a number of boarded-up houses, eyes peering suspiciously out at them through gaps in the planks, and the nightmare proved to be real once again.
_The world is doing a pretty good job of saving itself._
Even in these strangest and most desperate of times, some Maori protocol was observed. As a pakeha—a non-Maori—Rebecca needed permission from the tribal elders to enter the sacred ground of the Maori meeting ground. That duly came, although they dispensed with the traditional welcome. There was some discussion over Xena, but she, also, was eventually allowed to trot along beside Rebecca, holding her hand.
His father and mother were in the big meeting hall along with at least a hundred others of the tribe. Tane was acutely aware that Rebecca was the only pakeha there.
It was a tall, timbered, high-roofed hall, lined with traditional carvings, representations of their ancestors. It was dim inside, even under normal circumstances, but now the meager light of the white-hatted electric bulbs scattered around the ceiling was swallowed up by the black plastic sheeting that was taped around each of the windows.
It might stop the fog, Tane thought, but it wouldn't stop _them._
His mother screamed when she saw them. A mixture of fear, delight, and relief. She hugged Tane, and his father hugged Fatboy just as fiercely. Then she hugged Fatboy and Tane's father embraced him. Both his parents were crying and trying to talk about visits from the police and the fog all at once. Then his mother hugged Rebecca, and after an initial hesitation Rebecca's arms crept around his mother and held on tightly for a surprisingly long time.
Xena scrambled over to a chair at the side of the hall. A group of children surrounded her, laughing and playing with her.
After a few moments, the outpouring of emotion started to get a bit uncomfortable for Tane, but it was still another ten minutes before he could get a word in.
Rebecca joined them then, and his dad made a small circle of chairs at one end of the hall for them to sit in.
The others ignored them, lost in their own soft conversations and dramas.
Over the next hour, as the twilight dripped away into darkness, they told his father and mother everything. They started with Lake Sunnyvale and left out nothing of importance up to their trip home with Private Shaw.
There was silence for a long while after that. The light from the bulbs skimmed across the faces of the carvings cleaving deep shadows out of the somber expressions.
Fatboy and Tane knew not to speak, and Rebecca seemed to have nothing to say.
His dad drew in a long slow breath after a while and looked deeply into each of their eyes in turn.
The silence lengthened until Tane broke it. "There are places for you on the submarine. You can hide out with us until the fog passes over."
His father looked at his mother, and Tane saw a small shake of the head pass between them. His father gestured around at the room. Children jumped and danced around the chimpanzee, with happy faces, unknowing of the terror that approached. A young couple with a newborn baby sat, just out of earshot, lost in each other and the child. Three old women, dressed in black with matching black headscarves, sat a few feet away, toothlessly chewing up every word.
"You would have us leave, and yet our family, our _whanau,_ stay?" His father shook his head.
Tane said heavily, "Then you must run. The meeting hall won't protect you."
Fatboy said, "Get every car you can get, or buses if you can find them. Load everyone up and head south as fast and as far as you can."
"It's a sturdy old building," his father said doubtfully.
"We have seen these creatures up close," Tane said, struggling to stop himself from crying out in exasperation. "You have to run, or everyone here will die!"
His father closed his eyes. His mother reached out and took his hand in hers.
His father asked, "Can this thing be stopped? Now that it has started?"
Tane shook his head uncertainly, but Fatboy nodded.
"Rebecca thinks there may be a way," he said.
They all looked at her, her eyes on the floor, her shoulders hunched as if carrying a heavy load. She was, Tane thought. She had been carrying her burden for too long now.
"Maybe it's for the best. From a purely scientific point of view," she said, "once we humans are gone, this planet will be able to heal itself, and then when it is healthy once again, maybe in millions of years' time, the human race can start over. Like a forest fire, cleaning out the congestion and decay, so new life can sprout amid the ashes."
Tane started to argue, but his father held up a hand for silence. "You really believe we are a disease," he said.
Rebecca stared at the floor. "A biologist would describe us as a plague."
There was a piano against the far wall of the hall, near the entrance. Xena struggled for a moment with Rebecca to be let loose, and then ran across to the piano, with a crowd of children trailing her like the tail of a comet. She jumped up onto the seat and began to hammer tunelessly at the keys. She looked around as if expecting applause.
His father stood and crossed to Rebecca. He placed a hand under her chin and lifted her head up to meet his. "Maybe in this new age, what you say has some truth, but it was not always so."
"I know," Rebecca responded, a tear welling up in the corner of her eye.
His father watched her silently for a moment.
"Do you understand the meaning of 'Kaitiakitanga'?" he asked.
She shook her head. "Not really."
His father smiled. "You pakeha believe that land belongs to people. But we Maori believe that people belong to the land. We are _tangata whenua_ —people of the land. It is our privilege, not our right, and with it comes a great responsibility: Kaitiakitanga."
All were silent now, watching his father. Tane found his eyes wandering around the carvings of the ancestors that guarded the walls of the meeting house. He had a very real sense that they were watching him back.
His father said, "For thousands of years, we Maori have guarded and protected our environment. Replenished and replaced what we used. But then the pakeha came to our land. We _kaitiaki_ who should have stood up to the pakeha, who should have defended Papatuanuku—the Earth Mother—did not. Our voices fell silent."
"Then you agree," Rebecca said slowly. "Mankind must be destroyed before it destroys its host."
"No." His father's voice was soft, little more than the breath of an infant, and yet somehow carried such intensity that the carvings of the ancestors seemed to quiver and come to life, carrying his words to all corners of the room. "We are a part of nature, creatures of Papatuanuku. Greed and stupidity are the disease, not us."
The rest of the room had gathered around now, listening to the conversation.
"We can't go back!" Rebecca cried out against him. "You cannot reverse a mutation. Human beings can't go back to living in villages and farming _kumara_!"
"We cannot." His father smiled sadly. "But we can learn to live with the trees and the lakes, the mountains and the seas, the fish and the animals, as family, as _whanau,_ not as invaders, conquerors!"
There was a silence, and a breeze crept into the hall from outside, rustling the black plastic sheeting and reminding Tane that time was growing short.
"What can we do?" he asked quietly.
"What you know you must do," his father said, and repeated it. "What you know you must."
He sighed. "We live in a Western society, so we adopt Western ways, but we have never forgotten our culture."
"I have," Tane said painfully. "I have forgotten."
_My people. My culture. My_ whakapapa.
His father was silent for a moment. "No, son. You are Tane Williams, son of Rangitira Williams, grandson of Hemi Te Awa of the great Tuhoe tribe of Aotearoa. You have not forgotten your _whakapapa,_ because you cannot forget. You have merely closed your eyes for a moment. And now they are open." He placed a hand on Tane's shoulder. "You will face this challenge, find a way to defeat it, then show the rest of the world's people the way forward. The way of the _kaitiaki._
"You will lead the people, all of the world's people, into a new age. Te Kenehi Tuarua—the second genesis. You must teach the ways of _kaitiakitanga_ if the world is to survive. We must all become _kaitiaki._ "
He turned from Tane and addressed the room, "They call me a disease, but I am not. I am a child of the land. I am _tangata whenua._ I am a spiritual guardian of the Earth Mother. I am _kaitiaki_!"
There was a silence, and Tane felt that ancient spirits were repeating his father's words, whispering them to one another.
Fatboy rose and placed his hand on his father's arm. From an inside pocket in his jacket, he produced the _patu pounamu,_ the greenstone club their parents had given each of them at Christmas. He pressed it against his heart. "Neither am I an illness. I am _tangata whenua._ I am _kaitiaki._ "
There was a murmuring amongst the gathered crowd that subsided only when Tane rose, a little awkwardly, to his feet. He spoke quietly but his voice was clear.
"I, too, am _tangata whenua._ I am _kaitiaki._ " The room seemed to fade into blackness around him, and he looked only into the face of his father.
"I am Maori."
**N EW YEAR'S EVE**
**The first whispering tendrils of** mist crept over the ridges of the Albany hills just after nine p.m., sifting down through the tree blanket or slithering down the main highway that cut through the bush of the hillside.
It was New Year's Eve. The last day of the year. Some said the last day of all years.
That might be true, Crowe thought, watching the mist creep toward him on the video monitor, if they couldn't stop it here and now.
The mist flowed up to and over the camera, a small metal box stuck in the middle of the highway, near the top of the hill, turning everything to white. He switched to another camera, about halfway up the hill, and saw the mist just starting to writhe around a corner of the highway, far ahead.
In addition to the ground cameras, they had three helicopters operating, well above the puffy cumulus top of the fog, feeding images back to the control center.
Crowe picked up a radio. "The fog has crossed the hilltop," he said tersely. "Time to light it up."
Around him, the control center, just three hundred feet behind their main defensive line, was buzzing with activity. NZ Army and SAS officers were barking orders, running here and there, answering phones and radios. The battle for Auckland was commencing.
Lucy Southwell's voice came back to him on the radio. She sounded scared but calm. "Stony, we've had a lot of problems trying to evacuate Auckland. We are still trying to get people out. You have got to stop that fog, or at least slow it down. If it keeps going at the rate it's going, hundreds of thousands of people are going to die!"
Crowe turned and looked at Manderson standing next to him but said nothing.
Manderson smiled. "Let's show these fluffy white teddy bears who they're messing with."
Flight Lieutenant John Ramirez was already in the cockpit of his FA18 Super Hornet fighter-bomber with the canopy sealed when the order came through his headset. He acknowledged immediately and gave a brief wave to the ground staff who were preparing for takeoff. The rest of his wing were already lowering their canopies and would follow him off the deck at intervals of just a few seconds.
The USS _Abraham Lincoln_ was sailing a steady fifteen knots into the light breeze, to assist with the takeoffs. Once airborne, the flight to Auckland would take less than ten minutes.
On a hand signal from outside, he fired up his engines, turning night into day behind the jet, but held in place still by the steely grasp of the aircraft carrier.
All the planes had names. Some of the pilots named their planes after girls, like in the old bomber days. There was the _Mary-Lou_ and the _Barbara-Ann._ Others gave their planes macho names, full of bravado, like _Sky Warrior_ or _Grim Reaper._
Ramirez's plane was _Deus ex Machina._ Most of his fellow pilots had no idea what it meant. Some thought it was Hispanic, like Ramirez. It wasn't Spanish, though; it was Greek. _Deus ex Machina._ The God from the Machine.
Ramirez had majored in literature. In the ancient Greek plays, the hero would often get himself into all sorts of drama and strife, only to have it all solved, just in the nick of time, by a god, who would be lowered onto the stage by an elaborate piece of equipment. The God from the Machine would intervene, just when all looked lost, and save the day for the hero.
That was his role, Ramirez felt, and he had named his plane for it. When ground troops were under threat, his wing of close ground-support fighter-bombers would be called in to save the day.
Today he had a mixture of high-explosive and incendiary napalm bombs attached under his wings. Someone, or something, was in for a very hot time in Auckland tonight.
The launch officer raised a hand above his head, then brought it sweeping down. Ramirez punched in his after-burners as the launch wire exploded forward, catapulting the fighter down the short runway of the carrier.
The acceleration rammed him back in his seat. It was a volatile adrenaline thrill that no roller-coaster ride could ever come close to simulating. He would miss it, he thought, when he was too old to fly and rotated off onto some boring desk job somewhere.
The edge of the carrier flashed past, and the plane dipped fractionally, then caught itself and he arced upward to the left. The other five planes of his wing were off the deck in quick succession, forming up on his wing tips. The six planes seemed as one as they banked around toward the dark coast ahead of them.
_Tell me your troubles,_ Ramirez thought, _confess me your sins, for here comes the God from the Machine._
Fatboy's motorcycle was still parked on the gray concrete pad by the side of the house, where he had left it that morning. Had it been just one day? Tane realized that it had. It seemed like an eternity of time had passed since Fatboy had set out with the Chronophone.
The house was in darkness, except, unsurprisingly, for the flickering blue glow from an upstairs window.
They had driven back from the Marae in his dad's Jeep Cherokee. The roads were increasingly impassable, and the sturdy four-wheel-drive Jeep seemed like a better bet if they had to go across fields or shoulder other vehicles out of the way.
Rebecca hadn't said a word the entire trip. She had just sat there, thinking. Tane felt that she was making a momentous decision and left her alone to make it.
His own mind was filled with the image of his mother and father, standing at the carved wooden gates of the Marae, facing the fear, facing an uncertain future, in the embrace of their people, their _whanau._ He wondered if he would ever see them again and thought that he would not.
They entered the house quietly, so as not to alarm Rebecca's mum.
Rebecca's room was unchanged from when they had left it. The hole in the wall where the window used to be, the torn drapes, and the glass and wood splinters spread throughout the room. Even the paper was still on the printer.
**FTBYDNTGO.WTRBLSTMPS.DSVLETHM.**
**SLTABS.DNTABSRB.**
Rebecca looked at the message a long time, and finally sighed. "MPs, macrophages. ABs are antibodies."
"Water blast," Tane said. "High-pressure water."
"I think so," Rebecca said. "The macrophages are made of some kind of spongy tissue. Bullets don't affect them; they just punch straight through. But a pressurized jet of water would cut them to pieces and dissolve them."
"SLT?" Fatboy asked.
"My guess is salt," Rebecca replied. "I thought it meant that the antibodies can't absorb salt, but that makes no sense. But what we do know is that if you crush an antibody, it just gets absorbed back into the mist. Then the mist probably just makes a new one. So no matter how many you destroy, there are just as many attacking you a few minutes later."
"But salt stops them being absorbed," Tane worked out slowly.
"It's only a guess. But I suspect that salt, on that slimy surface, would alter the chemical structure of the creature, and that would stop it being absorbed."
Fatboy said, "So once you kill them, they stay dead!"
"Something like that."
Fatboy said, "We need to tell Crowe."
"What about the Chronophone?" Rebecca asked.
Tane asked, "What about the _Möbius_?"
"I can get back to the Skytower and install the Chronophone," Fatboy said. "I'll be able to squeeze into the city on the bike."
"No," Rebecca said. "I think you should both go to the Skytower. That has to take priority. I'll try and raise Crowe on the portable radio."
"Do you think he'll listen?" Tane said.
She just shrugged.
There was a long drawn-out scream from the skies above them, rising, then dropping away as the sound passed overhead.
Xena screeched in fright and leaped into Rebecca's arms.
Rebecca looked up. "What was that?"
"A jet," Tane said, "moving fast."
"More than one," Fatboy said. "Sounded like fighters."
"Oh crap," Tane breathed. "It's already started."
As if to confirm his analysis, a sound of distant thunder rolled in from the north. Through the smashed window, the skies lit up with brilliant flashes.
"They're bombing the hell out of something," Fatboy said.
"We're all out of time," Rebecca whispered. "We've got to get moving. We may already be too late."
"I'll get the Chronophone," Fatboy said. "Tane, find me a backpack of some kind to put it in. Rebecca, you've got to take the Jeep. Get to the Devonport Navy Base, find the submarine. I don't imagine it will be difficult to spot. Bring it across the harbor and we'll meet you down by the waterfront, at, say, the end of Princes Wharf. That's easy to find from the sea. On the way, try to raise Crowe on the radio."
"What about Mum?" Rebecca asked quietly.
"Take her with you. You can explain about her new home on the way."
Her new home. A little tin tube on the floor of the ocean.
Fatboy raced out to the Jeep to get the Chronophone and Rebecca disappeared upstairs.
Tane opened a few cupboards, trying to remember where he had seen Rebecca's schoolbag, a black backpack.
He had found it by the time Fatboy came in with the silver briefcase.
"Have we got time to run a test?" Tane asked.
"No. Where's Rebecca?"
"Still upstairs with her mum," Tane replied
"Go and hurry them up. She's got to get moving."
Before he could move, he heard Rebecca's voice, shrieking from above them.
Tane bounded up the stairs and down the hallway to Rebecca's mother's room. The door was wide open.
The television was on. Helicopter camera shots showed the fog creeping over the top of the Albany hills. The view cut to the black silhouettes of warplanes streaking overhead, just visible in the moonlight, then back to the hillside. Massive explosions rocked the camera, and the whole hillside shook in front of their eyes. Rivers of fire exploded in the treetops as breathless reporters tried to explain in voice-overs what was happening.
"Mum!" Rebecca shrieked once again. "You have to come with us. Now!"
"Don't be silly," her mother replied calmly, her eyes glued to the images of fire and fog. "This is the news. This is important."
Tane looked out the window. The sky to the north was ablaze, massive tongues of flames leaping up into the black air from the conflagration on the ground. There were more flashes, more thunder, and he saw the silver flash of a jet caught for a second in the moonlight.
Rebecca turned to Tane in anguish. "She won't come!"
"I'll get Fatboy," Tane said calmly. "We'll carry her out."
Rebecca had one last try. "Mum, if you stay here, you will die!"
"Ssshh," her mum said irritably. "I can't hear what they're saying."
"I'll get Fatboy," Tane said again, turning to go.
"No." Rebecca's hand was on his arm. "No. We don't have the time."
Her legs seemed to be unsteady, and Tane put his arm around her shoulders to support her.
She slowly backed out of the room, one small footstep after another, her eyes never leaving her mother, washed in the soft light from the television set.
Tane, by her side, had no words of comfort.
Rebecca said again, her voice just starting to crack, "We don't have the time."
**T HE BATTLE FOR AUCKLAND**
**Tane rode on the back** of Fatboy's motorcycle with the cool night breeze on his face and the touch of Rebecca on his lips.
She had been strangely calm as they had come down the stairs from her mother's room. Strangely accepting.
Fatboy had given him the backpack containing the Chronophone and handed him a spare helmet, but before he could put it on, Rebecca had been in his arms, her hands around his neck.
She wasn't crying. In fact, she had seemed stronger, more determined than before. She had spoken quietly, her lips right next to his ear. "What I said this morning, about not knowing you, it wasn't true. I wanted to hurt you. I was angry."
After a moment's hesitation, Tane had put his hands on her arms.
He'd said, "You were right to be angry. I was stupid."
"You weren't stupid; you were hurt. I knew that and I should have accepted it."
Tane had started to say something, but she had put a finger on his lips to hush him.
She'd said, "I've known you all my life. I've always known you. I will always know you. But I'm not strong."
Tane began, "You're the strongest—"
She hushed him again and said, "I'm not strong. But you make me strong."
Then her lips had brushed in passing against his, and the engine of the Jeep had roared and she was gone.
Tane had looked up a little guiltily at Fatboy, but there was no anger on his brother's face, just a quiet smile.
_I will always know you._
For the first time, right then, it had occurred to Tane that he might never see her again.
The engine of the Harley gave its usual throaty chuckle, and Fatboy swerved in between two deserted cars and up the motorway on-ramp, heading toward the city.
The motorway was clogged with cars, all of them deserted. The owners, unable to go forward or backward, had simply left their vehicles and their belongings and either started walking south or had gone back to their homes.
Fatboy had to pick his path carefully. The cars ramming their way down the motorway had not kept in nice straight lanes but had zigged and zagged all over the road.
A couple of times they found themselves in a dead end in the maze of vehicles and had to get off and push the heavy bike backward in order to try a different route.
It was eleven o'clock by the time they made it as far as spaghetti junction—the main motorway interchange.
The going was easier after that. Most of the traffic was heading south. Heading back north into the city, there were fewer cars. Bigger gaps. They made good time up between the lines of abandoned vehicles.
Not entirely abandoned.
From more than a couple of cars, frightened faces stared out at them as they glided past. Either not knowing what to do or too afraid to do it, or both.
As they got closer, they could see several small fires burning in the city center.
Once they were off the motorway and into the city, the avalanche of abandoned cars disappeared. This was the northern side of the city. Anyone in their right minds was on the southern side, trying to get on the Southern Motorway.
As they came off the flyover onto Nelson Street, a gang of drunken youths in a brand-new Mercedes convertible swung out of the darkness and tried to sideswipe them.
A skinny kid standing up on the backseat threw a bottle at them, but it just shattered on the road behind them.
Fatboy calmly accelerated away from them, swerving around abandoned vehicles, the side pegs scraping on the ground. The Mercedes engine roared as it gave chase.
A bus was parked sideways across the road at the intersection with Victoria Street, and two cars were tangled together in a smash on the other side of it. Fatboy gunned the bike through the gap, and Tane heard a satisfying screech of brakes behind them as the driver of the Mercedes realized his car couldn't fit where the bike could.
A small block of shops was burning fiercely just across the road from the Skytower, the flames brighter than the surrounding streetlights, burning back the darkness of the central city street. Looters, or vandals, Tane thought. It seemed that antibodies and macrophages were not the only dangers in Auckland city this night.
Left unattended, the fire would quickly spread to nearby buildings, and the whole city center could soon be ablaze. A massive funeral pyre for Auckland. But even as they swung around the corner by the entrance to the Skytower casino, they could see red flashing lights arriving from the opposite direction and a fire engine rolled smoothly to a stop in the middle of the road. It was a pumper, with a roof-mounted water nozzle. The crew quickly and efficiently set to work.
Fatboy pulled the Harley up outside the main doors to the casino. The huge glass doors were shut, and a quick shake by Tane ascertained that they were locked as well.
The casino never shut. But it was shut now. Inside, only the massive water feature, a thin sheet of water cascading two stories down a sheer glass plate, showed any signs of life. The lights that never went out, were out. The gaming rooms above them were silent.
Tane got back on and Fatboy gunned the bike past the busy fire crew, around the corner, and down the concrete ramp into the underground car park.
The barrier arms were down, but the Harley squeezed past, just, and Fatboy kicked down the stand of the bike right outside the main elevators.
"We have to go up to the lobby," he said, "then down the escalator to the base of the Skytower itself."
"I hope the elevators are still working," Tane said.
And they were.
The radio produced nothing but static.
Manderson had said it was tuned to the right frequency, but when she pressed the CALL button and released it, all she got was a loud squawk. Had they knocked the frequency knob somehow? She desperately tried every setting on the radio as she drove, her eyes flicking between the road and the radio. What she had to tell them would help; it might even turn the battle. But only if they had enough time to use the information.
It was slow going. It wasn't far to the naval base, as the crow flies, and if she'd had a boat, she could have cut straight across the harbor and been there in a matter of minutes.
But traveling by road meant heading north and then cutting back east through Greenhithe and Albany. And that proved to be a circuitous maze of blocked roads.
She played with the knobs on the radio as she maneuvered the Jeep through the constricted streets, and by the time she had reached the motorway overpass, she had managed to get through to the command center.
It was Manderson who answered the radio. "I thought I told you to hide away somewhere nice and deep," he drawled.
Rebecca stopped on the overpass and looked north toward the upper harbor highway. Along its length, Rebecca could see lights moving, vehicles, and soldiers with flashlights. There was no second line of defense. The line was just too long. It stretched from the beach at Mairangi Bay along through Albany and Greenhithe, all the way out to West Harbor.
If the line was breached, then Auckland was lost.
A flight of jets screamed just over her head, and the hillside in the distance, already blazing, flared like the sun for a brief second. A shock wave rattled the car windows a moment later. In that flash, though, she saw the reason for all the activity around her. The terror of Auckland.
In the flash of the incendiary bombs, she saw the long white cloud of the fog, stretching down the hillside, engulfing the North Harbor stadium and rolling forward toward them.
"I have to talk to Stony," she said urgently.
"He's busy," Manderson said. "If we don't stop this fog, or at least slow it down for a few hours, Auckland is going to become the biggest catastrophe the world has ever seen."
"I can help," Rebecca insisted.
"He really is too busy. We've had lab results come in on the chemical composition of the fog, so there's a lot of pressure to come up with some answers."
Rebecca could see that arguing was just going to waste a lot of time and produce no result. She said instead, "You can't shoot them. The...creatures."
"We know," Manderson said curtly.
"You can't shoot them and you can't blow them up."
"Not true. We've been knocking the stuffing out of those things for the last half hour. Claymore mines, rockets, mortars, you name it, we're chucking it at them."
"Hasn't stopped them coming, though, has it," Rebecca remarked.
"What are you saying?"
"You blow them up and the fog just reclaims them. They just get absorbed back, and the fog starts building another one. All you're doing is slowing them down."
There was a silence on the other end of the radio as Manderson thought about that. "You got a solution, or just a problem."
Rebecca said, "You need water, and lots of it. Water blasters. Fire hoses, if you can get them. They're soft. High-pressure water will cut right through them, and the water dissolves the material they're made of. It'll just run away into storm water drains and out into the ocean. That's the other thing. Salt. Spray salt, or even salt water on them and it alters the chemical structure of their bodies. Antib...the small ones as well as the big ones."
Manderson smiled and said, "You call them what you like. Are you sure about this?"
"Pretty much. Have you got any better suggestions?"
Mandy just said, "You just get somewhere safe. I'll get your information to the boss. He'll listen to me."
Rebecca said, "I hope so."
Tane and Fatboy emerged from the top of the last flight of stairs and stumbled out onto the observation deck of the Skytower. The tower elevators had been locked down. Protected by touch-pad security panels to which they did not know the combinations. The alternative was the stairs, and even that door had been locked, but a set of chairs had made a worthwhile battering ram, and they had been up the first flight in record time.
By the time they had reached the twentieth flight, the vigor was gone and it had just become a long hard slog. Made harder by the thought that this was just the start of the journey up to the peak of the tower. Fatboy took the heavy Chronophone, for which Tane was incredibly grateful. Even on the motorcycle, it had cut into his shoulders through the narrow nylon straps of Rebecca's schoolbag. Climbing twenty stories with that weight on your back seemed impossible, yet Fatboy carried it indefatigably.
From the observation deck, they could see the entire central city, the darkness, the scattered fires burning in vacant buildings. Here and there, the lights of a car roamed the urban wasteland.
In the north, across the harbor and behind a dark ridge in the distance, the furious fires of technology were burning against the inexorable forces of Mother Nature. The low clouds were punctuated again and again by the thunderclaps of bombs.
The moon was rising in the sky now, and from the height of the Skytower they could see the fog itself, smothering the hillside north of Albany. It stretched as far to the east, toward the east coast beaches, and as far to the west, toward West Harbor and Kumeu, as they could see.
It took them a while to find the next set of stairs, leading from the observation deck up to the Sky Deck, the upper observation deck. That door, fortunately, was not locked.
Tane offered to take the Chronophone up the next section, and to his surprise, Fatboy agreed. It must have been taking more out of him than Tane had thought. He hadn't complained, though.
The next ten flights of stairs were far more grueling with the weight on his back crushing down on his knees and his leg muscles with every step until finally they came out onto the Sky Deck.
Another door. Locked. Fatboy had brought a screwdriver and forced the lock with that. Tane shone a flashlight up inside the shaft. A metal ladder led straight upward.
The many flights of stairs had been lit by emergency lighting. Here there was none. The only light inside the narrow shaft was the dull glow of moonlight through the open door at the bottom of the ladders and from Tane's flashlight.
Fatboy lifted the backpack from his shoulders, and Tane breathed a small sigh of relief.
He gripped the ladder tightly. At least the staircases had been safe. Here it would take only one missed rung from one tired leg, and he would be lying at the bottom of the shaft with broken bones, or worse.
Above him Fatboy seemed as tireless as ever. Hand, foot, hand, foot. There seemed to be no end to the succession of metal rungs.
There was. But that only proved to be a small landing, and the start of the next ladder, on the opposite side of the shaft.
The sides of the Skytower were squeezing in on them all the time now as they neared the top.
Tane tried not to think too much of the last section. That would be the hardest. That was up another ladder. On the _outside_ of the tower.
**L INE OF FIRE**
**Ramirez looked down at the** path of destruction from the height of his FA18 Super Hornet. The line of burning trees, scrub, even some houses, extended from the coast into the far distance. It was a line of golden fire, eating a ragged path across this darkened country.
Six wings of Tomcats, some thirty-six planes in all, had worked for hours to create the flaming barrier. A fence of fire to keep the deadly fog trapped on one side.
It hadn't worked entirely. The fog rolled up to and over the burning hilltop. But it did have some effect. To the north of the line of fire, he could see dense banks of fog rolled up into tight blankets, painted in silver by the light of the moon. To the south, the fog was a light haze, and streetlights and some buildings were vaguely visible through the light gauze curtain.
To the south of that were the busy lights and black shapes of the men and machinery of the defensive line. He wished them luck and even said a quick prayer for them. He was glad to be in the embracing seat of the _Deus ex Machina,_ not dug into a hole in the dirt, waiting for the battle to arrive.
They had been ordered to stand off and to ascend, and now he saw why.
A river of dots showed up on his radar scope to the south, and when he sought visual confirmation, the moonlight showed a squadron of small aircraft approaching. They were just silver dots in the sky, too small to make out any features, but he knew what they would be from the radio traffic.
Just about every crop duster in the whole of the Waikato farming district, south of Auckland.
The fog was merely a few hundred yards from their line, Crowe realized. The massive searchlights mounted on the ridge behind him illuminated the front edge of the mist, making eerie patterns in the shifting clouds, in the darkness.
The moon helped, too, frosting the top of the mist and giving it a half-light glow.
_Where the hell were those fire trucks?_
Troopers had been dispatched to every fire station in the vicinity with orders to bring every machine they could find.
A smart supply sergeant had suggested a water blaster import company in West Auckland, and a small convoy of army supply trucks had been dispatched there immediately.
The first fire appliance arrived even as he was asking the question. A few more fire trucks, blood-red in the moonlight, were close on the heels of the first, and Crowe barked orders, positioning the trucks at intervals down the highway. They seemed few and far between.
A convoy of water tankers crested Sunset Ridge and made their way carefully down toward his position. Crowe swung up onto the running board of the first truck as it drew up near him.
"Seawater? You have a tank full of seawater?"
The trooper nodded. Crowe didn't ask where they had found the trucks and didn't much care. He jumped down and showed the trooper, with hand signals, where to go.
An explosion sounded, just over the highway. Then another, and another. The fog was getting closer and so were the blasts of the claymore mines as the creatures passed in front of their sensors. A new sound now. A low buzz from the sky. Many moving stars to the south. That would be the crop dusters, loaded, not with fertilizers or herbicides, but with water.
Salt. Water. Salt water. If they survived this long night, he was going to see that that girl got some kind of medal. They could make her the prime minister if they wanted; he'd support it. If she was right about the water, and the salt, then that might just turn the battle in their favor.
He hoped she was right. She could be right. It made scientific sense. But then, when you thought too hard about it, so did her explanation for the jellyfish and the snowmen. Certainly they had not yet come up with any other explanation for them. He shook his head violently, trying to shake out the ideas. Four of his men were dead. He wasn't going to let her call them germs.
Rebecca was less than ten miles from Devonport when she realized that she was not going to be able to get through. Devonport was at the end of a long peninsula, stretching from the North Shore like a finger pointing across the harbor. The road had clearly clogged up early in the day, and both sides of it were choked solid with empty vehicles.
She had avoided the motorway and come down the coast, on Beach Road. There had been fewer abandoned cars and blockages along that route, but here at Takapuna there was no way through. The center of the town was a tangled web of metal.
She stopped in the middle of the road, alongside a shiny silver Porsche sports car. A large theater complex, the Bruce Mason Center, lay to her right. To her left, a series of tall towers, apartment blocks, dominated the smaller shops and terraced houses on the other side of the street and the side roads.
Lights were on in many of the apartments. People who had been unable to escape and had decided to just wait it out.
The upper two floors were alive with light. People congregating as high as they could, she thought, hoping to get above the fog.
"What do I do now, Xena?" she asked the chimp, who was curled up, apparently sleeping, on the passenger seat. "And why aren't you wearing your safety belt?"
Xena said nothing.
She couldn't get through. But she had to get through. If she couldn't, then Tane and Fatboy would be stranded on the waterfront as the fog rolled down from the north.
There was a chance that Crowe and the army would stop it, but she didn't want to count on that.
To her left, a side road led down to the Takapuna boat ramp and onto Takapuna Beach.
The beach!
She spun the wheel of the Cherokee and bulldozed the expensive little Porsche out of the way. Rubber shrieked and metal crunched, and sparks began flying off something underneath the car, but then it was to one side and she was flying down the side road.
A trailer-sailer had been abandoned at the bottom of the boat ramp, but the nose of the Jeep turned that into match-sticks. The big tires dug into the sand, and she was moving once again toward Devonport. Toward the navy base and the _Möbius._ Toward sanctuary.
The tide was out and the hard-packed sand above the waterline made for easier going than the soft white sand at the top of the beach.
Rebecca kept her speed as fast as she dared on the beach, in the dark, not wanting to run into any driftwood or rocks, partially buried in the sand, that might damage the big tires, or throw the Jeep off balance.
Jeeps were made for this kind of driving, though, and it was almost a cruise until she got to the end of the beach and the rocky point that separated Takapuna Beach from the next bay beyond.
She gingerly let the wheels of the Cherokee pick their way up through the rocks, hoping that the Jeep's clearance would be high enough to pass over some of the larger rocks between her wheels.
There were harsh scraping sounds from underneath, but she kept moving forward. Then she was up over the crest of the point and bouncing over boulders down toward the sand on the other side.
Tane surveyed the battle from the ladder above the crow's nest, a sturdy ring of metal around the outside of the Skytower; it seemed to Tane no more wide or stable than a tightrope.
The countryside was alight. The fire was spreading, and the fog disappeared in those brightly burning patches. But all around the flames the whiteness gathered, moving always southward.
Looking to the south, they could still see long lines of taillights of thousands of cars, buses, and army trucks, jammed in a desperate race to get out of the city.
They hadn't counted on the wind. It was gusting now, in from the sea, not in a steady breeze that they could counter by leaning into, but in irregular puffs, sudden gusts that wrenched at their hands and tried to knock them from the ladder.
Technicians did climb this ladder. But they did so on calm days, with safety harnesses. Tane and Fatboy had no harnesses. And this was not calm. Or even daytime.
Already they were ten yards above the crow's nest. Around them now was a forest of satellite dishes and aerials, some large, some small.
Tane went up the ladder, shining a torch on the serial numbers stenciled in large black letters on the base of each dish. Fatboy was fastening the Chronophone to the metal grid of the crow's nest by winding some strong wire through loops built into the corners of the case.
The wind tore at him again, and he lost the grip of one hand, clutching desperately with his other hand and winding one knee around the ladder until the gust subsided.
He looked down and saw Fatboy spread-eagled over the top of the Chronophone. He got up, though, and nodded to show he was okay.
A few yards more and Tane found the dish they were looking for. It was massive. He didn't need to check the number; he had memorized that long ago.
"How the hell are we going to realign that monster?" he shouted over another gust of wind.
"No worries." Fatboy climbed up to him and passed him the end of a black cable. An identical cable was already plugged into the base of the satellite dish. Tane tried to unplug it, then realized that there was a round silver ring that had to be unscrewed first.
He let the original cable hang loose from its fastenings on the side of the tower and plugged their cable in its place, tightening up the silver ring until he could tighten it no more.
Fatboy was just below him now, and Tane swung to the side, hooking his leg around, hanging off the thin edge of a narrow metal ladder, hundreds of yards above the ground, to let his brother past.
Fatboy produced a ratchet wrench from a pocket in his leather jacket and fitted it onto a nut on the base of the dish.
"Watch the numbers," he shouted, the wind whipping his words out into the void around them.
The dish was calibrated. There were two adjustment bolts—one to control up and down, and one for left and right.
Tane looked down at the coordinates written on the back of his hand and watched as the mouth of the dish began to climb skyward.
"That's it!" he called when the red line matched perfectly with the correct white line.
Fatboy moved the wrench to the other bolt, almost dropped it, but regathered it in time, and began to turn.
"Try the other way," Tane yelled.
The dish began to align itself.
They were almost home.
**F ATEFUL LIGHTNING**
**Manderson's tall frame jutted out** from the foxhole to his right. Next to him, a fire crew, outfitted in biosuits, manned a couple of hoses in front of their truck. The rest of Crowe's men were spread along the side of the highway to his left. He had asked for volunteers from the Green, Orange, and Yellow Teams, and forty of the forty-eight members of those teams had flown in since the Christmas Day disaster, joining Red and Blue Teams, now dug in along the highway.
Already the streetlamps along the highway were clouded. They were on the edge of the mist.
Crowe raised his weapon. Not his usual XM8, but a common industrial water blaster. The gas-powered engine throbbed away behind him.
He made sure the setting was on its highest power and tried a short burst. The jet of water shot clear across the highway. Up and down the line, there were occasional bursts as other troopers tried out theirs.
Overhead, the tiny shapes of the crop dusters circled, waiting for instructions from the command center.
Crowe didn't want to use them, possibly his most potent weapon, until he was sure of his targets.
The streetlights disappeared into the thickening fog, yellow halos forming around the bright stars of the bulbs. The mist lapped at the edge of his foxhole, then drifted slowly down into it, wrapping around his legs and waist as he knelt in the shallow dugout.
The fear seemed to be a part of the mist. It was palpable, so real that he thought he would choke on it. He wanted to turn, jump out of the hole, and run for his life back down the motorway toward the city.
There was something about the mist that went beyond the usual fear of going into combat. It held the fear of the unknown, and something else, too. It was impossible to describe. It was as if, not just his mind, but his very body was afraid of the mist. As if the cells that formed the structures of his being were quivering at the approach of the creeping fog.
But he could not run. This was it. This was the moment of truth for Auckland, and possibly for the human race. Right here. Right now.
Crowe knew that if he was feeling the terror, then the rest of his men, and the Kiwis ready to fight alongside them, would be feeling it, too.
A voice sounded in his earpiece, Manderson's voice. He turned. His friend was singing. The song was an old one. Once the stirring hymn that had led the soldiers of the North against the Confederates in the American Civil War and later the battle cry of U.S. soldiers in the First and Second World Wars: "The Battle Hymn of the Republic."
"Mine eyes have seen the glory of the coming of the Lord. He is trampling out the vintage where the grapes of wrath are stored. He hath loosed the fateful lightning of His terrible swift sword. His truth is marching on."
Other voices were joining in now, from up and down the line. First the rest of the USABRF agents in their black combat biosuits. Then a few hesitant Kiwi accents.
He switched on his throat mike and added his voice to the chorus.
"Glory, glory! Hallelujah! Glory, glory! Hallelujah! Glory, glory! Hallelujah! His truth is marching on."
The fog swirled up around them, and the first jellyfish struck out of the darkness before the chorus had finished. The hymn trailed off into a ragged end as the men slapped at the creatures, whose tentacles were writhing into the fabric of the biosuits.
They came straight out of the dark fog. No searching, no seeking. Guided missiles locked onto their targets, arrowing through the mist, their fine tentacles trailing behind them.
Men hammered at them, squashing them, wrenching them off the arms and bodies of their biosuits. The air was suddenly thick with the swarming creatures.
Crowe twisted the nozzle of his water blaster around to "spray" and pulled the trigger, aiming in the air above them. Five or six of the jellyfish dropped out of the air and began squirming on the ground in front of him. The surface, the skin, if that was what it was, of the creatures was fizzing and bubbling. That was the salt, Crowe realized.
"Take that, you bloodsuckers!" Manderson cried to his right.
Crowe tried another spray and sent an instruction to the crews of the fire engines.
More of the jellyfish dropped to the ground, and he saw the fire crew to his right raise one of their nozzles. A curtain of salty water appeared in front of them. A mist of their own.
The jellyfish were dropping and fizzing by the hundreds, perhaps thousands. The roadway in front of them was covered in them, wriggling for a few moments, then lying still.
The ones he had sprayed earlier were still there, he noted. The outer surface had hardened, calcified into a pale white shell. _They had not absorbed back into the mist!_
"Here they come!" someone called in his earpiece, and he strained his eyes to see the first of the snowmen lumbering out of the fog toward them. The noise they made was no longer a hiss. Here in the thinned-out mist, they moved more slowly and the sound was like that of wind around a house on a stormy night.
At first he saw just a few, gradually solidifying out of the mist. Then more appeared behind them; and more behind them. Suddenly they seemed to be everywhere, marching across the gray asphalt toward them.
"My God!" Manderson murmured.
"Water blasters hold till they are closer," Crowe ordered. "Hose crews, fire at will."
The fire hoses to his right swelled and burst forth in two heavy streams of seawater. Long wet arms that reached through the fog across the highway to the hordes of creatures that approached.
Where the high-pressure water struck, the snowmen exploded, great globs of their substance flying into the air.
Crowe watched as the closest of the creatures were all but cut in half by the saltwater jets from the fire hoses.
And yet still they marched forward.
Crowe keyed his throat mike. "Bring in the dusters, right down the highway. Right over our heads."
There was a roar above him as the first plane made its low-level run, just above the fog, and the mist intensified as the salty sea spray drifted down around and in front of them.
Clouds of jellyfish dropped down through the mist, piling up on the highway, but the snowmen still marched forward. Their skin, too, was shriveling and bubbling with the salt, but it did not halt them.
The fire hoses were sweeping back and forth, ripping the creatures to shreds, but the fire crews were spread too thinly.
"Water blasters, pick your targets," Crowe yelled. "Make it count!"
More tankers were already on their way, but the water was still a limited supply. Thank God, or thank Rebecca, for the salt, which prevented the fog from reclaiming its own.
A white, sluglike creature lurched toward him. Crowe sighted along the barrel of the black plastic and metal rod of the water blaster and squeezed the trigger.
The jet of water shot out, and he directed it across the chest of the creature. It punched a hole straight through the spongy material, and Crowe sawed the blade of water back and forth, cutting the creature into pieces. Manderson was doing the same.
This close, and in the lighter mist, Crowe finally came face to face with the enemy. He could see why the first to sight them had referred to them as snowmen. They looked like puffed-up human beings, inflated somehow from within, and covered in a white, gelatinous substance. They had faces, he realized. Almost human, with mouths, noses, and even eyes, but all formed from the same white spongy substance. The eyeballs, the eyelids, were white. The eyelids did not blink. The terror that had welled up in him before intensified, but he stood his ground.
He cut another couple of the creatures down, taking the legs off one and slicing through the neck of another.
The creatures dropped.
The jet from the fire hose swept across in front of him then, and the ragged line of snowmen faltered for the first time. Crowe hacked at the creatures with the blast of water from his weapon and found he was whooping with exhilaration.
They were winning. The line was holding!
"The mist is thickening," Manderson called, next to him.
"Get those fighters in here," Crowe shouted. "Close support. Light up the other side of the road, see if we can thin out the mist a little."
The fog was indeed thickening, he realized, and the creatures were starting to move faster. He ripped another couple apart. The mass of the creatures' flesh was filling the roadway. Overhead, the crop dusters swooped, covering the bodies of the creatures with brine, shriveling and fizzing.
Across the highway, lightning flashed and he ducked down into his foxhole for a moment as the shock wave of multiple explosions shook the ground around them.
Lucy Southwell's voice on the radio then, from the command center, three hundred feet behind the line.
"Crowe, this is Lucy. The evacuation is nearly complete, but we've lost all contact with the easternmost sections of the line."
"My God!" Manderson said for the second time that night, only this time it sounded like a prayer.
**S ILENCE IN THE MIST**
**Ramirez pulled up and watched** below him as the last of his air-to-ground missiles impacted on the grassy strip, not even a hundred yards from where the troops cowered in their foxholes on the other side of the highway.
Now that was precision flying and precision targeting.
The crop dusters had left now, having exhausted their tanks. His aircraft were also out of bombs and missiles and were already heading back to the carrier to re-arm.
Ramirez alone remained over the battleground, circling, to feed information back to the carrier and to the troops on the ground.
The line was holding here at Albany, he saw, and also out to the west. But the east coast suburb of Mairangi Bay had been long swallowed by the dense white cloud, which had outskirted the defensive line, floating slowly out to sea and back in again behind it.
The frigates _Te Mana_ and _Te Kaha_ had been positioned in the bay, against just such an eventuality, he knew, but the _Te Mana_ was now resting, listing over, on the sands of Mairangi Bay beach, and the _Te Kaha_ was slowly grinding itself to pieces on the rocks of the head land. There was no movement on board either vessel.
The fog, apart from the holdup at Albany and farther west, was pouring down the east coast of the North Shore, spreading out behind the defense force at Albany and chewing its way across the affluent suburbs of Castor Bay, Campbells Bay, Milford, and on to Takapuna and Devonport.
He risked a low pass over the highway, trying to see the troops on the ground, but the mist was too dense.
Rebecca pulled up off the sand onto the grassy verge alongside Cheltenham Beach and thought she was lucky to have made it.
Some of the rocky promontories between the bays had been almost impassable. If it were not for the low tide, it would have been impossible.
She gunned the engine up past the navy training center and around onto the main road. Glancing to her right at the intersection, she realized with horror that the mist was barely a few hundred yards away and crawling rapidly forward along the road.
Only a few cars blocked the road here, and she swung from side to side, weaving in and out of them, the fog, omnipresent in her rearview mirror.
The final stretch, alongside the Devonport Golf Club, was clear.
Xena had woken now, if she had actually been asleep and not just resting. She was quiet, though, watching Rebecca drive with wise eyes.
Rebecca skirted the base of Mount Victoria on the long looping road and accelerated down the deserted main street of Devonport.
At the wharf, she turned right, heading along the breakwater toward the naval base.
The barriers were down at the entrance to the base, which didn't surprise her. What did surprise her, and perhaps shouldn't have, was the armed guard who stepped out of the security booth and waved her to a halt, the pistol held ready for use in his right hand.
"No admittance," the guard said, not at all calmly. "This is a military area."
Another guard stepped out of the booth then, and he had an automatic rifle held at the ready.
"I have to get through," Rebecca said urgently. "I have orders from Doctor Crowe and Doctor Lucy Southwell."
"No admittance," the guard repeated.
"Get out of here," the other guard growled.
Xena screeched, alarming the guards, who had not noticed her until then.
"What the hell?" the first guard said, looking at Xena.
"Oh hell!" The second guard said, looking where Xena and Rebecca were looking.
The fog was rolling rapidly down the slope toward the sea, swallowing building after building as it came.
Two more of the nightmarish white creatures hurled themselves out of the ever-thickening fog. Crowe cut a diagonal slash across them with the jet from his water blaster and they fell.
The crop dusters were gone now. So, too, were the fighter-bombers. The girl had been right. She had been right about the salt and the water, and everything else she had advised or suggested.
Another snowman reared up in front of them, but Manderson cut it open at the neck before Crowe could pull the trigger on his weapon.
_Was it possible that she was right about the creatures?_
The twin fire hoses next to them were silent now, and Crowe glanced across to see why. Had the water run out?
Where the crew had been, two men to a hose, four of the white sluglike creatures stood silently. Absorbing. Digesting.
"Stony," Manderson said urgently, looking behind them.
Crowe turned. The fog had come up behind them. It was closing in on their position as he watched. The front of the cloud was alive with antibodies, and behind them moved the dense shapes of the macrophages.
"Stony, we did it," Manderson began, with a quiet resignation in his voice. "We held out long enough..." But there was a hissing noise from the front, and Mandy disappeared, replaced by one of _them._ The white-lidded eyes stared unblinkingly at Crowe from where Manderson had crouched.
Crowe screamed and turned his water blaster onto the macrophage. It tore a jagged line across the creature, and the remains of Manderson's suit spilled out, hanging loosely out of the torn white flesh.
The fog was thickening all around them now. He looked to the left and right, but if any of his men were left, they were invisible in the fog. He tapped his microphone and called his team, but got only silence in return.
An antibody struck the faceplate of his helmet, covering his eyes. Crowe screamed again and slapped it away. He strode forward into the mist, shaking his head violently, erratically, from side to side. The hose of the water blaster pulled him back, tried to stop him, and he wrenched at it, felt something give, then strode forward again.
The barrel of the device, disconnected and useless in his hands, swung around as he aimed the empty weapon at the whiter-than-white shapes that appeared around him.
"Mine eyes have seen the glory of the coming of the Lord. He is trampling out the vintage where the grapes of wrath are stored."
His voice filled the suit, and he unsealed and flipped up his face mask to let the words out into the fog.
"Glory, glory! Hallelujah!"
And then there was silence in the mist.
**T HE GOD FROM THE MACHINE**
**It was faster going down** the ladders, but there were still a lot of rungs. The staircases, too, were faster; they bounded down them in the dim light of the emergency lamps, two, three, even four steps at a time.
But it still seemed to take them a long time to get down to the main observation deck.
Tane found the entrance to the many flights of stairs leading down the main shaft of the Skytower, but Fatboy said, "Wait."
He checked himself and backed up to where Fatboy was staring out of the huge toughened glass windows of the observation deck.
His heart began to pound and blood rushed to his ears with a thrumming sound. His legs felt unsteady and he leaned against a handrail for support. The world around them had turned to white.
Auckland was awash in a sea of mist. The entire city had disappeared. All they could see in the cold glow of moonlight was the top of a cotton wool cloud. Way below them, the fog flashed on and off, a dim red color, and Tane remembered the fire engine parked in the middle of the road.
"Did we give them enough time?" he asked, looking to the south.
"I think we did," Fatboy said.
"What about us?" Tane asked in a small, quivering voice.
"We're too late," Fatboy said steadily. "There's no way out. It's over for us now." He turned and looked at Tane. "But not for Rebecca. If she made it to the submarine, then she can still make it to the underwater cave. She can still send the messages."
"Not for Rebecca." Tane's voice was a distant echo. Rebecca alone would endure life in the submarine as the rest of the world disappeared above her. Rebecca alone would send the messages back to the past.
_The messages!_
"The messages were signed 'TR,'" Tane protested feebly. "'Tane and Rebecca.' Not just 'R'!"
Fatboy was quiet for a moment, staring out at the cloud. "If you were Rebecca, alone in a submarine, sending messages to the past, to herself and to you," he said slowly, "how would you sign them off?"
Tane realized with a cold heart that he was right. He looked at his brother and said nothing. He just looked, and Fatboy looked back at him without awkwardness. How wrong he had been, Tane thought, not to trust him from the beginning.
Fatboy said, "You and Rebecca—"
"Good mates," Tane said quickly.
Fatboy laughed. "She likes you, Tane. She really likes you. More than just good mates."
"No, really. We're just...what did she say?"
"Nothing," Fatboy said. "I doubt she would. But I could tell."
"You're wrong," Tane said.
"No, I'm right." Fatboy shook his head. "But it would have been wrong for me not to tell you now. Now that..." His voice trailed off, and his eyes drifted back to the rising fog around them.
"You always looked out for me," Tane said. "I should have—"
"You shouldn't have done anything different," Fatboy said. "I knew where you were coming from."
But this time Fatboy wouldn't be able to look out for him. Nor would he be able to look out for Fatboy. There was nothing anyone could do.
Tane looked at his brother and held out his hand. It was strange, but it seemed like the right thing to do. Fatboy took it and shook it, but then pulled him closer and pressed his nose and forehead to Tane's. Three times he pressed in the traditional _hongi._ Once for the person, once for the ancestors, and once for life in the world.
The _hongi_ was a greeting, but this, they knew, was a goodbye.
They descended into the stairwell, into the shaft of the tower.
"Get out of the way!" Rebecca screamed, and jammed the Jeep into gear.
It smashed into the barrier pole with a crunch and the sound of broken glass.
"Stop!" the first guard shouted, but he seemed uncertain, and his pistol wavered between Rebecca and the oncoming fog.
The barrier pole didn't break; it was metal and just bent a little.
Rebecca thrust the car into reverse and backed off a few yards.
"Stop!" the other guard shouted now, raising his weapon.
She ignored him and stomped on the accelerator. The Jeep rammed forward and the metal bar bent a little more. She kept her foot down hard and the tires began to smoke, the end of the Jeep sliding around as the bar prevented her progress.
There was a crack of a gun, and the windshield shattered. Xena and Rebecca screamed in unison.
She dived across the seat of the Jeep, catching the door handle on her first try and falling out of the door of the vehicle. She grabbed Xena by the hand and slung her up around her neck, running toward the edge of the breakwater, instinctively putting the Jeep in between herself and the guards to give herself time to get some distance before they could fire again.
It helped for only a few seconds. There was another crack then and a whistle past her ear. Xena squealed again.
Rebecca raced down the road leading into the small cluster of buildings that was the naval base. Old, weather-boarded buildings with tin roofs, dating from the 1940s or earlier.
There were no more shots, and when she glanced back, she saw why. The guard hut and the Jeep were shrouded in mist, and where the second guard had been, the white shape of a macrophage stood silently, motionless.
The first guard was lying on the ground, screaming noiselessly, with a dozen antibodies covering his arms, legs, and face.
The fog was rolling in over the dark buildings to her right, but she had no option except to follow the road.
And there it was. Moored to the side of a long jetty. The familiar, bulbous, warm-yellow shape of the _Möbius._ It was barely a hundred yards away, out along the jetty.
The fog clouded around her and she stumbled, tripped and fell, Xena rolling away from her across the tar-seal with a squawk.
Rebecca tried to get back up, but her leg didn't seem to be working properly. Confused, she looked down to see a glutinous shape, with short fibrous tentacles, latched onto her thigh.
She screamed in fear, and then screamed again in pain as the sting of the needlelike fibers reached her shocked brain.
Somehow she hauled herself to her feet and hopped forward, dragging the useless leg behind her.
"Come on, Xena!" she called, but the little chimpanzee just sat there, quivering with fright, and looked at her.
There was a stinging in her left arm now as two of the antibodies attached themselves to it; then her arm, too, went numb and useless.
"You come if you want to," she yelled at the chimp. "I ain't waiting for you!"
She had no chance of making the submarine—she knew that now—but if she could just make the edge of the jetty and the waters of the harbor...
She collapsed again, just a yard or so from the edge, from safety, and looked down at another antibody digging its way into the shin of her good leg.
"Get off me!" she screamed, and crawled forward as best as she could with one good arm and only a little movement in her right leg.
The fog swirled around her, and a shape moved in front of her. She raised her head to see a macrophage, tall and white in the early light before the dawn, standing between her and the ocean. Waiting for her.
"Leave me alone!" she screamed, and crawled forward another few inches.
The shadow of the creature approached, and Rebecca screamed one last time, except the scream wasn't hers. It wasn't her voice; it wasn't even a human voice. There was a blur of brown hair and the sensation of small feet on her back, and Xena leaped up from her shoulders straight at the macrophage, clutching around its neck, bending it backward, overbalancing, and then there was no macrophage in front of her, only a rising spout of seawater, and it seemed to be minutes later that she heard the splash.
"Xena!" she cried as the fog continued to make ghastly, ghostly patterns all around her.
The voice of his flight controller sounded in his headset, and Ramirez pulled his jet around in a tight bank. As far as he could see, there was only fog.
"Roger that," Ramirez replied. They wanted one last low-level pass to visually confirm the status of the ground troops.
He circled around once again and descended. Beginning his run from the north. Toward the tall spire of the Skytower, jutting up through the clouds in the distance in front of him.
He dropped down out of the sky and skimmed across the top of the mist like a stone skipping across water. He tried to peer down through the fog, but it had thickened and was now impenetrable.
He shook his head and keyed his radio to call base, but the words never left his mouth. The fog itself seemed to rear up in front of his aircraft.
The world went white, and there were banging noises against the cockpit and the body of the plane.
He hauled back on the stick and lifted the jet out of the rising mist, but it was already too late. There was a cough from his right wing and the jet flamed out. Something had been sucked through the engine. The left engine went two seconds later.
"Mayday, Mayday, Mayday," Flight Lieutenant Ramirez said urgently, but with professional calm, into his radio, "I have a double flameout. I am about to eject."
He kept the stick back, gaining as much height as he could, and then punched out of the cockpit with the two-handled ejection lever. The ejector seat kicked like a horse, and the jet hurtled onward in the sky, pilotless, toward the dark tower in the distance.
Ramirez's parachute opened with a grab on the back of his spine that wrenched at his insides. He felt sure he had broken something. But it didn't matter. By the time the parachute had floated gently to the earth, the harness and the flight suit were empty.
**H OBSON STREET**
**The shaft was clear of** mist. Free from antibodies and macrophages. They descended slowly, step by step, then faster.
Fatboy wore his full motorcycle leathers, but Tane had on just jeans and the fashionable leather jacket he had bought with Rebecca on their Lotto spending spree, a hundred and fifty years ago.
He thought the leather might be strong enough to keep out the antibodies, but he wasn't sure about the jeans.
At the very base of the tower, they had stowed their helmets, both full-face. It felt comforting to slide it on. It was at least some protection.
They emerged into a silent world at the top of the long staircase. The main foyer of the casino. The huge, weather-proof glass doors of the casino remained shut. Keeping the world, and the fog, at bay.
Fatboy moved toward the elevators but Tane put a hand on his arm.
"We'll never make it on the Harley," he whispered. "We need some protection from _them._ "
"What are you thinking?" Fatboy asked.
Tane nodded toward the huge glass doors of the foyer. Even as he did so, a shadow moved through the mist outside, silhouetted by the flashing red lights of the fire engine farther down the street.
They crouched hurriedly down inside the circular security desk near the casino entrance.
"The fire truck?" Fatboy asked with a frown. "They'll see us coming for miles."
"It's big and strong," Tane said. "And from here to Princes Wharf is all downhill, straight down Hobson Street."
Fatboy nodded his agreement. "It's worth a go."
More shapes moved past the doorway in stark contrast to his hopes. There was a sudden bang from one of the glass panels, and they both instinctively looked up but saw nothing.
Then came another loud bang, this time accompanied by the sound of cracking glass.
"I hope Rebecca is waiting for us when we get there," Tane said.
Fatboy smiled tightly in the mist. "How are we going to reach the fire truck?"
Outside, the intermittent red flash seemed like a bait to lure them out into the clutches of the macrophages.
"I say we make a run for it," Fatboy said. "Make a break for the fire truck, try to get inside before they can get to us."
"We'll never make it," Tane said doubtfully.
"Can't stay here forever," Fatboy countered.
Tane nodded. "Okay, but I think I need something a bit thicker than just a pair of jeans covering my legs if we're going out there."
He glanced around the foyer. There was a long reception counter on one side. A concierge desk in the center and on the other side a large café and a gift/souvenir shop. That seemed the most likely.
The shop door was locked, but the window surrendered easily to a blow from a trash bin, and they picked their way gingerly through the broken glass.
"A few plastic tikis are not going to keep those things away," Fatboy said.
There were baseball caps with New Zealand symbols, scarves, belts, woolen beanies, and T-shirts on a clothing rack in the center of the room. Tane was experimenting with the T-shirts, wrapping them thickly around his legs, when Fatboy said, "Over here."
Piled in a corner of the shop were sheepskin rugs, thick, woolly, and whiter than white. A popular souvenir for tourists of a country with 80 million sheep. Tane ran his fingers over the thick leather backing and nodded. "That should do it."
He wrapped a thick sheepskin around each leg, strapping them on with expensive leather belts. Another went around his midriff.
He pulled up the collar of his leather jacket and then tied a thick woolen scarf around his neck for further protection. A pair of leather gloves covered his hands.
"How do I look?" he asked.
"Baa," Fatboy replied.
Tane stopped what he was doing and considered that. He took another large sheepskin and draped it around his shoulders, fastening the corners around his neck with a large metal safety pin.
"Shape recognition," he said. "The antibodies recognize shapes. Remember Dr. Green's diagram with the circles and triangles and stalks? If we can change our shapes, then maybe they won't recognize us."
A moment or two later, Fatboy also was covered in fluffy sheepskins. If nothing else, they reasoned, it was more protection against the creatures.
They looked at each other for a long moment, then both burst out laughing, despite, or perhaps because of, the danger they faced.
The banging and crashing was constant now, from the toughened glass of the entranceway, and the ominous cracking noises were getting louder.
"We can't go that way," Fatboy said. "Maybe if we go back down in the elevator and come out through the car park on the side of the—"
Two of the huge glass panels shattered simultaneously, and fog poured into the atrium of the casino. With it came a terrible hissing sound, and the front of the fog came alive in front of them. Antibodies, hundreds of them, and behind them the larger shapes of macrophages.
"Get back," Fatboy yelled, spinning around and racing back into the interior of the casino.
Tane risked a glance backward. The casino was already full of a light mist, and it was thickening with every second. The macrophages were following them, but slowly, as if moving through water.
"They've slowed down," he shouted. "They can't move as fast where the fog is thin."
Tane looked over at the central row of elevators. The doors were open on the one they had used earlier.
They ran to it and Tane pressed the button for the first parking level. Nothing happened.
"Oh crap!" he said.
Outside the lift, the macrophages turned the corner of the central lift well and floated toward them, almost as if in slow motion.
Tane jammed his finger on the button again, and Fatboy stabbed at the CLOSE DOORS button.
The white shapes moved closer, blocking their exit. Tane cowered into the back of the elevator, holding his arms in front of him as if that would somehow protect him against _them._ Fatboy pulled himself up to his full height, folded his arms, and faced the approaching creatures. The closest of the creatures was just about at the doors of the lift when they slid smoothly shut.
The doors opened again with a _ping_ on a misty, murky, gray concrete parking level. Only a small stream of fog flowed slowly down the ramps from the upper levels and was swiftly dealt with by the heavy air conditioners, used to dealing with car exhaust fumes.
"This way," Fatboy called, pointing to a sign marked EXIT.
It was an eerie feeling running through the underground mist. A broken fluorescent light flickered nearby, bleakly strobing the thin vapor. Their running footsteps echoed off the hard walls of the parking level.
Tane expected antibodies or macrophages to come flying out of the mist at any moment, but none materialized. Here on the parking levels, the mist was simply too thin for them.
They ran up a long sloping ramp, and then another, toward a sign marked HOBSON STREET EXIT.
The exit proved to be a curving lane, taking them up and out into the thicker fog of the open air.
"Crawl, don't walk," Tane said with a sudden flash of inspiration. "We have to change our shapes as much as possible. And move slowly. Crowe said they can feel movement in the fog."
They dropped to their hands and knees and moved with languorous, careful movements, trying not to disturb the thickening fog. They bypassed the payment booths and barriers at the far end of the lane, to find themselves on the pavement outside.
The fire engine was barely fifty yards away, in the middle of the intersection, the twin red flashing lights at the front and rear of the machine the only thing they could see of it. There was no sign of the crew, but Tane had not expected there to be. In the distance, a dog was barking madly.
The city street itself might as well not have existed, for all they could make of it.
Tane touched his helmet to Fatboy's and whispered, "Don't speak if you can help it. They are attracted to sound as well."
From near the fire engine, there came a hissing sound, growing in intensity. Tane froze and saw Fatboy do the same. He drew a cross over the faceplate of his helmet, which Fatboy could just see in the thick fog. _Don't even breathe._
Two macrophages flew past them, moving fast, not stopping. They seemed not to sense Tane and Fatboy, motionless, on all fours, on the pavement at the top of the ramp. There was a whistling sound and a swirling in the air around them and a handful of antibodies drifted by.
Tane waited until the sounds had faded before tapping Fatboy lightly on the shoulder.
They stopped once more on the way to the fire engine, when a macrophage hissed to a halt right in front of them. Tane was sure it had seen them, and what it was waiting for he couldn't imagine. He shut his eyes and held his breath, waiting. What was it like, he wondered, to be ingested by one of the creatures? Did it hurt, or was it all too quick? He slowly opened his eyes, still waiting for obliteration; however, after a few moments, it slid away with a gradually increasing hiss.
Tane peered around the corner of the casino, trying to detect if there were any macrophages waiting by the machine. If there were any, he could not tell. They were invisible in the thick fog, and if they were not moving, they were silent.
The dog started barking again, very close by, although he could not see it.
There were hissing noises from the same direction, and the barking grew louder. The dog was quite safe, Tane thought bitterly. It was _human_ cells the macrophages were out to destroy.
He crawled a yard forward, then another, and suddenly there was a low shape running through the fog at him, barking madly. The dog stopped a yard away and growled viciously, snarling at him with its lips drawn back and teeth bared. He reared back and upward in an instinctive reaction.
Suddenly there were hissing noises converging on them from three different directions.
"Run for it!" Fatboy cried, jumping to his feet.
The door closest to them was open.
Tane was in through the door first and heard Fatboy slamming it behind him. The driver's window was down, and he groped for the handle in the dark and fog. His hand latched on to something and he wound it furiously. The window closed.
There was a bang from the door they had just dived through, but the metal and toughened glass held. He caught a glimpse of a bloated white shape outside but forced his attention back to what he was doing.
Through the front windshield three or four of the creatures were approaching.
"Drive!" Fatboy shouted, grabbing at controls on the passenger side.
_Drive? How?_ He had never driven a fire engine, or any kind of truck, before in his life. But then, neither had Fatboy. He found a key on the front dashboard, not on the steering column, and turned it. The engine roared into life.
Gearshift, where was the gearshift? The truck was automatic, he realized, and pulled the lever into DRIVE.
He found the handbrake and stomped on the gas pedal. The truck lifted and surged forward. There was a hard bang from the front of the machine, and two of the approaching macrophages disappeared, parts of their bodies flying out to the sides of the windshield. He spun the wheel to head down Hobson Street, toward the wharves. It was a one-way street, and they were going the wrong way, but that thought barely registered.
The thick wool of the sheepskins that covered him were smothered by antibodies, each Y-shaped creature fitting snuggly into the next, more and more of them, covering him like a hideous patchwork quilt. A few landed on his helmet visor, and he brushed those away but ignored the rest.
There were more bangs now, from both sides of the truck.
"Got it!" Fatboy shouted, hitting a button and grabbing at a large joystick. A jet of water shot out from over their heads. Fatboy wrenched at the joystick and the stream shifted in front of the truck. "Go for it," Fatboy yelled.
Tane was. His foot was on the floor, and the big red engine accelerated smoothly.
A group of macrophages reared up out of the mist in front of them, and Fatboy swept the water across them, slicing them into pieces.
The crashing noises were all around the truck now, from the roof as well.
A side window behind them shattered, and the main windshield cracked as a macrophage slammed into it, the body crushing itself against the toughened glass, before flying away to the side.
The Mercedes convertible they had seen earlier was parked haphazardly, sideways across the street. Its nose was a crumpled mess, rammed into a parked car.
Tane hit it near the rear end, and it spun around and up into the air, over the parked car, and landed on its roof on the pavement.
Fatboy swept the jet of water from side to side and in front of them, clearing a path through the charging macrophages.
They flew across the Fanshawe Street intersection, colliding with, and demolishing, a traffic light. It barely slowed them down.
Tane held the engine straight on course, across the overpass, and gunned the engine again as the Tepid Baths slipped past below to his left.
A gust of sea breeze, the same sea breeze that had nearly knocked him off the tip of the Skytower, swept in from the harbor, lifting and pushing back the fog for a moment.
In that tiny window of time, Tane saw their doom.
They would never make it to Princes Wharf, he realized. They would never make it to the sea.
In front of them, blocking their path, revealed just for a moment by the rising curtain of fog, dimly lit by the long line of streetlights, were macrophages. Row upon ragged row. Column upon column. Spread across Quay Street and out through the entranceway to the wharves. Thousands of them. At the same location where he and Rebecca had once congregated with hundreds of people to march for the whales, now the macrophages were massed to march against the human race.
Not even the charging fire engine would be able to cut a path through that number of the creatures.
"Tane!" Fatboy shouted, and Tane spun his head to see the white face of a macrophage inside the truck, just a yard from his own. On Fatboy's side of the machine, the rear door had lost the battle with the macrophages and, now just a twisted mess of metal, hung pathetically from a single hinge.
Tane screamed, and the fire engine veered toward the concrete base of the overpass.
It all seemed to happen in slow motion. One moment the creature was moving toward him, and the next Fatboy was there, a shiny object in his raised hand.
Tane couldn't understand what it was for a second, and then it flashed down at the macrophage, slicing into its flesh. The _patu pounamu,_ the greenstone club, slashing again and again at the creature.
It fell backward, and Fatboy thrust forward at it, the club a blur. He was shouting and chanting in Maori now, the blood of the warrior surging through his veins.
But the creature thrust forward again, its white flesh engulfing Fatboy's arm. He grunted a terrible, hollow sound, and the club dropped to the floor.
The truck veered into the concrete wall, sparks flying from the tortured metal. Tane wrenched his eyes back onto the road for a second, hauling at the wheel to keep the truck from smashing into the dirty gray concrete bridge support.
He looked back to see the creature, and Fatboy, disappearing out through the door of the machine.
Fatboy's voice, despairing but somehow undefeated, "Save the world, Tane!"
Then he was gone.
"Harley!" Tane screamed.
There was no time to grieve, no time to even accept the full enormity of the disaster that had just occurred. That would have to come later. There was a thunderclap above him, and he looked out through the shattered passenger window, through the thinned-out fog, to see a jet fighter aircraft clip the side of a skyscraper and plunge toward the ground in front of him.
The wreckage of the jet exploded along Quay Street in front of him, and there was no time to brake or take evasive action as the burning jet fuel and torn chunks of metal carved a long straight fiery scar across the city center.
The concussion blew out the window of the fire engine, and only Tane's instinctive reaction, ducking down behind the dashboard, saved him from being smeared over the back wall.
There was a wall of flames around him now, a barrier of fire, but then that was past, too, and he realized that the massive, fiery explosion had shattered the ranks of the macrophages, hundreds of them disintegrating in a single instant, many more being blown across the edge of the long wharf and into the ocean, where they shriveled and oozed. The seawater turned white.
Others disappeared under the thundering wheels of the fire truck, now out of control and tearing across the side of the wharf.
A small café, tables and chairs already scattered by the explosion, erupted under the charge of the fire engine. The truck veered to the left, but the metal safety railing coaxed it back onto a straight course.
Dazed, clinging to the doorframe, Tane could only watch as cafés, restaurants, and apartments flashed by the driver's window. The safety railing at the very end of the wharf was fast approaching.
The remaining macrophages were attacking now, flinging themselves at the broken body of the fire engine.
One reared up under the windshield and clambered over the edge of shattered glass and twisted rubber, reaching out toward Tane; then there was a thunderous crash, and Tane flew forward into the dashboard, blood pouring from his head, and the macrophage flew outward, spinning backward in midair as it fell toward the ocean.
Tane was almost aware of the safety rail shattering under the impact of the crash and the momentum of the fire engine, and then there was a strange silence, with just the screaming of the high revving engine and the ocean rushing up to meet the cabin of the truck.
**T HE DREAM**
**Images came in gasping blurs.**
The sudden rush of water into the cabin. He remembered that, he thought. Or was it just his imagination creating dreams and tricking him into thinking that it was memory?
Like Fatboy's voice as the creature dragged him—no, that wasn't right, as _he thrust the creature_ —out through the shattered door of the fire engine. Had that been real?
He thought he remembered the swirl of mud as the front of the truck nose-dived into the soft bottom of the ocean, at the end of the wharf. The patterns that appeared in the brown-gray whirls, the faces that tried to speak to him but just spiraled away into the water.
He remembered seeing the _Möbius._ The crazy yellow doll-like shape of the little submarine.
_Giddy arses blown will tea-coffee a waste-bin._
He didn't remember swimming, or latching on to the top of the sub, or finding the escape hatch and climbing inside.
He didn't remember any of that, and yet he must have done all of that, for his next coherent memory was that of Rebecca, weeping, one arm around his neck, the other hanging limply by her side.
And the stupid grinning face of that damn monkey.
**T E KENEHI TUARUA**
**Tane turned the outside lights** on for a while to watch the fish at play, twittering around beneath the ceiling of the cave. They fascinated him.
It had been three days.
Three days since the wild ride down Hobson Street in the runaway fire engine. Three days since his brother had given up his own life for his. Three days since the end of the world.
No, not the end of the world. Just the beginning of the end of mankind.
There were millions, billions, of life-forms on the planet Earth, this tiny rock hurtling around the sun. But the human species was the one that had overstepped the mark. Pushed the boundaries a little too much. Tried to conquer and dominate that which could be neither conquered nor dominated.
Pride cometh before a fall, they said, and mankind certainly had had pride. Building monuments and civilizations that they thought would last forever, yet were just a pinprick in the long roll of fabric that was the history of the planet.
Three days, and he had spent most of them sleeping, according to Rebecca, who had somehow cared for him, despite grievous injuries of her own. Concussion, she thought, but she wasn't a doctor, so who could really know.
She wasn't a doctor, and neither was Tane. There soon would be no more doctors. Nor lawyers, nor stockbrokers, nor movie stars, nor presidents.
The view from the periscope camera was always the same. Endless banks of white fog, stretching in all directions.
By now it might have covered the whole of the country. Perhaps already made its start toward Australia. The Pacific Islands, and on to the coast of South America.
They'd try to stop it. They might even get better and better at fighting it. But it was too big. Too hungry. And there wasn't enough time.
There was good news, though. Rebecca's arm had regained a little bit of movement. She still couldn't walk, but she could feel her toes again, so that indicated that her system was slowly recovering from the tentacles of the antibodies.
Not Fatboy, though. There was no coming back from what had got him.
Rebecca smiled up at Tane from the bunk as he entered with a can of Diet Coke.
"There you go," he said, handing it to her one good arm. "Ready for some exercise?"
"What do you have in mind?" Rebecca smiled.
"I thought a little aerobics, maybe followed by some kickboxing, and a little mountain biking to round it all off."
She laughed.
Tane said, "Or we could just do the stretching exercises, like yesterday."
Xena flipped herself down from the topmost bunk as Tane grasped Rebecca's right ankle and bent her knee, pushing her ankle up to her backside. He held it there for a moment, then repeated the process on the other leg.
"Progress," he said cheerfully. "You were helping a bit that time. I could feel it."
She smiled again. "Every day it gets a little better. I don't think it's a permanent thing."
"Thank God."
"Indeed."
He stretched and bent her right leg a few more times.
Rebecca said wistfully, "I had my whole life mapped out, you know. Ever since I was twelve. I knew what university I wanted to go to, what I was going to major in. Where I was going to work. Everything like that. But life kept changing things. First my dad died. But after a while I thought I could cope with that and still stick to my plans. Then there was that thing with Mum. But I was on top of that, too, I thought."
Tane thought he caught a glimpse of a tear, but nothing more.
She said, "Then all this happened. Now nothing is the same. Nothing is ever going to be the same."
Tane changed legs. "Not me. I didn't know where I'd be, one year ahead. Most of the time I didn't know where I'd be one day ahead. But whatever I expected to come from my life, I sure as hell didn't expect to be sitting on the bottom of a cave in the Hauraki Gulf in a yellow submarine, drinking Diet Coke during the apocalypse."
Xena grabbed the can out of Rebecca's hand and wandered off toward the cockpit, chattering happily. She had quite a taste for it, they had discovered.
A brightly colored school of fish surrounded the port-hole for a moment, as if fascinated by what they saw inside. Here, we are the goldfish bowl, Tane thought, but didn't say it out loud. It took a bit of getting used to, this life under the sea.
"Did we save them?" Rebecca asked. "Do you think?"
Tane smiled and took her hand. "I think we bought them a little time. We slowed down the fog enough to let them evacuate Auckland. But the fog won't stay still. It'll keep moving south, and so will the..." He paused, searching for the word.
"Refugees," Rebecca contributed, and she was right, but it seemed an odd word to be using for the population of Auckland.
"And pretty soon they'll run out of room to run."
Rebecca clasped her other hand over his and he felt the tension in her grasp.
"Is there any way to stop it? If only we—"
"I thought we agreed never to say 'if only.'" Tane forced a smile.
"Yeah, but if only—"
"If only this, if only that. If only we'd done things a little differently, we might have stopped the Chimera Project. You can never say 'if only.'"
"I know."
Tane closed his eyes for a moment, trying to get it out of his head. _If only..._
He looked up to see Rebecca smiling at him.
Tane fingered the _patu pounamu_ hanging from a cord around his neck. He hadn't remembered grabbing it, but Rebecca had apparently had to prize it out of his fingers after she had got him on board.
"You think we're supposed to start over?" Tane asked.
She nodded. "I think so. Once it is all over and the clouds of fog have slowly dissipated. Maybe months, maybe years from now. I think we get a new beginning. This time we will do it like your father said, as family with the world around us, not as conquerors of it."
He took her hand and held it, thinking about her words.
He tried to imagine it but a flood of images kept intruding. Fatboy in the door of the fire engine. His mother and father waving goodbye at the entrance to the Marae. Rebecca's mother, watching events on her television that were unfolding outside her window.
Xena sat in the corner of the cabin and slurped at her Coke and grinned at him.
"No," he said finally.
"No?" Rebecca's eyes were wide, questioning.
"Too many people died," Tane said. "Too many people died."
"But Kaitiakitanga," Rebecca protested. "Building a new race, teaching them to live in peace with the planet..."
"All of that I agree with," Tane said. "The teachings of the ancestors, the responsibility, the guardianship of all that lies around us. Kaitiakitanga."
He clasped her hand in both of his.
She said nothing, but he didn't need her to.
"You asked if there was any way to stop it, and I think there is. Nobody has to die."
"Nobody has to die." She twisted the words around in her mouth. "What do you mean?"
"The only way to stop it is to stop it from starting."
"Stop it from starting?"
She looked at him blankly and he asked, "Where's that notebook of yours with all the messages in it?"
"It's in my pocket. Why?"
"Those aren't the messages we're going to send."
She caught her breath, catching on to his meaning. "We change the messages!"
"Nobody has to die," Tane said. "If we change the messages, we can change the past."
"If we change the past," Rebecca breathed, "we change the future!"
"Our present." Tane smiled. "We can change our _now._ "
"Then nobody has to die. Not Fatboy, not your mum and dad, my mum. Not even Zeta."
"Not even Zeta." Tane smiled. "I'm not sure about Grandad, though."
Rebecca laughed, and said, "Just when I was getting used to the idea of being stuck in a submarine with you for the rest of my life."
Tane laughed, but then his face grew serious. "But we can't make the same mistakes we did last time."
He paused and looked her straight in the eye before continuing. "So this time...let's get it right."
**T HE BEGINNING**
_They took all the trees_
_Put 'em in a tree museum_
_And they charged the people_
_A dollar and a half just to see 'em._
_Don't it always seem to go_
_That you don't know what you've got_
_Till it's gone_
_They paved paradise_
_And put up a parking lot._
_—Joni Mitchell, "Big Yellow Taxi"_
**The saving of the world** started quietly enough for Tane Williams and Rebecca Richards, lying on their backs on a wooden platform on Lake Sunnyvale. Which wasn't really a lake at all.
Sunnyvale School was set in a small valley. A nice little suburban valley. A hundred years ago, it had been a nice little swamp where Pukeko and Black Stilts had competed for the best nesting positions, and croakless native frogs had snared insects with their flicking tongues. But now it was a nice little suburban valley, surrounded by nice little homes belonging to nice little homeowners who painted their fences and paid their taxes and never gave any thought to the fact that when it rained, all the water that ran through their properties also ran through the properties below, and the properties below those, and so on until it reached the lowest point of the valley floor. Which happened to be Sunnyvale School.
As a consequence, Sunnyvale School had to have very good drainage. When it rained hard, as it often did in the west of Auckland, an awful lot of that rain made its way down from the hillsides surrounding the school and ended up on the playing fields and netball courts of the small but cheerful school.
The stars above shone down with a piercing intensity that penetrated the haze of lights from the suburban homes around the valley. The moon, too, was lurking about, turning the weathered wood of the small platform to silver. From an open window in a house somewhere on the surrounding slopes, an old Joni Mitchell folk song reached out plaintively across the water to them.
Tane and Rebecca lay on their backs on the small wooden platform in the center of the two main playing fields and looked up at the stars, for the rain had stopped many hours ago, and the night was clear and beautiful....
**A CKNOWLEDGMENTS**
Many thanks to Mere Whaanga, who provided great wisdom and advice on the cultural aspects of this story, and to Dr. Roger Booth, associate professor of immunology and health psychology at the University of Auckland, whose vivid descriptions of the immune system at work provided the basis for much of the science in this book and made a technical and complex subject highly entertaining. Thanks also to Creative New Zealand for their generous funding, which allowed me time to work on this book.
**A BOUT THE AUTHOR**
Whether it's an undersea encounter with a moray eel or a ride on an elephant with a mind of its own, Brian Falkner often finds himself in strange and exciting situations—just the sorts of adventures he enjoys writing about! While Tane and Rebecca don't run into any eels or elephants, characters in Brian's future books had better be prepared for anything....
Brian studied computers in college but left school to pursue his love of writing. After training as a journalist, he worked as a reporter, an advertising copywriter, a radio announcer, and an Internet developer before his first children's book was published. Now he lives on the North Shore of New Zealand's largest city, Auckland, with his wife, their two kids, and two dogs.
To learn more about Brian's adventures—and his books!—visit his Web site at www.brianfalkner.co.nz.
This is a work of fiction. Names, characters, places, and incidents either are the product of the author's imagination or are used fictitiously. Any resemblance to actual persons, living or dead, events, or locales is entirely coincidental.
Text copyright © 2008 by Brian Falkner
Grateful acknowledgment is made to the following for permission to reprint previously published material:
Alfred Publishing Company, Inc.: Lyrics from "Big Yellow Taxi," words and music by Joni Mitchell, copyright © 1970 (renewed) by Crazy Crow Music. All rights reserved by Sony/ATV Music Publishing. Lyrics from "Santa Claus Is Comin' to Town," words by Haven Gillespie and music by J. Fred Coots, copyright © 1934 (renewed) by EMI Feist Catalog, Inc. Rights for the extended renewal term in the United States are controlled by Haven Gillespie Music and EMI Feist Catalog, Inc. Exclusive worldwide print rights administered by Alfred Publishing Company, Inc. All rights reserved.
Sony/ATV Music Publishing: Lyrics from "Yellow Submarine," words and music by John Lennon and Paul McCartney, copyright © 1966. All rights reserved.
All rights reserved.
Published in the United States by Random House Children's Books, a division of Random House, Inc., New York.
Random House and colophon are registered trademarks of Random House, Inc.
Visit us on the Web!
www.randomhouse.com/teens
Educators and librarians, for a variety of teaching tools, visit us at
www.randomhouse.com/teachers
_Library of Congress Cataloging-in-Publication Data_
Falkner, Brian.
The tomorrow code / Brian Falkner.—1st American ed.
p. cm.
Summary: Two New Zealand teenagers receive a desperate SOS from their future selves and set out on a quest to stop an impending ecological disaster that could mean the end of humanity.
[1. Science fiction. 2. Environmental disasters—Fiction. 3. New Zealand—Fiction.] I. Title.
PZ7.F1947To 2008
[Fic]—dc22
2007036607
Random House Children's Books supports the First Amendment and celebrates the right to read.
eISBN: 978-0-375-89233-2
v3.0
|
{
"redpajama_set_name": "RedPajamaBook"
}
| 8,444
|
Q: Fatal Error in SQL Query: LOAD DATA LOCAL INFILE My system is Ubuntu 18.04.2 LTS running Lubuntu.
When trying to load a text into Learning With Texts language program i get a Fatal Error in SQL Query: LOAD DATA LOCAL INFILE error. The program works perfectly except this prevents the text from becoming loaded.
Tried to change the my.cnf files by enabling local-infile, etc. and that parameter is currently set to on.
Error Code & Message: [0]
Backtrace:
#0 do_mysqli_query(LOAD DATA LOCAL INFILE '/tmp/tmpti.txt' INTO TABLE temptextitems FIELDS TERMINATED BY '\t' LINES TERMINATED BY '\n' (@c) set TiSeID = @b, TiCount = (@d:=@d+CHAR_LENGTH(@c))+1-CHAR_LENGTH(@c), TiOrder = if(@c like "%\r",case when (@c:=REPLACE(@c,"\r","")) is NULL then NULL when (@b:=@b+1) is NULL then NULL when @d:= @e is NULL then NULL else @a:=@a+1 end, @a:=@a+1), TiText = @c,TiWordCount=('¶.!?:;' not like concat("%",@c,"%")) and (@c rlike '[\'a-zA-ZÀ-ÖØ-öø-ȳЀ-ӹ]+')) called at [/var/www/html/lwt/utilities.inc.php:3296]
I am expecting to be able to load a simple text into a table to be displayed on Learning With Texts, where i can then read in a foreign language. Everything on Learning with Texts works as expected except for the text being displayed. I instead get the above error message.
Even 'unsecure' solutions are fine as this is just a personal database which isn't important for anything so even if it got hacked or corrupted it wouldn't worry me.
A: There is a setting in the server (mysql) as well as a setting in the client (php).
The client setting I needed to change to get this working was in the php.ini file. phpinfo() should show the current value. Uncommenting the following line in php.ini and then restarting the relevant web server / module (e.g. php-fpm in my case) worked:
mysqli.allow_local_infile = On
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 5,669
|
Q: ScrollToRowAtIndex equivalent in Flutter Background:
I recently started development using flutter/dart, and as you can guess I came from iOS background. It was very easy to scroll in a UITableView to some given indexPath (section, row) in iOS, and I believe similar is the case in Android, but it has been a real pain to achieve the same in Flutter.
Questions:
Can someone please point me to a solution where, in a ListView, I can scroll to a given index smoothly?
What I have tried:
I have been trying different packages like (https://pub.dev/packages/flutter_section_table_view) because I needed a section-wise list view, it comes with an animate to section/row, but it depends on the accurate height of each row and creates a map to jump or animate the scroll view to that height value. It is not easy to achieve given that the items in that list view are dynamic and can expand or shrink depending on some actions. I tried to handle that as well by calculating heights and all that, but that was impacting the performance of scrolling. That is why I am looking for a solution with some precalculated map that can scroll to the tag or hash just like we have in web pages.
A: You can copy paste run full code below
You can use package https://pub.dev/packages/scroll_to_index
You can wrap widget with AutoScrollTag and call controller.scrollToIndex
code snippet
await controller.scrollToIndex(98,
preferPosition: AutoScrollPosition.begin);
...
Widget _wrapScrollTag({int index, Widget child}) => AutoScrollTag(
key: ValueKey(index),
controller: controller,
index: index,
child: child,
highlightColor: Colors.black.withOpacity(0.1),
);
working demo
full code
import 'dart:math' as math;
import 'package:flutter/material.dart';
import 'package:scroll_to_index/scroll_to_index.dart';
void main() => runApp(MyApp());
class MyApp extends StatelessWidget {
// This widget is the root of your application.
@override
Widget build(BuildContext context) {
return MaterialApp(
title: 'Scroll To Index Demo',
theme: ThemeData(
primarySwatch: Colors.blue,
),
home: MyHomePage(title: 'Scroll To Index Demo'),
);
}
}
class MyHomePage extends StatefulWidget {
MyHomePage({Key key, this.title}) : super(key: key);
final String title;
@override
_MyHomePageState createState() => _MyHomePageState();
}
class _MyHomePageState extends State<MyHomePage> {
static const maxCount = 10000;
final random = math.Random();
final scrollDirection = Axis.vertical;
AutoScrollController controller;
List<List<int>> randomList;
@override
void initState() {
super.initState();
controller = AutoScrollController(
viewportBoundaryGetter: () =>
Rect.fromLTRB(0, 0, 0, MediaQuery.of(context).padding.bottom),
axis: scrollDirection);
randomList = List.generate(maxCount,
(index) => <int>[index, (1000 * random.nextDouble()).toInt()]);
}
@override
Widget build(BuildContext context) {
return Scaffold(
appBar: AppBar(
title: Text(widget.title),
),
body: ListView(
scrollDirection: scrollDirection,
controller: controller,
children: randomList.map<Widget>((data) {
return Padding(
padding: EdgeInsets.all(8),
child: _getRow(data[0], math.max(data[1].toDouble(), 50.0)),
);
}).toList(),
),
floatingActionButton: FloatingActionButton(
onPressed: _scrollToIndex,
tooltip: 'Increment',
child: Text(counter.toString()),
),
);
}
int counter = -1;
Future _scrollToIndex() async {
setState(() {
counter++;
if (counter >= maxCount) counter = 0;
});
//await controller.scrollToIndex(counter, preferPosition: AutoScrollPosition.begin);
await controller.scrollToIndex(98,
preferPosition: AutoScrollPosition.begin);
controller.highlight(counter);
}
Widget _getRow(int index, double height) {
return _wrapScrollTag(
index: index,
child: ListTile(title: Text('index: $index, height: $height')));
}
Widget _wrapScrollTag({int index, Widget child}) => AutoScrollTag(
key: ValueKey(index),
controller: controller,
index: index,
child: child,
highlightColor: Colors.black.withOpacity(0.1),
);
}
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 9,471
|
{"url":"https:\/\/ltwork.net\/in-a-direct-democracy-people-vote-on-issues-themselves-rather--8024153","text":"# In a direct democracy, people vote on issues themselves, rather than electing representatives to decide\n\n###### Question:\n\nIn a direct democracy, people vote on issues themselves, rather than electing representatives to decide for them. Group of answer choices\nTrue or False ?\n\n### Reflect across the y=x. Fill in the new coordinates below:\n\nReflect across the y=x. Fill in the new coordinates below: $Reflect across the y=x. Fill in the new coordinates below:$...\n\n### Which equation represents a population of 250 animals that decreases at an annual rate of 21%\n\nWhich equation represents a population of 250 animals that decreases at an annual rate of 21%...\n\n### The sum of a number and its square is 90. Find the number.\n\nThe sum of a number and its square is 90. Find the number....\n\n### Which of the following methods can be used by officials to solve the problem of water pollution? use\n\nWhich of the following methods can be used by officials to solve the problem of water pollution? use scientific solutions. stop droughts and floods. increase number of factories. dispose of chemical wastes in dustbins....\n\nPlease tell me what method your using and show your work. $Please tell me what method your using and show your work.$...\n\n### Find the coordinates of the midpoint of the segment with the given endpoints 3 and 5\n\nFind the coordinates of the midpoint of the segment with the given endpoints 3 and 5...\n\n### How can you solve problems involving percents\n\nHow can you solve problems involving percents...\n\n### The diagram shows a 6 ft student standing near a tree. The shadow of the student and the shadow of the tree ends at point\n\nThe diagram shows a 6 ft student standing near a tree. The shadow of the student and the shadow of the tree ends at point A. What is the height of the tree? $The diagram shows a 6 ft student standing near a tree. The shadow of the student and the shadow of$...\n\n### The owens family has saved up $50,000 for a down paysaved up$50,000 for a down payment on a house.\n\nThe owens family has saved up $50,000 for a down paysaved up$50,000 for a down payment on a house. they aregoing to use this money as the 10% down payment their lender is reqmake to purchase the house. what is the price of the neeune house. what is the price of the house that the owens familyis buy...\n\nIf someone could please help that would be great \u200b $If someone could please help that would be great \u200b$...\n\n### Can help cross word puzzle\n\nCan help cross word puzzle...\n\n### Primare Corporation has provided the following data concerning last month\u2019s manufacturing operations. Purchases of raw materials\n\nPrimare Corporation has provided the following data concerning last month\u2019s manufacturing operations. Purchases of raw materials $30,000 Indirect materials included in manufacturing overhead$ 5,000 Direct labor $58,000 Manufacturing overhead applied to work in process$ 87,000 Underapplied ...\n\n### In which two sentences does the author use an ironic tone towards the subject\n\nIn which two sentences does the author use an ironic tone towards the subject...\n\n### When jamie's car moves forward such that each tire makes one full rotation, the car has traveled 72\n\nWhen jamie's car moves forward such that each tire makes one full rotation, the car has traveled 72 inches. how many full rotations will the tires need to make for jamie's car to travel 10 yards?...\n\n### \u00bfAlguien ser\u00eda mi novia? <3\n\n\u00bfAlguien ser\u00eda mi novia? <3...\n\n### Free points cause i feel nice :)\n\nFree points cause i feel nice :)...\n\n### Which of the following is an EXPLICIT formula for the sequence 12, 15, 18, 21...Help ASAP please!!\n\nWhich of the following is an EXPLICIT formula for the sequence 12, 15, 18, 21... Help ASAP please!! $Which of the following is an EXPLICIT formula for the sequence 12, 15, 18, 21... Help ASAP please$...\n\n### What is an irregular tessellation? How many irregular tessellations are possible? How can one create an irregular polygon\n\nWhat is an irregular tessellation? How many irregular tessellations are possible? How can one create an irregular polygon that tessellates? (5 points)...","date":"2023-01-27 16:54:51","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 2, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.3763250410556793, \"perplexity\": 1614.8038658452858}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2023-06\/segments\/1674764495001.99\/warc\/CC-MAIN-20230127164242-20230127194242-00599.warc.gz\"}"}
| null | null |
If you have a product rather than a service, then you may already know the value of a solid product review. You know that a strong product review will greet potential buyers and those buyers will be influenced, one way or another, by what they read. Do you know what makes a good product review? Do you know how long that review should be or the structure that grabs attention? If you are having problems selling your product and feel it relates to the product reviews, here are some tips to change that.
A good product review will reveal the good and the bad of the product. This doesn't mean the bad of the product has to be awful. It can simply be a minor issue that someone had with the product. The pros and cons should be reasonable. The wording should also be reasonable. For example, you do not want to read how great and wonderful the product was through words like "awesome" and "incredible." You want to read how it was awesome and incredible. Specifics in pros and cons are the key to a solid product review.
If you are marketing to a specific niche group, then speak to that group through the product review. Have your content manager or creator touch on the points the audience needs to know. For example, if you are selling a product to make a new moms life easier, then you want to convey how it does that. Have the pros push how the product gave the mom more freedom, less stress, or how it made a task shorter and easier. Have the con convey how the product could have helped in other ways, but go back to how the help it did provide being the key reason the reviewer would buy it again.
The length of the review should be easily scannable. This means a 500 word review is overkill and you will lose the reader completely. A 250 to 300 word review is a solid option. It offers enough words to get the point across without getting wordy and losing the reader.
You want to make sure that every review has a pros sections, cons section, description of the product, a customer service review if applicable, and a price section. Knowing these details helps the reader determine the answer to several aspects of the buying process. People also want to know product dimensions, material, and where the product was made.
Including all of this information is difficult for business owners to develop using content software. If you are still using content software, despite the pitfalls and danger of losing your client base, then consider hiring a content specialist for the reviews. Make sure they speak and write native English content. It is a huge turn off to readers to find a product review written with poor English and grammar. It's difficult to read and the potential customer will move on to another review or product.
|
{
"redpajama_set_name": "RedPajamaC4"
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| 5,669
|
{"url":"https:\/\/optimization-online.org\/tag\/semidefinite-representations\/","text":"## On semi-infinite systems of convex polynomial inequalities and polynomial optimization problems\n\nWe consider the semi-infinite system of polynomial inequalities of the form $\\mathbf{K}:=\\{x\\in\\mathbb{R}^m\\mid p(x,y)\\ge 0,\\ \\ \\forall y\\in S\\subseteq\\mathbb{R}^n\\},$ where $p(X,Y)$ is a real polynomial in the variables $X$ and the parameters $Y$, the index set $S$ is a basic semialgebraic set in $\\mathbb{R}^n$, $-p(X,y)$ is convex in $X$ for every $y\\in S$. We \u2026 Read more\n\n## Invariance and efficiency of convex representations\n\nWe consider two notions for the representations of convex cones: $G$-representation and lifted-$G$-representation. The former represents a convex cone as a slice of another; the latter allows in addition, the usage of auxiliary variables in the representation. We first study the basic properties of these representations. We show that some basic properties of convex cones \u2026 Read more","date":"2022-08-14 09:58:41","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 2, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.8344772458076477, \"perplexity\": 242.89171251810563}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.3, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2022-33\/segments\/1659882572021.17\/warc\/CC-MAIN-20220814083156-20220814113156-00077.warc.gz\"}"}
| null | null |
package template_metrics
import (
"fmt"
metrics "github.com/sonots/go-metrics" // max,mean,min,stddev,percentile
"sync"
"time"
)
type Metrics struct {
name string
timers map[string]metrics.Timer
}
func newMetrics(name string) *Metrics {
return &Metrics{
name: name,
timers: map[string]metrics.Timer{},
}
}
var mutex = sync.Mutex{}
//print the elapsed time on each request if Verbose flag is true
func (proxy *Metrics) printVerbose(elapsedTime time.Duration, base string) {
fmt.Printf("time:%v\ttemplate:%s\tbase:%s\telapsed:%f\n",
time.Now(),
proxy.name,
base,
elapsedTime.Seconds(),
)
}
func (proxy *Metrics) printMetrics(duration int) {
for base, timer := range proxy.timers {
count := timer.Count()
if count > 0 {
fmt.Printf(
"time:%v\ttemplate:%s\tbase:%s\tcount:%d\tmax:%f\tmean:%f\tmin:%f\tpercentile95:%f\tsum:%f\tduration:%d\n",
time.Now(),
proxy.name,
base,
timer.Count(),
float64(timer.Max())/float64(time.Second),
timer.Mean()/float64(time.Second),
float64(timer.Min())/float64(time.Second),
timer.Percentile(0.95)/float64(time.Second),
float64(timer.Sum())/float64(time.Second),
duration,
)
proxy.timers[base] = metrics.NewTimer()
}
}
}
//measure the time
func (proxy *Metrics) measure(startTime time.Time, base string) {
elapsedTime := time.Now().Sub(startTime)
if Summary {
if proxy.timers[base] == nil {
mutex.Lock()
if proxy.timers[base] == nil {
proxy.timers[base] = metrics.NewTimer()
}
mutex.Unlock()
}
proxy.timers[base].Update(elapsedTime)
}
if Verbose {
proxy.printVerbose(elapsedTime, base)
}
}
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 543
|
Q: How would I take this .scrollTop code and modify it to pause it's function on Mouse Hover or Scrolling? <script>
window.setInterval(function() {
var elem = document.getElementById('onDiv');
elem.scrollTop = elem.scrollHeight;
}, 5000, 'swing');
</script>
<div id="onDiv" align="center" style="width:100%; height:250px; overflow:auto; padding-left: 2px;">
<table id="asdf" cellspacing="0" cellpadding="0" border="0" width="100%" style="min-height:250px; overflow:auto; max-height:250px; padding-left: 2px;" >
<tr>
<td height="250" valign="top">Content</td>
</tr>
</table>
</div>
The code above is set to scroll down to the bottom every 5 seconds per new content added to the div.
I'm relatively new to using jQuery so there may be some incorrect formatting here. Just a heads up.
Thanks Guys/Gals!
This code that reyaner gave seems to be what I want to do, however I can't get it to work in the current declaration.
<script>
var IV;
function setInterval(){
IV = window.setInterval(function() {
var elem = document.getElementById('onDiv');
elem.scrollTop = elem.scrollHeight;
}, 1000, 'swing');
}
setInterval();
("#onDiv").hover(function(){
clearInterval(IV);
}, function(){
setInterval();
});
</script>
A: Does this work?
var IV;
function mysetInterval() {
IV = setInterval(function() {
var elem = document.getElementById('onDiv');
elem.scrollTop = elem.scrollHeight;
}, 1000);
}
mysetInterval();
$(function() {
$("#onDiv").hover(function() {
clearInterval(IV);
}, function() {
mysetInterval();
});
});
A: Something like this: (copied your code in func..)
var IV;
function setInterval(){
IV = window.setInterval(function() {
var elem = document.getElementById('onDiv');
elem.scrollTop = elem.scrollHeight;
}, 5000, 'swing');
}
setInterval();
$("element").hover(function(){
clearInterval(IV);
}, function(){
setInterval();
});
A: Take a look at this:
http://jsfiddle.net/balintbako/Xn9gd/
var interval;
$("#onDiv").hover(function () {
$(this).stop(true);
clearInterval(interval);
}, function () {
queue();
});
function queue() {
interval = setInterval(function () {
$("#onDiv").animate({
scrollTop: $("#onDiv").prop('scrollHeight')
}, 1000);
}, 500);
}
queue();
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 3,768
|
Q: Jasmine - JavaScript: How to check array response by validating one of the array returned data? What if I want to validate a result from an array response!
Response:
({ id: '612d56952ca01806398dac1f', lastUpdated: '2021-08-30T22:07:17.099Z', userToken: 'public', data: Object({ createdOn: '2021-08-30T22:07:16.916Z', lastUpdatedBy: null, userId: '60f469cf784379051298e96d', displayName: 'Nadia', postText: 'post text sample', postImages: 'image test', pluginInstance: Object({ pluginInstanceId: '1627334094776-047554258642281355', pluginInstanceTitle: 'communityFeedPlugin' }), isPublic: true, _buildfire: Object({ index: Object({ array1: [ Object({ string1: 'userId_60f469cf784379051298e96d' }), Object({ string1: 'displayName_nadia' }), Object({ string1: 'pluginTitle_communityfeedplugin' }), Object({ string1: 'isPublic_1' }) ] }) }) }) }), Object({ id: '612d564dfabce50627bdb5db', lastUpdated: '2021-08-30T22:06:05.699Z', userToken: 'public', data: Object({ createdOn: '2021-08-30T22:06:05.500Z', lastUpdatedBy: null, userId: '60f469cf784379051298e96d', displayName: 'Nadia', postText: 'post text sample', postImages: 'image test', pluginInstance: Object ...
Tried the following but did not work:
it('Get all posts' , function(done){
Posts.getPosts('publicPosts' ,(err,resp) =>{
expect(resp).toContain(jasmine.objectContaining({
id: '612d50bafabce50627bdb538',
postText:'post text sample',
userId: '60f469cf784379051298e96d',
isPublic: true
}));
done();
});
});
A: this did not work too:
it('Get all posts' , function(done){
Posts.getPosts({publicPosts : true},(err,resp) => {
expect(resp.data).toEqual(jasmine.objectContaining({
userId: authManager.currentUser._id,
isPublic: true
}));
done();
});
});
with resp.data I am getting undefined , and without data it will fail
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 9,677
|
\section{Introduction}
The Marinatto-Weber (MW) idea of quantum $2\times2$ games
introduced in \cite{marinatto} has found application in many
branches of game theory. The MW approach to evolutionary games
\cite{aspect} and Stackelberg equilibrium \cite{iqbalbackwards}
are merely two of many applications. In the papers
\cite{fracornormalrepresentation} and \cite{imperfect} we have
shown that the MW idea is applicable as well to finite games in
extensive form. Consequently, this scheme of playing quantum games
can be applied to many other game-theoretical problems. In this
paper we deal with the problem of quantization of twice repeated
$2\times2$ games. Since a finitely repeated game is just
a~particular case of a finite extensive game, we apply the method
based on \cite{fracornormalrepresentation} and \cite{imperfect} to
play the repeated game in the quantum way. The idea of quantum
repreated games was first introduced in \cite{iqbalrepeat}, where
the Authors adapt the MW scheme for the twice repeated Prisoner's
Dilemma. Then, they investigate if results that are unavailable
when the game is played classically can occur in the quantum area.
The point of the paper \cite{iqbalrepeat} is to provide sufficient
conditions for players' cooperation in the Prisoner's Dilemma
game. We examine the idea of \cite{iqbalrepeat} before we define
our scheme. Firstly, we study the problem of cooperation
considered by the Authors of \cite{{iqbalrepeat}} and we prove
that player's cooperation in the game defined by the protocol
proposed in \cite{iqbalrepeat} is not possible. Secondly, we check
whether that scheme is actually in accordance with the concept of
repeated game. As we will show, the discussed scheme does not
include the classical twice repeated Prisoner's Dilemma, hence it
cannot be the quantum realization of this game in the spirit of
the MW approach. To support our arguments we propose the new
protocol for a twice repeated $2 \times 2$ game and prove that our
idea generalizes the classical twice repeated game. Our paper also
contains the proof of the advantage of the quantum scheme over the
classical one: We prove that both players can benefit from playing
game via our protocol. Moreover, we show that contrary to the
situation encountered in the classical game, the cooperation of
the players is possible for some sort of Prisoner's Dilemma games
played repeatedly when our quantum approach is used. \vspace{12pt}
\noindent Studying our paper requires little background in game
theory. All notions like extensive game, information set,
strategy, equilibrium, subgame perfect equilibrium etc. used in
the paper are explained in an accessible way, for example, in
\cite{osborne} and \cite{peters}. The adequate preliminaries can
also be found in the paper \cite{fracornormalrepresentation},
where quantum games in an extensive form are examined.
\section{Twice repeated games and the Prisoner's
Dilemma}\label{sectiondwa} The Prisoner's Dilemma (PD) is one of
the most fundamental problems in game theory (the general form of
the PD according to \cite{rapoport} is given in
Fig.~\ref{figure1}(a).
\begin{figure}[t]
\centering
\includegraphics[angle=0, scale=0.85]{pd.eps}
\caption{The strategic a) and extensive b) form of the Prisoner's
Dilemma.} \label{figure1}
\end{figure}
It demonstrates why the rationality of players can lead them to an
inefficient outcome. Although the payoff vector $(R,R)$ is better
to both players than $(P,P)$, they cannot obtain this outcome
since each player's strategy $C$ (cooperation) is strictly
dominated by $D$ (defection). As a result, the players end up with
payoff $P$ corresponding to the unique Nash equilibrium $(D,D)$.
A similar scenario occurs in a case of finitely repeated PD. The
concept of a finitely repeated game assumes playing a static game
(a stage of the repeated game) for a fixed number of times.
Additionally, the players are informed about results of
consecutive stages. In the twice repeated case it means that each
player's strategy specify an action at the first stage and four
actions at the second stage where a particular action is chosen
depending on what of the four outcomes of the first stage has
occurred. It is clearly visible when we write twice repeated game
in the extensive form (see Fig.~\ref{figure2}).
\begin{figure}[t]
\centering
\includegraphics[angle=0, scale=0.85]{epd.eps}
\caption{The extensive form for a twice repeated Prisoner
Dilemma.} \label{figure2}
\end{figure}
The first stage of the twice repeated PD in the extensive form is
simply the game depicted in Fig.~\ref{figure1}(b) where the
players specify an action $C$ or $D$ at the information set 1.1
and 2.1, respectively (the information sets of player 2 are
distinguished by dotted line connecting the nodes to show lack of
knowledge of the second player about the previous move of the
first player). When the players choose their actions, the result
of the first stage is announced. Since they have knowledge about
the results of the first stage, they can choose different actions
at the second stage depending on the previous result, hence the
next four game trees from Fig.~\ref{figure1} are required to
describe the repeated game. The game tree exhibits ten information
sets (five for each player labelled $1.\cdot$ and $2.\cdot$,
respectively) at which each of the players has two moves. Thus,
each of them has $2^5 = 32$ strategies as they specify $C$ or $D$
at their own five information sets.
To find the Nash equilibrium in finitely repeated game it is
convenient to use the property that the equilibrium profile always
implies the Nash equilibrium at the last stage of the game.
Therefore, to find the Nash equilibrium in the twice repeated PD
it is sufficient to consider strategy profiles that generate the
profile $(D,D)$ at the second stage. Then it follows that $D$ is
the best response for players at the first stage as well. By
induction it can be shown that playing the action $D$ at each
stage of finitely repeated PD constitutes the unique Nash
equilibrium. It is worth noting that if a single stage of repeated
game has more than one equilibrium, different Nash equilibria may
be played at the last stage depending on results of previous
stages. For example, let us consider the Battle of the Sexes (BoS)
game given by the following bimatrix:
\begin{equation}
\begin{array}{l} ~~~~~~~~~~~~~ \quad O \qquad~~ F \\
\Gamma: \begin{array}{c} O \\ F \end{array}
\left[\begin{array}{cc}
(\alpha,\beta) & (\gamma,\gamma) \\
(\gamma,\gamma) & (\beta,\alpha) \\
\end{array}\right], \quad $where$ \quad \alpha
> \beta > \gamma.
\end{array}
\end{equation}
It has two pure Nash equilibria, namely, $(O,O)$ and $(F,F)$. Let
us examine now the twice repeated BoS. Obviously, its game tree is
the same as one in Fig.~\ref{figure2}. Let us assign appropriate
sum of two stage payoff outcomes to each possible profile (like it
has been done in the case of the twice repeated PD). Then we find
many different Nash equilibria. One of these is to play the Nash
equilibrium $(O,O)$ at the first stage, keep playing $(O,O)$ at
the second stage if the outcome of the first one is $(O,O)$ or
$(O,F)$, otherwise to play stage-game Nash equilibrium $(F,F)$.
\section{Comment on `Quantum repeated games' by Iqbal and Toor \cite{iqbalrepeat}}
Let us remind the MW approach to playing the PD repeatedly
introduced in \cite{iqbalrepeat}. According to this concept the
two-stage PD is placed in $\mathscr{H} =
\left(\mathds{C}^2\right)^{\otimes4}$ complex Hilbert space with
the computational basis. The game starts with preparing 4-qubit
pure state represented by a unit vector in $\mathscr{H}$. The
general form of this state is described as follows:
\begin{align} \begin{split}
&|\psi_{\mathrm{in}}\rangle = \sum_{x_{1}, x_{2}, x_{3}, x_{4} =
0,1} \lambda_{x_{1}, x_{2}, x_{3},
x_{4}}|x_{1},x_{2},x_{3},x_{4}\rangle,\\
&\mbox{where} \quad \lambda_{x_{1}, x_{2}, x_{3}, x_{4}} \in
\mathds{C} \quad \mbox{and} \quad \sum_{x_{1}, x_{2}, x_{3}, x_{4}
= 0,1} |\lambda_{x_{1}, x_{2}, x_{3}, x_{4}}|^2 = 1.
\label{drinitialstate} \end{split}
\end{align} Players' moves are identified with the identity operator $\sigma_{0}$
and the bit flip Pauli operator $\sigma_{1}$. Player 1 is allowed
to act on the first and third qubit, and player 2 acts on the
second and fourth one.
In the first stage of the game the two first qubits
are manipulated. Let $\rho_{\mathrm{in}}$ be the density operator
for the initial state (\ref{drinitialstate}). Then the state
$\rho$ after the players' actions takes the form
\begin{align} \label{samoro} \begin{split}
&\rho = \sum_{\kappa_{1}, \kappa_{2} =
0,1}p_{\kappa_{1}}q_{\kappa_{2}}(\sigma^1_{\kappa_{1}} \otimes
\sigma^2_{\kappa_{2}}) \rho_{\mathrm{in}}
(\sigma^1_{\kappa_{1}} \otimes \sigma^2_{\kappa_{2}}),\\
&\mbox{and} \;\; \sum_{\kappa_{1} = 0,1}p_{\kappa_{1}} =
\sum_{\kappa_{2} = 0,1}q_{\kappa_{2}} =1, \end{split}
\end{align}
where $p_{\kappa_{1}}$ ($q_{\kappa_{2}}$) is the probability of
applying $\sigma^1_{\kappa_{1}}$ ($\sigma^2_{\kappa_{2}})$ to the
first (second) qubit. Next, the other two qubits are manipulated
which, according to Iqbal and Toor, corresponds to the second
stage of the classical game. The operation $\sigma^3_{\kappa_{3}}$
on the third qubit with probability $p_{\kappa_{3}}$ and operation
$\sigma^4_{\kappa_{4}}$ on the fourth qubit with probability
$q_{\kappa_{4}}$ change the state $\rho$ to
\begin{align}\label{finalro} \begin{split}
&\rho_{\mathrm{fin}} = \sum_{\kappa_{3}, \kappa_{4} =
0,1}p_{\kappa_{3}}q_{\kappa_{4}}(\sigma^3_{\kappa_{3}} \otimes
\sigma^4_{\kappa_{4}}) \rho(\sigma^3_{\kappa_{3}} \otimes
\sigma^4_{\kappa_{4}}),
\\ &\mbox{and} \quad \;\; \sum_{\kappa_{3} = 0,1}p_{\kappa_{3}} =
\sum_{\kappa_{4} = 0,1}q_{\kappa_{4}} =1.
\end{split}
\end{align}
The next step is to measure the final state $\rho_{\mathrm{fin}}$
in order to determine final payoffs. The measurement is defined by
the four payoffs operators $X_{i.j}$, $i,j=1,2$ associated with
particular: player $i$ and stage $j$. That is
\begin{align}\label{drpayoffoperator} \begin{split}
X_{1.1} = (R|00\rangle \langle00| + S|01\rangle \langle01| +
T|10\rangle \langle 10| + P|11\rangle \langle 11|) \otimes
\mathds{1}^{\otimes 2};\\
X_{1.2} =\mathds{1}^{\otimes 2} \otimes (R|00\rangle \langle00| +
S|01\rangle \langle01| + T|10\rangle \langle 10| + P|11\rangle
\langle 11|);\\ X_{2.1} = (R|00\rangle \langle00| + T|01\rangle
\langle01| + S|10\rangle \langle 10| + P|11\rangle \langle 11|)
\otimes \mathds{1}^{\otimes 2};\\
X_{2.2} = \mathds{1}^{\otimes 2} \otimes (R|00\rangle \langle00| +
T|01\rangle \langle01| + S|10\rangle \langle 10| + P|11\rangle
\langle 11|). \end{split}
\end{align}
Then the expected payoff $E_{i.j}$ for player $i$ at stage $j$
when player 1 chooses strategy $(\sigma^1_{\kappa_{1}},
\sigma^3_{\kappa_{3}})$ and player 2 chooses
$(\sigma^2_{\kappa_{2}}, \sigma^4_{\kappa_{4}})$ is obtained by
the following formula:
\begin{align}
E_{i.j}\left((\sigma^1_{\kappa_{1}}, \sigma^3_{\kappa_{3}}),
(\sigma^2_{\kappa_{2}}, \sigma^4_{\kappa_{4}})\right) =
\mathrm{tr}(X_{i.j} \rho_{\mathrm{fin}}). \label{drpayoff}
\end{align}
\noindent The authors took up the issue of cooperation in
two-stage PD. Given the initial state
\begin{align} \label{initialstateiqbal}
|\psi_{\mathrm{in}}\rangle = \lambda_{0000}|0000\rangle +
\lambda_{0011}|0011\rangle + \lambda_{1100}|1100\rangle +
\lambda_{1111}|1111\rangle
\end{align}
and fixed payoffs
\begin{align}\label{pdpayoff}
T = 5, \quad R = 3, \quad P = 1, \quad S = 0,
\end{align}
they identify $\sigma_{0}$ and $\sigma_{1}$ as actions of
cooperation and defection, respectively, and claim that conditions
\begin{align}
|\lambda_{0000}|^2 + |\lambda_{0011}|^2 \leq \frac{1}{3}, \quad
|\lambda_{0011}|^2 + |\lambda_{1111}|^2 \leq \frac{1}{3}
\label{wrongcondition}
\end{align}
are sufficient to choose $\sigma_{0}$ by both players (thereby
cooperating) at the first stage given that the players have chosen
$\sigma_{1}$ at the second one. We raise below two objections
concerning the results of the paper \cite{iqbalrepeat}.
\subsection{The incompatibility of the protocol (\ref{drinitialstate})-(\ref{drpayoff}) and theory of repeated games}
The main fault of the protocol
(\ref{drinitialstate})-(\ref{drpayoff}) is that the twice repeated
game cannot be described in this way. In fact, it quantizes the
game PD played twice when the players are not informed about a
result of the first stage. It is noticeable, for example, when we
re-examine the way of finding the optimal solution provided in
\cite{iqbalrepeat}. The authors analyze the game backwards, first
by focusing on the Nash equilibria at the second stage. They set
condition for the profile $(\sigma^3_{1}, \sigma^4_{1})$ to be the
Nash equilibrium at the second stage. Next, given that
$(\sigma^3_{1}, \sigma^4_{1})$ is fixed, they determine the set of
amplitudes for which the profile $((\sigma^1_{0}, \sigma^2_{0}),
(\sigma^3_{1}, \sigma^4_{1}))$ is the Nash equilibrium of the game
implied by (\ref{drinitialstate})-(\ref{drpayoff}). This method to
find the Nash equilibria is not correct since it doesn't include
the possibility that players make their actions depending on a
result of the first stage. Although the problem seems to be
insignificant where a stage of a repeated game has unique Nash
equilibrium, it becomes visible in remaining cases. Let us
consider the initial state (\ref{initialstateiqbal}) satisfying
the requirement
\begin{align}
2(|\lambda_{0000}|^2 + |\lambda_{1100}|^2) = |\lambda_{0011}|^2 +
|\lambda_{1111}|^2
\end{align}
and let us take (\ref{pdpayoff}) to be PD's payoffs. Then, the
expected payoffs for the players at the second stage of the game
defined by the scheme (\ref{drinitialstate})-(\ref{drpayoff}) are
as follows:
\begin{align}\label{2stage}\begin{split}
&E_{i.2}\bigl((\cdot, \sigma^3_{0}), (\cdot, \sigma^4_{0})\bigr) =
E_{1.2}\bigl((\cdot, \sigma^3_{1}), (\cdot, \sigma^4_{0})\bigr) =
E_{2.2}\bigl((\cdot, \sigma^3_{0}), (\cdot, \sigma^4_{1})\bigr) =
5|\lambda|;\\
&E_{1.2}\bigl((\cdot, \sigma^3_{0}), (\cdot, \sigma^4_{1})\bigr) =
E_{2.2}\bigl((\cdot, \sigma^3_{1}), (\cdot, \sigma^4_{0})\bigr) =
10|\lambda|;\\
&E_{i.2}\bigl((\cdot, \sigma^3_{1}), (\cdot, \sigma^4_{1})\bigr) =
7|\lambda|,
\end{split}
\end{align}
where $|\lambda| = |\lambda_{0000}|^2 + |\lambda_{1100}|^2$ and $i
= 1,2$. Results of (\ref{2stage}) imply continuum of Nash
equilibria in the second stage (it is easy to note, for example,
when we draw a $2\times2$ bimatrix with entries defined by
(\ref{2stage})), among them $\left((\cdot, \sigma^3_{0}),(\cdot,
\sigma^4_{1})\right)$ and $\left((\cdot, \sigma^3_{1}),(\cdot,
\sigma^4_{0})\right)$. Bearing in mind the remark in Section
\ref{sectiondwa} about possible profiles in the BoS game, the
correct protocol for quantum repeated games should be able to
assign a payoff outcome (by the measurement
(\ref{drpayoffoperator})) to a strategy profile, where different
Nash equilibria are played at the second stage depending on
actions chosen at the first one. However, an example of a profile
where the players play $\left((\cdot, \sigma^3_{0}),(\cdot,
\sigma^4_{1})\right)$ at the second stage if a result of the first
stage is $\left((\sigma^1_{0},\cdot),
(\sigma^2_{0},\cdot)\right)$, and they play $\left((\cdot,
\sigma^3_{1}),(\cdot, \sigma^4_{0})\right)$ in other cases cannot
be measured by the scheme (\ref{drinitialstate})-(\ref{drpayoff}).
Since there is two qubit register allotted to the second stage, it
allows to write only one pair of actions $(\sigma^3_{\kappa_{3}},
\sigma^4_{\kappa_{4}})$ before the measurement is made.
An argument against the scheme in \cite{iqbalrepeat} can be
expressed in another way. Namely, all results included in
\cite{iqbalrepeat} can be obtained by considering simplified
protocol (\ref{drinitialstate})-(\ref{drpayoff}) where the
sequential procedure (\ref{samoro}) and (\ref{finalro}) for
determining the final state $\rho_{\mathrm{fin}}$ is simply
replaced with
\begin{align}
\rho_{\mathrm{fin}} = \bigotimes^4_{j=1}\sigma^j_{\kappa_{j}}
\rho_{\mathrm{in}}\bigotimes^4_{j=1}\sigma^j_{\kappa_{j}},
\label{drfinalstate2}
\end{align}
In this case, the first and the second player simultaneously pick
$(\sigma^1_{\kappa_{1}}, \sigma^3_{\kappa_{3}})$ and
$(\sigma^2_{\kappa_{2}}, \sigma^4_{\kappa_{4}})$, respectively,
having essentially only four strategies each. However, as we
mentioned in the previous section, each player has 32 strategies
in the classical twice repeated game. As a result, the protocol
(\ref{drinitialstate})-(\ref{drpayoff}) cannot coincide with the
classical case if $|\psi_{\mathrm{in}}\rangle = |0000\rangle$.
Despite the fact that the Authors assume that a player knows her
opponent's action taken previously, the scheme
(\ref{drinitialstate})-(\ref{drpayoff}) does not take it into
consideration. In consequence, a game being quantized by
(\ref{drinitialstate})-(\ref{drpayoff}) differs from the game in
Fig.~\ref{figure2} in that the nodes 1.2, 1.3, 1.4 and 1.5 (2.2,
2.3, 2.4 and 2.5) lie at the same information set (are connected
with dotted line).
\subsection{The misconception about the cooperative strategy in the PD played via the MW approach}
The another fault, we are going to discuss, is based on
misinterpreting the operators $\sigma_{0}$ and $\sigma_{1}$ as
cooperation and defection in the protocol given by
(\ref{drinitialstate})-(\ref{drpayoff}). Let us consider the
initial state $|\psi_{\mathrm{in}}\rangle$ where the two first
qubits associated with the first stage are prepared in the state
$|x_{1},x_{2}\rangle$, for $x_{1},x_{2} \in \{0,1\}$. Then the
first stage of the game given by
(\ref{drinitialstate})-(\ref{drpayoff}) is isomorphic to the
classical PD game. When the initial state is $|00\rangle$ then
$\sigma_{0}$ corresponds to the action~$C$ and $\sigma_{1}$
corresponds to $D$. However, when the initial state is
$|11\rangle$, the action `cooperate' are identified with
$\sigma_{1}$ and the action `defect' with $\sigma_{0}$ since by
putting $\rho_{\mathrm{fin}} = (\sigma^1_{\kappa_{1}} \otimes
\sigma^2_{\kappa_{2}}) |11 \rangle \langle 11|
(\sigma^1_{\kappa_{1}} \otimes \sigma^2_{\kappa_{2}})$ into the
formula (\ref{drpayoff}) we have
\begin{align}
(\mathrm{tr}(X_{1.1} \rho_{\mathrm{fin}}), \mathrm{tr}(X_{2.1}
\rho_{\mathrm{fin}})) =
\left\{\begin{array}{lll} (R,R), & \mbox{if} & (\kappa_1, \kappa_2) = (1,1);\\
(S,T), & \mbox{if} & (\kappa_1, \kappa_2) = (1,0);\\
(T,S), & \mbox{if} & (\kappa_1, \kappa_2) = (0,1);\\
(P,P), & \mbox{if} & (\kappa_1, \kappa_2) = (0,0).
\end{array}\right. \label{prostyprzyklad}
\end{align}
That is, the outcome of the game does not depend intrinsically on
the operators but depends on the initial state and on what the
final state $\rho_{\mathrm{fin}}$ can be obtained through the
available operators. Thus identification of operators with actions
taken in classical game without taking into consideration the form
of the initial state is not correct. The misidentification assumed
in \cite{iqbalrepeat} implies that the condition
(\ref{wrongcondition}) cannot solve the problem formulated in this
paper. It is clearly visible when we take, for example, the
initial state $|\psi_{\mathrm{in}}\rangle = |1100\rangle$. It
satisfies the inequalities (\ref{wrongcondition}) thus,
$\sigma_{0}$ is optimal at the first stage for each player. In
fact, $\sigma_{0}$ is the action `defect' as it is shown in
(\ref{prostyprzyklad}). Note also that the payoff corresponding to
the profile $(\sigma_{0}, \sigma_{0})$ at the first stage and
$(\sigma_{1}, \sigma_{1})$ at the second one is $2P$ for each
player - total payoff for the defection. Thus, the condition
(\ref{wrongcondition}) does not ensure the cooperation at the
first stage.
Quite the opposite, it turns out that the players never cooperate
when they play the game defined by
(\ref{drinitialstate})-(\ref{drpayoff}). Let us consider any
initial state (\ref{drinitialstate}) in which the first and the
second qubit are prepared in a way that for $(s_{1},
s_{2})=((\sigma^1_{\kappa_{1}}, \sigma^3_{\kappa_{3}}),
(\sigma^2_{\kappa_{2}}, \sigma^4_{\kappa_{4}}))$ we have
\begin{align}
(E_{1.1}(s_{1}, s_{2}), E_{2.1}(s_{1}, s_{2})) =
\left\{\begin{array}{lll} (R',R'), & \mbox{if} & (\kappa_1, \kappa_2) = (0,0);\\
(S',T'), & \mbox{if} & (\kappa_1, \kappa_2) = (0,1);\\
(T',S'), & \mbox{if} & (\kappa_1, \kappa_2) = (1,0);\\
(P',P'), & \mbox{if} & (\kappa_1, \kappa_2) = (1,1).
\end{array}\right. \label{prostyprzyklad2}
\end{align}
where the values $T', R', P', S'$ meet the requirements of the PD
given in Fig.\ref{figure1}(a), so the operators $\sigma_{0}$ and
$\sigma_{1}$ can be regarded as cooperation and defection,
respectively. Next, let us estimate the difference
\begin{align}
E_{1}((\sigma^1_{1}, \sigma^3_{0}), s_{2}) - E_{1}((\sigma^1_{0},
\sigma^3_{0}), s_{2}) \quad \mbox{for any}\quad s_{2} =
(\sigma^2_{\kappa_{2}}, \sigma^4_{\kappa_{4}}),
\label{drdifference}
\end{align}
where $E_{1} = E_{1.1}+E_{1.2}$. Since the same actions are taken
on the third and the fourth qubit, we have $E_{1.2}((\sigma^1_{0},
\sigma^3_{0}), s_{2}) = E_{1.2}((\sigma^1_{1}, \sigma^3_{0}),
s_{2})$, therefore, the value $E_{1}$ depends only on $E_{1.1}$,
Thus, for $s_{2} = (\sigma^2_{\kappa_{2}},
\sigma^4_{\kappa_{4}})$, we obtain from (\ref{prostyprzyklad2})
that
\begin{align}
0<E_{1}((\sigma^1_{1}, \sigma^3_{0}), s_{2}) -
E_{1}((\sigma^1_{0}, \sigma^3_{0}),
s_{2}) =\left\{\begin{array}{lll} T'-R', & \mbox{if} & \kappa_{2} = 0;\\
P'-S', & \mbox{if} & \kappa_2 = 1.
\end{array}\right.
\end{align}
In similar way we can prove that the strategy $(\sigma^1_{0},
\sigma^3_{1})$ of player 1 is strictly dominated by
$(\sigma^1_{1}, \sigma^3_{1})$. As a result, we conclude that
$\sigma^1_{1}$ is the best response of player 1 at the first
stage. Symmetry of payoffs in PD implies that strategy
$(\sigma^2_{0}, \sigma^4_{0})$ of player 2 is strictly dominated
by $(\sigma^2_{1}, \sigma^4_{0})$, as well as $(\sigma^2_{0},
\sigma^4_{1})$ is strictly dominated by $(\sigma^2_{1},
\sigma^4_{1})$. Thus, there is no Nash equilibrium in which the
players choose $\sigma_{0}$ (cooperation) at the first stage.
\section{The MW approach to twice repeated quantum games}
In this section we propose a scheme of playing a twice repeated
$2\times2$ quantum game that is free from the faults we have
pointed in the previous section. Our construction is based on the
protocol that we proposed in \cite{fracornormalrepresentation}
where general finite extensive quantum games were considered.
Since a repeated game is a special case of an extensive game, we
can adapt this concept. Next, we examine what results can be
obtained from such protocol. In particular, we re-examine the
problem of cooperation studied in \cite{iqbalrepeat}.
\subsection{Construction of a twice repeated $\mathbf{2\times2}$ quantum game via the MW protocol}
\noindent Let us consider a $2 \times 2$ game defined by the
outcomes $O_{\iota_{1}, \iota_{2}}$, $\iota_{1}, \iota_{2} = 0,1$.
The twice repeated $2\times2$ quantum game played according to the
MW approach is as follows:
\noindent Let $\mathscr{H} = \left(\mathds{C}^2\right)^{\otimes
10}$ be a Hilbert space with the computational basis
$\{|x_{1},x_{2},\dots,x_{10}\rangle\}$, $x_{j} = 0,1$. Then the
initial state of the game is a ten-qubit pure state represented by
a~unit vector in the space $\mathscr{H}$:
\begin{align}
|\psi_{\mathrm{in}}\rangle =
\sum^{2^{10}-1}_{x=0}\lambda_{x}|x\rangle, \quad \mbox{for} \quad
\lambda_{x} \in \mathds{C} \quad \mbox{and} \quad
\sum_{x}|\lambda_{x}|^2 = 1, \label{drpsi}
\end{align}
where the sum is over all possible decimal values of $x =
(x)_{10}=(x_{1}x_{2}\dots x_{10})_{2}$. The players are allowed to
apply operators $\sigma_{0}$ and $\sigma_{1}$. The qubits with odd
indices are manipulated by player 1 and the qubits labelled by
even indices are manipulated by player 2. Such assignment implies
32 possible strategies for each players as they specify five
operations $\sigma^j_{\kappa_{j}}$ (where $j$ and $\kappa_{j}$
indicate qubit number and operation number, respectively) on their
own qubits. We denote a player $i$'s strategy by $\tau_{i} =
(\sigma^i_{\kappa_{i}}, \sigma^{i+2}_{\kappa_{i+2}},
\sigma^{i+4}_{\kappa_{i+4}},
\sigma^{i+6}_{\kappa_{i+6}},\sigma^{i+8}_{\kappa_{i+8}}),$ where
$i=1,2$. The profile $\tau = (\tau_{1}, \tau_{2})$ gives rise to
the final state:
\begin{align}\label{drpureinitialstate}
|\psi_{\mathrm{fin}}\rangle =
\bigotimes^{10}_{j=1}\sigma^j_{\kappa_{j}}|\psi_{\mathrm{in}}\rangle.
\end{align}
If the players each take $\tau^t_{1}$ and $\tau^{t'}_{2}$ with
probability $p_{t}$ and $q_{t'}$, respectively, that corresponds
to the state $|\psi_{\mathrm{fin}}^{t,t'}\rangle$ (defined by
(\ref{drpureinitialstate})) with probability $p_{t}q_{t'}$, then
the final state is the density operator associated with the
ensemble $\{p_{t}q_{t'}, |\psi_{\mathrm{fin}}^{t,t'}\rangle\}$.
That is
\begin{align}
\rho_{\mathrm{fin}} =
\sum_{t,t'}p_{t}q_{t'}|\psi_{\mathrm{fin}}^{t,t'}\rangle \langle
\psi_{\mathrm{fin}}^{t,t'}|.
\end{align}
Till now, a difference between the concept in \cite{iqbalrepeat}
and our protocol lies in the dimension of the space $\mathscr{H}$.
The next difference is clearly visible in a description of
measurement operators. The measurement on $\rho_{\mathrm{fin}}$
that determines an outcome of the game is described by a
collection $\{X_{1}, X_{2.00}, X_{2.01}, X_{2.10}, X_{2.11}\}$,
where its components are defined as follows:
\begin{align}
X_{1} &= \sum_{x_{1}, x_{2} \in \{0,1\}} O_{x_{1},x_{2}}
|x_{1},x_{2}\rangle \langle x_{1},x_{2}| \otimes
\mathds{1}^{\otimes 8}; \label{drduzyiks}\\
\begin{split}
X_{2.00} &= \sum_{x_{3}, x_{4} \in \{0,1\}} O_{x_{3},
x_{4}}|00\rangle \langle 00| \otimes |x_{3}, x_{4} \rangle \langle
x_{3}, x_{4}| \otimes \mathds{1}^{\otimes 6};\\
X_{2.01} &= \sum_{x_{5}, x_{6} \in \{0,1\}} O_{x_{5},
x_{6}}|01\rangle \langle 01| \otimes \mathds{1}^{\otimes 2}
\otimes |x_{5}, x_{6} \rangle \langle x_{5}, x_{6}| \otimes
\mathds{1}^{\otimes
4};\\
X_{2.10} &= \sum_{x_{7}, x_{8} \in \{0,1\}} O_{x_{7},
x_{8}}|10\rangle \langle 10| \otimes \mathds{1}^{\otimes 4}
\otimes |x_{7}, x_{8} \rangle \langle x_{7}, x_{8}| \otimes
\mathds{1}^{\otimes
2};\\
X_{2.11} &= \sum_{x_{9}, x_{10} \in \{0,1\}} O_{x_{9},
x_{10}}|11\rangle \langle 11| \otimes \mathds{1}^{\otimes 6}
\otimes |x_{9}, x_{10} \rangle \langle x_{9}, x_{10}|.
\label{drduzeiksy}\end{split}
\end{align}
Then the expected outcomes: $E_{i.1}$ at the first stage and
$E_{i.2}$ at the second stage are calculated by using the
following formulae:
\begin{align}
E_{i.1} = \mathrm{tr}(X_{1} \rho_{\mathrm{fin}}), \quad E_{i.2} =
\mathrm{tr}\left(\sum_{\iota_{1}, \iota_{2}}
X_{2.\iota_{1},\iota_{2}} \rho_{\mathrm{fin}}\right).
\label{dreprotocol}
\end{align}
Let us give justification of our construction. Notice that
$2^{10}$ is a minimal dimension of the space $\mathscr{H}$ in
order to play the twice repeated $2\times2$ game. Since a player's
strategy in a twice repeated $2\times2$ game specifies action at
the first stage and at each of four subgames fixed by the outcome
of the first stage, the quantum protocol needs a five-qubit
register to write a player's strategy. The first two qubits are
used to perform operations at the first stage of the repeated
game. Then given the form of $X_{1}$ and strategies of players
restricted to manipulate the first and the second qubit, in fact,
the protocol (\ref{drpsi})-(\ref{dreprotocol}) coincides with the
MW scheme of playing $2 \times 2$ quantum game \cite{marinatto}.
The remaining eight qubits are used to define players' moves at
the second stage. That is, by pairing consecutive qubits from the
third qubit onwards,
actions at the second stage are defined on appropriate pair of
qubits depending on the outcome at the previous stage. For
example, given the outcome $O_{10}$ has occurred at the first
stage (that is the outcome 10 on the first two qubits has been
measured), the expected outcome $E_{i.2}$ depends only on
operation on $x_{7}$ and $x_{8}$, i.e, $E_{i.2} =
\mathrm{tr}(X_{2.10}\rho_{\mathrm{fin}})$. Then the players play
the second stage in the same way as in the protocol
(\ref{drinitialstate})-(\ref{drpayoff}). However, contrary to the
previous idea, each player specifies her move for each possible
outcome
$O_{x_{1},x_{2}}$.\\
\noindent A game generated by our scheme naturally coincides with
the classical case when appropriate initial state is prepared. We
prove this fact by means of a convenient sequential approach to
(\ref{drpsi})-(\ref{dreprotocol}) provided in the next section.
\subsection{Extensive form of a quantum twice repeated
$\mathbf{2\times2}$ game}
\noindent The protocol (\ref{drpsi})-(\ref{dreprotocol}) allows to
put a game into an extensive form by using a similar method to
what was described in \cite{fracornormalrepresentation}. The
extensive form is obtained through sequential calculating the
final state $\rho_{\mathrm{fin}}$ according to the following
procedure. At first the players manipulate the first pair of
qubits. Then the measurement in the computational basis is made on
these qubits (as a result, an outcome $O_{\iota_{1}, \iota_{2}}$
of the first stage is returned). The measured outcome is sent to
the players. Depending on the measurement outcome $\iota_{1},
\iota_{2}$ that occurs with probability $p(\iota_{1},\iota_{2})$
the players act on the next pair of qubits: if $\iota_{1},
\iota_{2}$ is observed then player 1 and player 2 manipulate qubits $2\iota + 3$ and $2\iota + 4$, respectively, where $\iota = (\iota_{1}
\iota_{2})_{2}$ is a decimal representation of a binary number
$\iota_{1}\iota_{2}$. The procedure can be formally described as
follows:
\begin{algorithm}\label{algorithmostatni}
\end{algorithm}
\newcommand*{\dupaI}{1.}
\newcommand*{\temphead}{$(\sigma^1_{\kappa_{1}}\otimes\sigma^2_{\kappa_{2}})|\psi_{\mathrm{in}}\rangle = |\psi\rangle$}
\newcommand*{\pustyI}{}
\newcommand*{\TempHead}{\footnotesize The players perform their operations $\sigma^1_{\kappa_{1}}$ and $\sigma^2_{\kappa_{2}}$ on the initial state $|\psi_{\mathrm{in}}\rangle$.}
\newcommand*{\dupaII}{2.}
\newcommand*{\tempdrugi}{$\to \displaystyle\frac{M_{\iota_{1},\iota_{2}}|\psi\rangle}{\sqrt{\langle \psi|M_{\iota_{1},\iota_{2}}|\psi \rangle}} = |\psi_{\iota_{1}, \iota_{2}}\rangle$}
\newcommand*{\pustyII}{}
\newcommand*{\Tempdrugi}{\footnotesize The first two qubits in the state $\rho$ are measured. The measurement is described by a collection $\{M_{\iota_{1}, \iota_{2}} \colon M_{\iota_{1}, \iota_{2}} = |\iota_{1},\iota_{2} \rangle \langle \iota_{1},\iota_{2}| \otimes \mathds{1}^{\otimes 8},\iota_{1}, \iota_{2} = 0,1\}$.}
\newcommand*{\dupaIII}{3.}
\newcommand*{\temptrzeci}{$\to \{p(\iota_{1},\iota_{2}), (\sigma^{2\iota+3}_{\kappa_{2\iota+3}}~\otimes~\sigma^{2\iota+4}_{\kappa_{2\iota+4}})|\psi_{\iota_{1},\iota_{2}}\rangle\}$ $~~~~~~~p(\iota_{1},\iota_{2}) =
\langle \psi|M_{\iota_{1},\iota_{2}}|\psi \rangle$}
\newcommand*{\pustyIII}{}
\newcommand*{\Temptrzeci}{\footnotesize Given that the outcome $\iota_{1},\iota_{2}$ has been observed,
players 1 and 2 perform operations
$\sigma^{2\iota+3}_{\kappa_{2\iota+3}}$ and
$\sigma^{2\iota+4}_{\kappa_{2\iota+4}}$ on the post-measurement
state.}
\newcommand*{\dupaIV}{}
\newcommand*{\tempczwarty}{}
\newcommand*{\pustyIV}{}
\newcommand*{\Tempczwarty}{}
\setlength{\tabcolsep}{-5mm}
\renewcommand*{\arraystretch}{1.5}
\begin{tabular}{p{2cm}p{6cm}p{3cm}p{7.4cm}}
\dupaI & \temphead & \pustyI & \TempHead\\
\dupaII & \tempdrugi & \pustyII & \Tempdrugi\\
\dupaIII & \temptrzeci & \pustyIII & \Temptrzeci\\
\dupaIV & \tempczwarty & \pustyIV & \Tempczwarty
\end{tabular}
\noindent It turns out that we can prove \vspace{12pt}
\begin{proposition}
The density operator $|\psi_{\mathrm{fin}}\rangle \langle
\psi_{\mathrm{fin}}|$ associated with state
(\ref{drpureinitialstate}) and the density operator for the
ensemble $\{p(\iota_{1},\iota_{2}),
(\sigma^{2\iota+3}_{\kappa_{2\iota+3}} \otimes
\sigma^{2\iota+4}_{\kappa_{2\iota+4}})|\psi_{\iota_{1},\iota_{2}}\rangle\}$
in Algorithm~\ref{algorithmostatni} determine the same outcomes
$E_{i.1}$ and $E_{i.2}$ with regard to the measurement
(\ref{drduzyiks})--(\ref{drduzeiksy}).\end{proposition}
\vspace*{12pt}
\begin{xx}
Let us put $\rho = |\psi\rangle \langle \psi|$. Given that
$|\psi_{\iota_{1}, \iota_{2}}\rangle \langle \psi_{\iota_{1},
\iota_{2}}| = M_{\iota_{1},\iota_{2}}\rho
M_{\iota_{1},\iota_{2}}/p(\iota_{1},\iota_{2})$ the state
$\rho'_{\mathrm{fin}}$ can be written as:
\begin{align}
\rho'_{\mathrm{fin}} = \sum_{\iota_{1}, \iota_{2} = 0,1}
\sigma^{2\iota+3}_{\kappa_{2\iota+3}} \otimes
\sigma^{2\iota+4}_{\kappa_{2\iota+4}} M_{\iota_{1},\iota_{2}}\rho
M_{\iota_{1},\iota_{2}}\sigma^{2\iota+3}_{\kappa_{2\iota+3}}
\otimes \sigma^{2\iota+4}_{\kappa_{2\iota+4}}.
\end{align}
Since the first and the second qubits are measured, any operation
$\sigma^j_{\kappa_{j}}$ for which $j \ne 1,2$ does not influence
the measurement. Therefore we have
\begin{align}
\rho'_{\mathrm{fin}} = \sum_{\iota_{1}, \iota_{2} = 0,1}
M_{\iota_{1},\iota_{2}} \sigma^{2\iota+3}_{\kappa_{2\iota+3}}
\otimes \sigma^{2\iota+4}_{\kappa_{2\iota+4}} \rho\,
\sigma^{2\iota+3}_{\kappa_{2\iota+3}} \otimes
\sigma^{2\iota+4}_{\kappa_{2\iota+4}}M_{\iota_{1},\iota_{2}}.
\label{drequality}
\end{align}
Note that $X_{\iota'_{1},\iota'_{2}}M_{\iota_{1},\iota_{2}} =
\delta_{\iota,\iota'}X_{\iota'_{1},\iota'_{2}}$, where
$\delta_{\iota,\iota'}$ is the Kronecker's delta, and $\iota
=(\iota_{1},\iota_{2})_{2}$, and $\iota'
=(\iota'_{1},\iota'_{2})_{2}$, Using the form (\ref{drequality})
of $\rho'_{\mathrm{fin}}$ we have
\begin{align}
\mathrm{tr}\left(\sum_{\iota'}X_{2.\iota'_{1}, \iota'_{2}}
\rho'_{\mathrm{fin}}\right) =
\mathrm{tr}\left(\sum_{\iota}X_{2.\iota_{1},
\iota_{2}}\sigma^{2\iota+3}_{\kappa_{2\iota+3}} \otimes
\sigma^{2\iota+4}_{\kappa_{2\iota+4}} \rho
\sigma^{2\iota+3}_{\kappa_{2\iota+3}} \otimes
\sigma^{2\iota+4}_{\kappa_{2\iota+4}} \right). \label{drequation2}
\end{align}
For each $\iota$ the trace of each term of the sum on the
right-hand side of equation (\ref{drequation2}) depends only on an
operation $\sigma^j_{\kappa_{j}}$ on a qubit $j$, where $j \in
\{1,2,2\iota+3,2\iota+4\}$. Thus, the equation (\ref{drequation2})
holds when also the rest of operations $\sigma^j_{\kappa_{j}}$ are
added:
\begin{align}
\mathrm{tr}\left(\sum_{\iota}X_{2.\iota_{1}, \iota_{2}}
\rho'_{\mathrm{fin}}\right) = \mathrm{tr}\left(\sum_{\iota}
X_{2.\iota_{1},\iota_{2}}\bigotimes^{10}_{j=1}\sigma^j_{\kappa_{j}}\rho_{\mathrm{in}}\bigotimes^{10}_{j=1}\sigma^j_{\kappa_{j}}
\right). \label{drequation3}
\end{align}
As a result, the left-hand side of (\ref{drequation3}) is equal to
the expected outcome $E_{i.2}$ associated with the final state
$|\psi_{\mathrm{fin}}\rangle \langle\psi_{\mathrm{fin}}|$. To
prove that $\rho'_{\mathrm{fin}}$ also determines the expected
outcome $E_{i.1}$ let us see that $X_{1}$ and
$\{M_{\iota_{1},\iota_{2}}\}$ are the same projective measurement
up to the eigenvalues. Hence
\begin{align}
\mathrm{tr}\left(X_{1}\rho'_{\mathrm{fin}}\right) =
\mathrm{tr}\left(X_{1}\sigma^{2\iota+3}_{\kappa_{2\iota+3}}
\otimes \sigma^{2\iota+4}_{\kappa_{2\iota+4}} \rho
\sigma^{2\iota+3}_{\kappa_{2\iota+3}} \otimes
\sigma^{2\iota+4}_{\kappa_{2\iota+4}} \right).\end{align} Since
$\rho = \sigma^1_{\kappa_{1}} \otimes \sigma^2_{\kappa_{2}}
\rho_{\mathrm{in}} \sigma^1_{\kappa_{1}} \otimes
\sigma^2_{\kappa_{2}}$, we obtain
\begin{align}
\mathrm{tr}\left(X_{1} \rho'_{\mathrm{fin}}\right) =
\mathrm{tr}\left(
X_{1}\bigotimes^{10}_{j=1}\sigma^j_{\kappa_{j}}\rho_{\mathrm{in}}\bigotimes^{10}_{j=1}\sigma^j_{\kappa_{j}}
\right). \label{drequation4}
\end{align}
Equations (\ref{drequation3}) and (\ref{drequation4}) show that
the state determined by the sequential procedure and state
(\ref{drpureinitialstate}) set the same outcomes $E_{i.1}$ and
$E_{i.2}$ for $i=1,2$. Using the same way as above and the
linearity of the trace it can be proved that the equivalence is
true if the players pick nondegenerate mixed strategies as well.
\end{xx}
\noindent Having a sequential approach that is in conformity with
protocol (\ref{drpsi})-(\ref{dreprotocol}) we are able to analyze
a quantum repeated game through an extensive form. It can
facilitate the work significantly bearing in mind $32\times32$
bimatrix associated with the normal representation of twice
repeated $2\times2$ game. Let us study the game tree drawn from
the sequential procedure if the initial state (\ref{drpsi}) takes
the form
\begin{align}
|\psi_{\mathrm{in}}\rangle = \lambda_{0}|0\rangle^{\otimes 10} +
\lambda_{1}|1\rangle^{\otimes 10}. \label{drsplatany}
\end{align}
Let us use the sequential procedure step by step. At first the
players manipulate $\sigma^1_{\kappa_{1}}$ and
$\sigma^2_{\kappa_{2}}$. Hence we obtain the following state:
\begin{align}
\sigma^1_{\kappa_{1}} \otimes
\sigma^2_{\kappa_{2}}|\psi_{\mathrm{in}}\rangle =
\lambda_{0}|\kappa_{1},\kappa_{2}\rangle |0\rangle^{\otimes 8} +
\lambda_{1}|\overline{\kappa}_{1},\overline{\kappa}_{2}\rangle
|1\rangle^{\otimes 8}, \label{dr2state}
\end{align}
where $\overline{\kappa}_{j}$ is the negation of $\kappa_{j}$. A
game tree at this phase is just the game tree corresponding to
a~$2\times2$ game (see Fig.~\ref{figure1}(b)), where
$\sigma^j_{\kappa_{j}}$ for $j=1,2$, $\kappa_{j} = 0,1$ are
associated with respective branches of that game tree. After a
sequence of actions $(\sigma^1_{\kappa_{1}},
\sigma^2_{\kappa_{2}})$ the measurement $\{M_{\iota_{1},
\iota_{2}}\}$ is made. Let us focus on the cases when the
measurement outcome 00 or 11 has been observed. The form of
(\ref{dr2state}) tells us that the measurement outcomes 00 and 11
are possible only if the profile at the first stage takes the form
of $(\sigma^1_{\kappa}, \sigma^2_{\kappa})$, where $\kappa = 0,1$.
Then, the probability $p(00)$ ($p(11)$) that the measurement
outcome 00 (11) will occur is equal to $|\lambda_{\kappa}|^2$
($|\lambda_{\overline{\kappa}}|^2$). Thus, the game tree is
extended to include random actions 00 and 11 with associated
probabilities after the both histories $(\sigma^1_{\kappa},
\sigma^2_{\kappa})$. Since further moves of the players depend
only on the measurement, the pair of histories $(\sigma^1_{0},
\sigma^2_{0},00)$, $(\sigma^1_{1}, \sigma^2_{1}, 00)$ and the pair
$(\sigma^1_{0}, \sigma^2_{0},11)$, $(\sigma^1_{1}, \sigma^2_{1},
11)$ constitute two separate information sets. Next, given that 00
(11) has occurred, following the sequential procedure, the players
manipulate third and fourth (ninth and tenth) qubit at the second
stage. Therefore another extensive form of $2\times2$ is added to
each sequence $(\sigma^1_{\kappa}, \sigma^2_{\kappa},\iota\iota)$,
where $\kappa,\iota = 0,1$. In consequence we obtain a game tree
shown in Fig.~\ref{figure3} (a part of the game tree after
histories of $(\sigma^1_{\kappa}, \sigma^2_{\overline{\kappa}})$,
$\kappa = 0,1$ is similar).
\begin{figure}[t]
\centering
\includegraphics[angle=0, scale=0.85]{epdq.eps}
\caption{The extensive form for a twice repeated Prisoner Dilemma
played through protocol (\ref{drpsi})-(\ref{dreprotocol}) when the
initial state is on the form of (\ref{drsplatany}).}
\label{figure3}
\end{figure}
Each outcome associated with a terminal sequence are determined by
a pure state from the ensemble given by the sequential procedure.
For example, after sequence $(\sigma^1_{1}, \sigma^2_{1})$ the
post-measurement state takes the form of $|0\rangle^{\otimes
2}|1\rangle^{\otimes 8}$ (up to a global phase factor) with
probability $|\lambda_{1}|^2$, and the players choose sequence
$(\sigma^3_{\kappa_{3}},\sigma^4_{\kappa_{4}})$. Then the total
outcome $E_{i}\mathrel{\mathop:}=E_{i.1} + E_{i.2}$ associated
with a sequence $(\sigma^1_{1}, \sigma^2_{1}, 00,
\sigma^3_{\kappa_{3}},\sigma^4_{\kappa_{4}})$ is calculated
according to formulae (\ref{dreprotocol}):
\begin{align}
E_{i} = \mathrm{tr}\left(\left(X_{1}+ \sum_{\iota_{1},\iota_{2}}
X_{2.\iota_{1},\iota_{2}}\right)(|0\rangle \langle 0|)^{\otimes
2}|\overline{\kappa}_{3},\overline{\kappa}_{4}\rangle \langle
\overline{\kappa}_{3},\overline{\kappa}_{4}|(|1\rangle \langle
1|)^{\otimes 6}\right).
\end{align}
The extensive approach allows us to see directly that our scheme
coincides with the classical twice repeated $2\times2$ game when
$|\psi_{\mathrm{in}}\rangle = |0\rangle^{\otimes 10}$. Without
loss of generality, let the outcomes $O_{\iota_{1},\iota_{2}}$ be
the payoff outcomes corresponding to the PD game. Then putting
$|\lambda_{0}|^2 = 1$ in (\ref{drsplatany}) and assuming
$\sigma^j_{0} \mathrel{\mathop:}= C$, $\sigma^j_{1}
\mathrel{\mathop:}= D$ the game in Fig.~\ref{figure3} depicts
exactly the classical twice repeated PD game (compare Fig.
\ref{figure2} and Fig. \ref{figure3}).
\subsection{Twice repeated PD game played by means of the protocol (\ref{drpsi})-(\ref{dreprotocol})}
Let us study the twice repeated PD game played with the use of our
scheme. Analysis of our protocol with the general form initial
state (\ref{drpsi}) is a laborious task and it deserves a separate
paper to report about. Nevertheless, we can derive many
interesting features with less effort considering the initial
state of the form
\begin{align}
|\psi_{\mathrm{in}}\rangle =
\bigotimes^5_{j=1}|\varphi_{j}\rangle, \;\mbox{where}\;
|\varphi_{j}\rangle\; \mbox{is a state of $2j-1$ and $2j$ qubit}.
\label{drprostyinitialstate}
\end{align}
Let us consider first the problem of optimization of the
equilibrium payoffs,
given a~space of initial states as a domain.
\vspace*{12pt}
\begin{proposition}
There are infinitely many settings of the initial state
(\ref{drpsi}) for which the twice repeated PD game played with the
use of the protocol (\ref{drpsi})-(\ref{dreprotocol}) has a unique
subgame perfect equilibrium with the equilibrium payoff $(2Q,2Q)$
such that $Q>P$. \label{drproposition}
\end{proposition}
\vspace*{12pt}
\begin{xx}
Let us put the initial state (\ref{drprostyinitialstate}) into the
protocol (\ref{drpsi})-(\ref{dreprotocol}) assuming that
$|\varphi_{j}\rangle = |\varphi\rangle$ for any $j$. Then, the
measurement $\{M_{\iota_{1},\iota_{2}}\}$ on the first pair of
qubits does affect others qubits. Moreover, given that the outcome
$O_{\iota_{1},\iota_{2}}$ has occurred, the expected outcome
$E_{i.2}$ depends only on manipulating on one pair of qubits
$|\varphi\rangle$ due to the form of (\ref{drduzeiksy}).
Therefore, regardless of the first stage outcome
$O_{\iota_{1},\iota_{2}}$, the players are faced with a~$2\times2$
quantum game at the second stage (played via the MW approach).
That is, the players are faced with the problem
\begin{align}
\left(|\varphi\rangle \langle \varphi|, \{\sigma_{0},
\sigma_{1}\}, X'_{i}\right), \label{drpdmw}
\end{align}
where player 1 and 2 apply operators from the set $\{\sigma_{0},
\sigma_{1}\}$ on the first and the second qubit of
$|\varphi\rangle$, respectively. The outcome operator $X'_{i}$
takes the form
\begin{align}
X'_{i} = \sum_{y_{1},y_{2}=0,1} O_{y_{1},y_{2}}|y_{1},y_{2}\rangle
\langle y_{1},y_{2}|,
\end{align}
and the expected outcome is equal to
\begin{align}
E_{i}(\sigma^1_{\kappa_{1}}, \sigma^2_{\kappa_{2}}) =
\mathrm{tr}\left(\sigma^1_{\kappa_{1}} \otimes
\sigma^2_{\kappa_{2}}|\varphi\rangle \langle
\varphi|\sigma^1_{\kappa_{1}} \otimes \sigma^2_{\kappa_{2}}
X'_{i}\right).
\end{align}
Obviously, the first stage game is also described exactly as the
triple (\ref{drpdmw}). Since a~quantum game according to the MW
approach is a game expressed by a bimatrix, it leads us to the
conclusion that protocol (\ref{drpsi})-(\ref{dreprotocol}) with
the initial state $|\varphi \rangle^{\otimes 5}$, in fact, can be
treated as
a twice repeated bimatrix game generated by (\ref{drpdmw}).
\noindent Let us substitute $O_{y_{1},y_{2}}$ for the payoffs of
the PD game in the game (\ref{drpdmw})
and examine it towards uniqueness of Nash equilibria. Putting a state
$|\varphi\rangle = \lambda_{0}|00\rangle + \lambda_{1}|11\rangle$,
for which the amplitudes of $|\varphi\rangle$ satisfy the
condition:
\begin{align}
0<|\lambda_{0}|^2 < \frac{\min\{T-R, P-S\}}{T-R+P-S}
\label{drconditionend}
\end{align}
the inequalities
\begin{align}\label{inequalities}
E_{1}\left(\sigma^1_{0},\sigma^2_{\kappa_{2}}\right) >
E_{1}\left(\sigma^1_{1},\sigma^2_{\kappa_{2}} \right) \quad
\mbox{and} \quad
E_{2}\left(\sigma^1_{\kappa_{1}},\sigma^2_{0}\right)
> E_{1}\left(\sigma^1_{\kappa_{1}},\sigma^2_{1} \right)
\end{align}
are true for any $\kappa_{1}, \kappa_{2} = 0,1$. Inequalities
(\ref{inequalities}) imply the unique Nash equilibrium
$(\sigma^1_{0},\sigma^2_{0})$. Moreover, the first inequality of
condition (\ref{drconditionend}) ensures that \begin{align}
E_{1}\left(\sigma^1_{0},\sigma^2_{0}\right) = |\lambda_{0}|^2R +
|\lambda_{1}|^2P
> P.
\end{align} Since the game constructed in the proof can be regarded as a classical
twice repeated game, we are allowed to use all facts of classical
repeated game theory. One of these tells us that a unique
stage-game Nash equilibrium implies, for any finite number of
repetitions, a unique subgame perfect equilibrium in which the
stage-game Nash equilibrium is played in every stage. This
completes the proof.
\end{xx}
Of course, the protocol (\ref{drpsi})-(\ref{dreprotocol}) can be
re-formulated for any finitely repeated $2\times2$ game and then
statement analogical to Proposition \ref{drproposition} can be
articulated. Unfortunately, the number of qubits required in our
protocol grows exponentially with number of stages. For example,
in the case of a game repeated three times, the protocol
(\ref{drpsi})-(\ref{dreprotocol}) needs next 32 qubits to describe
the third stage. In general, the number of $\sum^n_{j=1} 2^{2j-1}$
qubits is required for a $2\times2$ game repeated $n$ times.
\vspace*{12pt}
\noindent We shall re-examine now the problem of cooperation
considered in \cite{iqbalrepeat}. We demonstrated in Section 3
that the cooperation at the first stage is not possible in the
game defined by the Iqbal and Toor scheme. However, we also showed
that this protocol does not take into consideration a player's
move at the second stage as a function of the first stage result.
Therefore, in fact it does not allow to study the cooperation
problem in a~proper way. The following example proves that the
cooperation of players is possible if the twice repeated PD game
is played via our scheme. \vspace{12pt}
\begin{example}
\textup{Let us set the PD game with payoff vectors
\begin{align} O_{00} = (4,4),\; O_{01} = (0,5),\; O_{10} =
(5,0),\; O_{11} = (1,1) \label{drpayoffvectors}\end{align}
inserted in (\ref{drduzyiks}) and (\ref{drduzeiksy}). Let us also
assume that the initial state (\ref{drpsi}) takes the form
\begin{align}
|\psi_{\mathrm{in}}\rangle = |0\rangle^{\otimes
2}\left(\sqrt{0,6}|0\rangle^{\otimes 2} +
\sqrt{0,4}|1\rangle^{\otimes 2} \right)|0\rangle^{\otimes 6}.
\label{drstateexample}
\end{align}
A game specified in this way differs from the classical one only
in the subgame following the outcome $O_{00}$ of the first stage
because then $E_{i.2}$ depends on operations on entangled third
and fourth qubit. Since two first qubits in the state $|00\rangle$
imply the classical PD game at the first stage, we are permitted
to identify the action `cooperate' and the action `defect' with
$\sigma_{0}$ and $\sigma_{1}$, respectively, assuming
$C\mathrel{\mathop:}=\sigma_{0}$ and $D \mathrel{\mathop:}=
\sigma_{1}$. Moreover, the quantum measurement after the first
stage is trivialized in this case and it coincides with the
classical observation in an extensive game. It follows that both
the game defined by (\ref{drpsi})-(\ref{dreprotocol}),
(\ref{drpayoffvectors}), (\ref{drstateexample}) and the classical
game can be represented by the same game tree as well as the same
payoff values except when 00 has been measured on the first pair
of qubits after the first stage. Let us determine now the payoff
outcomes at the second stage given that the post-measurement state
of the first pair of qubits is $|00\rangle$ (in other words, when
player 1's strategy is $\tau_{1} = \left(\sigma^1_{\kappa_{1}},
\sigma^3_{\kappa_{3}}, \sigma^5_{\kappa_{5}},
\sigma^7_{\kappa_{7}}, \sigma^9_{\kappa_{9}}\right)$ and player
2's strategy is $\tau_{2} = \left(\sigma^2_{\kappa_{2}},
\sigma^4_{\kappa_{4}}, \sigma^6_{\kappa_{6}},
\sigma^8_{\kappa_{8}}, \sigma^{10}_{\kappa_{10}}\right)$, which
makes the strategy profile in the form $(\tau_{1}, \tau_{2}) =
\left(\left(\sigma^1_{0}, \cdot, \cdot, \cdot,
\cdot),(\sigma^2_{0},\cdot, \cdot, \cdot, \cdot\right)\right)$).
Given the initial state (\ref{drstateexample}) and the form of
operators (\ref{drduzeiksy}), the payoff outcome $E_{i.2}$ for
each $\kappa_{3}, \kappa_{4} \in \{0,1\}$ and $i=1,2$ is as
follows:
\begin{align}
E_{i.2}\left(\left(\sigma^1_{0}, \sigma^3_{\kappa_{3}}, \cdot,
\cdot, \cdot),(\sigma^2_{0},\sigma^4_{\kappa_{4}}, \cdot, \cdot,
\cdot\right)\right) = 0,6O_{\kappa_{3},\kappa_{4}} +
0,4O_{\overline{\kappa}_{3},\overline{\kappa}_{4}}.
\end{align}
The extensive form of the game with expected payoffs $E_{i.1 } +
E_{i.2}$ given by (\ref{drpayoffvectors}) is shown in Fig.~\ref{figure4}.
\begin{figure}[t]
\centering
\includegraphics[angle=0, scale=0.85]{examplepd.eps}
\caption{The extensive form for a twice repeated Prisoner Dilemma
(\ref{drpayoffvectors}) played through protocol
(\ref{drpsi})-(\ref{dreprotocol}) with update on the initial state
(\ref{drstateexample}).} \label{figure4}
\end{figure}
Let us examine this game for subgame perfect equilibria. Such
profile has to induce the Nash equilibrium in any subgame fixed by
an outcome at first stage. In our case, it is a profile in which
both players take $\sigma_{1}$ on qubits from the third qubit
onward. Consequently, in quest of subgame perfect equilibria, we
take only the following profiles into consideration:
\begin{align}
(\tau_{1}, \tau_{2}) \in \left\{\sigma^1_{\kappa_{1}}\times
\sigma^2_{\kappa_{2}} \times \prod^{10}_{j=3}
\sigma^j_{1}\right\}.
\end{align}
Then it turns out that the noncooperative
subgame perfect equilibrium is still preserved. If one of the players picks $\sigma_{1}$ at the first stage,
the best response of the other one is to pick $\sigma_{1}$ too.
Therefore, the profile $(\tau'_{1}, \tau'_{2}) =
\prod^{10}_{j=1}\sigma^j_{1}$ constitutes a subgame perfect
equilibrium. However, contrary to the classical twice repeated PD, there is another
subgame perfect equilibrium $(\tau''_{1}, \tau''_{2})$ in which
each player chooses $\sigma_{0}$ (cooperates) at the first stage i.e., $\tau''_{1} = \left(\sigma^1_{0}, \sigma^3_{1}, \sigma^5_{1}, \sigma^7_{1},
\sigma^9_{1}\right)$ and $\tau''_{2} = \left(\sigma^2_{0}, \sigma^4_{1}, \sigma^6_{1}, \sigma^8_{1},
\sigma^{10}_{1}\right)$. Moreover, only the latter equilibrium is reasonable since it yields the payoff 6,2 instead of
2 for each player.} \label{drexample}
\end{example}
\vspace{12pt} Example \ref{drexample} shows that the cooperation
of players is possible when the twice repeated PD game is played
according to our scheme. Unfortunately, the example does not solve
this problem for any PD game. The condition $2R
> T+S$ imposed on the payoffs allows to select an arbitrary large finite number $T$ (if a sufficiently small number $S$ is selected). We suppose that an appropriately large $T$
may convince the players to defect even if the game is played in
quantum domain.
\section{Conclusion}
Our paper proves that repeated games can be quantized. That is, we
have shown that appropriately modified the MW scheme for
$2\times2$ quantum games can indeed generalize a twice repeated
game. In addition, such quantized game can be further analyzed by
strategic as well as extensive form games. Our results also
indicate (with the use of the twice repeated Prisoner's Dilemma)
that playing repeated games in the quantum domain can give
superior results in comparison with the classical ones. At the
same time we have answered why the previous approach
\cite{{iqbalrepeat}} cannot be treated as a correct protocol for
quantum repeated games. The main objection is that the protocol
\cite{{iqbalrepeat}} is unable to consider a full set of
strategies available to players. In contrary to the Iqbal and
Toor's scheme, the protocol defined in this paper is free from the
mentioned fault.
\section*{Acknowledgements} \noindent The author is very grateful to
his supervisor Prof. J. Pykacz from the Institute of Mathematics,
University of Gda\'nsk, Poland for his great help in putting this
paper into its final form.
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"redpajama_set_name": "RedPajamaArXiv"
}
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The event - co-hosted by the APA, Contagious and the IPA - provided a whistle-stop tour of the most important trends and technologies affecting our industry, with top agency bods on hand to provide case study examples and explain the most profound shifts in their business models.
Suitably caffeined-up, the audience was then treated to a campaign case study from Tribal DDB creative director Simon Richings, who talked through the agency's immersive - not to mention devilishly useful - 'True Life Costs' website, built for Volkswagen. He explained the rich and carefully constructed site, saying: 'Consumers often equate the quality of the online experience with the quality of the product that's being talked about.' He was followed by Tom Thirlwall, CEO of production company Bigballs and founder/director Luke Taylor who introduced their stunning new interactive football game, I AM PLAYR, which plunges the player into the life of a fictional footballer - sordid affairs and air rifle accidents included... They believe that the premise of building a social game is that 'you are building a service, not a product'.
The penultimate speaker was VCCP head of strategy and innovation, Amelia Torode. She brought to the audience's attention a beautiful analogy by comparing Detroit - a city which failed by working towards one end product with an out-dated model - to the advertising industry, emphasising the importance of growth, change and innovation. Her 'four pillars' to consider are People, Data, Value and Attitude. Things were then wrapped up in a whirlwind fashion by Contagious Insider director Jess Greenwood, who working against the clock, whipped through the most exciting trends and future-facing marketing strategies identified by Contagious in the last six months - illustrated with a gaggle of truly knock-out case study examples.
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"redpajama_set_name": "RedPajamaC4"
}
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I was in Joann's one day and I couldn't resist taking a walk through the aisles of fabrics. I spotted this imitation snake skin in grays and black. I have seen so many cute snake skin skirts on other people's blogs that I thought I would try it.
I made a muslin of this skirt in a wool that I have in my stash. It's nice but since learning it was not my color, I have relegated it to muslin fabric. (It was on sale when I bought it so in spite of the fact that it is nice 100% wool, it makes a luxurious and economical muslin fabric.) It fit very nicely.
So I made up the skirt in the snakeskin fabric and put a lining in using some thicker polyester lining fabric I had in my stash. Well, I am not too happy with the resulting garment. It is a little too tight, probably because the polyester has less give than the wool woven I used for the muslin. Moreover, the styling of the pattern does not correspond to the snakeskin vibe. It looks like Old Grandma Hipster.
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"redpajama_set_name": "RedPajamaC4"
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<manifest package="com.github.vmironov.jetpack.arguments">
<application/>
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"redpajama_set_name": "RedPajamaGithub"
}
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For the 7th year in a row, we are hosting our massive Holiday Extravaganza at the brewery, this year on December 9th from 1-6pm.
As always, we will have a ton of rare and limited releases on draft, specialty can offerings, guest taps, and a few surprises. Also joining the festivities will be food trucks, DJs, live performances, face painting, photo booth, games and more.
General Admission tickets are $10 and include admission, a custom glass and your first drink ticket.
There is also a VIP Diamond Experience available that includes early admission to the festival and preffered access to super limited draft offerings (12pm entrance), 4 additional drink tickets, a special limited beer release, and a custom glass.
We are limiting the number of tickets for this years festivities in order to allow everyone easy access to beers and cut down on congestion throughout the event and live performances.
This event has sold out every year since inception, and we expect more of the same this time around, so don't wait to pickup tickets as there will be No Door Sales.
We'll see you at the party!!!!
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"redpajama_set_name": "RedPajamaC4"
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'This book packs an extraordinary amount of insight into a deceptively small space' Stephanie Flanders, _Financial Times_
'Mr Kay is a brilliant writer.' _Wall Street Journal_
'It will probably be the cheapest and most valuable investment IFAs can make, both for themselves and for their clients.' Russell Taylor, _Money Management_
'Beautifully written and immensely readable.' Jeremy Peat, formerly chief economist, Royal Bank of Scotland.
'I am in awe' Andrew Hilton, Director, CSFI
'John Kay is one of the most engaging and accessible writers on economics in Britain today' _Morning Star_
**JOHN KAY** is 'both a first-class economist and an excellent writer' ( _Financial Times_ ). He is a visiting professor at the London School of Economics, a Fellow of St John's College, Oxford and director of several public companies. He contributes a weekly column to the _Financial Times_. He chaired the UK government review of Equity Markets which reported in 2012 recommending substantial reforms. He is the author of many books including _Obliquity_ [9781846682896] and _Other People's Money_ [9781781254455] both published by Profile Books.
ALSO BY JOHN KAY
_Obliquity
Other People's Money_
**the long and the short of it**
a guide to finance and investment for normally intelligent people who aren't in the industry
_Second Edition_
_John Kay_
This edition published in Great Britain in 2016 by PROFILE BOOKS LTD
3 Holford Yard
Bevin Way
London
WC1X 9HD
_www.profilebooks.com_
Copyright © John Kay, 2009, 2016
The moral right of the author has been asserted.
All rights reserved. Without limiting the rights under copyright reserved above, no part of this publication may be reproduced, stored or introduced into a retrieval system, or transmitted, in any form or by any means (electronic, mechanical, photocopying, recording or otherwise), without the prior written permission of both the copyright owner and the publisher of this book.
A CIP catalogue record for this book is available from the British Library.
eISBN 978 1 78283 269 0
**Contents**
_Preface_
_Acknowledgements_
**1: SENSE AND THE CITY**
Can you be your own investment manager?
From gentlemen to players
**2: BASICS OF INVESTMENT**
Before you begin
Review financial risks outside your investment portfolio
Investment choices
Conventional investment strategies
Table 1: Percentage distribution of net assets by long-horizon funds, 2015
Realistic expectations
Benchmarking performance
Table 2: CalPERS returns by asset class (% p.a.)
The effects of compounding
**3: INVESTMENT OPTIONS**
Sense about shares
Basics of bonds
Realities of real estate
Facts about funds
**4: EFFICIENT MARKETS**
Efficient markets and asset values
Illuminating but not true
Information asymmetry
Strategic trading, moral hazard and market manipulation
Investment strategies for imperfectly efficient markets
**5: THE MIND OF THE MARKET**
Market psychology
When the market loses its mind
The power of conventional thinking
The prevalence of short-term thinking
The malign effect on corporate performance
The intelligent investor's advantage
**6: IN SEARCH OF FUNDAMENTAL VALUE**
Accounting for earnings
Table 3: Current PE ratios, various US sectors, January 2016
Table 4: CAPE in different equity markets, July 2015
Creative accounting
Cash is king
Competitive advantage
**7: RISK AND REWARD**
Risk and uncertainty
Subjective expected utility
Think probabilities
Comparing distributions
Table 5: The cumulative frequency distribution from tossing a coin
My ship came in
Diminishing marginal utility
Mind your portfolio
Table 6: Value of alternative portfolios depending on the fate of the shipbuilding nationalisation bill of 1976 (£)
The capital asset pricing model
**8: A WORLD OF UNKNOWNS**
Unknown unknowns
Narratives and patterns
Mirages, systems and hot hands
Table 7: Percentage of US mutual funds in each performance quartile, 2009–12, which subsequently achieved above-median performance in 2012–15
Models and their limits
Forecasts and their limits
The China effect is 'in the price'
Approaches to risk
**9: MODERN DEVELOPMENTS IN FINANCIAL MARKETS**
Derivatives
When bonds were no longer boring
Structured products
Hedge funds
Other alternative assets
**10: THE CONVENTIONAL INVESTOR**
Establish your portfolio
Table 8: Save now, spend later
Pay less
The conventional investor's portfolio
**11: THE INTELLIGENT INVESTOR'S STRATEGY**
Developing a strategy
Reviewing your goals
Diversification is the key to managing risk
The illusory security of cash and bonds
Contrary thinking
Market timing
Asset allocation
**12: INTELLIGENT INVESTMENT CHOICES**
Diversify
Fund selection
Property
Stock picking
Stock selection
**13: THE CUSTOMERS' YACHTS**
_Glossary_
_Bookshelves and Bookmarks_
_Bibliography_
**Preface**
The purpose of this book is to give you the information you need to be your own investment manager. It is a natural companion to _Other People's Money_ (Profile and Public Affairs, 2015), which analyses the process of financialisation that culminated in the global financial crisis of 2008. The critique of the finance sector developed in _Other People's Money_ raises two questions: what should be done to protect the public interest, and what should I do to protect my own interests? The concluding chapters of _Other People's Money_ attempt to answer the first of these questions. _The Long and the Short of It_ attempts to answer the second. In one respect, my task here is easier. To reform the financial sector, we need to persuade politicians, regulators and the public. To reform your financial affairs, the only person I need to persuade is you.
In this book I'll describe the investment options available, and the institutions that will try to sell them to you. I'll explain the principles of sound investment, and introduce you to the research that supports these principles. Sound investment is based on the returns from productive assets, and in a modern economy these are mostly owned by companies. So I'll describe how businesses succeed, and fail, in generating value for their shareholders, and how to distinguish fact from fiction in what they tell you.
Profitable investment involves risk, and the returns from investment are uncertain. Amateurs and professionals alike are bad at managing investment risks, although for different reasons. I'll discuss why, and explain the theories that try to clarify how we assess risks and the evidence that provides partial, but only partial, justification for these theories. I'll go on to develop a practical investment strategy for the intelligent investor, based on three fundamental principles – pay less, diversify more and resist conventional thinking – and give you the specific information you need to implement that strategy.
I'll describe the sophisticated innovations of the modern financial system. The same innovations that led directly to the global financial crisis of 2008. The world I will describe is complex and sophisticated, but greedy, cynical and self-interested. The only way to cope is to acquire your own knowledge and form your own judgement. You cannot, unfortunately, trust people who offer financial advice. If you can trust me, it is because the only product I am trying to sell you – this book – is one you have already bought. Even so, I need to insert the caveat that I am not offering advice that is specific to your circumstances, that I am not recommending any particular investment or investment product to you and that I accept no liability whatever for the consequences of your investment decisions. When they work out badly – and some of them will – it will be your fault, and when they work out well – and I hope some of them will also – you will certainly take the credit. I will explain that even a decision that works out badly may have been a good decision. But that is not a proposition that ambulance-chasing lawyers, crusading journalists or politicians wise after the event find easy to accept. So I am not going to offer advice, only suggest how you should make your own decisions.
Most investment books advise you how to trade; I suggest you trade as little as possible. Most investment books advise you to get to know the mind of the market; I suggest you think for yourself instead. Most investment books take for granted that your search for new investments is a search for stocks that are likely to go up. While I'm certainly not going to dissuade you from that, I suggest you give equal weight to a different question: is this investment different in character from those I already own?
In this book I shall describe the principles of intelligent investment that lead to these, and other, unconventional conclusions. But you will probably want to begin with a more conventional stance, and I would encourage you to do so. The conventional investor follows the average of what professional investors do. The power of conventional thinking in the City is so pervasive that this is, in reality, what the vast majority of professional investors do themselves. But you can follow that consensus with the aid of publicly available information and the properties of efficient markets. Instead of paying heavily for conventional thinking, you can use conventional thinking for free.
The conventional investor is in awe of those who have a deep understanding of what the market thinks. He should be: he is typically paying enough for the privilege. The education of the conventional investor begins with forming a sceptical view of financial market expertise. The intelligent investor doesn't care what 'the market thinks', save to the extent that its mistakes and irrationalities create opportunities. For the conventional investor, risk is being out of step; for the intelligent investor, risk is losing money.
But it is uncomfortable to be out of step. And the conventional investor will do as well as the average of professional investors, which isn't bad, and better than the vast majority of retail investors. The strategy I recommend is that you begin as a conventional investor, and as you gain experience and confidence, you devote an increasing fraction of your portfolio to intelligent investment. The intelligent investor has a mind of his or her own, can take a sceptical view of market wisdom and make his or her own risk evaluation. Such detachment enables the intelligent investor to earn better returns with less risk of loss.
The term 'intelligent investor' originates with Benjamin Graham, who wrote a book with that title. Graham's modern disciple is Warren Buffett, the most successful investor in history. He has claimed that
> Observing correctly that the market was frequently efficient, they (the academics – and many investment professionals and corporate managers) went on to conclude incorrectly that it was always efficient. The difference between these propositions is night and day. (Cunningham, quoting Buffett, 2002)
I'll paraphrase Buffett's remark by saying that markets are 80 per cent efficient, but the profits from investment are mostly to be found in the 20 per cent that is not.
There is more to this comment – a lot more – than meets the eye. Your immediate, and understandable, reaction might be to ask 'tell me about the 20 per cent'. But I want first to raise the more general philosophical issue. What does it mean to say that a theory is 80 per cent true? We're used to the idea that theories are either true or false. There isn't room for theories that are partly true, or mostly true.
But we have to find room. The world of business and finance is comprehensible only with the aid of theories, such as market efficiency, which are illuminating but not true. It would be easy to pursue this issue into the philosophy of science, but this is intended to be a practical book. I shall simply make the pragmatic assertion that you cannot be an intelligent investor if you believe either that markets are always efficient or that they are not mostly efficient – if you believe that the efficient market hypothesis is true, or if you believe it is false. The same is true of two other theories – the capital asset pricing model and the approach to risk analysis that I describe as 'subjective expected utility' – theories that are central to modern finance theory and to the risk control models widely used in financial institutions.
The 80/20 per cent hypothesis – that the world of business and finance is best understood with the aid of models that are partly true, partly false, is at the heart of this book. You need to understand both the 80 per cent and the 20 per cent. It is necessary both to be familiar with the ideas of efficient markets and subjective expected utility, and yet not to take either notion too seriously. Intelligent laypeople, who approach finance and investment without preconceptions, may find this easier to accept than professionals who have been forced to take one side or another in a debate.
The target readers of this book have to deal with issues of finance and investment because they have been successful in some other, unrelated, activity. These readers expect the same level of intellectual seriousness as from a book of popular science or popular history, and that is the level at which this book is written.
Many issues of finance and investment are, necessarily, technical, and their explanation involves jargon. I've provided at the end of the book a Glossary, which offers quick definitions of many terms that may be unfamiliar. The jargon is the language of the financial professional. No apology for jargon should be required if the explanation of its meaning needs to be a good deal longer than the word or phrase itself. There is one conspicuous exception to that principle – the use of the term 'high-net-worth' for 'rich'. I hope there is none of the jargon of the management world, in which new terms are coined to conceal emptiness of thought.
But this book is not an academic monograph. I have kept references to a minimum. I give references to enable the reader to check the sources of claims made. In a final section on bookmarks and bookshelves I have indicated some books and web sites that readers can use to pursue issues further. I have not thought it necessary to provide links to the commercial web sites of product providers – these are very easy to locate – but I have included details of some useful web sites that provide information or comparisons useful to intelligent investors.
**Acknowledgements**
Many people assisted in the preparation of this book. I would like to thank those who have helped me understand the financial services industry over the past twenty-five years. This book is critical of that industry, and of what it offers to small customers. Yet while there are people in the sector whose ethics and behaviour are beyond contempt, there are also many who consistently display exceptional intelligence and unquestionable integrity. I have been privileged to work with, and learn from, some of them.
I am grateful to Robert Metz and Liz Bates for research assistance on this edition, and to Jo Charrington, who has not only organised the process from first draft to finished book but who has also, by keeping my life in order, made the whole venture possible. And an especial thank you to Mika Oldham. She suggested the idea of such a volume in the first place and has been a source of encouragement throughout.
1
**Sense and the City**
**Can you be your own investment manager?**
Many people think that successful investment is about hot tips. If someone rings you with a hot tip, ask yourself, 'Why is he calling me?' and put down the phone. If you act on a hot tip from a friend, you may lose a friend and some money. If you act on a hot tip from a stranger, you will just lose some money.
Books tell you how to become rich from stocks. Software programs and training courses claim to help you trade successfully. Their authors assert that, with their assistance, you can make a comfortable living playing the market. Before you succumb, ask the following question: if I had a system that held the secret of lazy riches, would I publicise it in a book from which I will earn only modest royalties? Writing a book is hard work, believe me.
You might already have asked a similar question: why would anyone who could write a book like this one choose to do so? I am one of a minority, perhaps eccentric, who find the study of financial markets intellectually engaging. And I enjoy writing, and hope that you enjoy reading this book as much as I have enjoyed writing it.
My target reader wants to make good returns on investments without worry. He or she probably thinks that managing money is a chore, and that people obsessed by the stock market are sad. My target reader's financial objective is to have sufficient financial security not to have to worry about money. My target reader would be happy to go on holiday, even for months, and not look at his or her portfolio. My target reader is willing to take risks, but only with small amounts.
My target reader is a private investor, a term by which I simply mean someone who is investing his or her own money, or would like to do so. The purpose of this book is to help such a reader become an intelligent investor who can be his or her own investment manager. If you are hesitant about taking on that responsibility, you should, by the end of this book, be able to ask penetrating questions of anyone who offers you financial advice. This book is not for people who want to become professional traders, but for those who want to sleep securely in the knowledge that their portfolio is in the most trustworthy of hands – their own.
A book that told you how to be your own doctor or lawyer would be an irresponsible book. 'The man who is his own lawyer has a fool for a client.' Is it possible to be your own investment manager? The financial services industry attracts many smart people, and certainly employs many of the best-paid people in the world. Financial centres accommodate thousands of professionals. Traders spend long days dealing in securities, with access to unlimited computing power and extensive data resources. How can you compete with them? You can't, and you shouldn't. But you can hold your own in their world. There are reasons why DIY investing is possible, even necessary, unlike DIY law or DIY medicine.
An obvious and depressing reason for relying on your own judgement is that most people who offer financial advice to small and medium investors aren't much good. A doctor or lawyer may not always get it right, but you can be confident that their opinions are based on extensive knowledge derived from a rigorous training programme with demanding entry requirements. You can also expect that the doctor or lawyer will have real concern for your interests, not just his or her own.
Traditionally, financial advisers were neither expert nor disinterested. People who called themselves financial advisers were salesmen (overwhelmingly they were men) remunerated by commissions and selected for bonhomie and persuasiveness rather than financial acumen. They were financial advisers in the same sense that car dealers are transport consultants. Most of these financial advisers knew little that their customers did not know – except one piece of information which they did not share: how much the adviser would be paid to make a sale.
In some ways, things are getting better. Regulation has driven many ill-equipped financial advisers from the market. The large conglomerates that dominate the finance sector spend billions of dollars on training and regulatory compliance. But at the same time they have been obliged to pay billions of dollars in fines and compensation to people whose trust they have abused. The Dodd–Frank legislation in the United States, and the consequent establishment of the Consumer Financial Protection Bureau, have given new emphasis to the interests of customers, although these measures have been vigorously resisted by industry lobbyists. In Europe, however, the extensive new rules introduced since the global financial crisis of 2007–8 are as much aimed at protecting established financial institutions as at protecting their customers.
So there is still a long way to go. A 'bias to activity' is intrinsic to the processes of financial advice; few people will pay much to be told to do nothing, even though that is often wise counsel. Training and compliance are largely concerned with the process of selling and the mechanics of regulation rather than financial economics. After reading this book, you will understand the principles of investment better than most people who offer financial advice to retail investors. You do not have to worry about 'knowing your customer' if you are your own customer. You need not fear that the advice you give yourself will be biased by the prospect of a fat commission.
You can also benefit from the wider set of opportunities the internet has given the individual investor. Financial services can not only be bought and sold electronically, but can also be delivered electronically. From home or office, you can now obtain a wide range of information, buy and sell securities very cheaply, and access many investment products that did not exist two decades ago. Comparison sites enable you to scan a range of providers.
The internet will never replace the truly skilled intermediary, just as it will never replace the doctor or lawyer, though it may change the roles of these intermediaries. But the search engine and the comparison site can now do much of what intermediaries would once have done. Unfortunately, the search engine and the comparison site are also corruptible. Like the sales people, they are paid by product providers. What these sites display may not be comprehensive; what they highlight is not necessarily what is best for you.
The divergence between your interests and the interests of those who would sell you financial products is pervasive. One of the oldest anecdotes in the financial world tells of a visitor to Newport, Rhode Island, weekend home of American plutocrats, who is shown the symbols of the wealth of financial titans. There is Mr Morgan's yacht, and there is Mr Mellon's yacht. But, he asks, where are the customers' yachts? The question is as pertinent today.
A sideshow of the global financial crisis was the exposure of some of the largest frauds in history. Bernard Madoff and Allen Stanford, who had each stolen billions of dollars from their customers, ended up facing extended terms of imprisonment. Such crude theft is, fortunately, rare in the financial services industry, although many practices came close to what an ordinary person might describe as robbery. But the lawful earnings of many people in the industry seem ludicrous to an ordinary person, and are. In the course of this book you will learn how activities of little social value are so profitable for the individuals engaged in them.
No complex analysis is required to see that every penny that people take home from finance is derived from fees, commissions and trading profits obtained from outside finance. This is a simple matter of accounting. You and I, and people like us around the world, pay the large salaries and bonuses of people who work in the financial sector. We do so in our various roles as investors, as prospective pensioners, as customers of financial institutions and as consumers of the products of businesses that use financial services.
The massive rewards available in financial services are sometimes defended as the result of competition to attract talented people. The observation is true. Finance recruits many of the cleverest graduates from leading universities. In my experience, only a few top academics and lawyers rival the best minds in finance for raw intelligence. The mechanism that achieves this result is indirect. Massive rewards attract greedy people. If the number of greedy people is large, then financial services businesses can select the most talented among them. Within finance you find many who are greedy and talented, many who are greedy and untalented but few who are talented but not greedy. Interest in ideas is generally secondary to interest in money. That is why these people are in finance. So it is, unfortunately, necessary to be suspicious of the motives of everyone who offers you financial advice.
There are people in the financial world whose concerns are primarily intellectual. You will find some in the finance departments of universities and business schools. Others are behind the scenes in banks and financial institutions, where they are described as 'quants' or 'rocket scientists'. Modern financial markets are sufficiently complex, and the relevant analytic tools sufficiently sophisticated, for some people to find observation of these markets interesting for its own sake.
The good news is that the private investor can take a free ride on all these skills and activities. You can be a beneficiary of the efficiency of financial markets. Efficiency has both a wide and a narrow interpretation. Financial markets, though costly and imperfect, proved to be a more effective mechanism for promoting economic growth than central direction of production and investment. But in investment circles market efficiency has a specific technical meaning.
That meaning relates to the 'efficient market hypothesis' (EMH), the bedrock of financial economics. Much of this book will be concerned with the implications of that hypothesis, and its limitations. The professional expertise of everyone in financial markets is focused on the value of stocks and shares, bonds, currencies and properties, and on advising on when to buy and sell. Market prices reflect a consensus of informed opinions. The information that Apple builds great products or that the economy of Venezuela is in a mess is known to everyone who trades Apple stock or Venezuelan bolivars.
The efficient market hypothesis posits that all such information is absorbed in the market-place – it is 'in the price'. The market is a voting machine in which the opinions of all participants about the prospects of companies, the value of currencies and the future of interest rates are registered, and the result is publicly announced. The corollary of the efficient market hypothesis is that the results of the painstaking research of everyone in the financial sector are available to you for free.
If that conclusion seems startling, and it should, then imagine going to an auction – a fine-wine auction, for example – dominated by professionals. At first, you might be intimidated by the assembled expertise. But if you behave prudently, the dominance of professionals ensures you can't go too far wrong, because their bids will be the main influence on the price you pay.
You may not be convinced by this analogy. You may fear that there will be collusion among the dealers at the auction, that the market is rigged against the little guy. You may well be right. Fifty years ago, you would have been justified in having similar suspicions about securities markets. But, over recent decades, extensive public resources have been devoted to securing the integrity and transparency of financial transactions.
These regulatory provisions don't work perfectly, and never will. When you trade, your broker must normally get you the best price available in the market. The reality is that a bank dealing on its own account will often do better. But not so much better. The edge that the skilled and experienced buyer may have can be more than offset by the advantages you have in trading for yourself. You have greater knowledge of your own needs; you know that you can trust yourself. Best of all, you don't have to pay yourself. Your bonus is already in your pocket.
**From gentlemen to players**
The efficient market hypothesis describes how the market handles information. Information has always been the life-blood of markets, but the manner in which information is handled has changed. Finance was once based on relationships: the community bank manager, the locally based insurance agent and the gentlemanly stockbroker knew their clients, often socially as well as professionally. Investment bankers nurtured a long-term association with big and small companies, investing time in acquiring knowledge of the business, in the expectation that they might occasionally be rewarded by commission on a new issue or fees on an acquisition.
The atmosphere of today's financial markets is very different. Most stock exchanges no longer have a trading floor where buyers and sellers meet. Now, the large investment banks have their own raucous trading floors, which may contain hundreds of desks, each linked to the market via a screen. The visual display has several parts, designed to convey the impression of an unmanageable flow of new information.
The modern financial institution encompasses a mixture of people and approaches: the urbane sophistication of the investment bankers who plan new issues and plot mergers and acquisitions; the rocket scientists and the quants – frequently intellectually sophisticated but often lacking in common sense; the traders – some of them graduates of leading business schools, some with no higher education at all – demonstrating the aggression and ethics of the market stall. The organisations that combine these functions are an explosive mix which frequently does detonate. The overall change is from benign amateurism, much of it based on what would now be called insider trading, to specialised professionalism, based on sophisticated analytic tools.
Technology, product innovation, globalisation and changes in regulation drove this transition. The development of quantitative techniques in investment analysis goes back more than a century. The first index of stock prices – an average of the prices of twelve leading stocks – was created in the US in 1896 by Charles Dow. A few years later the Frenchman Louis Bachelier presented a thesis on the mathematics of securities prices that is generally celebrated as the foundation of mathematical finance. Bachelier encountered resistance, both from practitioners, who ignored his work, and from his examiners, who gave it the modest accolade of 'honourable' – insufficient to enable the author to pursue an academic career. For half a century his work would be ignored.
Other seminal contributors to quantitative finance – such as Harry Markowitz, Fischer Black and Myron Scholes – would also initially encounter negative reactions. I'll describe their theories and these responses in later chapters. Their work, like that of Bachelier, is now fundamental not just to the analysis but also to the operation of modern financial markets. Much of this work originated at the University of Chicago, and the city of Chicago was also where new derivative markets were first established. Both the graduates and the techniques were taken up by Wall Street.
All kinds of financial institution – retail banks, investment banks, insurance companies – have broadened the range of their activities. Retail banks have become financial conglomerates, with a wide range of investment products as well as their traditional roles of providing payment services for large and small customers and lending, both to households and to corporate clients. The rationale of such conglomeration lies in cross-subsidy and cross-selling. Cross-subsidy involves selling some products below cost to enhance the sales of other more profitable lines; cross-selling involves use of the customer list for one group of products to promote others. Both practices generally operate to the long-term, and frequently immediate, disadvantage of customers. I'll discuss the conflicts of interest between investment banking and retail banking, and advise you to resist cross-selling.
Many commentators have anticipated the consolidation of retail banks into a smaller number of global firms, and the demise of the bank branch. Both of these developments will probably happen, but more slowly than has been generally supposed. The traditional bank manager was an independent financial adviser. He was an active participant in the local community, to which he would usually be attached for many years.
Personalised banking is today prohibitively costly for a mass market. While the cost of processing transactions has fallen with the development of information technology, the cost of employing knowledgeable people to handle them has risen rapidly; the scope of the financial services industry has become such that people of mediocre abilities now command large salaries. A vestige of relationship-based advice survives in private banking for high-net-worth individuals. But the modern bank branch is a shop and, like other shops, is staffed by pleasant sales assistants with limited knowledge and training. The physical environment has been remodelled accordingly.
Despite the efforts of designers, the bank branch is less inviting than most shops – security precludes too inviting a display of the goods. Few products are more suited to online retailing than financial services, and that is how I suggest you buy them. Still, the branches remain busy, as you discover if you try to visit one at lunchtime. Many people dislike dealing with money and managing financial services and need personal reassurance when they do. And more than a few readers will continue to feel this need for reassurance at the end of this book. Take advice, then, but do not pay much for it. Adopt the same sceptical attitude towards your advisers as you would towards a sales assistant in a shop.
While the passing of amateur finance was inevitable, there were losses as well as gains. The old world of finance was characterised by a certain rigid integrity. It wasn't exactly that people wouldn't steal from or cheat each other, rather that they would do so only in certain well-defined and tolerated ways. This carefully modulated self-regulation could not survive the globalisation of financial markets. Globalisation, professionalisation and the rise of financial conglomerates made new and different forms of regulation of financial services inevitable.
Although regulation of financial services is extensive, even intrusive, regulation has plainly not served consumers such as the readers of this book well. Individual savers, and the institutions through which they invested, suffered in the New Economy bubble in the 1990s. The promotion of complex packaged securities in the years after 2003 was followed by a near-meltdown in the global financial system following the collapse of the market for these products in 2007. The reckless lending of financial institutions in the north of Europe to borrowers in the south has created economic tensions that threaten social stability and the very survival of the European project. The common characteristics of all these episodes has been that individuals who promoted them became very rich and the public at large – taxpayers and users of financial products – was left poorer.
The fundamental reason for the repeated – and continued – failure of regulation to meet the needs of private investors is the prevalence of what economists characterise as 'regulatory capture'. Regulation operates, in large part, in the interests of the firms which are regulated rather than the customers of these firms. This is not because regulators are corrupt; most of the administrators engaged in the regulatory process are honest people performing a difficult job to the best of their abilities. In the United States, the degradation of the political process by the mechanisms of campaign funding is the primary cause of regulatory failure. In Europe, the problem is more an instinctive corporatism which tends to equate the national interest with the interests of large national firms. We have held, and continue to hold, unrealistic expectations of what regulation might achieve.
Scepticism and wariness are needed in dealing with even the most apparently reputable financial institutions. The intelligent investor needs to develop his or her own strategy. This book is intended to help the intelligent investor do that.
2
**Basics of Investment**
**Before you begin**
I'll start with preliminaries that you should consider before you even think about investment principles or investment options. Compile a list of your financial assets and liabilities. Do it with your spouse or partner. Whether you manage your financial affairs separately or collectively, plan together. The amount by which your assets exceed your liabilities is your net worth. 'High-net-worth' is the new euphemism for rich, but what people mean by 'high-net-worth' varies. You are well off, it is often said, if you earn more than your brother-in-law. Many people will find that their house, with the associated loan, dominates the calculation. I suggest that you put this in a separate column.
But don't put house and mortgage out of your mind altogether. A housing loan is the only form of borrowing appropriate for anyone planning an investment portfolio.
Later in this chapter I will suggest you aim for a target rate of return of 8 per cent, before tax and inflation. Even if you fall short of this, mortgage borrowing is so cheap that it is realistic to expect to earn more on your investment portfolio than the cost of the loan. Many conservative people want to reduce their mortgage as quickly as they can, but such a strategy isn't necessarily wise.
Although the mortgage is a product widely sold in all advanced countries, there are considerable differences internationally in the form of the typical mortgage contract. For example, US mortgages generally have interest rates fixed for the whole term of the mortgage, but the borrower has an option to repay, and even for someone who does not move house, refinancing may be attractive if rates have fallen. A German mortgage also has a fixed rate, but repayment may entail a penalty reflecting the difference between historic and current interest rates. Most British mortgages, by contrast, normally have variable rates of interest which the lender can raise or lower according to market conditions. Mortgage interest is tax deductible in some countries – the US and the Netherlands, for example – but not in most. American mortgages are generally 'non-recourse' – the borrower can 'hand back the keys' and walk away from the debt – but in most European countries the lender can pursue the borrower for any outstanding amount even after repossessing the house and selling it.
You will often be better off if you re-mortgage your property, and you should certainly take the opportunity to find out what deals may be available to you. You may be able to increase the amount of the debt, and this may be worth considering as you establish an investment portfolio. This option is particularly attractive in countries such as the Netherlands and the United States, where interest is tax deductible. The net cost of the mortgage is then significantly less than the return you can reasonably hope to earn on your investments. However, most countries no longer allow such tax deductions. In any event, before taking out any home loan be clear what the upfront fees are, what the rate of interest is and what the mechanism is through which the debt will be repaid.
If you have any debts other than a mortgage, repay them before you undertake any investment. In particular, repay outstanding credit card balances. The interest rate on almost all credit cards is well above 8 per cent, and certainly above the return you can realistically expect on your investments.
You will pay income tax on the yield from your portfolio. The rate will depend on the country in which you live and your personal circumstances. Work out the marginal rate of tax you pay – the proportion of any extra money you earn that will be levied in income tax. In many countries there is a difference between tax rates for dividends or other savings income and the rates which apply to earnings. You may also be liable for capital gains tax. Most countries also have gift or inheritance taxes. Take the opportunity to clarify these tax issues.
In particular, you will want to consider how your assets are distributed between yourself and your spouse or partner, whether you want to make gifts to your children and whether you are taking full advantage of reliefs for pension savings and any other tax concessions for investment. A financial adviser can help you with this but will almost certainly take advantage of the opportunity to sell you products. If you need assistance, I strongly recommend you nevertheless undertake your own internet research first. As I will show below, tax can have a very substantial impact on the value of your investment portfolio.
You need a current bank account, a savings or money market account, a payment (credit or debit) card and a mortgage. Many people buy all or most financial products from the same firm and have been with the same provider for years. This is usually a mistake. Financial institutions rely on inertia and often give introductory incentives to buy what later become uncompetitive products. The problem originates with consumers rather than with institutions. Banks or insurers who provide everyday good-value pricing find their customers drift away to the introductory bargains offered by other companies.
If you are willing to be a 'rate tart', sitting regularly at your computer pursuing the best deals, you can benefit permanently from introductory incentives. Most readers have better things to do. I would prefer to find a provider who offers competitive terms and good service, and so, I suspect, would you.
The best buys are often from 'monoliners' – firms that specialise in a single product group – because their future depends on their reputation in these products. Financial services conglomerates, especially the major banks, will try to cross-sell you products. They are not very successful at this, but they emphasise this business strategy to themselves and their shareholders. The products they cross-sell are rarely competitive, and the sales pitch will waste your time.
Since the modern bank branch is a shop selling products you probably don't want to buy, you do not need to pay fees – in the form of charges for a branch-based current or savings account – to go there. An internet account will operate on the platforms of the major banks so that processing of everyday transactions should proceed in exactly the same way. Many banks seeking new current accounts now offer services designed to make switching accounts easy.
The benefits of getting these preliminaries of banking arrangements and mortgages right are so large that anyone who fails to do so isn't serious about managing money. This exercise is likely to be the best paid few hours' work you will undertake in your lifetime. Obtaining a better deal may involve moving away from an established high-street provider (although frequently moving to a subsidiary of an established high-street provider). You don't need to be a rate tart; a comprehensive review of your arrangements every few years will be enough.
**Review financial risks outside your investment portfolio**
Households generally cover some risks by insurance – risks such as domestic fire or theft, car accidents, travel disruption. I'll come back to insurance in Chapter 11, after discussing the ways people think about risk. Most people take out insurances they shouldn't, and fail to take out insurances they should. Here are a few preliminary pointers.
If you have dependent children, you should consider insurance on the life of both parents. The low take-up of such policies is a legacy of the sale of life assurance, costly savings products with a small element of life insurance, by sales people on commission. The policyholders could rarely afford the premiums, so most policies lapsed, leaving children without protection and creating a well-founded perception that life insurance was expensive.
A more appropriate form of insurance – life insurance that has no savings component but lasts for the period of the child's dependency – is cheap, since young parents rarely die, although the financial consequences can be extremely serious if they do. Policies that cover costs and loss of earnings from long-term illness or disability are also worth considering, although far from cheap.
Do not buy insurance from someone who offers it when you are buying something else. You will be invited to purchase extended warranties on domestic appliances and to take out travel insurance when you buy a holiday. Because the purchaser has little opportunity to compare prices, or even to consider the purchase carefully, these policies are usually expensive. In Britain, payment protection insurance, pressed by banks on customers who took loans from them, was such scandalously bad value that its sale was effectively banned. When you refuse these policies, you are, unfortunately, depriving the sales person of income, since he or she receives large commissions for selling these policies. That is why the policies are expensive and the seller is persistent. If you want such insurance – mostly you don't – buy it elsewhere.
Insurance can cover some of the financial risks of everyday life: accidents at home and on the road, theft, sudden death. But the most serious financial risks that most households face come from redundancy and unemployment, old age and chronic illness, marriage and relationship breakdown. Such risks cannot be insured effectively and can be mitigated only through careful money management and successful investment. That's the main subject of this book.
**Investment choices**
When people discuss investment in bars and lunch rooms, they generally mean investment in securities – stocks and shares, bonds and deposits. But investment finds another meaning in the creation of the physical infrastructure of the economy.
Companies invest in plant and machinery. They must finance inventories and pay for work in progress. They operate from offices and shops. The government invests in schools and hospitals, bridges and roads. People used to make a sharper distinction than they do now between savings – putting money aside for later – and investment – putting that money to use. But, perhaps significantly, these words have lost this differentiated meaning. I'll talk instead about financial investment and productive investment. The return on financial investment must, ultimately, be generated by the return on productive investment. The quest for financial perpetual motion machines is unending, but the basic principle of bookkeeping – money that goes out equals money that comes in – is as immutable as the natural laws of thermodynamics. Financial investments fund physical investments, and the two principal means by which they do so are shares and loans.
A stock or share (an equity) gives you a portion, for better or worse, of earnings and realisations from the productive activities of a company. A secured loan – such as a mortgage – gives the lender the right to seize some or all of the borrower's assets if the borrower fails to pay agreed interest and the original money lent. Unsecured loans give the lender only a legal claim against the borrower for payment, and that claim will be met only after secured creditors have realised the value of the assets pledged to them.1
Physical objects, such as offices and machinery, are not the only productive assets. Many of the assets of modern companies are intangible. Companies own brands, and build reputations. They acquire licences and intellectual property, and they invest in internal systems and organisational structures. These assets earn a return, just as plant and buildings earn a return; many such assets can be bought and sold, and can even provide security for loans.
Some financial assets are not directly associated with productive assets. Governments discovered – hundreds, even thousands, of years ago – that they could borrow to finance wars and profligate consumption because lenders knew that states could raise taxes to repay the loans. (This did not always mean they would, as lenders continue to rediscover.) Individuals can, more modestly, borrow to spend now on the strength of their ability to earn income in future. So some investments are rewarded by what economists coyly call 'time preference' – the premium that impatient or hard-pressed people will pay to spend now rather than later.
Lending to government, or a credit card borrower, must compete with lending to a company. So the returns you can earn on these financial investments are ultimately governed by what productive investments can earn. None of the complexity of modern financial markets, which package and repackage securities in ever more elaborate ways, can alter that essential truth. A loan that is tradable is called a bond.
While investors mostly hold indirect claims on productive assets, through shares and loans, investors can also own a direct claim on the productive investments of businesses, governments and households. They sometimes do so by owning the buildings – shops, offices, warehouses, factories and houses – from which businesses, governments and households operate. Or they may lease equipment to them.
Shares, loans and property are the main investment choices. Most investors will also require some cash for immediate needs, and to exploit investment opportunities as they arise. In gloomy economic circumstances, cash may be an investment class in its own right.
What choices should you make? The strategy I will propose in Chapters 10 to 12 is that you should begin as a conventional investor – relying on the illumination of efficient markets to provide you with the assembled wisdom of the financial services industry. Recognising that this efficient market theory is only illuminating, not true, you will want, over time, to become an intelligent investor, employing your own judgement – to pay less, diversify more and take contrarian stances.
**Conventional investment strategies**
The conventional investor follows the average of what professional investors do. The power of conventional thinking in the finance sector is so pervasive that this is, in reality, what the vast majority of professional investors do themselves. You can follow that consensus with the aid of publicly available information and the properties of efficient markets. Instead of paying heavily for conventional thinking, you can use conventional thinking for free.
What do conventional investors do? One starting point is to look at the largest investment portfolios in the world. These funds publish their strategies in considerable detail. At the end of 2015 three of the biggest funds were the Norwegian government oil fund, with assets of around $900 billion, the Dutch pension fund ABP ($400 billion) and the California Public Employees' Retirement Scheme (CalPERS), which has built up a portfolio worth $300 billion.
All of these funds have long time horizons, and their scale gives them access to whatever investment advice they feel they need. The Norwegian fund is managed directly by Norges Bank Investment Management (NBIM), a division of Norway's central bank. ABP uses both in-house and external expertise. The assets of the Californian fund are distributed among hundreds of asset managers, each responsible for part of the fund and utilising their specialist expertise. Table 1 below describes how the three schemes invested their money.
Table 1: **Percentage distribution of net assets by long-horizon funds, 2015**
Source: NBIM Government Pension Fund Global Annual Report (at 31 December, Table 8), ABP Quarterly Report (at end-Q3) and CalPERS Comprehensive Annual Report (at 30 June, pp. 55 and 101).
For each, shares are the largest asset class, with the equity component (including private equity) ranging from 28 per cent at ABP to 61 per cent for NBIM. The Californian scheme has about one quarter of its assets in bonds, but for both the Dutch and Norwegians the figure is closer to 40 per cent. ABP has a particularly large commitment to inflation-linked securities (I will explain these further in the next chapter). CalPERS and ABP each have around 10 per cent of their assets in real estate.
If you were to take the average of the three asset allocations of these global investors, you would put 51 per cent in shares, 35 per cent in bonds, 7 per cent in real estate and have 7 per cent available for liquidity and other opportunities. That is typical of the asset allocation strategy a prudent, conventional adviser would recommend for a long-term investor, and that is how these funds came to adopt these policies.
There are other options. Some large educational endowments in the United States – particularly Harvard and Yale Universities – pioneered the use of a wider range of investment categories to diversify their holdings. Yale's investment manager, David Swenson, attracted a following, first in the United States and then internationally, for an innovative approach which eschewed bonds and emphasised 'alternatives', notably hedge funds and private equity – which I will describe in Chapter 9. After the New Economy bubble burst in 2000, many investors, having learned that shares do not always go up, also turned their attention to alternative investment possibilities.
What was the next new thing? If you couldn't rely on equities any more, what was the new source of easy wealth? This search would be the dominant investment theme of the first years of the twenty-first century. Like a caravan of prospectors huddled together for mutual support and security, investment institutions moved from one asset class to another, bidding up each – property, minerals, emerging market bonds, energy and foodstuffs – to new highs. An older meaning of the term 'alternative investment' also described investments in objects such as fine art, vintage cars or wine, and in commodities such as gold, copper or coffee. The most widely favoured alternative assets today are private equity (shares not quoted on a stock exchange, which few institutions supported before the 1990s) and hedge funds.
These alternative asset classes were supported partly for their novelty and partly in the belief that a wider range of investment classes provided greater diversification and security of return. But after the global financial crisis of 2008 almost all these asset categories fell in value. Panicked investors, desperate to secure cash and deprived of access to funding, sold what they could sell almost regardless of the underlying character or value of the investment. And the golden reputations of the Harvard and Yale endowment managers lost much of their glister.
**Realistic expectations**
An earlier generation of investors was taught (in some cases compelled) to match their spending to their income – the interest and dividends they received each year from their portfolio. But when, as in the 1970s, inflation rates were much higher than interest rates, anyone who spent their income was eroding the real value of their capital. Conversely, if your investments have a low yield but continue to generate capital gains – as with many investments in the last two decades – then it may be sensible to treat some of that growth as available for current expenditure.
A modern maxim is 'Think total return', which means regarding both income and capital gain as part of the return on your investment. The CalPERS fund has a target total return of 7.5 per cent and over the last twenty years has achieved a total return of 8.5 per cent per annum. With inflation in the US averaging 2.4 per cent over the last twenty years, this corresponds to a total return in real terms of 6.0 per cent.
Some readers will think such figures are disappointing. They have heard many stories about people who have made a killing on the stock market through hot tips and inspired timing. If you click on internet bulletin boards, you will read about the search for 'ten and twenty baggers' – stocks that will rise, or have risen, to ten and twenty times their initial value. If you open an online stockbroking account – and you should – you will soon be seduced by bulletin boards and offered hot tips. The stocks promoted there may be 'concept stocks', based on unproven technologies or bright ideas. Penny shares are priced so low that any movement seems to have a large effect on the wealth of the holders.
When you begin investing, you may be inclined to back some of these tips, and I wouldn't discourage new investors from doing so with a very small proportion of their available funds. It is a cheap means of beginning to learn about the mechanics and the psychology of the market. A few people, possessed of ill-founded self-confidence or books or courses on how to trade, set out to make a fortune in this way. The New Economy bubble saw the emergence of 'day traders', who bought and sold stocks several times a day. Stockbrokers established offices to service them.
As with many forms of professional gambling, there do seem to be small numbers of people who are good at it. To good luck they add exceptional psychological insight and rigorous self-control. But surveys of day traders show that after a few months most give up, or are obliged to do so by their families. Not all. Besides the few who develop successful careers, a larger group of people become addicts, whose financial stability and family lives may be destroyed by their obsession.
Financial innovations, especially in derivative markets, have greatly increased the opportunities and risks of market-related gambling for both amateurs and professionals. In securities markets many people are in a position to gamble, legally, with their employers' money.
Some become hooked; a few corporate and municipal treasurers have lost large sums, and a personal tragedy has become an expensive disaster for shareholders or taxpayers. The most sophisticated punters are managers of hedge funds and operators in the proprietary trading operations – 'prop desks' – of investment banks. Even there, there is a heady mix of real skill, addiction, obsession and self-delusion on the part of both individuals and organisations – a self-delusion which in 2008 brought some of the largest and longest- established financial institutions in the world – Royal Bank of Scotland, Union Bank of Switzerland and Citigroup – to the point of collapse. If you are tempted to make a living from stock market speculation, this is not the book for you. There are many other books aimed at the seekers after twenty baggers. The target reader of this book is concerned, as are the global investment funds in Table 1, with the careful and responsible stewardship of assets. They are responsible investors of other people's money. You are a responsible investor of your own.
**Benchmarking performance**
Table 2 looks in more detail at CalPERS' performance. The data is a warning that, even over a five-year time horizon, returns can be volatile.
CalPERS' earnings on its global equity portfolio in the five years from 2004 to 2009 were virtually zero, and this was also true of its real estate holdings between 2009 and 2014. Shares were the best performer in the second period and the worst in the first; for property the opposite was true. And when I wrote the first edition of this book, slightly more than five years ago, this statement was precisely reversed. There is a marked tendency for periods of underperformance to be followed by periods of outperformance, and vice versa. I will come back to this phenomenon, called mean reversion, in Chapter 4.
You could have become rich by holding shares in the 1990s and disposing of them to buy property in 1999, selling that property in 2007 and putting your wealth back into shares in 2009. But neither you nor I, and not even the well-advised folk at CalPERS, had the foresight which hindsight now tells us would have been so rewarding. Most people thought the opposite, which is why prices moved as they did; shares were in fashion in the 1990s, and property unappreciated. In that new virtual world of 1999, economic power would lie with the young titans who had been the first to 'get it', and bricks and mortar would play a minor role in commerce. When this fantasy was dispelled, or at least postponed, it was obvious to all that low and falling interest rates and readily available finance for property purchases meant that real estate values would always rise. Until they didn't. Conventional wisdom dominates asset prices, but you would have been better off – over the long run – with a sceptical stance.
Table 2: **CalPERS returns by asset class (% p.a.)**
(Total return including income, rents, dividends) (%)
Source: CalPERS, author's calculations
Over a longer period, returns are less volatile, as the ten-year CalPERS figures show. (CalPERS' real estate performance is wretched because the fund made some big and unsuccessful transactions in residential development during the US housing bubble.) But in other asset categories, CalPERS has performed pretty much in line with its benchmarks, ahead of them in the two categories of bond and behind in equities.
What are these benchmarks? Benchmarks play a central role in the lives of professional investors. These investors are judged relative to the performance of their peers, and therefore tend to focus not on absolute return but on relative return. It is not enough to have earned 20 per cent if the benchmark index has returned 25 per cent (as it often will, over a short period, and as it did over longer periods in the 1990s). It is defensible (even if uncomfortable) to have lost 15 per cent if the benchmark has lost 20 per cent. The manager can blame 'the market' for his poor performance.
These benchmarks are indices designed to measure the overall performance of an asset class and are widely – almost obsessively – used by asset managers and investment consultants. Despite the plethora of advice available to a fund the size of CalPERS, it has not done significantly better than its benchmarks. Nor is this true on a sustained basis for other, similar funds. How could such funds outperform, since these funds dominate the markets from which the benchmarks are compiled?
Large funds do not do better than their benchmarks, but most retail investors do worse – much worse. Charges are part of the explanation. Mutual fund investors pay not only management fees but also trading costs within the funds they hold, and may suffer a spread of prices when they buy and sell, which they do too often. But the principal explanation is bad timing. Retail investors buy high and sell low. They are late into fashionable sectors and late out of unfashionable ones. There is probably no worse investment strategy than following the conventional wisdom with a time lag, but that is precisely what many small investors do – often with the encouragement of their advisers.
Realism can be depressing. You may want to close the book at this point, and either continue, alone, the search for the twenty bagger, or decide that you might as well leave your money in the bank. Both decisions would be mistakes. There is no reason why you cannot do as CalPERS and other large investment institutions do, and consequently do more or less as well as these institutions. You can follow their asset allocation. And you can buy the benchmarks which they follow. I will explain in more detail in later chapters how to do this using low-cost funds which track these benchmark indices. In this way you can more or less guarantee yourself a return, which may be good or bad, but which will not be very different from that obtained by the world's largest and most sophisticated investors.
**The effects of compounding**
Albert Einstein is reported as saying that compound interest is the most powerful force in the universe. There is no evidence that Einstein said this, and it is unlikely that he did, but the sentiment has some truth. The reinvestment of returns over a long period has dramatic consequences.
Over forty years, €100 invested at an 8 per cent rate of return becomes €2,172. However, if fees, charges and poor investment strategy reduce that return to 3 per cent, an underperformance typical of retail investors, the sum would be only €326. Your nest egg would be 85 per cent less. The power of compound interest means that halving the return considerably more than halves the value of your accumulated savings.
Let's be optimistic. An average return of 8 per cent a year before tax and before inflation is a demanding, but not impossible, target for the intelligent investor. If inflation and tax reduce that return by 3 per cent, then a corresponding real after-tax return is around 5 per cent. An 8 per cent return will turn €100 into €215 in ten years, and even a 5 per cent real return will add over 60 per cent to the purchasing power of your savings in a decade. While 8 per cent may seem a disappointing rate of return to locker-room braggarts, viewed as a return on productive investment, rather than on financial investment, 8 per cent appears high. While businesses often target higher returns on investment than this, they rarely achieve them.2
Look at it in another way. The return on secure assets such as cash and government bonds is today between 0 and 3 per cent depending on the length of time for which you are willing to invest your cash. If you succeed in earning an 8 per cent rate of return, that means you are receiving an additional 5 to 8 per cent per annum – the risk premium – for adopting a more adventurous strategy. This is a handsome reward. Nevertheless, it is not unrealistic.
This generous level of return for investing in more volatile assets has proved achievable over very long periods – a century or more. The question of why the risk premium is so high has puzzled financial economists, who call it the 'equity premium paradox'. I'll come back to the equity premium paradox in Chapters 7 and . The size of the premium is a reason why most investment institutions allocate very little to cash. You should do the same. (There are stronger reasons for making allocations to indexed bonds.)
If you set yourself a target rate of return of 8 per cent, you are already differentiating yourself from many conventional and most professional investors. I have described the significance benchmarks have for professional investors. Benchmarks don't have the same relevance to you. When you become your own investment manager, the incentives and consequences are different. The money you manage is your own, and while you can blame 'the market' for disappointing returns, the exercise is pointless. Relative performance may pay the salary and bonus of a fund manager, but relative performance doesn't pay the bills of the investor. What pays bills, in the long run, is total return, in absolute terms, after tax and expenses. In the next chapter I'll review the ways in which shares, bonds and property can contribute to that objective.
**Footnotes**
1The distinctions between shares, equities and secured and unsecured loans have become much less clear-cut than formerly as a result of financial innovation. A non-recourse loan gives the lender rights over the asset on which the borrowing is secured, but not over the other assets of the borrower. Junior, mezzanine or subordinated debt will be paid only after other lenders have been repaid. Such debt may, in recompense, enjoy some share of the profits.
2The return on equity they say they earn is often much higher. After reading Chapter 6, you should understand why.
3
**Investment Options**
**Sense about shares**
You might buy a share in an initial public offering (IPO), when a company raises capital by selling stock to the public. It is much more likely that you will buy on a market that trades shares in established businesses. Exxon Mobil, the oil giant, and Berkshire Hathaway, the investment vehicle of the legendary Warren Buffett, are the largest companies listed on the 'big board' of the New York Stock Exchange (NYSE), the venerable institution which was founded in 1789 under the buttonwood tree at 68 Wall Street. But Alphabet (the holding company for Google) and Apple, today the most valuable companies in the world, are traded on NASDAQ, the upstart rival to the NYSE. NASDAQ was established in 1971 as an electronic alternative to the trading floor.
The stock price of these large companies varies every minute of the trading day. You can get this price from a wide variety of internet sources (possibly the price as it was fifteen or twenty minutes ago). The real-time price is considered valuable information and is typically available only to people who pay for the information or are clients of stockbrokers. As I write this in June 2016, one Berkshire Hathaway share costs $211,940 to buy and can be sold for $211,820. Berkshire Hathaway is unique in having a share price so high that one share in the company is equivalent in value to a modest house. Apple stock is at $99, while Exxon Mobil stands at $89. The difference of $120 between the buying and selling price of Berkshire stock – the spread – is only 0.05 per cent of its value, and for Apple and Exxon the spread will only be a few cents, because the volume of trade in all these companies is large. The difference between buying and selling price for a small, lightly traded stock will be many times higher. The spread, and the broker's commission, is the price you pay every time you trade.
In 2015 Exxon made a profit after tax of $3.85 per share, from which it paid dividends of $2.88. The directors of the company decide how much will be paid to shareholders in dividends and how much will be retained for investment. The company doesn't have to pay the shareholders anything. What the directors decide to pay can be, and generally is, varied in line with the performance of the business. But if the company does declare a dividend, all shareholders have an equal right to receive it. If the company is wound up, the shareholders are entitled to whatever is left – normally very little – when all the liabilities of the company have been met. Most companies are worth more than the realisable value of their assets if they remain a going concern, and the usual reason a company is wound up is that the business is failing. But the possibility that the value of the business could be distributed to the shareholders influences the value of the stock, even if such a distribution rarely happens in practice.
These entitlements are universal, but the status of shareholders depends on the company law of the country in which the firm is incorporated. People often say that shareholders 'own' the company. They don't, as you will find out if you turn up at Apple's spectacular new headquarters campus at Cupertino or Berkshire's small office suite in Omaha, Nebraska, to assert your 'ownership'. What shareholders own is their shares, and ownership of shares confers a variety of rights. The value of a share is the value of these rights.
A key right conferred by a share is the right to vote. But what you are entitled to vote on depends on the constitution of the company and local law and regulation. At annual general meetings of companies, shareholders may be able to elect directors and can pass, and sometimes propose, resolutions to determine the policy of the company. In practice, these meetings are typically agreeable but insignificant occasions attended by professional advisers and a few, mostly retired, small shareholders.1 All nominations and resolutions are put forward by incumbent management and approved by overwhelming majorities. The process resembles elections in totalitarian states. In recent years, however, there have been the beginnings of shareholder dissent, mostly over excessive executive remuneration.
Most people find it surprising that shareholder rights are more extensive in the UK than in the United States, where there is a continuing battle over 'proxy access' – the ability of shareholders to put their own resolutions, including nominations of directors, to the shareholder meeting. One reason is that in the US there is competition between individual states to attract company registrations. The competition has largely been won by Delaware; the state has more incorporations than people, and 1209 North Orange Street, Wilmington, houses the registered office of around 300,000 corporations. This is not because Wilmington, Delaware, is a hub of industrial activity but because Delaware law is particularly favourable to the interests of incumbent management (which in practice makes the decision about where the company will be registered).
The 2015 financial statement of Exxon Mobil values the corporation's assets at $335 billion, of which $250 billion is the value attributed to the company's infrastructure – its oil wells, refineries and tankers. The balance is made up of stocks of oil, cash and other short-term assets, and investments in other companies – mainly joint ventures with smaller or local partners. At $335 billion, the book value of the company (the asset value recorded in the accounts) is similar to the $370 billion market capitalisation of the company – the value reached by multiplying the share price of $89 by the 4.2 billion shares in issue.
However, Exxon Mobil is unusual among modern companies, both in having very substantial physical assets and in having a market capitalisation in line with the book value of these assets. The accounts of Apple show a very different picture. At end-2015 the total value of Apple's physical assets – property, plant etc. – was only $22 billion, less than one-tenth of the figure for Exxon Mobil, and less than Apple's own long-term debt. Apple had an incredible $200 billion of cash on deposit, the result of the company's exceptional profitability – the $53 billion recorded in 2015 is the highest annual return any company has recorded. With 5.8 billion shares in issue, Apple's earnings per share were $9.28, out of which it paid a dividend of $1.98. (The company financed that dividend through long-term debt because its cash is largely outside the United States and there would be a tax charge on profits remitted stateside. However, its shareholders are predominantly in the United States.) At $99 per share in June 2016, the market capitalisation of Apple – the total value of its shares – is $550 billion.
Two key metrics used in share valuation are the price:earnings ratio and the dividend yield. For Exxon Mobil, with a stock price of $89, earnings per share of $3.85 and a dividend of $2.88, the price:earnings ratio is 23 ($89 ÷ $3.85) and the yield 3.2 per cent ($2.88 ÷ $89). The equivalent figures for Apple are a price:earnings ratio of 11 and a dividend yield of 2.0 per cent.
Lower price:earnings ratios and higher dividend yields are generally treated as indication of slower growth prospects for the company (and hence slower growth of both earnings and dividends). And a year earlier, the relative price:earnings ratios of these two companies would have been more or less exactly reversed. But the collapse of oil prices in 2015 halved Exxon Mobil's profits, although its share price increased; Apple increased its already record-breaking profits by 30 per cent, yet its share price fell.
The fortunes of companies go up and down, and so do their share prices, but few companies have experienced twists and turns of fortune as extreme as those of Apple. At the time of Steve Jobs's return to the company in 1997, the company's stock was worth little more than $1. This had increased to $10 by the time the New Economy bubble burst in March 2000. But even after the iPod was successfully launched in 2002, the stock price had declined to $2 by 2004.
The stock had recovered to $20 by the time the iPhone was launched in 2007: it lost almost half that value in the global financial crisis, so that in 2009 you could have picked up Apple stock for little more than $10. But by the time of Jobs's death in 2011 the shares were worth $60. They continued to rise to over $100 in 2012. In the year that followed they halved in value, before recovering to their all-time high of $132. But then the stock price fell again to below $100, and Alphabet overtook Apple as the world's most valuable company (and Warren Buffett had taken the opportunity to acquire a stake).
Some of these wild gyrations were in response to genuine changes in the trading prospects for the company. Others – the precipitate rise in 1997–2000 and fall to 2004, and the collapse of the share price in 2008–9, reflect overall market conditions that had little or no relevance to Apple. Some fluctuations – the sharp decline in 2013 and subsequent recovery – do not appear to have any rational explanation at all. During the week I am writing these paragraphs, the difference between the highest and lowest price for Apple shares was $10 – equivalent to a difference of more than $50 billion in the overall value of the company.
People in markets offer 'explanations' for these constant price fluctuations. Sometimes these accounts relate to broad economic events – the release of new data, or a Delphic utterance from the chairman of the Federal Reserve Board. Others may be company-specific – such as rumours about new products. The idea that these events can add or subtract $50 billion to or from the value of Apple's assets and earnings makes no sense. These share price fluctuations are what natural scientists call 'noise': random interference in physical processes. Developing this analogy, financial economists describe people who trade on hunch and gossip, with little skill or knowledge of what they are doing, as 'noise traders'. Most tip sheets, and many of the books on how to trade the market, are written for noise traders.
While day-to-day share price fluctuations dominate short-term returns on investment, in the long run dividends represent an important part of total return. For example, if you had held Exxon Mobil stock for the last ten years, and reinvested the dividends as they fell due, your initial $58 investment would now be worth $114 rather than the $89 current stock price. If you had bought Apple shares for $1 or $2, of course, your experience would have been very different, and virtually all your return would have been the result of capital gain.
But a lot more people owned Exxon Mobil at $58 than Apple at $2, and you will be fortunate indeed if you are the person who bit into Apple when it appeared poisonous. If you have been a shareholder in Apple, it is difficult to have lost money in the long run – as a committed holder over a five-year period. But not impossible – if you had bought in 1999 and sold in 2004, you would have seen most of your investment wiped out. If you had followed prevailing market opinion, you might have done just that.
The value of shares in Apple, Exxon Mobil and other companies depends partly on the fundamentals of the company – its assets, earnings, cash flow and dividends, its brand, competitive position and management strengths – and partly on market sentiment towards it. Returns from shares are the result of the interacting influences of the company's fundamentals and 'the mind of the market'. These issues will be the subject of the next three chapters.
**Basics of bonds**
In 1888 Britain ruled the waves, and Queen Victoria was on the throne. The British government consolidated many of its debts into a security – known thereafter as Consols – which would pay 2½ per cent interest each year, for ever, on every £100 subscribed. Bonds have always been part of the conventional investor's portfolio. The signature bond today is the bund, securities with maturities of ten and thirty years issued by the Federal Republic of Germany, perhaps the most secure investment on the planet. A €100 bund from the current thirty-year issue will pay you €2.50 every year till 2046.
This return is guaranteed only if you hold the bond until 2046. The German government may not and will not pay your capital back a day earlier than the maturity date of 15 August 2046. If you need to realise your investment before then, you will have to sell it to someone else, and the price you receive will depend on the level of interest rates at the time. This is interest rate risk, and accepting it is part of any investment in bonds; the later the repayment date of the bond, the greater the interest rate risk. When interest rates in Britain reached almost 20 per cent in the 1980s, the price of Queen Victoria's 2½ per cent Consols fell to only £15.
The 2.5 per cent bund maturing in 2046 is today on sale not for its repayment value of €100 but for €147. That means that the yield is not 2.5 per cent but 1.7 per cent (€2.50 per €147 invested). And since the German government will still only return €100 in 2046, you will have to factor into your calculations a loss of €47 over the thirty-year life of the bond. This loss, which will accumulate year by year, reduces the effective rate of return on your investment to only 0.7 per cent. This effective rate of return is known as the gross redemption yield (GRY).
Interest rates differ depending on the length of maturity of the bond. The normal shape of the yield curve is upwards, and that is its shape today. A five-year German government bond is priced at €102 and pays an interest rate of 0.25 per cent. The effective return on this investment is nothing at all.
You can do a bit better in other countries and other currencies. A thirty-year Treasury security from the US government currently (June 2016) yields 2.5 per cent (in dollars) and a thirty-year gilt from the UK yields 2.1 per cent (in sterling). An equivalent Japanese bond, however, offers only 0.3 per cent (in yen). Over five years a bond will return about 0.75 per cent per annum in the UK and 1.2 per cent in the US, but in Japan, as in Germany, you will earn nothing at all. In 2015 the Swiss federal government issued a ten-year bond with a negative yield. You do not have to pay the Swiss government on a year-by-year basis. In fact they will pay you some interest: they will just return rather less than you paid for the bond. If you wonder why anyone would invest at a negative interest rate, the answer is that while you can keep 1,000 Swiss Francs under the mattress, you cannot sleep safely or comfortably with 10 million Swiss Francs under the mattress. Entrusting it to the Swiss government is simply the most economical way of safeguarding your cash.
The differences between the interest rates available on bonds of different governments reflect expectations about what will happen to currencies rather than perceptions of the creditworthiness of the borrower. Bonds issued by the governments of major, well-managed economies, such as Britain, Germany, Japan, Switzerland and the United States, carry virtually no credit risk – risk that the states concerned will fail to repay. Government bonds usually offer the lowest interest rates available in the currencies in which they are denominated. (Governments other than the US and outside the eurozone may often borrow in dollars or euros.)
But Puerto Rico is currently failing to service bonds, Greece has effectively defaulted, and Latin American states have routinely done so. Significantly, when the US Congress was gridlocked by grandstanding Republicans attempting to derail the federal budget and debt ceiling, the rates required from the highest-quality private-sector borrowers – firms such as Exxon Mobil – fell below those available to the US Treasury. Bonds from countries with unstable governments, or from companies less asset-rich and soundly financed than Exxon Mobil, naturally carry higher interest rates to reflect their credit risk. Ford and General Motors were once among the most respected companies in the world. But by 2006 fears over these companies' obligations and their weak current trading meant their bonds were rated as junk. In 2009 GM filed for bankruptcy and its bondholders were forced to accept a writedown of the value of their holdings.
Inflation after 1888 led not only to a reduction in the price of Consols due to the associated rise in interest rates, but also to further loss of value through rising prices. While £100 in 1888 might have bought a modest house for one of Queen Victoria's subjects, in 1974 the £15 value of the investments would not have been sufficient for a replacement of the front door. Conventional bonds provide certainty of cash return. Only indexed bonds – first introduced in the 1980s – provide certainty of real return.
However, in Britain and Germany this certainty is obtained at the price of a accepting a real return that is actually negative. A five-year indexed bond in either country will erode the real value of your investment by almost 1 per cent per year. A thirty-year security also offers negative yields. United States Treasury Inflation Protected Securities (TIPS) look slightly better. While they will still lag inflation over five years, they provide a 0.8 per cent a year real return over thirty years.
**Realities of real estate**
If you own real estate, you own both a building and the land on which it stands. If the property is in a large city, the land may account for most of its value: my own central London house has a market value more than five times its value for insurance purposes, which is an estimate of what it would cost today to build a similar property. An office block in central London or Hong Kong may now have a life of only twenty years. Because it is the location that is valuable, rather than the building, it may often be profitable to replace the structure with one that is more attractive to the changing needs of tenants. Some London office blocks are being demolished to build flats because the land is today more valuable in residential than in commercial use.
But there are also properties which, did they not already exist, would not be built, or would not be built in a similar style. Houses in depressed regions sell for less than their replacement cost. You can buy a French château, wholly impractical for modern living, for less than the price of a modest apartment in the centre of Paris. The historic building is a liability, not an asset. An industrial warehouse on the fringes of a town is, in contrast, little different from any other item of plant and machinery. It will depreciate in use over twenty or thirty years until the structure is replaced, and the land on which it stands has little value.
If you draw up a personal balance sheet in the way a company would, you are very likely to find that your house is your most valuable asset by far. Yet your house is not quite like your other investments. You bought your house in order to occupy it, and for the rest of your life you have to live somewhere. That doesn't, however, mean you are indifferent to its value. When house prices rise substantially – as they did in many countries in the years between the turn of the century and the inception of the global financial crisis – many people discover that they are earning more each day from the appreciation of the value of their property than by working. This gives rise to a lot of confused discussion.
As a homeowner, you are both an investor in housing and a consumer of housing services. When you are young, your investment in housing is probably modest, and your prospective consumption of housing services extensive. As you age, your investment in housing is greater and the date at which the graveyard caters inexpensively for your accommodation needs draws closer. Young people want housing to be cheap; older owner-occupiers can benefit from its being expensive. People in middle age may intend to move to a smaller property when their children leave the nest. Retirees may use equity release schemes to realise some of the value in their houses.
Much of the value of an owner-occupied house will accrue to the heirs of the purchasers. Only the owners themselves can decide how they view that legacy. For all these reasons your house, and the associated mortgage, cannot be treated in the same way as the assets you own for purely investment motives. And housing and house prices are inevitably a politically charged issue because they have considerable influence on the distribution and transfer of wealth between generations.
Most people first encounter the concept of leverage when they buy a house. Suppose you borrow €150,000 of the €200,000 you need to purchase a property. If the value of the property rises to €300,000 – a 50 per cent increase – then the difference between the property value and the loan is now €150,000. Your initial investment of €50,000, often called your equity in the property, has trebled in value.
If leverage sounds too good to be true, it is, because it operates in both directions. In the years before the global financial crisis, many people thought that 'flipping properties' was an easy path to riches. Books elaborating this proposition reached the bestseller list, seminars were packed with expectant – and greedy – participants. If you borrowed 90 per cent of the cost of a new €200,000 apartment, and were able to sell it for only 10 per cent more than you had paid for it, you could double the value of a €20,000 investment in only a few months.
But if prices instead fell by 10 per cent, your savings would be wiped out. And when the housing boom stalled and the credit crunch developed, many people found that the properties they had hoped to flip quickly at a profit could not be sold at all. Folk wisdom has it that house prices always go up, at least in the long run. The truth is that experience is varied. In parts of Germany, Switzerland and Japan property values are lower today than they were twenty-five years ago. In most of the rest of the world prices have risen over time, but with substantial short-term volatility. Scandinavian house prices plunged in the early 1990s; in this century Spain, Ireland and some parts of the United States, such as Florida and Las Vegas, have experienced spectacular booms and busts. For owner-occupiers, the timing of an initial purchase and any significant subsequent 'trading up' has often been as important as career choice in determining personal and household wealth. Like share prices, property values go up and down. The history of the property market is full of stories of well-known names who have discovered to their considerable cost that leverage can work both ways. Donald Trump has been to the bankruptcy court four times: three times in failed casino ventures and once after his highly leveraged acquisition of New York's grandest old hotel, the Plaza.
'Safe as houses' is true only if you plan to live in one. The positive experience of owner-occupiers has encouraged many individuals to buy houses and flats to let. Although an investment property of reasonable quality is likely to cost €100,000 or more, gearing means that such a property can be bought with a deposit of €30,000 to €40,000. While residential property is less volatile than most other asset classes, leverage multiplies the risk. Buy-to-let is a low-risk strategy only for someone rich enough to make such investment a modest part of a larger portfolio.
While individuals have recently been active in buy-to-let investment properties, there has been little new institutional investment in residential property. Insurance companies and pension funds prefer commercial property – shops, offices and industrial premises – which only rich individuals can hold directly. Small investors can access these properties through funds, and they generally should.
Ownership of property was once bound up with power and status – landed estates were associated with aristocratic titles. In the early days of industrialisation the capitalist owned the premises – the mill, the shop, the factory – where the workers laboured. Today great estates are liabilities, not assets, and businesses rarely own the premises they occupy. Financial markets have become more sophisticated, and premises have become less specialised. Once, a textile business or a steel company would build a mill or a plant to its own particular requirement. Modern companies will often rent space on an industrial estate, in a unit adaptable to many alternative uses. Much retail trade has moved to warehouses out of town. The specifics of the terms of leases, the conventions of the property market and the legal framework are vital to property investors, and vary substantially from country to country. For these reasons – and because of tax complications – it is much more difficult to buy overseas property than overseas shares. But funds allow individuals with small sums to invest in foreign property.
Owning and letting a residential investment property is an investment option for anyone with a portfolio of reasonable size. (I shall discuss what this means in Chapters 10 and 11.) While it was once true that institutions owned large amounts of residential property, this sector declined as a result of a history of rent control and the growth of owner-occupation among middle- class households. Rental property today is mostly owned by social landlords – national or local governments or not-for-profit agencies – or by private individuals. Institutional investors today generally prefer commercial property – shops, offices and industrial premises. Since these normally represent larger units, only investors with substantial assets can hold them directly, but everyone can access them through investment funds.
**Facts about funds**
There are two basic kinds of investment fund – open-end funds and closed-end funds. Open-ended funds are usually called mutual funds in the United States, unit trusts or OEICs in Britain, SICAVs in much of continental Europe. Each investor owns a fraction of a portfolio of the assets of an open-ended fund. The portfolio is regularly revalued, and units of the fund can be bought or sold at a price based on that regular revaluation. Open-ended funds grow or shrink as holders add to or redeem the units they have. Very occasionally an open-ended fund may close to either new money or redemption. Most of the investment funds marketed to small savers are open-ended funds.
A closed-end fund is usually a company with shares quoted on a stock exchange: in this respect, like Apple or Exxon Mobil. But instead of operating a business, the closed-end fund holds an investment portfolio. Berkshire Hathaway is a closed-end investment fund, although a rather unusual one, because of the size of its holdings in portfolio companies (100 per cent of many of them). There are some investment companies with focused portfolios similar to Berkshire Hathaway in Britain and the United States, rather more in continental Europe.
But the more common type of closed-end fund is the investment trust. Investment trusts holding portfolios of company securities were widely distributed in the early years of the twentieth century, but they lost popularity in the United States after they were at the centre of many investment scams in the years before the Wall Street crash. UK funds were more conservative, and many are still in operation, but restrictions on listed companies marketing their own shares, or paying commissions, led most sales efforts to be devoted to the marketing of open-ended funds. A new generation of US closed-end funds came into being in the 1960s, when legislation permitted real estate investment trusts (REITs), and the innovation has been copied elsewhere.
While the size of open-ended funds expands or contracts with purchases and redemption, the size of the closed-end fund is determined by the amount initially subscribed, and the subsequent investment performance of the portfolio. You can dispose of your share of the portfolio in a closed-end fund only by selling it to someone else. Traditionally, this meant a closed-end fund could be more adventurous and invest in assets that were not easily realisable, which made it an appropriate vehicle for real estate and private equity funds. A closed-end fund can also use leverage, which is generally not permitted to an open-ended fund.
The value of your share in a closed-end fund can be more or less than the value of your share of the underlying assets in the portfolio. The possibility of variability in this discount or premium to asset value compounds the investment risk. If property is currently a fashionable investment, then real estate investment trust shares may sell at a premium to their asset value – the value of the shares is greater than the value of the properties the company owns. If emerging market stocks are unfashionable, it may be possible to buy shares at a large discount in investment companies specialising in emerging market stocks. You can sometimes buy out-of-favour funds at substantial discounts. Closed-end funds involve extra risks but also extra opportunities.
Most funds, open- or closed-end, are actively managed: the fund manager selects a portfolio of stocks and buys and sells in the hope – not often fulfilled, as I shall describe in Chapters 8 and 10 – of outperforming the market. Passive funds simply replicate the performance of an index. A computer buys and sells shares to minimise the tracking error – the difference between the index and the value of the assets the fund holds. The exchange-traded fund (ETF) is a more recent innovation. The mechanics of ETFs are more complex – a hybrid of open- and closed-end fund – but the objective is that the investor can buy or sell a share that will reproduce almost exactly the performance of an index – the FT all-share index, an index of Brazilian stocks or a basket of commodities.
In the first UK edition of this book I strongly recommended exchange-traded funds as a tool for intelligent investors. But, as so often, the greed of financial innovators messed things up. Complex ETFs, 'synthetic' funds based on derivatives and supported by collateral drawn from the dustbins of the sponsoring banks, proliferated. While many transparent and honest ETFs remain, it is more difficult for the intelligent investor confidently to separate the wheat from the chaff. I'll make some more specific suggestions in later chapters, since ETFs continue to be a useful – and globally available – vehicle for investment.
In this chapter I've described the three main asset categories available and the choice of direct or indirect investment. In the next three I will review the relationship between prices and values – the fundamental issues of intelligent investment. In Chapters 10 to 12 I'll explain how to build these different assets into an investment portfolio.
**Footnote**
1The Berkshire Hathaway AGM, at which Buffett holds court in front of 35,000 admiring shareholders in the Omaha Arena, is a conspicuous exception.
4
**Efficient Markets**
**Efficient markets and asset values**
Few of the many jokes about economists are funny. One tells of the finance professor walking down the street with his wife. 'There's a $10 bill on the pavement,' she says. 'Don't be silly,' he replies, 'if there was, someone would already have picked it up.'
The joke refers to the efficient market hypothesis (EMH), but the joke is on the teller. There are few $10 bills on the street, for the reason the professor elucidates. People rarely drop $10 bills, and when they do, the notes are quickly picked up. Most pieces of litter that look like $10 bills are, indeed, litter. The professor's theory of why there are no $10 bills on the pavement is illuminating. It will save considerable time for anyone who plans to make a living searching for bills on the streets. The theory is not, however, true. There are occasional discarded $10 bills. There are also people who make profits by picking them up. Otherwise, $10 bills in the street would be easier to find.
In a securities market the $10 bill is the difference between the price and the value of an asset. Since the efficient market hypothesis is illuminating but not true, such differences occur, but rarely, and they are hard to identify with confidence. So what determines the relationship between price and value?
An economist, another joke goes, knows the price of everything and the value of nothing. The relationship between price and value is a subject with a long history – particularly among accountants.
There are two basic approaches:
* the value of an asset is what someone is willing to pay for it;
* the value of an asset is the cash it will generate over its life.
I'll call these the 'mark-to-market' principle and the 'fundamental-value' principle. Almost all valuation methods are based on one or both of these principles. If you are appraising a property, you might call in a valuer. The valuer will have experience of transactions in similar properties in the area and should have a good idea of what the building will fetch. The mark-to-market principle is easy to apply in this case.
The fundamental-value principle looks instead at the cash generated over the life of the asset. Begin with the flow of quarterly rents from a property, taking account of possible increases at future rent reviews, then deduct any expenses you might need to incur to preserve the value of the property.
Perhaps the building has a finite life, like an industrial warehouse or modern office block; you will need to consider what the property will be worth when the structure needs to be replaced. Other buildings can, with some attention, maintain a satisfactory flow of rental income for ever.
The fundamental value of a company is calculated in a similar way. Look at either the path of dividends or the stream of earnings. If you focus on dividends, observe the current dividend level and conjecture how rapidly the company will increase that dividend in future years. If you focus on earnings, you need to estimate their probable growth, bearing in mind the growth rate of the economy, the industry and the specific prospects of the business.
The mark-to-market principle and fundamental-value principle are certainly different, and seem incompatible. They involve different styles of thinking and lead to different approaches to investment. The mark-to-market principle is easier, and less speculative. But it is a common mistake to emphasise what you can measure at the expense of more important things that you can't. It is generally better to be approximately right than precisely wrong.
Which of these two principles, price or value, should be applied? Market prices can't be completely independent of fundamental value because opinions about fundamental value influence market prices. This is the metaphor of markets as 'voting machine'. Everyone who trades registers an opinion. The results of this survey of opinion are reported in the financial papers every day. The metaphor contains a possible reconciliation of the mark-to-market and fundamental-value principles: the market price is the result of a plebiscite on fundamental value.
The market is democratic in the sense that everyone is allowed to vote, but it is not a fair election. The weight given to your opinion depends on the amount of money you can put behind your vote. The votes of a large fund management group such as BlackRock or an investment bank such as Goldman Sachs count more than the votes of you and me. Perhaps this is as it should be. If you are really confident of your opinion, you can make that opinion count by putting a lot of money behind it. I shall describe how George Soros pulled off a spectacular coup by doing exactly that.
The concept of the market as voting machine provides a justification for my claim that every investor can access, without charge, all the expertise that the financial services industry can bring to bear. That idea leads to the efficient market hypothesis. If the market price were an average of informed estimates of fundamental value, markets would be efficient not only in the narrow technical sense that all available information is in the price but also in a broader sense – market prices provide signals that guide the efficient allocation of investment funds.
In this chapter I'll focus on the narrower interpretation of market efficiency, the one relevant to investors. But there are three sub-interpretations, because there are three variants of the efficient market hypothesis involving progressively more demanding assumptions:
* the weak form – past movements of prices convey no information about future price movements;
* the semi-strong form – all public information about securities is already reflected in their price; and
* the strong form – everything that could be known about securities is already reflected in their price.
Louis Bachelier, the little-honoured Frenchman whose PhD thesis on 'The Theory of Speculation' proved to be the foundation of finance theory, discovered that the data he compiled about stock price movements resembled the observations of natural processes, such as the movement of small particles suspended in a fluid. Physicists call this behaviour Brownian motion. Financial economists, more engagingly, employ the term 'random walk'. A random walk is directionless. Every step is independent of every previous step and just as likely to be in one direction as in another. All three versions of the efficient market hypothesis claim that securities prices will follow a random walk. The random walk is a mathematical concept inseparable from the EMH.
Most of the evidence observed by early researchers was favourable to the random walk theory. The earliest evaluations were undertaken by statisticians – until the 1950s, Wall Street and the City of London were innumerate and there was no serious finance theory in business schools. At Stanford University, Holbrook Working showed that long series of commodity prices followed a random walk. The British statistician Maurice Kendall analysed long series of stock prices and more than a hundred years of cotton price data. He concluded that 'The Stock Exchange, it would appear, has a memory lasting less than a week.' While none of the three versions of the efficient market hypothesis is true, each offers its own shafts of illumination. Markets may have no memory, but commentators on markets do, and the random walk theory suggests that most of their observations and analyses are of little use. 'Shares are 20 per cent lower than a year ago – this is a buying opportunity.' 'The market needs to pause for breath after its recent rise.' 'The dollar is experiencing a technical correction, which will continue.'
You regularly hear statements like these in popular discussion. The weak form of the efficient market hypothesis claims that no such statement can ever be justified because prices move without knowledge of their history. The probability that they will rise next year, or next month, or today, is unaffected by whether prices rose yesterday afternoon, or last week, or in the first quarter of 2011.
The weak EMH is a blow for people who rely on projections of historical trends, or identification of recurrent patterns, to make their investment decisions – most of all for chartists, whom I'll describe in the next chapter. But the weak version does nothing to discourage people from looking at fundamental values. Rather the opposite – the weak EMH encourages you to believe that you might make money at the expense of noise traders by paying close attention to fundamentals.
There is a considerable practical difference between weak and strong EMH. Strong EMH – which postulates that everything that could be known about securities is already in their price – essentially rules out any possibility of investment skill. Legendary investors such as Buffett have simply been lucky. No research or analysis can be useful, because its results will be reflected in the market. Put like this, the theory seems absurd.
Indeed, the strong version of EMH contains an inherent contradiction. If it were the case that research and analysis could never be profitable, because the result of that research and analysis would always be 'in the price', why would anyone undertake such research and analysis? If prices were always efficient, what could be the process by which they become efficient?
The semi-strong version of the efficient market hypothesis is less extreme, claiming only that information is 'in the price' if publicly available. Thus the semi-strong version does not exclude the possibility that insightful new analysis, unpublished or not yet widely circulated, might be valuable to those who possess it. It certainly permits possibilities of profit from trading on private information. The investment banker who knows of a planned takeover and the corporate executive who knows that the coming results will be worse than is generally expected can both deal advantageously on the basis of this inside information.
Such insider trading was once the daily practice of brokers and fund managers, and put small investors at a substantial disadvantage. But most countries have made it illegal for people who have inside knowledge to deal. These laws do not cover all forms of trading on private information, nor can they ever be perfectly effective, but they have probably reduced the incidence of these practices.
Outside the United States, few people are convicted of insider trading, and not many there. This is because evidence is hard to obtain, not because the activity is rare. Cases that have come to court show that criminal insiders often made little profit from their unlawful activities because their actions moved market prices against them. Martha Stewart received a five-month prison sentence and suffered incalculable reputational damage for a $45,000 profit. Information held by insiders does not have to be announced to percolate into the market-place.
That observation has important implications for intelligent investors. Be very wary of sustained individual price movements that seem to have no explanation. A falling share price may be the result of inside knowledge; a rise that cannot be accounted for may herald a possible takeover. Perhaps information the investor does not know, and may never know, is already reflected in the market. Be suspicious of advisers whose predictions are based on extrapolations from past price movements, or who claim to see patterns in data.
**Illuminating but not true**
The efficient market hypothesis has a sufficiently strong hold on the investment world – particularly among finance academics – for deviations still to be described as 'anomalies'. Over the last two decades evidence of anomalies has accumulated steadily. One collection of essays is provocatively titled _A Non-Random Walk down Wall Street_ (Lo and MacKinlay, 1999), in deliberate contrast to Burton Malkiel's popular introduction to efficient market theory in _A Random Walk down Wall Street_ (Malkiel, 1973). Robert Shiller, who achieved wide attention by publishing _Irrational Exuberance_ (Shiller, 2000) at the peak of the New Economy boom, has become the intellectual leader of the sceptics. In a bizarre conjunction, the 2013 Nobel Prize in economics was shared between Eugene Fama, for his work in developing the efficient market hypothesis, and Shiller, for his work in refuting it.
There are many investment maxims, mechanical rules that promise superior returns. 'Sell in May and go away.' Benefit from 'the small company effect' – consistently higher returns on smaller companies that may be the subject of less intensive research. Buy 'the dogs of the Dow' – the worst-performing large companies of the previous year. These ideas rarely work. There are several problems. It is always possible to find some such pattern by studying the past; that doesn't mean that the same pattern will persist in future. Even if the scheme appears profitable, profits may be inadequate to offset the costs of turning over a portfolio. And the very fact of publishing these seemingly attractive investment ideas may lead others to desist from the behaviour that gave rise to the so-called anomalies. The efficient market hypothesis strikes back.
So, is the empirical claim that prices follow a random walk correct? You may still be puzzled by the equivalence of the weak EMH and the random walk theory. Why does the assumption that all information is 'in the price' imply that prices follow a random walk? Bachelier's empirical observation of the random walk, and Kendall's supporting data, long preceded a comprehensive explanation of the relationship.
The definitive account of why price movements in an efficient market would appear random came in 1953 from the Nobel Prize-winning economist Paul Samuelson, whose introductory textbook is familiar to everyone who has taken a basic economics course. Return to that metaphor of markets as voting machine. The result of every election depends on the question you put to the voters. The question that is put to the voters here is not 'What do you think the Federal Reserve should do?' It is not even 'What do you think the Federal Reserve will do?' The question is 'How do you think the bond market and the currency market will react to what the Federal Reserve will do?'
Suppose people think that the Federal Reserve Board will raise American interest rates or that Apple will announce good results. These expectations lead these people to sell American bonds and buy dollars, or to buy Apple shares. Views about the future are therefore built into the level of prices we see in the market today. If the Federal Reserve Board is expected to act, bond yields and currencies will already reflect this, and will move only if these expectations change. If Apple's business is improving, the market price will already acknowledge the improvement. The price will move only if people think that outcomes will be better still, or that these outcomes will not, after all, be as good as previously expected. Traders are not voting on what Apple's results will be, or whether Apple is a good business; they are voting on what they think will happen to Apple's share price. They vote 'up' when they buy, and 'down' when they sell. John Maynard Keynes described this process through the analogy of a newspaper beauty contest. Competitors were asked to choose the most beautiful faces, and those who selected the most popular faces, according to the majority choices of all competitors, were the competition winners. As Keynes explained, what thoughtful entrants are trying to do is not to choose the most beautiful but to decide what they think others will find most beautiful, or even what they think people will think other people will find most beautiful – and so on.
Samuelson elaborated on this idea. Everyone in the market knows that prices will change in future as sentiment changes. If more people are expecting favourable changes in opinion than unfavourable, they will buy. If they think these yet unknown market events are more likely to be adverse, they will sell. So at any time prices will be bid up or down until the expected impact of future changes is equally likely to increase values as to reduce them. That is why, Samuelson claimed, properly anticipated prices fluctuate randomly.
Market reactions to events and to changes in the trading environment are not easy to predict and may seem disproportionate to the events that trigger them. Certainly they may be larger than can possibly be explained by reference to assessments of fundamental value. The worst day in the history of modern stock markets was 19 October 1987, when the Dow Jones Industrial Average, the most widely followed US stock index, dropped 22 per cent. The Brady Commission, established by the government to investigate, concluded that the fall was triggered by a higher than expected trade deficit and a rumour of higher taxes, magnified by the use of portfolio insurance – computerised trading systems that sold large quantities of stocks when markets fell.
Perhaps. In any event, none of these factors was directly relevant to stock markets outside the United States, which fell by almost as much. Within a few days, and for no obvious reasons, market sentiment switched from sanguine optimism to unreasonable pessimism. On days like 19 October 1987 market movements can be explained only by reference to market psychology; attempts to explain them by reference to changes in fundamental value are simply false rationalisations. Prices rise or fall in the short run on ephemeral changes in sentiment, with little if any basis in fundamental value.
In some markets, such as that for Old Master paintings, concepts of fundamental value are themselves elusive. These market prices are the result of the whims of a small number of very rich men. The principle that an asset is worth what someone will pay remains dominant in this market. But it is implausible that the prices of bonds, shares or properties can forever remain unrelated to fundamental value. Prices move around fundamental value – sometimes violently so, as on 19 October 1987 or in the New Economy bubble. Constantly changing, prices are influenced, but not determined, by a fundamental value that is itself unknown and constantly changing.
From time to time the path of market prices strays far from the path of fundamental value, but these divergences are ultimately corrected. Prices display 'short-term positive serial correlation' – if you examine prices over a day, a week or a month, then upward price movements are slightly more likely to be followed by further upward price movements. Downward price movements are also more likely to lead to further downward price movements. This feature of short-term price change is called 'momentum'.
There is also evidence of 'long-term negative serial correlation'. If you look at prices over much longer periods – three or five years – upward movements are more than averagely likely to be followed by downward movements, while periods of underperformance of this length are more than averagely likely to be followed by periods of outperformance. This feature of long-term price movements is called mean reversion. Like the efficient market hypothesis with which it is closely associated, the random walk model is illuminating but not true.
**Information asymmetry**
Market prices measure prevailing market sentiment, incorporating all the analysis, prejudice, skills and irrationality of market players. What is general knowledge will normally be 'in the price' and of little value to investors. Information such as 'General Electric is a well-managed business', 'Warren Buffett is an investment genius' or 'Households will always need to buy electricity', though true, does not provide a reason to buy related stocks. This information is in the market.
'General knowledge' includes such knowledge as we have of the future as well as knowledge of the present and past. The term 'consensus forecast' is used to describe common expectations about economic prospects. There usually isn't a very wide spread of opinions. The canard that economists always disagree is much exaggerated. (Although there are many other, well-founded, criticisms of economists, especially of zealots who embrace the EMH with fundamentalist fervour.) You don't have to pay for the consensus forecast, or spend time reading about it – it's already 'in the market'.
The glossy descriptions of economic prospects that financial advisers, investment managers and banks circulate to clients may be illuminating (I rarely find them so), but not as a guide to investment decisions. You don't have any valuable insight into the future – and nor do they. I've sometimes asked people in business and finance, who crave accurate economic forecasts, 'If I gave you the exact figure for national income three years from now, what would you do with the information?' Useful economic knowledge is usually specific rather than general.
This central role of information, and the dependence of trade on differential information, distinguishes securities markets from other markets. Trade in modern economies is mostly the result of specialisation. I grow apples, you grow pears, and each of us gains by swapping one for the other. We call on the plumber for his specialist skills. We buy from Volkswagen because the company is organised to produce cars, and we are not. America sells computers in return for Saudi oil. All these exchanges normally benefit both parties to the trade.
Sometimes trade in financial markets has a similar character. An importer buys foreign currency because he and his customers are based in Germany, and his supplier is based in China. The commercial transaction requires a foreign exchange transaction. But most trade in securities markets – even in foreign currency markets – is not like that. Both parties to the transaction have the same need – to make as much money as possible; they just have different ideas about how to do it. Most foreign exchange trading is speculative. I think the dollar will rise in value; you think it will fall. I think Apple shares are underpriced; you think they are overvalued. In each of these trades one of us will be proved right, the other wrong.
In any market in which there is wide and irresolvable uncertainty, in any market where participants have different information and beliefs, many trades will be the result of mistakes. And in financial markets uncertainty and differential information are endemic. A brilliant exposition of the problems this would create for market efficiency – in both a narrow and a wide sense – was provided in 1970 by George Akerlof, who would receive the Nobel Prize for his analysis. Akerlof used the metaphor of the market for lemons. Some cars are 'lemons', prone to faults. The seller knows whether the car is a lemon; the buyer does not. If the price of the used car reflects the average probability that the car is a lemon, that is a good price for owners of lemons and a bad price for everyone else. Sellers will be inclined to sell lemons, but not good cars. The proportion of cars on used car lots that are lemons will be much greater than the proportion of lemons in the population as a whole. As buyers realise this, the price they will pay for any second-hand car, lemon or not, will fall. The result of this will be to reduce still further the proportion of good cars on sale. The outcome is a market characterised by mutual mistrust, low prices, low-quality products and buyer dissatisfaction.
If you fail to understand the problem of imperfect information, you may fall victim to the 'winner's curse': you think you have won, only to discover subsequently that you wish you hadn't. This problem was first identified when the US government auctioned offshore oil blocks. The bidders in all cases were the same large oil companies. Both the government and the companies quickly realised that the successful bids were higher than had been expected, or than were justified. All the companies wanted the oil and had similar access to funds. Their preferences and resources were the same, but their assessments differed. All judgements were to some degree in error, but the successful bidder was – by the nature of the auction – the one who bid highest. The most likely reason one estimate was higher than the rest was that the winning company's geologists had made a mistake.
You might think that the inevitable errors made by market participants with imperfect information would tend to cancel out. They don't. Assets will always tend to be held by people who are too optimistic about their value rather than people who are too pessimistic. This gives rise to systematic, and destabilising, asset mispricing. The growing complexity of the products that are sold in financial markets means that this problem recurs with increasing frequency. In Chapter 9 I will look at some of the consequences – and how information asymmetry has been central to the recent recurrent financial crises.
**Strategic trading, moral hazard and market manipulation**
In an experiment, Professor Joe Stiglitz would invite a class to bid for the money contained in his wallet. An obvious strategy would be to guess how much might be in the wallet, and offer a bit less. But that would be a mistake. In fact, to participate at all would be a mistake. If a bidder offers less than the amount in the wallet, the professor can refuse to trade. If the bid is more than that sum, it will be accepted. If the seller is willing to sell, the buyer is foolish to buy. Why do I want to buy what they want to sell? In such a market, you never do. Anyone tempted to purchase a complex financial product should ponder the lesson of the wallet auction.
Problems which have these structures give rise to a problem known as 'moral hazard'. Wherever there are imperfections of information, people with superior information can use that knowledge to sell products for more than they are worth, or to rid themselves of risks which others underestimate. As the lemon example illustrates, the result is a decline in the average quality of products and risks. Moral hazard has always been evident in insurance and credit markets (there is no divorce insurance, because only unhappy couples would pay the premium) and is widespread in financial services. The confirmation in 2008 that governments would not allow large banks to fail compounded the problem of moral hazard revealed in the banks' aggressive assumption of risk.
Sellers of securities usually have the advantage over buyers. The people who run a company know more about it than do the people who plan to invest in it. The people who issue a bond know more about the issuer than the buyer does. The people who construct a complex derivative understand its characteristics better than the people who buy these securities. Some of the derivatives and structured products of modern finance can only be valued – if at all – by the banks that devised them. This not only leaves the purchasers vulnerable but leaves the banks themselves at the mercy of the few employees who understand the models.
Once oil companies understood the winner's curse, they adjusted their bids downwards. Markets characterised by differential information are reflexive, because prices reveal information about the behaviour and the expectations of other parties. These expectations may in turn affect assessments of fundamental values by other potential buyers or sellers. If Exxon Mobil think this is an attractive block, perhaps that company knows something we at Shell don't. This interdependence of expectations may give rise to strategic behaviour. If offers to buy or sell are perceived as revealing what other people think, traders may deal with a view to influencing the expectations of others, hence creating profit opportunities in future. This strategic thinking would be refined in the algorithms devised by high-frequency traders – the businesses which use pre-programmed computers to trade with each other and which now account for much of the trading volume on London and New York stock exchanges.
In 2006 the hedge fund manager John Paulson recognised that the marketing of subprime mortgages to indigent house buyers in the United States would end in tears. He persuaded the investment bank Goldman Sachs to put together securities whose value was based on the performance of particularly egregious subprime mortgages and to find buyers for these products. The transaction had no underlying purpose other than to enable Paulson and his investors to benefit from the inevitable defaults on the mortgages. When the market did indeed collapse in 2007, the outcome for Paulson was described, with some exaggeration, as 'the greatest trade ever' (Zuckerman, 2009). Although Paulson was not sufficiently communicative to be featured in Michael Lewis's _The Big Short_ , and hence escaped being played by Brad Pitt, his was almost certainly the largest of the bets against the US housing market which that film described. Goldman Sachs would pay a fine of $550 million, without admitting legal liability, for its part in the affair.
The most entertaining form of market manipulation is 'the corner', in which a bidder attempts to dominate the whole of the available supply and force buyers (especially) to pay whatever price the bidder chooses. In 2008–9 there was a battle for control of Volkswagen within the Porsche and Piëch families (descended respectively from the son and daughter of Ferdinand Porsche). Some hedge funds believed (correctly, as it turned out) that Wolfgang Porsche and his chief executive at the eponymous luxury car maker, Wendelin Wiedeking, would not succeed in taking over the much larger VW company. These hedge funds acted as 'short sellers', vendors of shares they did not own (I'll explain this more fully in Chapter 9), in the hope of making profits when the bid failed and VW shares fell. But they misunderstood the effect of the complex shareholding structure of Volkswagen; almost all the voting stock was controlled either by Porsche or by the state of Lower Saxony. When VW shares rose in value, the short sellers discovered that there were no shares available in the market. Obliged to buy to meet their obligations to return the stock they had sold, they found themselves having to bid for VW stock at any price. The stock has typically sold for between €100 and €200; as a result of the 'corner', some shares were sold at prices around €1,000.
**Investment strategies for imperfectly efficient markets**
The efficient market hypothesis, although illuminating, is not true. Momentum rules in the short run mean reversion in the long run. A day is short run; five years is long run. If there were a means of telling just when the short run becomes the long run, there would be a sure-fire route to making money: ride the wave, jump off before it breaks.
This isn't possible. The two phenomena of momentum (short-run positive serial correlation) and mean reversion (long-run negative serial correlation) map into two basic investment techniques. One is understanding the vagaries of market sentiment; the other is analysing the sources of fundamental value.
The first strategy seeks to understand the mind of the market and the psychology of its participants – exploit momentum, buy into market rises, sell ahead of market falls. The alternative approach ignores these fluctuations, focuses attention on fundamental value and anticipates that, in the long run, truth will out through mean reversion. The first of these strategies can be associated with George Soros, the second with Warren Buffett. These two men are the best-known and most successful investors of recent decades. Both are now over eighty years old, having begun their public investment careers in the 1960s.
Soros left Eastern Europe as a child refugee. A devotee of Karl Popper, he would prefer to be remembered for his philosophy rather than his fortune. But Soros will go down in history as the man who broke the Bank of England in 1992. His massive bet against sterling on 'Black Wednesday' – another contender for the title of 'the greatest trade ever' – proved decisive in forcing Britain out of the European Monetary System. Soros's Open Society network has made large philanthropic contributions to the promotion of education and democracy in post-Communist Eastern Europe. His Quantum Fund – an early example of what is now known as a hedge fund – returned an average of 30 per cent per annum to its investors over the period from 1970 to 2000. The Quantum Fund traded actively, buying and selling currencies or commodities and any other assets that appealed, or failed to appeal, to Soros and his colleagues.
Berkshire Hathaway is Buffett's investment vehicle, and was a textile company when Buffett took control. Insurance is now Berkshire's largest business. Insurance premiums are received well before claims are paid, so insurance generates a large float of investable cash. This facilitates investment, and Berkshire owns many companies – from the private jet charter business Netjets to See's Candies, famous for making the world's largest lollipop – and large stakes in businesses such as Procter & Gamble and Heinz. Buffett's investment success has taken him from modest beginnings to vying with Bill Gates (to whose foundation he is donating most of his fortune) as the world's richest man. Despite that, Buffett still lives in the bungalow in Omaha that he bought fifty years ago, and regularly enjoys a meal at a local steak house washed down with a glass of Cherry Coke (he switched brands after buying a 10 per cent stake in the Coca-Cola Corporation).
There will always be individuals who have outperformed the market – just as there will always be winners of the lottery. I'll observe in Chapter 8 how exceptional investment performance rarely persists. Most people who do well are lucky rather than clever. But Buffett and Soros not only established outstanding records; they also continued to demonstrate remarkable performance long after they had been widely recognised as the most talented managers of their generation. To win the lottery once may be evidence of either luck or skill; to win it repeatedly is evidence of skill.
You might think that the strategies of the folksy Buffett and the philosophical Soros would be the subject of intense study by students of finance and investment professionals. You would be wrong. These individuals receive a lot of journalistic attention but no academic attention. Since Buffett and Soros cannot exist, according to the strong efficient market hypothesis, they are treated as if they don't. A lot of ink has been spilt on the proposition that what plainly exists in practice can't exist in theory.
In turn, Buffett and Soros are open in their contempt for most academic work in finance. They can afford that contempt. But their contempt is not simply the practical man's disdain for the intellectual – they are both smart and well-read people. They simply believe that much of this academic work is misdirected, engaged in obsessive pursuit of narrow ideas of limited application. The distinction between the market as a mechanism for voting on events and the market as a mechanism for voting on market reactions to events was understood and expressed generations ago by Keynes and Samuelson – and by Benjamin Graham, the first intelligent investor. But the distinction is not understood by many commentators, who regard market judgements as a repository of wisdom not just about the market judgements of other market participants but about the real economy.
The distinction between the anticipation of events and the anticipation of beliefs about events is elided by most finance academics. They understand that there might be such a distinction, but have developed a group of arguments – known, in extreme form, as 'rational expectations' – to suggest that the difference does not matter. The rational expectations school is associated with the strong form of the efficient market hypothesis – all information that is capable of being known is already incorporated in prices.
But the distinction between market judgements and underlying realities, the distinction between prices and values, and the distinction between the mind of the market and economic and business fundamentals manifestly do matter. By exploiting divergences between them, Buffett and Soros have made billions of dollars. Both individuals recognise that such divergences are the basis of their success. Buffett shouts from Omaha that markets are only imperfectly efficient. Soros, quoting Keynes's metaphor of the beauty contest, writes that 'The fact that a theory is flawed does not mean that we should not invest in it as long as other people believe in it and there is a large group of people left to be convinced... we are ahead of the game because we can limit our losses when the market also discovers what we already know' (Soros, 2003, p. 25).
The essence of Soros's investment strategy is to read, and anticipate, the changing mind of the market more successfully than other traders. The essence of Buffett's investment strategy is to emphasise fundamental values and use the volatile mind of the market as an opportunity to buy assets that are underpriced relative to their fundamental value. I'll discuss these two broad approaches in the two chapters that follow.
5
**The Mind of the Market**
**Market psychology**
People in finance use phrases such as 'The market thinks', 'The view of the market is' or even, if the person is very senior, 'We could ask the market'. Of course, the market does not think; only people can think. But the anthropomorphic view, which treats 'the market' as if it were a person, is so pervasive that the metaphor is part of everyday language. Benjamin Graham, the original intelligent investor, formulated the anthropomorphic analogy when he wrote of the moody, volatile Mr Market. Mr Market has grown in influence in subsequent decades. Mr Market features regularly in Warren Buffett's homilies.
The anthropomorphic metaphor may be contrasted with the metaphor of market as voting machine, which aggregates and weighs the different views of different players. Although there is some similarity between the two, there are also fundamental differences. The market as voting machine is democratic – the analogy supposes a diversity of views, in which all judgements are relevant (if not necessarily equal). And, as in a democratic election, the result is definitive.
The analogy of Mr Market is hierarchical – the market view can be ascertained and interpreted, but not questioned by market participants. Financiers talk of 'confidence', by which they mean the standing of an individual, organisation or opinion in the eyes of the market – in much the same way as commentators identify the confidants at the court of an autocratic politician or chief executive. Such confidence needs to be earned, and can easily be forfeited. And, as at the royal court or within the sycophantic C-suite, the autocrat can only be mocked by those outside the inner circle; the king or CEO cannot be questioned but can be outsmarted, and that is exactly how Graham and Buffett viewed Mr Market.
Of course, the person who inspires such deference, whom it is perilous to defy, has no tangible existence. There is no Mr Market. 'The mind of the market' is what people believe the mind of the market to be, and the evolution of what 'the market thinks' is a process of convergence of these common beliefs. Understanding 'the mind of the market' – a deep appreciation of the psychology of Mr Market – is a potentially rewarding investment strategy, and some people are good at it. Not, however, as many as those who think they are good at it.
Trading once took place in physical markets, where buyers and sellers would meet to exchange securities and to exchange information. Some of the information might have been true, other parts of it false; some information would be contained in what they said, some of it contained in the way they behaved. In the crudest versions of face-to-face market-making, such as the 'pit' of the Chicago Mercantile Exchange, the physical capacity of the traders to elbow aside their colleagues would contribute to their success. This short-term trading environment was a testosterone-laden world. Technology transferred trading from meeting places to screens.
The rise of financial conglomerates means that dealing rooms are now the places where the traders of Goldman Sachs or Morgan Stanley sit beside each other and make electronic contact with their competitors at other firms. The atmosphere is still frenzied – and still very masculine, even though there are now some women traders. You will still hear frequent obscenities on trading floors; many financial firms have faced claims of sex discrimination and harassment; and lap-dancing clubs can be found around the fringes of areas where traders congregate.
Most traders are now employees of major investment banks. These conglomerate institutions manage assets for investment clients, execute deals for fund managers and provide advice to major corporations, as well as trading on their own account. Banks are required to maintain 'Chinese walls' in order to prevent people in these different departments communicating with each other. There are differing views on the effectiveness of the Chinese walls. Within these institutions are assembled expertise and information, unparalleled elsewhere, about all aspects of securities markets and, indeed, about wider economic and geopolitical events.
While most trades are made on behalf of clients, the bank is itself acting as principal in these transactions – the intermediary is more like the used car dealer, who buys and sells vehicles from his own stock, than the real estate broker, who is facilitating a meeting between a buyer and a seller. When the 'Volcker rule' limiting the scale of proprietary trading – dealing purely on the banks' own account – was introduced in the United States after the 2008 crash, banks claimed that such trading constituted only a small part of their overall activities. However, trading profits account for most of the reported revenues of investment banks such as Goldman Sachs and a significant part of the revenues even of organisations which are principally commercial banks.
Traders use the capital, credit and reputation of their employers. They take a significant share of the profits from their activities, but not the losses. The penalty for making large losses is being fired. This asymmetric structure of incentives encourages trading strategies and styles that return regular profits while countenancing occasional exceptional losses that may, over time, swamp the profits. These incentive structures play a central role in modern finance, and I'll come back to them in Chapter 8.
If markets are characterised by weak positive short-term serial correlation – momentum – and weak negative long-term serial correlation – mean reversion – then it ought to be possible to analyse the processes that give rise to these 'anomalies'. George Soros describes the boom-and-bust process associated with momentum and mean reversion. He illustrates how in a variety of markets, from conglomerate acquisition to real estate investment trusts, a momentum-driven boom creates the seeds of its own destruction and leads to an inevitable downturn.
But describing the cycle is not at all the same as picking accurately the turning point of the cycle. Soros did, indeed, identify correctly the broad shape of some large swings in behaviour during the 1970s and 1980s to profit both on the upswing – following the momentum-driven herd – and on the downswing – benefiting by anticipating a period of mean reversion. These judgements proved very profitable. But such success became harder to achieve.
Can mathematical techniques help with understanding past and future cycles? Many amateur speculators rely on charts or other forms of what is known as technical analysis. This approach to investment analysis involves the identification of recurrent patterns in price series. The technical analyst looks at a chart of recent prices and identifies trends, believing that it is possible to identify buy and sell signals from inspection of the chart.
Purists among chartists do not care what the chart records – whether it is the stock price of Apple, the exchange value of the dollar or the price of an index-linked bond. Chartists look for pictures in the data – the 'head and shoulders' is an especially popular image. They draw trend lines through market rises and declines and horizontal resistance levels across more stationary ones. They see recurrent cycles. The Kondratieff fifty-year cycle has been discussed since the 1920s, and there is just enough evidence of long-term fluctuation to make continued attention to it plausible. The Elliott 'long wave', the discovery of an accountant and amateur investor, is a favourite of many chartists. Among the community of finance academics in universities, technical analysis has a terrible reputation. Most textbooks simply refuse even to acknowledge that the technique exists. But chartism continues to have a cult-like following, and market prognostications based on technical analysis are regularly found in the financial press and business journals.
There are good reasons for professional disdain. Many of the statements made by chartists resemble those of astrologers. The analytic component is ascientific and the predictions sufficiently ambiguous not to be falsified by any likely events. Chartists make use of the natural human inclination to see patterns in data – the same tendency that leads us to see images in ink blots and interpret rocks as sculptures.
Technical analysts are often associated with investment gurus, more common in the United States, where there are many published investment newsletters, or tip sheets. The guru makes an outlandish prediction. Sometimes the prediction comes true, so the newsletter attracts subscriptions, the seminars are oversubscribed. The guru continues to make predictions – often the same predictions – but when these are proven to be false, the audience and reputation gradually drift away. The relatively few gurus who have a long-term track record of success – or, indeed, of employment – are generally based in large institutions with substantial resources devoted to fundamental research.
Chartists, gurus and people with proprietary and secret systems are, overwhelmingly, charlatans. But the story is not quite so simple. A central tenet of this book is that the efficient market hypothesis, although illuminating, is not true. And so it is not good enough to assert that since the EMH is true, no claim that is made by chartists, gurus or for mechanical rules or trading systems can ever be valid. At most, what can be said is that there is little evidence that any individuals or systems use these methods with sustained success.
Perhaps the difficulty of identifying patterns in historic price series is the result not of the absence of such patterns but of the absence of techniques sufficiently powerful to identify them. At some level this must be correct. No one really believes that price movements are truly random, although explanations of these price movements may be so complex and so varied that we can never do better than to describe them as a random process.
Much of this book describes the theory of finance that originated in Chicago around fifty years ago and has been developed by economists since. Today that body of analysis is subject to a pincer movement, pressured from one side by behavioural economics – the applied psychology that emphasises the ways in which the beliefs of traders influence the determination of prices – and from the other by higher-powered mathematics and the computers used by high-frequency traders which analyse 'the mind of the market' by constantly offering to buy and sell. (Most of the offers to buy and sell which these computers make do not result in trades.)
Techniques brought to bear on financial markets more recently push towards, even beyond, the frontiers of applied mathematics. These approaches use models to reproduce what the most skilled traders may be able to do instinctively. In what is superficially a paradox, soft revisions from psychology and hard revisions from mathematics turn out to be two sides of the same coin. The mathematics of complex systems may help to explain the formation of expectations and the dissemination of beliefs in markets. If it does, then the ways in which expectations are formed and beliefs disseminated will change.
In the first chapter I suggested that the intelligent investor could fend for him- or herself in a world populated by financial professionals. But in the world I have described in this chapter the intelligent but amateur investor cannot fend for him- or herself. A minority of successful traders in hedge funds and investment banks have an intuitive feel for market psychology. These few individuals have honed their skills over years of experience and are exposed to a wide range of market information every minute of the trading day.
The mathematicians who build trading systems have access to the best brains and most powerful computing resources available. Successfully riding market momentum involves frequent trading, and investment banks have much lower trading costs than retail investors. Even with all these advantages it is not at all clear that over the long run the activity is profitable for investment banks. You need to set against their reported profits the losses from supposedly unforeseeable events and allegedly unauthorised trading. We simply don't know whether over a period of years banks make money out of these market judgements (as distinct from making money out of their customers' transactions), and nor do they. Most people who are described as successful traders blow up after a long lucky run.
The David and Goliath notion that with a home computer, a proprietary software package and a book of trading rules you are likely to succeed where the best-resourced institutions in the world have largely failed is laughable. Most people who claim to trade successfully in this way are themselves on a lucky run in a rising market, or engaged in self-delusion, or both.
The intelligent investor cannot match the professionals in understanding market psychology. But he or she does have advantages when it comes to fundamental value. In the remainder of this chapter, and the next one, I'll explain why and how.
**When the market loses its mind**
There have been asset bubbles so long as there have been financial markets. In the Dutch tulip mania of 1636–7 a single prized bulb was supposedly valued at the price of twelve acres of land, or ten times the annual earnings of a skilled craftsman.
In the preceding chapter I described the two enduring general principles of asset valuation: an asset is worth what someone is willing to pay for it; and the value of an asset is the cash it will generate over its life. The characteristic of a bubble is that the first of these principles overwhelms the second. Assets are bought with little or no regard for their fundamental value because people believe that, whatever their fundamental value, someone else will pay more for it. That was why a tulip bulb was, briefly, equal in value to twelve acres of land.
Most bubbles have their origins in some real event – an economic, technological or geopolitical event that changes the fundamental determinants, though not the fundamental principles, of asset valuation – the spread of railways, the establishment of market economies in the developing world, the exploitation of the internet. Whenever there is such a change, there is a human tendency to enthusiasm, to exaggerate the pace and significance of that change, and that tendency is increased if the enthusiasm is profitable as well as fashionable.
So the effects spill into asset markets. And as bubbles expand, the belief that assets can always be sold at a higher price, that someone else will pay more, seems to be borne out by events. Someone else really is willing to pay more. Those who point to the absurdity of prices unrelated to fundamental value are exposed as fools who have passed up the opportunity of an easy profit. Many of them will finally join the party. That was the fate of the scientist Isaac Newton, who a century after the tulip mania was more than intelligent enough to see that the South Sea bubble was a fraud. But after watching on the side-lines, he could not resist the prospect of an easy profit, and was finally persuaded to invest. 'I can predict the motion of heavenly bodies, but not the madness of crowds,' he (allegedly) lamented as he counted his losses when the bubble burst.
When I was a student, I read J. K. Galbraith's book _The Great Crash, 1929_ , about the speculative bubble of that era. The boom and then the ensuing depression had been followed by fifty years of historically exceptional stability in financial markets. How could people have been so stupid, I wondered, as I learned of the leveraged piles of junk which investment banks had put together and their customers had rushed to buy. I was soon to learn.
The first great modern bubble occurred in Japan in the 1980s. Japan's post-war economic development had transformed the country from a primitive society to a new industrial state in two generations. The growth rate this implied, projected into the future, suggested that Japan would soon dominate the economic world. Demand for Japanese assets from foreign investors and domestic speculators fuelled each other, and the share prices of Japanese companies and the value of Japanese property exploded, as did prices. At the peak it was widely claimed that the grounds of the Imperial Palace in Tokyo were worth more than the entire state of California. The Nikkei index of Japanese share prices peaked at 38,957. Twenty-five year later, it is still less than half that value.
The New Economy bubble began to inflate in 1995, with the initial public offering of Netscape, a company whose internet browser made access to the worldwide web widely available. The company, founded less than two years earlier, was valued at $3 billion, and the share price doubled on the first day. (Netscape's browser would in due course be crushed by Microsoft's Internet Explorer.) A report by Mary Meeker, 'the internet goddess', pointed to other companies that might attract similar enthusiasm.
In the five years that followed, the phenomenon of Netscape was repeated again and again. A newly established business with little revenue and no profit would be launched on the market, and its price would rise on a wave of enthusiasm. When I led Oxford's business school between 1997 and 1999, I found many students had no interest in the traditional interviews with McKinsey and Goldman Sachs because they expected to be internet millionaires within the year. Many of the increasingly ridiculous businesses that were launched had no real purpose other than the raising of funds from gullible investors. It became common to see business plans that extended only to an IPO.
Few European internet companies made it through to an IPO before fundamental values reasserted themselves. The aptly named lastminute.com came to market as the boom was subsiding. Boo.com, a fashion site and perhaps the most absurd of all internet businesses (its founders blew over £100 million on air travel, hotels and parties before the business collapsed) sold few clothes and no shares to the public. In the spring of 2000 the bubble burst.
Bubbles should be distinguished from asset mispricing. Asset mispricing occurs when market participants, faced with imperfect information and an uncertain future, get values wrong. Such mispricing occurs all the time and provides opportunities for intelligent investors. In bubbles few traders are concerned with fundamental values, and they generally hope to sell on their securities, soon, at a higher price. Intelligent investors can do nothing in bubbles.
In Chapter 9 I will describe how another bubble, in parts of the US housing market, contributed to the financial crisis of 2007–8. That bubble was promoted by brokers who encouraged borrowers to take out mortgages they could not afford in the expectation that the brokers would be able to refinance the mortgages for a higher value, and further commission, within a year or two. The structure was predicated on the assumption that house prices would rise continuously, so that even a pause would bring that structure crashing down. That pause eventually happened.
As a student, I had wondered how the markets of 1929 could have accommodated such folly. In 1999–2000 I had learned how people could indeed be so foolish; in 2007–8 I learned that such foolishness was ineradicable in modern financial services markets. The origins of these events lie not so much in innate stupidity as in the capacity of clever people, especially when blinded by the prospect of financial gain, to fall victim to the power of conventional thinking.
**The power of conventional thinking**
In the New Economy and credit bubbles conventional, yet magical, thinking created its own reality. It is hard to overstate the power of group thinking in business and finance. People repeat to each other the same transitorily fashionable views in mutual reinforcement. The power of conventional thinking asserts itself in the reiteration of banal opinions on fashionable issues. Today these include the excitement and paranoia over China's economic development, the obsession with climate change and the scramble for alternative assets in hedge funds and private equity.
Both truths and falsehoods spread contagiously, forming what Galbraith called 'the conventional wisdom'. These processes of contagion can lead people to believe absurd things – that Salem was besieged by witches, that the US was endangered by Communist subversion, that boo.com would dominate its market and that house prices can only go up. It is not only common but reasonable to believe what is widely believed, especially among business and political leaders, who have little time for reflection. I believe the world is round, not because I have myself verified it, but because the general belief is that the world is round. If I had lived a thousand years ago, I would have believed the earth was flat, and for the same reasons. The court of Mr Market creates and experiences its own reality.
In the complex and uncertain world of modern finance, such behaviour takes us far away from market efficiency and fundamental value. In all market bubbles – tulips or the South Sea, the New Economy or the housing bubble – distortions are supported by commercial interests that benefit from their promotion. There was always a kernel of truth in the exaggerated propositions. Speculators in the South Sea bubble were on the verge of an industrial revolution and an explosion of world trade. Railways, electricity and modern information technology were transforming discoveries. Financial innovation did create opportunities for better risk management. The creation of the eurozone did promote economic convergence within the European Union. The people who lost money in these bubbles were not mistaken in their basic thesis, but they greatly overestimated both the pace of change and the extent to which individual companies would benefit from it.
The mind of the market became its own reality. Investors sought insight into 'the mind of the market', and that was what they received – and continued to receive. What was true was what was believed, and what was believed was true. And when the bubbles burst, what had once been believed and was therefore true was no longer believed and therefore no longer true. The dot.com shares that were once stars were now dogs; there was nothing more to be said. Practical men of the financial world, who would be appalled at the suggestion that they might be influenced by French philosophy, are the most determined of postmodernists. In financial centres across the world truth is in the eye of the observer. Jean Baudrillard notoriously remarked that 'the (first) Gulf War took place only on television'; in much the same sense, the New Economy was observed only on Wall Street.
**The prevalence of short-term thinking**
The misapprehensions of the conventional wisdom – true or false – affect prices and may even affect fundamental values themselves. George Soros describes this phenomenon as 'reflexivity'. Although the development of the internet was not an event of great significance for Walmart, the belief that it was a significant event had a large influence on its businesses. There was an even larger effect on the investment climate within which such companies operated. The New Economy bubble led to large overinvestment in telecommunications capacity, which itself made nemesis more likely. The massive overpayments by European telecoms companies for mobile phone licences were the result of misperceptions of reality generated by the New Economy bubble.
The senior executives of these businesses were captured by these misconceptions and could not have escaped them even if they had been more thoughtful people. Five years later the senior executives of banks were in the same position. In the distorting mirror provided by ostensibly sophisticated risk management models, they saw irresistible profit opportunities. There was a new paradigm for banks, as there had been a new paradigm for retailers and phone companies, and the winners would be the ones to pursue their fantastic vision most vigorously. The price of contradicting the mind of the market was to lose one's job.
The divergence between the interests of the investor and the interests of the businesses through which the investor's funds are transmitted runs all along that chain of intermediation. Risk for a fund manager is not volatility but underperformance relative to an index. It is, as Keynes observed, better for a career to be conventionally wrong than unconventionally right, and disastrous to be unconventionally wrong. The control of the manager's perceived risk leads to the practice of 'closet indexation'. Although the fund manager is paid for stock selection, the composition of his fund mirrors closely the composition of the index. Most fund managers receive daily reports of their overweight and underweight positions, relative to the benchmark index. As a result, many allegedly 'actively managed' funds virtually replicate an index. The successful fund manager is one who can stay close to his benchmark index but be consistently slightly ahead of it.
You might think that this could be achieved by close attention to fundamental value. If performance were measured every five years, when mean reversion would have asserted itself, such a policy might well succeed. But performance is measured not every five years but much more often – commonly every three months, often more frequently. Over such short periods momentum dominates mean reversion – this quarter's outperforming stocks are more than averagely likely to have been last quarter's outperforming stocks. Knowing the mind of the market trumps fundamental value.
If you were a professional investment manager, and you had made the – good – call that shares in 1996 were overvalued relative to property, you would almost certainly have been fired. From 1996 to 2000 shares rose steadily while property stagnated. The reversal came afterwards. If you are managing other people's money, three years is a long time – too long to be, or to appear to be, wrong.
Managers who are doing markedly worse than their peers for as long as three years will find their jobs in jeopardy. 'Where have all the geniuses gone?' _Fortune_ asked in 1999. Warren Buffett 'didn't get it', and shares in Berkshire Hathaway fell by almost 50 per cent that year. George Soros failed to read the timing of boom and bust correctly, and retired from active involvement in managing other people's money, content to devote himself to the philanthropic management of his own.
Keynes reportedly said (there is no evidence that he did) that the market can be wrong for longer than you can stay solvent, warning of the risks associated with ignoring the mind of the market in favour of fundamental value. Writing today, he would say that the market can be wrong for longer than a contrarian fund manager can hold his or her job. And so it has proved.
While three years is a long time-scale for an investment manager, it isn't a long time-scale for the intelligent investor. This difference in the time-scale on which you are judged gives a big advantage to the DIY portfolio manager. Patience pays.
**The malign effect on corporate performance**
The emphasis by institutional investors on quarterly figures has led analysts and fund managers into a symbiotic relationship with corporate executives known as 'earnings guidance'. In pre-war days the great figures who headed large companies, such as Pierre du Pont of the chemical firm bearing his family name or Alfred Sloan of General Motors, had virtually no contact with shareholders (although they were significant shareholders themselves). The emergence of the hostile takeover was probably the most important single factor in changing this behaviour. Senior managers of large companies began to pay closer attention to the share price – their jobs depended on it. The growth of stock options as a means of payment reinforced the trend. It provided an additional reason for executives to pay close attention to the share price – their wealth depended on it.
Now, most large companies will have an investor relations department or employ a specialist firm to handle these issues. The chairman, chief executive and chief financial officer will expect to devote a substantial proportion of their time to investor relations. That means pitching the company to analysts and talking to the investment managers who are the principal shareholders. Sometimes analysts will benefit from lavish hospitality in the course of learning about the company's affairs.
The market responds not to good and bad news from companies but to news that is better or worse than market expectations. Therefore many businesses want to manage both outcome and expectation themselves. They seek steady earnings growth just a little faster than the market is anticipating, and manage their quarterly reporting in order to achieve this. I will describe in the next chapter how much scope finance directors have for keeping their reported earnings on a steady path. Often the main criterion by which analysts judge the quality of management is whether executives do the things they say they will do. This mutually rewarding process of earnings guidance has only the loosest of connections with how the business is actually performing. Eventually, of course, fundamental value breaks through. But this process may take time. The favoured analysts of companies such as WorldCom and Enron – for example Grubman and Curt Hamner of CSFB (Credit Suisse First Boston) – continued to puff these stocks until the companies collapsed.
A major part – in many cases much the largest part – of the remuneration of senior executives is now related to movements in the company's share price. That makes it inevitable that the mind of the market takes precedence over fundamental value. In tribute to the power of Mr Market, many companies display the constantly fluctuating share price in their reception area. Some chief executives have it on their desks, and it is not unknown for them to check it several times a day. Their business plans are heavily influenced by the conventional wisdom of their peers, analysts and bankers, rather than their own judgement. Success, as measured by the market, is being able to anticipate the new fashion slightly more quickly than other people. Advance insight into the conventional wisdom is the commodity that many consultants, business gurus, journalists and investment analysts now sell.
**The intelligent investor's advantage**
The unproductive contact between finance directors and analysts provides an opportunity for the individual investor – the rewarding opportunity to ignore it. You aren't party to earnings guidance, but it doesn't matter, because the information such guidance provides has little to do with the substance of the business. You don't have to remain popular with company management to do your job. You don't have to worry that you will be fired if you underperform the market in one quarter, or several. Being close to the market is not necessarily an advantage. Occasionally fund managers have developed such a strong reputation with investors that they can ignore the braying of analysts and the pressures of the dysfunctional cycle of earnings management and guidance. While Buffett was criticised for refusing to participate in the New Economy bubble, he could laugh all the way to the bank, and the steak house. In Omaha, Nebraska, far away from Wall Street.
Best of all, you can emphasise absolute, rather than relative, returns. That emphasis enables the intelligent investor to focus on the analysis of fundamental value, rather than the mind of the market. For the intelligent – and therefore patient – investor, there is a simple reconciliation between the mark-to-market and fundamental-value principles: an asset is worth the higher of its fundamental value and its market price.
If the market price is above fundamental value, an intelligent investor can sell for the market price and look for something else. If the market price is below fundamental value, an intelligent investor can continue to hold and enjoy the benefit of the projected stream of cash returns.
This freedom gives the intelligent investor an immediate advantage over the majority of professional fund managers, bound by the routine of quarterly performance measurement. For the professional fund manager, the mark-to-market principle rules – an asset is worth what someone is willing to pay for it.
Being able to take a detached view of fundamental value is a big advantage. But it is not easy. Estimating fundamental value requires a view of the long-term prospects of the company. The information you need is extensive, difficult to obtain and changes constantly. The analysis of fundamental value is speculative, and different people are likely to come up with different answers. In the next chapter I'll look at what is required.
6
**In Search of Fundamental Value**
**Accounting for earnings**
Fundamental analysis of a company begins with its profits. In 2015 Exxon Mobil earned $3.85 per share, and the corporation's stock price was $89. The price:earnings ratio – the number of years of current earnings needed to earn the current stock price – was therefore 23. The price:earnings ratio is the most common single metric used in stock valuation.
Table 3 shows the price:earnings ratio of some sectors of the US stock market at January 2016.
The wide variation across industries is the result of different perceptions of growth prospects. In 2014 Exxon Mobil earned $7.60 per share, and in June 2015 the stock price was $82, equivalent to a PE ratio of 11. The market was already discounting lower profits as a result of falling oil prices. The highly rated pharmaceutical sector, however, includes many companies with considerable earnings growth potential. Profits from software are mostly in the future rather than the present. These future differences are reflected in current ratings.
The ratio between the price:earnings ratio and the growth rate of earnings (the PEG) is an indicator popular with many investors. Their reasoning is that a low PEG ratio suggests that future growth can be bought at a relatively modest price. It's not that easy: a measure such as PEG is at best a simple screen for identifying companies whose fundamentals deserve further examination.
Table 3: **Current PE ratios, various US sectors, January 2016**
Banks (money centre) | 17.4
---|---
Automobile | 14.0
Pharmaceutical | 32.1
Oil and gas (integrated) | 27.0
Retail (general) | 22.1
Tobacco | 35.0
Utility | 19.3
Software (systems and applications) | 127.7
Source: Stern School, New York
Just as price:earnings ratios vary across companies and business sectors, so they also vary over time. The cyclically adjusted price:earnings ratio (CAPE) is a measure popularised by Robert Shiller. This measure looks at the ratio of the current market price to average earnings (adjusted for inflation) over the previous ten years.
For the US, Shiller has calculated the long-run historical average CAPE (over more than a century) for the market as a whole as 16.5. A 'normal' range has been between 10 and 25. Price:earnings ratios also vary considerably across countries. Table 4 shows how at July 2015 CAPE was low in many European markets, while for the US the figure is close to the top of its historical range.
If you measure performance over ten-year periods, the outcome is generally better if the CAPE is at the bottom end of its normal range than near the top. And on the three occasions when the US market has broken through the top end of this range – the late 1920s, the mid-1960s and the late 1990s – the event has been followed by a major stock market decline. (The crashes of 1929 and 2000–1 are well known; there was no 'crash' in 1966, but the Dow Jones index did not regain its 1966 level until 1982.) CAPE is not a method of timing the market, but the apparent predictive value of CAPE, resulting from the power of long-run mean reversion, is nevertheless relevant to the patient investor.
Table 4: **CAPE in different equity markets, July 2015**
Source: Wellershoff and Partners
But less relevant to the herd. When the price:earnings ratio rises outside its conventional range, analysts opine that historical experience is no longer applicable. When Japanese share prices soared in the 1980s, they advised that Japanese companies should be viewed differently from companies elsewhere. A decade later, many financial commentators argued that the New Economy and the economic and political environment that followed the end of the Cold War had changed the basic principles of stock valuation. In the noughties, new tools of risk management had tamed financial risk, and a 'great moderation' was in progress.
Once again, the world had not changed, only opinion. Long-term mean reversion struck, and PE ratios returned to more normal levels. There is a saying that the most expensive words in investment are 'It's different this time'. Sceptics who were reluctant to believe the rules had changed avoided the bubbles in Japan, the New Economy and the credit boom to their ultimate benefit.
For a stock such as Exxon Mobil with a price:earnings ratio of 23, the earnings attributable to each share are equal to a little over 4 per cent of the share price. This ratio, the inverse of the PE ratio, is known as the earnings yield. If a normal PE ratio is between 10 and 25, a normal earnings yield lies between 4 and 10 per cent. Since shareholders do not receive the company's earnings, the earnings yield is not directly relevant to the shareholder. However, it is indirectly relevant, since earnings determine the dividends the company can pay. The earnings yield is a guide to the long-run rate of return that it is reasonable to expect from owning shares.
**Creative accounting**
But the value of such a guide depends on the value of the information on which it is based. The collapse of Enron brought to the attention of a wider public what market professionals and companies have always known: companies have a lot of scope to make their reported earnings and assets what they want them to be. What is the profit of a company? It all depends on the meaning of 'is', as Bill Clinton famously explained. Creative accounting is the quantitative equivalent of the modern politician's lie – a statement that is in some narrow technical sense true but is substantively false.
The jury decided that the practices at Enron constituted fraud, and its former chief executive, Jeff Skilling, and former chief financial officer, Andy Fastow, went to jail for their part in one of America's most spectacular corporate collapses. But the methods that Skilling and Fastow employed were only aggressive versions of financial techniques used by more respectable companies.
Wall Street wanted corporations to report a steady stream of increased earnings (part of the process of managing investor expectations that I described in the previous chapter). So that is what corporations reported. How were companies able to manage, not their businesses, but their declared earnings in this way?
A prudent business that acquires another will take a careful look at the assets it has acquired. So will an imprudent one. The imprudent one writes down the value of these assets to low levels and then attributes to its own superior management skills the gains apparently made when these undervalued assets are sold or used in the business. The 'restoration' of value can be used to enhance profits for many years. This practice is called 'acquisition accounting', and is a relatively simple device. Recent innovations in creative accounting have been much more complex.
Most large businesses are, from a legal perspective, a group of companies under common ownership. Consolidated accounts add the individual accounts of all these companies and report the overall result as if the business were a single entity. Since shareholders are, indirectly, shareholders in all the companies within the group, this is the only sensible way to present the figures in a manageable, comprehensible form.
But companies often give stakes in associated businesses to outsiders. In a joint venture between two or more companies, each cooperating company will usually have a share of the legal entity that conducts the business. A firm establishing a new business venture may want to give shares to key individuals. Or a company may simply own shares in another business as a strategic investment.
The general principle – and it is a sensible one – is that if a company manages and controls a subsidiary or associate company, the consolidated accounts of the parent should include the relevant share of assets and earnings of the subsidiary or associate. If the parent doesn't manage and control the other company, then its interest should be treated as if it were an investment. The asset is the value of the shares, and the income is the dividend.
So far so good, if not so simple. But what exactly is meant by management and control? US accounting standards – known as Generally Accepted Accounting Principles (GAAP) – try to pin this down. But whenever standards try to pin down a concept, the creature flies away. Creating subsidiaries that you do, in fact, manage and control, but which you do not manage and control for the purposes of generally accepted accounting principles, has many advantages for the creative accountant. You can use transactions between the parent and subsidiary to create profits, or to make them disappear. Enron created hundreds of these subsidiaries. Fastow helpfully took stakes in some of them for himself. If you went through the detail of the company's accounting filings, you would discover all this. But to do so would involve burying yourself in thousands of pages of figures. Some analysts did work it out. I'll come back to them in Chapter 9.
Some companies of the New Economy era would simply exchange long-term contracts to supply each other with the same thing, enabling both to credit turnover and profits even though no real transaction took place – or ever would. Such manipulation doesn't work for ever; sooner or later these manoeuvres will have to be reversed. But you can make it later rather than sooner. And remember, in the long run we are all dead. Or, in some cases, in the federal penitentiary.
Another fraudster, Bernie Ebbers of WorldCom, reduced the slick presentation to bare essentials. He would arrive at meetings and simply point to a graph of the company's rising share price before inviting questions. Any more extensive account of what the company was doing might have taxed his business knowledge and financial expertise. The accountants of both Enron and WorldCom, encouraged to support that rising share price, took full advantage of opportunities provided by the complexity of modern accounting. As Fastow explained his activities at Enron to his son: 'Every way I could find ways to follow the rule technically but undermine the principle of the rule, I did it. And because I did that, it caused a lot of harm and that's why I should be in prison.'
A long-term contract or project will incur costs and accrue revenues over several years. A conservative approach recognises the profit when the books have closed and the money is banked, acknowledging only the bird in the hand. An optimistic frame of mind takes credit for the anticipated profit on the deal the moment it is signed, chalking up many birds in the bush as soon as the hunter's gun is primed. A true and fair view would strike a position somewhere in between. The right balance requires careful, and subjective, judgement. But the exercise of careful judgement was not the activity in which self-aggrandising executives, such as those of Enron and WorldCom, were engaged. Jeff Skilling notoriously celebrated permission from the Securities and Exchange Commission for Enron to mark its gas contracts to market with champagne for his colleagues. The agreement enabled the corporation to bring many years of potential profit into this year's accounts.
Advisers and accountants connived in these schemes. It is impossible to believe that sophisticated, intelligent investment bankers did not realise that Kenneth Lay, Enron's chairman, was an affable, skilful political operator with little grasp of business detail, that Jeff Skilling, the chief executive, was as corrupt as he was brilliant, or that Bernie Ebbers had none of the attributes needed to run a global telecoms company. But Enron's complex financial transactions and WorldCom's constant acquisitions generated large fees.
Accountants were once the butt of jokes – stuffy people with a rigid professional ethic. In the 1980s the large global firms that audit virtually all major corporations in Britain and the United States consolidated into what is now the big four of KPMG, PwC, EY and Deloitte. Their partners became hungry for consulting revenue as fee levels from their traditional audit business came under pressure.
Pleasing corporate clients received greater emphasis. 'Eat what you kill' was an increasingly common slogan in law and accountancy firms. Partners would take home the revenues they had personally earned rather than a share of the collective profits. The characterisation of customers as prey is revealing. Arthur Andersen, always the most commercial of the major firms, would pursue new clients, and the interests and wishes of those clients, most aggressively – and would be destroyed by its acquiescence in Enron's frauds.
The main counteracting force to the effect of competitive pressures in reducing standards is public regulation of accounting standards and the sale of securities. The most important accounting standards bodies are America's Federal Accounting Standards Board and the global International Accounting Standards Board. The latter body determines International Financial Reporting Standards, the analogue of GAAP for Europe and much of the rest of the world. All countries have financial regulators, and America's Securities and Exchange Commission is the most formidable.
These organisations have done much to rein in abuses and, as a result, formal accounting statements now contain much less rubbish than they did even a decade ago. Still, the presentation of company accounts remains confusing. Many companies highlight partial or misleading information, and commercial pressures on analysts to join in corporate self-congratulation and facilitate the process of earnings guidance remain strong.
Beware of EBITDA (Earnings Before Interest, Tax, Depreciation, Amortisation), a measure devised in the 1990s to enable technology start-ups to flatter their accounts. Interest and tax are costs. Be particularly suspicious of 'pro forma earnings' or any numbers that include or exclude supposedly non-recurring items. These figures that company executives present to the public are sometimes described as 'earnings before bad stuff'.
The issue of revenue recognition – how to account for profit from projects with a long time-scale – is fundamental to an understanding of corporate accounts. Depreciation and amortisation are costs: the terms are more or less interchangeable, although depreciation is most often used in relation to fixed assets and amortisation in relation to intangible assets. Look closely and sceptically at amortisation of intangible assets, which may often be related to acquisition accounting.
But depreciation of fixed assets is generally real enough; your car loses value as you drive it and wear it out, and loses further value every year whether you drive it or not. But this cost is not money out of your pocket in the way the fuel you buy to fill the tank is a cost. Money leaves your pocket when you have to shell out for a new car. Because expenditure on fixed assets is lumpy, your cash flow profile differs from your income and expenditure profile – I will say more about income, expenditure and the prudent spending rate in Chapter 11.
In a similar way, the contrast between the erratic pattern of capital expenditure in a business and the smoothed path of accounting depreciation means that the stream of properly calculated profits differs from the cash flow. Even Apple does not build a new $5 billion headquarters campus every year, nor Exxon Mobil open a new oilfield. A growing business will often have profitability running ahead of cash flow, while a declining one will be generating more cash than profit. 'Free cash flow' is the difference between net revenues from operations and necessary expenditure on productive investment.
But you may need to look carefully not only at cash flow and profits but also at the underlying business activities in order to determine whether a business is really investing for the future or is effectively running itself down. Managers given a target of finding 5 or 10 per cent cost savings, a frequent corporate practice, can usually deliver. Sometimes this can be done with little effect on the efficiency of the business or the quality of its product. We have all encountered people who reduce, rather than increase, the effectiveness of the organisation that employs them, and whose services can profitably be terminated. You can always spend less on maintenance, or on customer service, or press employees to work harder. These actions will have an impact on future profits, but the consequences will be some time ahead, and may prove to be more or less expensive than the initial cost savings.
Most large companies have engaged in successive rounds of these cost savings in the last decade, either to enhance measured efficiency or to produce the 'synergies' that supposedly justify their merger activity. Partly as a result, corporate earnings have grown faster than revenues or the economy as a whole. Some of these cost savings represent real efficiency gains; others enhance current profits at the expense of future profits. Often, only time will tell.
In the first edition of this book I illustrated the issue with the example of BP, the British-based oil company which had taken cost-cutting too far, especially in the US Amoco business which it had acquired. One result had been a major explosion at its Texas City refinery in 2005. But much worse was to come. A year after publication of that first edition, the Deepwater Horizon rig in the Gulf of Mexico caused the largest oil spill in the history of the industry, costing BP tens of billions of dollars in fines and compensation.
But BP is not the only case where the pursuit of short-term earnings damaged the company's performance in the longer run. Banks sold their customers payment protection insurance on which many of the policyholders stood no chance of a successful claim until regulators forced them to return billions in compensation to their customers. Pharmaceutical companies stressed marketing and cost reduction and spent their ample profits on buying each other rather than on new research, leaving their pipelines of new drugs empty and endangering the fragile implicit contract between the industry and the public.
Despite these deplorable examples, many businesses are better and more tightly run today than two decades ago. Almost every company can reduce cost at the expense of future revenues and, if analysts project this growth of earnings into the future, the stock price gets a double bonus. The key issue in every case is to look at the sustainability of the business strategy and, above all, the sustainability of the firm's competitive advantage. Competitive advantage can be understood only by going behind the earnings statement.
One clear pointer to the problems at Enron was that the company, while supposedly very profitable, was not generating the cash that matched its reported earnings. Cash is a good reality check. A common feature of all the accounting wheezes described above – dubious transactions between the company and its subsidiaries, premature revenue recognition, inappropriate writing down of acquired assets – is that increased reported earnings have no cash counterpart. Tax is also a clue to what is really going on. Revenue authorities have many faults, but they do not often levy tax on profits that do not exist. High profits and a low tax charge may be explained by clever tax planning, or by large investment programmes that produce large deductions. Another possible explanation is that profits are not what they seem.
There are ways of piercing the fog created by earnings reports. One is to focus on the ability of a business to generate cash. Another is to look, usually in a more qualitative way, at the inherent capabilities of a company's business – its competitive advantages. I'll look at each of these approaches in turn.
**Cash is king**
Most thoughtful analysts look at cash flow and assets rather than reported earnings when they try to assess the fundamental value of a business. The general principles of asset valuation apply just as much to a company, and to the assets of a company, as to a security:
* an asset is worth what someone is willing to pay for it; and
* the value of an asset is the cash it will generate for its owner over its life.
Applying the second of these principles requires a process for translating a stream of cash that accrues continuously over time into a value at a particular point in time. The cash flow will come from the dividends on a share or on the earnings of a company. The standard method is called 'discounted cash flow' (DCF), the first and indispensable tool for any quantitative investment analyst.
Compound interest at 8 per cent per year means that $100 now will be worth $108 a year from now, $117 in two years' time and so on. If $100 is worth $108 a year from now, then $100 a year from now is today worth $93 ($100 ÷ 1.08). A similar calculation would make $100 two years from now worth $86 today ($100 ÷ 1.17). The method of discounted cash flow derives in this way a set of conversion factors, like exchange rates, which can be used to convert cash at any future date into an equivalent value of cash today ($100 in 2018 corresponds to $86 in 2016). A promise to pay $300 in three equal annual instalments, starting now, is worth not $300, but $100 + $93 + $86, or $279. All you need do is choose an appropriate discount rate. The choice of an appropriate discount rate is not a minor problem, as we shall see.
The method of DCF valuation can be applied either to the cash flow accruing to the shareholder – the dividends to be paid – or to the cash accruing to the business itself. I'll look in turn at each of these approaches.
Suppose a company, such as Exxon, is paying a dividend of $2.88 per share. We need to estimate the likely growth of that dividend. If dividends grow at 5 per cent, next year's dividend might be $3.02, rising to $3.17 the year after and so on. All we need to do is write down that stream of cash flows, discount it at, say, 8 per cent, and we can calculate the fundamental value.
If 5 per cent growth is the long-term average, there is a simplifying trick. Subtract 5 per cent, the growth rate, from 8 per cent, the discount rate. The difference, 3 per cent, should be the dividend yield on the share, as in this case it more or less is (3.2 per cent). It is easy to work through the mathematics, but you may prefer just to believe the result. The fundamental value that emerges from this calculation is $96, close to the current share price.
The DCF approach requires knowledge of the appropriate discount rate and of the appropriate cash flows. The usual approach is to say that future cash flows should be discounted at the rate the company has to pay to raise money – the cost of capital. In the calculation above I used an arbitrary 8 per cent – the target rate of return of Chapter 2 – as the discount rate (a common procedure), but if the technique is to have scientific value, the number requires a more principled basis.
One common practice would be to take the yield on long-dated bonds, currently 2.5 per cent for a thirty-year US Treasury bond, as a starting point. Payments of interest and principal by the US government are secure. But since the future values, profits and dividends of Exxon Mobil are speculative, it is common practice to use a much higher discount rate to reflect the greater volatility associated with investment in shares. This allowance for business risk is the 'equity risk premium'. The simplest and commonest way of measuring the equity risk premium is the average historical difference between returns on shares and the return on safe assets such as US Treasury bonds. The data in Chapter 2 might justify a figure of about 5 per cent, which is broadly consistent with an 8 per cent figure for the cost of capital, equal to the intelligent investor's target rate of return. I'll come back to the calculation of this risk element of the cost of capital in the next chapter.
The alternative approach to DCF appraisal of the cash that a corporation will return to its shareholders is to look at cash flows generated within the business itself. The 'free cash flow' approach starts from the cash generated by a company's operations – the difference between net revenues from operations and necessary productive investment, as defined above. If we had a sufficiently long series of future free cash flows, the DCF technique would enable us to calculate the fundamental value of the business. But that is a big if.
A successful business may have negative free cash flow because the company is spending more on building the business than it is currently earning. Most new companies are in this position. Only after several years will projected cash flows become positive. Many years of data are therefore required before any clear assessment of fundamental value can be made.
A further difficulty then emerges. As illustration, go back to the DCF valuation of Exxon and use the dividend growth approach. Over the next five years the DCF value generated from dividends will be around $15, less than 20 per cent of the hypothetical value of the share. Most of the projected cash flow lies in a far distant future.
Worse still, the result of the calculation is extremely sensitive to the assumptions. Suppose the dividend growth projection is 6 per cent rather than 5 per cent, and the discount rate is 7 per cent rather than 8 per cent. These small adjustments raise the projected share price to $300. Or suppose you are more pessimistic. Put dividend growth at zero and the discount rate at 10 per cent; then the shares are worth only $30.
At this point you may be inclined to give up on the use of DCF measures of fundamental value. Yet the DCF technique doesn't create the problem – it reveals it. Most of the value of today's major businesses really does lie in a future beyond the five or ten years that seem reasonably foreseeable. Small changes in assumptions, compounded far into the future, have large effects on current values. DCF calculations are illuminating, but don't make the mistake of believing that any particular DCF calculation is true.
There is a partial solution to these problems, which is to cut off the cash flow analysis after a period – five years is a common choice – and make an assumption about the value of the business at that date. This is not a very satisfactory answer. It resolves the difficult question posed initially – What is the fundamental value of the business now? – by requiring an answer to a more difficult question – What will be the fundamental value of the business five years from now? At this point the first principle of asset valuation – the value of an asset is the cash it generates for its owner – becomes confounded with the second – an asset is worth what someone is willing to pay for it.
That emphasis on asset value suggests another approach to valuing a business. Look not at the earnings but at the assets the business owns. Every company in which you might invest produces a balance sheet, an accounting statement of the value of the assets it owns. Traditional accounting practice held that the balance sheet should report the lower of the amount the company paid for an asset and the value of the cash the asset will generate over its life. While this formulation closely mirrored the general principles of asset valuation, it did so in a deliberately cautious way. This caution arose, first, from insisting that the lower of the possible answers was used, and, second, by substituting the amount the company paid for the asset for the amount someone else would be willing to pay for it.
The pioneers of securities analysis, such as Benjamin Graham and David Dodd, operated successfully by simply having the skills and energy to penetrate accounting statements. Since the assets reported in the balance sheet were generally estimated conservatively, shares whose prices were below the reported asset value were generally good buys. Some investors became rich by identifying companies where the value of the assets the company owned – shops or land – was greater than the value of the company as a trading entity.
Those days of easy returns have long gone. The techniques Graham and Dodd employed were possible because they were early to realise that business assets had become less specialised. The value of a railway or brewery was inseparable from the value of the company that ran the trains or brewed the beer. But the modern office or shop has many potential tenants; the trucks and the distribution warehouses can distribute many different products. Today most companies have stripped their own assets. When assets have a value separable from the company that uses them, companies have generally realised that value.
The important assets of modern businesses are no longer their buildings, plant and machinery but their intangible property – their reputation with customers, their relationships with suppliers. When pompous executives proclaim, 'Our people are our greatest asset', what they say is often true. Most other assets are likely to be owned by, or pledged to, the financial sector.
Financial companies directly or indirectly own much of the plant or the property occupied by other industrial and commercial businesses. For banks and insurance companies and for property and investment companies it remains important to understand the structure of their assets. There are a few sectors, such as oil and house-building, where the market position of the company will depend on its control of future reserves of petroleum or of land. But while company accounts report a figure for the net assets of the company, the asset value is usually now a poor guide to the value of its shares.
Investment analysts used to distinguish value and growth investing. Broadly speaking, value focused on companies with strong tangible assets, growth on companies with strong intangible assets. Nowadays these categories have become too blurred to be useful. Some people think that more systematic attempts should be made to include the value of intangible assets in company accounts.
Accounting theorists want to revive the balance sheet for a modern era; creative finance directors want to present the most favourable impression of the company's affairs. But the goodwill you find on corporate balance sheets is usually an accounting fiction that arises when a company pays, or overpays, in the acquisition of another business. These numbers should be disregarded. Corporate accounts reveal imperfectly, or not at all, the most important determinant of a company's fundamental value – its competitive advantage.
**Competitive advantage**
In the nineteenth-century businesses that Karl Marx described, the capitalist owned the factory and the plant in which workers produced the goods, and that state of affairs defined both the economic and the political relationship between them. The analytic framework of that era lingers, although the modern business environment is quite different. The bosses of large corporations have authority because of their job titles, not their wealth (though they are increasingly inclined to use the former to secure the latter). They don't own the place where they work, and nor, as a rule, does the company that employs them.
The modern company is best viewed as a collection of capabilities. It is defined by its brands and its products, through its internal systems and its relationships with customers and suppliers, by the resources to which it has access and the operating licences it holds, and by its reputation with the public, governments and prospective employees. We have capitalism without capital.
When the company's most valuable assets descend in the lift at the end of the month, they take their pay cheques with them. The ability of the company to generate returns for shareholders depends on its ability to add value to its employees through its own capabilities. In a competitive market-place a business can add such value only if there is something distinctive about these capabilities that yields a competitive advantage.
A distinctive capability becomes valuable when it yields a sustainable competitive advantage. That is what Warren Buffett understood when he bought stakes in Coca-Cola, Gillette and the _Washington Post_. He didn't think he was buying a head office in Atlanta, a razor blade manufacturing line or a printing press. He was buying powerful brands and irreproducible market positions.
Let's look at the competitive advantages of some of the largest companies in the world. Exxon Mobil is the largest of the successor companies which were established on the break-up of John D. Rockefeller's Standard Oil in 1911. The sustainability of its competitive advantage is demonstrated by the durability of the 'seven sisters' which ruled the world industry from that break-up to the rise of OPEC in the 1970s. The seven have been reduced to four by mergers. Exxon acquired Mobil, while Chevron is the combination of Gulf Oil, SoCal and Texaco. Alongside these were the European leaders BP and Shell. Although the power of the oil majors has been somewhat reduced by the greater influence of the governments of oil-producing states and the national champions they have created, the skill base that originated in Rockefeller's day remains distinctive and indispensable.
It is hard to imagine that any firms outside that small group could replicate their integrated systems or ability to attract and develop strong teams of managers and engineers. The well-resourced national oil companies of major producing states such as Norway and Saudi Arabia have not really tried, while Russia's predatory Gazprom continues to depend on Western expertise and distribution systems. The distinctive capabilities of these businesses extend from privileged access to oil reserves through their integrated production and distribution systems to their retail brands.
HSBC has an even longer history than Exxon, although its emergence as a major international bank is recent. As the Hong Kong and Shanghai Banking Corporation, the company facilitated trade and investment in the Far East in the heyday of the British empire. In 1980, with China still emerging from the rule of Mao and the status of Hong Kong uncertain, the future of the business seemed in doubt. But the company's fortunes have since been transformed by a programme of acquisitions around the world and the emergence of China as an economic powerhouse. Well-established banks have a series of competitive advantages: a name that others trust, although they have squandered that asset in pursuit of short-term profits; a customer base of corporate and private accounts; and branch networks. The company has attempted to develop HSBC as a global retail brand, re-badging the local banks it has bought in many countries.
Novartis is the combination of the pharmaceutical businesses of three of Europe's largest established chemical companies – Ciba, Geigy and Sandoz. Although the company's headquarters is located in Switzerland, the company undertakes research, manufacturing and sales around the world. Its chief executive and the head of its pharmaceutical division are both Americans, and the US is by far the company's largest market.
Novartis's competitive advantage depends in significant part on the intellectual property it derives from drug patents. Even though these patents have a limited life, drugs continue to have brand value even in the face of the generic competition that emerges when such protection expires. As Europe's leading pharmaceutical company, Novartis enjoys a privileged position in association with the strong academic science base of Britain and Switzerland.
But original research in pharmacology is moving from large pharmaceutical businesses towards a mixture of publicly funded basic research and smaller specialist businesses financed by venture capital. The era of blockbuster drugs, which alleviate but do not cure the chronic illnesses of the well-to-do, is coming to an end. Companies like Novartis are increasingly organisations focused on marketing and the management of the complex processes of regulation.
Nestlé, also headquartered in Switzerland, is the world's largest food company. Its nineteenth-century foundations lie in the discovery of patented processes from the manufacture of condensed milk and milk chocolate. The company's brand names include not only the ubiquitous Nescafé but also Carnation, Gerber, Maggi and Purina. Even more than Novartis, Nestlé is a genuinely global company. Its current CEO is Belgian, and his predecessor, now chairman, came from Austria.
These companies illustrate common types of competitive advantage. Generally, the distinctive capabilities that create competitive advantages fall into four broad categories:
* Brands and reputation – the essence of Nestlé. Also important for Exxon Mobil, both for the retail brands that attract its customers and the reputation that encourages governments to select the firm as a partner in exploiting oil reserves. Brand and reputation are also a strength of HSBC, although the value of financial brands has been much reduced by recent misconduct and incompetence.
* Strategic assets – exclusive access to specific resources, such as the licences of HSBC and the reserves controlled by Exxon.
* Proprietary architecture – systems and structures, such as those of the oil companies and banks.
* Innovation and intellectual property – vital to Novartis.
The value of competitive advantages depends on the extent to which they are appropriable and sustainable. A competitive advantage is appropriable only if the company can defend it against suppliers and customers. An asset is not appropriable if it can go down in the lift – much of the value created by financial firms is paid out in bonuses to employees. A competitive advantage has continuing value only if it is hard to replicate.
In a slow-moving industry like food, a brand like Nescafé can be sustained, with appropriate management, indefinitely. If the market can be developed through geographic expansion – introducing Nescafé to China – or product extension – such as the creation of Nespresso – the value of this competitive advantage may continue to grow.
In a fast-moving market, such as consumer electronic goods, a brand can maintain its value only if the name can be attached to a stream of innovative products. Apple has enjoyed astonishing success in this. The company's technology is not, in itself, remarkable – the underlying electronics is available to everyone. But the company has repeatedly shown itself able to package these capabilities in products which meet consumer needs – often consumer needs which the consumers did not know they had.
Apple's first breakthrough was the graphical user interface (GUI), which allowed small computers to be used by people who understood nothing of DOS or programming. But this innovation – which had in fact been pioneered by Xerox – was soon imitated by Microsoft with Windows. The idea of a GUI could not be protected by patent or copyright, while the computer code of the Microsoft operating system did enjoy such protection. But Apple could repeat its innovative performance, more sustainably, with its music and video player, the iPod, and then with the iPad. The market leadership these products delivered proved sustainable. Although the concept of the smartphone could be imitated – and was – the operating systems, linked to Apple computers, were intellectual property. These various contrasts raise questions about the economic effectiveness of such legislation.
The value of such innovations depends critically on their timing and presentation. The tablet computer had been invented several times before – even by Apple itself. But only when the range of mobile services had reached a certain stage of development did this idea appear irresistible to customers. Innovation is not only, or primarily, about the application of advanced science. Commercial success in innovation is largely about marketing and timing. Pharmaceuticals is one of relatively few industries in which intellectual property protection is sufficiently powerful to make innovation appropriable.
Patents and other strategic assets, like those of Exxon or HSBC, are powerful and valuable capabilities. But these depend on governments, and what governments give they can also take away. The Russian government expropriated part of BP's assets in that country, while Exxon's Russian assets were much diminished in value by sanctions imposed after Russia's invasion of Ukraine. Western governments rarely expropriate but can, and do, set and alter the terms on which monopoly franchises are offered, as demonstrated in the US government response to BP's failures. Regulatory authorities can reduce the value of existing franchises by issuing more of them, as with airline liberalisation. More sustainable strategic assets are found where the monopoly is intrinsic rather than government-conferred, as with Heathrow and other London airports (owned by BAA, now a subsidiary of the Spanish company Ferrovial) or the _Financial Times_ (now owned by the Japanese company Nikkei), which operates in a niche profitable for one newspaper but unlikely to be profitable for two.
Valuable competitive advantages are sustainable over time, appropriable for the firm that holds them and defensible against pressure from competitors, suppliers and customers. Is it possible to use these principles to compute, numerically, the fundamental value of a business? It is certainly possible to sketch the outlines of a calculation. Such an exercise may have value as a reality check – could these competitive advantages equal, or be much more or less than, the value implied by the current market price?
You should treat such calculations as illuminating rather than true. I'll discuss bogus quantification further in Chapter 8. The analytic framework described here, like many others, is better used as a set of questions. What are the sources of competitive advantage? Are they replicable, sustainable, defensible, appropriable? The calculations are a better guide to relative valuations than to absolute ones.
Value investing was once the purchase of tangible assets at levels below their market value. Value investing today is buying sustainable competitive advantages at a good price. Charlie Munger, Buffett's partner, has recalled how Berkshire's purchase of See's Candies in 1972 marked that transition. The opportunity to acquire such competitive advantage arises most often because the mind of the market is distracted by essentially short-term considerations: a poor economic outlook for the industry, a strategic mistake by the business, weak management or simply negative momentum.
These are the signals for the intelligent investor to make contrarian purchases. Buffett began his legendary run by purchasing a large stake in American Express when the share price was depressed by a large, but ultimately inconsequential, fraud involving salad oil.
7
**Risk and Reward**
**Risk and uncertainty**
The first share I bought was in a small shipyard called Robb Caledon. The purchase was both a rewarding investment and a rewarding lesson in investment.
You might naturally begin to establish an investment portfolio with a selection of blue chip stocks, such as Apple and Exxon Mobil. Such a strategy sounds much less risky than a stake in Robb Caledon. But when I bought Robb Caledon, I was learning about the capital asset pricing model (CAPM) – the dominant theory of modern financial economics. That theory suggests the opposite conclusion.
Fast-forward more than two decades. I am at a meeting between British government economists and senior executives from defence contractors. The largest of these firms, BAE Systems, had recently suffered huge losses from cost overruns on a major project. Its business was in turmoil, its shares depressed, and loss of confidence and aversion to risk had affected the whole sector.
The government economists, who had recently studied the CAPM at the best graduate schools, explained that the cost of capital to these companies was very low – on a par, in fact, with the government bond rate. They repeated themselves several times. They needed to – the business people reacted as they would have reacted to aliens describing life on another planet. The two sides meant completely different things when they talked about risk. In this chapter I'll explain the CAPM and illustrate the different interpretations attached to the term 'risk' by business people and financial professionals.
During the New Economy bubble, stocks carried large risk for investors. Speculators hoped for – and sometimes made – large profits, and ultimately suffered very large losses. The principal risk faced by investment managers, confronting the same opportunities, was that they would lose the support of their customers by failing to match the performance of their rivals. The risk perceived by the fund manager was different from the risk faced by the customer.
Common sense tells us that flying is risky. Travelling in a metal tube at high speed and an altitude of 10,000 metres is intrinsically dangerous. But precisely because flying is potentially dangerous, planes are scrupulously maintained, pilots are rigorously trained, and safety precautions are extensive. Fewer than 1,000 people per year die in commercial plane crashes (and in 2014 and 2015 the four most serious air accidents were all caused by criminal activity, either by aircrew or from the ground, rather than by pilot error or equipment failure). Falling down the stairs is a more common cause of death.
Worldwide, 1.2 million people die each year in road accidents. But the death rate from motor accidents in Sweden is less than one-fifth the global average, despite high levels of vehicle ownership and use. Fewer children are killed on the roads in Britain today than in the 1920s. The roads are more dangerous, but, because parents recognise this, their children are safer. The statistics do not lie, nor do our assessments of risk. The dangers that risks pose to our health – and our financial well-being – are the product both of the intrinsic riskiness of an activity and of the measures, both social and individual, that we take to counteract the dangers. The figures showing that flying is safe and the intuition that flying is risky are both correct. It all depends on what you mean by risk.
Until recently, households found security in large families. In many parts of the world they still do. Infant mortality was high, and only in a big family was it probable that some children would live to become providers and carers. Few people in rich countries now think in this way. Large families are more exposed financially and run more serious risk of grief through accident or serious illness. What is risky in one context may be prudent in a different one, and vice versa.
The government economists who thought some risks could be completely diversified used a different frame of thought from the business people who worried that a risky project might bankrupt their companies. The investor who worried about losing his shirt and the fund manager who worried about losing his job had different perceptions of risk. The nervous passenger and the student of accident statistics see flight risk differently. Some parents think it would be improvident to have a large family; others think it would be improvident not to.
The antithesis of risk is security. Much of the ambiguity in the meaning of risk results from different concepts of security. Do not confuse security with certainty. Certainty is knowing what is going to happen, but perhaps what is going to happen is not very satisfactory. The man who knows he is going to be executed tomorrow has certainty but not security. More generally, low aspirations give a degree of certainty but rarely security. People who crave certainty frequently achieve its appearance by relinquishing control over their lives, only to find that there are no certainties. People who seek careers that offer lifetime employment only to see their employers renege on the implicit promise. People who make career and retirement plans and are suddenly diagnosed with a grave illness. The lowest percentiles of the risk distribution are generally the consequence of wholly unanticipated events.
The citizens of East Germany functioned in a predictable environment at the price of a dull life and a modest standard of living. Many were willing to accept that deal, only to witness the collapse of the regime which had made the promise. To take control of one's own life or financial affairs may seem to create uncertainty, but it is a prerequisite of security in a world that is inescapably uncertain. The most attractive financial goal for most people is to have enough money not to worry about money.
Security is a high degree of confidence that one's reasonable expectations will be met. Not 100 per cent confidence, because in a real world – as distinct from a model – there can be no such thing. For the prudent investor, security means confidence that investment performance will allow the achievement of realistic goals, such as the standard of living that might be financed from a real return of 4 per cent; risk is failure to achieve such a goal. For the investment manager, security means earning a large bonus for outperforming the market or attracting new business, rather than being fired. All these perceptions are subjectively valid.
These multiple perspectives lead to confusion in the ways people describe and interpret risk, and to ambiguity in their use of the word 'risk'. When you and your investment manager talk about risk, you probably mean different things. These differences of definition mean that misunderstandings and errors are frequent. There are consequential opportunities to gain, and to lose, substantial amounts of money. Much financial services activity relies for its profitability on confusions and inconsistencies in attitudes to risk and interpretations of risk.
**Subjective expected utility**
Yet among academics who teach students the theory of finance – and I have been both student and teacher – there is one, and only one, correct way of thinking about risk: the way of thinking about risk I had in mind when I bought Robb Caledon.
The system that I will call the theory of 'subjective expected utility' (SEU) is so universally accepted in financial economics that any other behaviour is described as irrational. SEU is the basis of virtually all the sophisticated models and quantitative techniques used in financial markets today. These models are both normative and positive – they claim that people should think about risk in SEU ways and derive conclusions about market behaviour from the assumption that they actually do behave according to the precepts of the model.
Although this SEU approach dominates today, it was not always so. Early in the twentieth century two alternative ways of describing risk vied for supremacy. On both sides were an urbane Englishman from Cambridge and an assertive American from Chicago. The winners were Frank Ramsey and Jimmie Savage.
Ramsey was a Cambridge philosopher who made important contributions to economics in his spare time. He died at the age of twenty-six, but his brother, perhaps the less talented sibling, went on to become Archbishop of Canterbury. Savage was labelled Jimmie by a nurse since his parents, not expecting him to live, did not bother to name him. (He survived another fifty-three years.) Ramsey and Savage, the founders of SEU, proposed a means of taking the mathematical tools of probability theory far beyond their initial field of application – the gaming saloon – into the boardroom and onto the trading floor. They are the founders of modern techniques of risk management.
On the other side of the debate were John Maynard Keynes and Frank Knight. Keynes's fellowship thesis – in effect, his doctorate – was not directly to do with economics. (Keynes, one of the great polymaths of the twentieth century, had taken a degree in mathematics.) His subject was probability theory, and his thesis was eventually published in 1921, more than a decade after its submission. The First World War – and the peace conference at Versailles which followed, and about which Keynes would write so memorably – got in the way.
In the same year Frank Knight, a farmer's son from the American Midwest (who would come to loathe Keynes and all he represented), also published a book, _Risk, Uncertainty and Profit_ , written in a different style but with a related argument. Keynes and Knight emphasised the uncertainty that arose from the necessarily imperfect nature of human knowledge. The future was not just unknown but unknowable.
Donald Rumsfeld expressed the difference between risk and uncertainty with uncharacteristic clarity. Rumsfeld famously distinguished 'known unknowns – the things we know we do not know' from 'unknown unknowns – the things we do not know we do not know'. This chapter will mainly be about risk – the things we know we do not know. The next chapter will deal with uncertainty – the things we do not know we do not know.
The claim made by the SEU school is that analytic tools can enable us to cut through much of this uncertainty with probabilistic reasoning and formal modelling. These approaches would provide the necessary underpinning for the growth in derivative markets after 1970 and the explosion of credit after 2000. They originated in the work of Ramsey and Savage, and received major extension in the modern portfolio theory of Markowitz and the capital asset-pricing model of Sharpe. In the remainder of this chapter I'll make the case for this approach, ahead of a critique in Chapter 8. Let's understand the claims of SEU by returning to Robb Caledon.
**Think probabilities**
Robb Caledon was bankrupt. The year was 1976, and a bill to nationalise British shipbuilding was going through Parliament. If the bill became law, the assets and liabilities of the yard would be assumed by the government, and the shareholders would receive around £1 per share. If the bill failed, the shares would be worthless. They were selling in the market at around 40p.
I had to be persistent to buy these shares. The brokers I called told me that the Labour government was incompetent, and that nationalising shipyards was a silly idea. They were right. They told me that the shipyard was on the point of financial collapse and yet the share price was twice what it had been two months before. They were right about that too. They told me that I risked losing everything I invested. Again, they were right.
All these observations were justified, but none seemed relevant. With youthful enthusiasm, I was applying the lessons I had been taught. Think probabilities – plan for different scenarios, think about the consequences of each and assess their likelihood. Be detached – don't be caught up in events and confuse what you want to happen with what you expect to happen. Mind your portfolio. The risk that is relevant to you is the overall risk of your whole investment portfolio, not the risk associated with the individual securities it contains.
Probabilities were central to my approach. Probabilities are essential to gamblers, though frequently ignored by them. The probability that a toss of a fair coin will show heads is one-half. This is a statement about frequency: if you toss a fair coin many times, approximately half the calls will be heads. If you can be bothered to toss the coin many times, you will verify this statement.
The ancient Greeks laid the foundations of modern mathematics. Although they gambled, they never discovered the – elementary – mathematics of probability, which was not developed until the seventeenth and eighteenth centuries. Two elements of probability theory and the methods of classical statistics based on it are particularly important in understanding the principles of investment – statistical distributions and expected values. Statistical distributions describe the probabilities of different outcomes of repeated random events. If you throw a fair coin twice, the probability that you get one head and one tail is one-half, and the probability of each of two heads or two tails is one-quarter. After many throws, approximately half the calls will be heads, and the distribution of outcomes is described by the most famous of all statistical distributions – the normal distribution – with its familiar 'bell-shaped' curve. The normal distribution is ubiquitous – it describes a wide variety of both social and natural phenomena, from the distribution of voting intentions to the heights of the voters. The assumption that the normal distribution can be applied to the distribution of daily movements in securities prices is commonly made in financial economics. It is an assumption that is illuminating, but not true.
The second key concept from probability theory is expected value. The expected value of a gamble is measured by multiplying the values of the outcomes of a gamble by their probabilities. If a coin-tossing game will pay you €1 if you win, and nothing if you lose, then its expected value is 50 cents. Despite the name, the expected value is not what you should expect. You will definitely not get 50 cents from a single toss of that coin.
But the more repeated events there are, the more likely it is that the outcome will approach its expected value. If instead of gambling to win €1 on a single toss, you were to gamble to win 50c on each of two consecutive tosses, the distribution of outcomes is less extreme. There is now a one in four chance of winning €1, from two heads, and a one in four chance of winning nothing, from two tails. Half the time, one coin will come up heads and the other tails, and you will indeed receive the expected value of 50c. If you gamble for a penny on each of 100 throws of the coin, the expected value is still 50c, and again you will probably receive something close to this figure.
**Comparing distributions**
In investment matters you will be concerned not just with the expected value but with the overall distribution of gains and losses. In particular, you will want to know what could go wrong. You might ask, 'What is the worst that could happen?' The worst that can happen is that you will be convicted of a grave crime of which you are entirely innocent, or that you will be murdered by a passing psychopath. But these events are extremely rare. What you really want to know about is outcomes that are bad but not completely unlikely. Contemplate, for example, the worst 1 per cent of outcomes, and then imagine the best of these conceivable outcomes. This is to take a gloomy view, but not so gloomy that you spend your life in bed. (Though note the observation above that more people die from falling down stairs than in aircraft accidents. There is no escape from risk and uncertainty.)
An outcome that will be exceeded 99 per cent of the time is called the first percentile of the frequency distribution of outcomes. It is not the worst that could happen, but the best of the worst 1 per cent of possibilities. Similarly, an outcome that will be exceeded 90 per cent of the time is the tenth percentile, and you might look at this percentile if you don't want to be quite so pessimistic. Table 5 shows the first percentile of the cumulative frequency distribution of outcomes for the coin-tossing game. On a single toss of a coin, zero is a very likely outcome. The first percentile of the distribution is zero – and indeed zero is the value of all percentiles up to the 50th. If you spread your bets across 100 tosses, however, the first percentile is 38p – on only 1 per cent of such trials will you win less than 38p, and on 99 per cent of occasions you will win more.
The value-at-risk models used by most financial institutions today seek to measure and control risks by setting values for these low percentiles of the frequency distribution of outcomes. The bank asks 'How much could we lose on a bad (say, one in a hundred) day?' The first percentile of the distribution of daily outcomes gives the answer to that question.
Table 5: **The cumulative frequency distribution from tossing a coin**
The same method can be used to describe the upper tail of the distribution. You will obtain the 90th percentile of a distribution at least 10 per cent of the time. So the 90th percentile on a single toss is €1. Just as every percentile below the 50th is zero, every percentile above the 50th is €1. For a gamble spread over 100 tosses, however, the 90th percentile is only 56c. You can expect to make more than 56c only 10 per cent of the time. To decide whether one distribution is preferable to another, it isn't really sufficient to make comparisons of a single percentile. The first percentile is a rather unlikely outcome. The prudent person, rather than the committed pessimist, might focus on the 10th percentile. You would have to be very gloomy to ignore altogether the good outcomes at higher percentiles, and you will want to look at the 50th percentile, the median outcome. In banks the traders tend to look only at the high percentiles, the risk managers only at the low ones. This creates tension between them which is sometimes constructive, often fatal.
The examples with a relatively small number of trials (one and two tosses) have lower pay-offs at low percentiles, and higher pay-offs at high percentiles. The worse outcomes are worse, and the better outcomes are better, in games of one or two tosses than those of games with twenty or a hundred tosses. That is what we mean by saying a strategy is riskier: low percentiles have worse outcomes, high percentiles have better ones.
In the coin-tossing game, the 50th percentile of all games – whether two tosses, twenty tosses or one thousand tosses – is the same, at 50c. (The single toss is different, because an outcome of 50c – the expected value – is impossible; the 50th percentile is either zero or €1, depending on whether you are a glass-half-empty or glass-half-full type of person). The median (the 50th percentile) is the expected value.
But this equivalence of median and expected value holds only if the probability distributions are symmetric. In coin-tossing, good and bad outcomes are equally likely. On a hundred tosses, you suffer a 10 per cent chance of falling more than 6c short of the expected value of 50c, and also a 10 per cent chance of getting more than 6c above the expected value. Many risk distributions, however, are asymmetric. The tailgating driver mostly gets to his destination a few seconds quicker, but occasionally dies in a car crash. The 10th and higher percentiles of these distributions are all good; the first percentile is terrible. The median outcome – the 50th percentile – is much better than the expected value, since the expected value must factor in the catastrophic losses that occasionally occur.
The outcomes of a lottery are also asymmetric, but in the opposite way. The lottery ticket buyer usually loses, but might, with low probability, win the jackpot. The 50th percentile, even the 80th percentile, of the outcomes of purchasing a single lottery ticket are lousy – you lose your stake – but the 99.9999th percentile makes you a millionaire. Tailgating and lottery tickets have highly asymmetric distributions.
The coin-tossing game has another important property: the most likely outcome is that you win 50c. Even on a game of two tosses, there is a 0.5 probability that you win 50c as against a one in four probability of either nothing or €1. The most likely outcome is called the mode of a distribution and many – though not all – symmetric distributions have this property.
Thus there are three different ways of measuring what statisticians call the central tendency of a distribution. The expected value is the mean of the distribution. The median is the 50th percentile: you are equally likely to do better or worse than the median. And the most likely outcome is the mode. The coin-tossing game has the property that mean, median and mode are all, at 50p, the same.
But this is true only because the coin-tossing game has a particular – though common – structure. The outcomes follow what is called a binomial distribution, which becomes the normal distribution when the number of trials is large. These distributions describe what is likely to be observed if the result of a process is the product of adding up the consequences of a large number of repeated but independent events – there is an obvious affinity between the normal distribution and the outcome of a random walk. Many real-world phenomena are like this, which is why the normal distribution is so widely encountered.
But many are not. And theorems that apply to the outcomes of repeated trials do not help us much in thinking about one-off events to which SEU asks us to apply subjective probabilities. If I had thought the subjective probability of Robb Caledon being nationalised was 0.7, then the (subjective) mode and median of the distribution of outcomes was £1 and the expected value 70p. If my subjective probability had been 0.3, the (subjective) mode and median would be zero and the expected value 30p. If your commute to work is scheduled for forty minutes, forty minutes is probably the mode. The most likely – though perhaps not very likely – outcome is that the train arrives on time. The median will be a bit more – the train arrives late more often than it arrives early. And the expected value will be greater still – the train will sometimes be very late but never very early.
Few people without some training in statistics distinguish carefully between the three measures of 'central tendency': expected value – the average of outcomes; median – the representative outcome; and mode – the most likely outcome. And one of the reasons for the popularity of assuming a normal distribution is that you don't have to, because the normal distribution has the property – unusual even among statistical distributions – that all three are the same. But failure to understand these distinctions is potentially highly misleading in financial markets, where many distributions are asymmetric and discontinuous. There is not much exaggeration in saying that such failure was a major contributor to the global financial crisis.
The method of comparing distributions by comparing different percentiles of outcomes is quite general. One distribution is riskier than another if it yields worse returns at lower percentiles, and better returns at higher percentiles. There will be a crossing point. The lower the crossing point, the more attractive is the riskier distribution. If one distribution has worse outcomes in its lowest five percentiles and better returns at all other percentiles, it is more likely to be acceptable than one for which the crossing point is at the 50th percentile.
A common technique simplifies the comparison of distributions with the aid of a mathematical trick. Many statistical distributions, including the normal distribution, are completely described by two parameters. One parameter is the mean or average, of the distribution – what is the average height of a population? The other is a measure of variability – how many people are over six feet or below five feet? If you know the average and the variability, then you know everything about the normal distribution. If two distributions have the same expected value, then a comparison of a single percentile – any percentile – will be enough to tell you which is riskier.
The usual statistical measure of variability is the variance or standard deviation. The Greek letter sigma is commonly used for the standard deviation. This term has entered popular usage, so that 'a six sigma event' is an event that is, or should be, very improbable. The Sharpe ratio, which is the ratio of expected return (relative to a benchmark) to standard deviation, is widely used by quantitative investors. If the analysts' distribution of returns were normal, or followed sufficiently closely another of the family of common statistical distributions – and if the parameters of these distributions were known – then techniques like value at risk and Sharpe ratios would be extremely valuable. The mathematics of probability, which opens the way to the application of powerful statistical tools, could be applied to the analysis of unique political and economic events, such as the failure of legislation and the future of a shipyard. To see some of the possibilities, let's once more go back to Robb Caledon.
**My ship came in**
The fate of Robb Caledon was a risk – a known unknown. People sometimes make probability statements about known unknowns, as when they say, 'The probability that Parliament will approve the shipbuilding nationalisation bill is one-half.' Such a statement is called an expression of personal, or subjective, probability.
When I talked about tossing a coin, I used the terms probability and frequency interchangeably. But I can't do that for one-off political events. Someone who makes that statement about the shipbuilding bill is not saying, 'If Parliament considers the bill one hundred times, the bill will be passed on fifty occasions.' They are making a statement of personal opinion. Or perhaps, in this case, an absence of opinion. It is common to say, 'It's fifty-fifty,' meaning, 'I just don't know.' These expressions sound like statements about probabilities. But can they, should they, be interpreted in this way?
The idea that risk and uncertainty can be handled with the aid of personal probabilities is often called Bayesian, after the Revd Thomas Bayes, an eighteenth-century clergyman who made important contributions to probability theory. Bayesians believe that each of us has a mental probability distribution of likely and unlikely outcomes, which we constantly revise as new information becomes available. Bayesians further believe that the mathematical rules for calculating compound probabilities (and hence statistical distributions) are universal. These rules can be applied to personal probabilities in the same way as they can be applied to frequentist probabilities. There are some obvious difficulties in the use of rules that are based on the observation of repeated events – like tossing a coin – in the interpretation of one-off events – such as shipbuilding nationalisation.
Many people do not find it easy to think probabilistically about what interest rates will be, whether Robb Caledon will be nationalised or what they will be doing in retirement. They ask instead, 'What is going to happen?' A committed Bayesian believes that, however we express our attitudes to risk and uncertainty, personal probabilities govern our decisions. We can discover these buried probabilities by asking questions such as 'How much would you be willing to bet on interest rates being above 5 per cent in twenty years' time?'
Securities markets elicit information by constantly offering such gambles. 'How much would you be willing to pay for a share in Robb Caledon which will pay £1 if the nationalisation bill goes through?' Mr Market asks that question every day when he gives you the chance to buy, or sell, Robb Caledon shares. He asks you to bet on future interest rates every time he quotes a price on a bond.
To make this link from probabilities to market prices, we have to assume that personal probabilities can be used to calculate expected values. If my personal probability that the shipbuilding nationalisation bill would pass was 0.5, and the potential pay-out was 100p per share, my subjective expected value of a purchase was 50p. The implicit rule is to compare the expected value of 50p with the market price of 40p. Since the expected value – given my subjective probability – was above the market price, the stock was a good buy.
While the assertion that 'a fair coin falls heads with probability of one-half' seems to be an objective statement about the world, and can be verified by experiment, personal probabilities are unavoidably subjective. One day the dust will have settled, and Parliament will either have approved the nationalisation bill, or not. Even then no one can say whose subjective judgement, whose personal probability, was right and whose was wrong.
The issue went to the wire. In 2012 the vote on the bill was the dramatic highlight of 'This House', a play by James Graham at London's National Theatre. The House of Commons approved shipbuilding nationalisation by a majority of one, and I duly received the payment of £1 per share. I was pleased with myself, but that doesn't mean my personal probability was right. Nor, if the bill had failed, would that necessarily mean my prior judgement of subjective probability had been wrong. A Bayesian would say that the stockbrokers I consulted had personal probabilities different from mine and that was why they had not recommended the shares.
Does the price of 40p, against a possible pay-off of 100p, mean that 0.4 was the average of everyone's personal probability? No, it doesn't. Many people, like those stockbrokers, attached low personal probabilities to nationalisation – if they had ever thought in this Bayesian way, which is doubtful – and many more people had not considered the matter at all. The price of 40p indicates that the number of people who found the investment attractive was sufficient to hold all the available Robb Caledon shares. Since the shipyard was small, this number did not have to be large (the company never had more than a few hundred shareholders). Only a few people need have personal probabilities in excess of 0.4 to establish a price of 40p for the shares.
Similar reasoning is sometimes used to infer probabilities from market prices. On a Bloomberg screen you can find implied probabilities that the European Central Bank will change interest rates. Commentators sometimes say '"the market" is assuming a 30 per cent chance of an interest rate cut'. But such anthropomorphisation should be viewed here, as everywhere, with caution.
Still, there seems to be a profit opportunity if your subjective probability differs from that of Mr Market. That is what I was exploiting with Robb Caledon. With an expected value of 50p and a share price of 40p, there was an expected profit of 10p. But, just as you will never receive the expected value of 50p when you toss a coin once, the expected value was not what I would receive from my Robb Caledon purchase. I would either have lost 40p or gained 60p. If I had lost, I would not even have had the consolation that if I had bought the share a hundred times, I would almost certainly have made a profit. Coin-tossing is a repeated event. The proposal to nationalise Robb Caledon was unique.
Still, if you always approach one-off risks in this probabilistic manner, while you will lose some and win others, over the long run you will probably come out ahead. That's a good argument, though not a conclusive argument, for the SEU approach. You will need to be sufficiently detached to ride both the swings and the roundabouts with equanimity. But it does not make the Bayesian approach to risk assessment objective or value-free. Frequentist probabilities and subjective probabilities are different. Frequentist probabilities are facts; personal probabilities are opinions. Subjective probabilities are not objectively right or wrong, even with hindsight. The passage of the nationalisation bill did not vindicate my decision to buy. A decision that has a good outcome is not necessarily a good decision, and vice versa. When you can comfortably say to yourself, 'Buying Robb Caledon shares was the right decision even though I lost my entire investment,' you have learned to think probabilistically.
You have also learned why most people – and especially most large organisations – find probabilistic thinking difficult. But you have also learned why individuals who have learned to think probabilistically – such as Buffett and Soros – have an advantage over others whose ability to take risks is constrained because they will be judged by superiors who will certainly exploit the benefit of hindsight. I will elaborate on the benefits of that for the intelligent investor in Chapters 11 and 12.
**Diminishing marginal utility**
The principle that risky outcomes should be judged by their expected value was one of the earliest notions in probability. Daniel Bernoulli, an eighteenth-century mathematician, who played a major role in developing this theory, quickly identified a difficulty. Even if people who calculated subjective expected values would frequently end up better off, expected values were not what seemed to govern their attitudes to risk. Many people would prefer a 50 per cent chance of winning €1 million to a 10 per cent chance of winning €5 million, and the certainty of €500,000 in cash to either. But the expected value of all of the outcomes is €500,000. The example Bernoulli used to illustrate this problem became known as the 'St Petersburg paradox'.
In the St Petersburg game a coin is tossed repeatedly. If the first toss of the coin is a head, the player receives €1. If the first toss is a tail, the prize is doubled to €2 and the coin is tossed again. If the prize is not won, it is doubled again, and so on until a head appears and the game ends. There is therefore a 50 per cent probability that the game ends with a single throw and the player gains €1. There is a much lower probability of winning €4 on the third toss. Simple mathematics shows that the expected value of the game is infinitely large.
If you think this is not a very realistic game, you should know that two and a half centuries later investment bankers would invent 'payment-in-kind securities', loans on which payment would increase if the quality of the underlying credit deteriorated. In essence, these were St Petersburg games and, like St Petersburg games, not a good idea for either banker or player.
No one will pay more than a few pounds to play Bernoulli's St Petersburg game. Its large expected value depends on multiplying large pay-offs by their small probabilities. The game might run for forty throws, by which time you would be richer than Warren Buffett. But few people seem willing to pay much for that opportunity. The resolution of the St Petersburg paradox that is most widely accepted today was known in Bernoulli's time. Jimmie Savage would articulate the reasoning most clearly, and in a manner that would have great influence on financial economics. The St Petersburg paradox, he suggested, is the result of what economists call 'diminishing marginal utility'. The first million is a lot better than the second million, or at least that is what people think before they've got their first million. By the time you have $50 billion, as Warren does, the prospect of another million is of little consequence. So $50 billion is valued at much less than 50 million times $1,000, and the expected _utility_ of the St Petersburg game is correspondingly low even if its expected _value_ is high.
People who prefer a sure thing to a gamble with the same expected monetary value – people who prefer €500,000 in cash to a 50 per cent chance of €1 million and who are not impressed by the St Petersburg game – are described as risk-averse. The standard hypothesis is that most people are risk-averse when they invest. They are also risk-averse when they insure – they accept a certain loss in return for giving up the gamble that their house will burn down or that they will have to pay unexpected medical bills. The insurance premium will generally be greater than the expected value of the claims; otherwise the insurance company would be unprofitable. Expected value is relevant to the insurance company, which can use its portfolio of policies to think frequencies rather than subjective probabilities.
Diminishing marginal utility can account for risk aversion. Such risk aversion can be significant only when the amounts you gain or lose are large relative to your wealth. Think of a gamble in which you can win, or lose, €1, and suppose you already have €14,265. If you refuse the bet, you will still have €14,265, while if you accept it you will have either €14,264 or €14,266. Few people would think there was much difference between these outcomes. Fewer still would think, 'I'd much rather see my wealth increase from €14,264 to €14,265 than see it increase from €14,265 to €14,266.' If the amount you are gambling is relatively small, then you should simply assess small bets by reference to their expected value.
This conclusion has major implications for intelligent investors who follow SEU. Since most individual investments will be small relative to your overall portfolio, far less your overall wealth, expected values are usually the right way to deal with risk. That leads to another, and perhaps the most significant, implication of thinking SEU: always look at risk in the context of your portfolio as a whole, rather than its individual elements. Focus on the outcome, not the process. The 'mind your portfolio' principle is fundamental. To see its consequences I'll go back yet again to Robb Caledon.
**Mind your portfolio**
In 1976 the Labour government promoting the shipbuilding nationalisation bill was struggling and unpopular. Margaret Thatcher had recently become Conservative leader, and her party was gaining support. If the shipbuilding bill had been lost, the government would have faced a 'no confidence' vote in the House of Commons and might have been forced to call an election, which the Conservatives would probably win. This happened, on a different issue, in 1979, and Thatcher then became Prime Minister.
A political development such as this was the main risk faced by holders of Robb Caledon shares, since a Conservative government would abandon plans to nationalise shipbuilding. While Labour's fall would have been bad news for Robb Caledon, it would have caused delight around the City, and the wider stock market would have risen immediately. Let's suppose that stocks would have been 5 per cent higher as a result of Labour's defeat. And suppose that you had £5,000 to invest.
Consider three possible portfolios:
* put all your money into Robb Caledon;
* put all your money into blue chip shares; or
* buy 250 Robb Caledon shares for £100 and put the rest of your money into blue chips.
Here is the value of your portfolio if the bill is passed, and if the bill fails.
Table 6: **Value of alternative portfolios depending on the fate of the shipbuilding nationalisation bill of 1976 (£)**
Which portfolio would have been best? With hindsight, of course, you should have gone for the first, and dived into Robb Caledon's docks. But it is easy to be a successful investor with hindsight. If you felt sure the Labour government would fall, you should have gone for the portfolio of blue chips. The first portfolio is extremely risky, suitable only for the most hardened speculator. The paradox is that the least risky portfolio is not the second, which focuses on safe assets, but the third, which includes a small holding of Robb Caledon shares.
In this example, adding a small investment in a very risky asset reduces the overall risk of the portfolio. You are sure to end up with around £5,150. The risk associated with a portfolio is not measured by adding up the risks of each individual element. A combination of risks may be less risky in whole than in part.
I thought Robb Caledon was an interesting speculation for two reasons. My personal probability that the bill would pass was a good deal higher than Mr Market's. How much higher it would have had to be to have made the purchase worthwhile would seem to depend on how risk-averse I was. But I had also made the kind of calculation illustrated in Table 6. All three portfolios would have had the same expected value if the probability that the bill would be passed were slightly more than 0.4. The 'mind your portfolio' principle transforms the way you approach this problem. If I had bought some blue chips, then however risk-averse I was, Robb Caledon would have been worth buying because its shares would go down when other shares would go up and vice versa. An asset with this property is called a hedge, and adding a hedge to a portfolio reduces overall portfolio risk, even if the hedge is, in itself, a speculative investment – like Robb Caledon. So the shipyard was doubly attractive.
Assets don't have to be hedges to reduce risk. It is sufficient that individual risks are not perfectly correlated with each other. If you bet €1 on a single toss of a coin, you win €1 or zero: if you bet 50c on each of two different throws, you win 50c half the time: if you bet 1c on each of a hundred throws, what you will win, as Table 5 shows, will be very close to 50c. This is the power of diversification. A small share in a large number of independent risks is much more certain than the outcome of any of the risks taken separately.
Hedging is the purchase of assets whose returns are negatively related to the returns on assets you already own. Diversification spreads risks across many securities whose individual risks – like the outcome of the toss of a coin – are unrelated to each other. The risk and reward of a portfolio, and the relationship between them, are determined not just by expected values and the variability of return. The portfolio risk also depends on the extent to which returns from different assets are related to each other. A risk-averse individual can build a low-risk portfolio from a collection of risky assets if the risky assets are appropriately selected. It is impossible to overstate the importance of this idea for intelligent investment.
The variability of a statistical distribution can be measured by its variance or standard deviation. A similar measure – called 'covariance' or 'correlation' – describes the relationship between the variability of two different distributions. Similar assets – if one rises in value, the other is likely to rise in value – have a high covariance. Hedges display a negative covariance. Wholly unrelated variables have a covariance of zero. In a coin-tossing game the covariance between the outcomes of successive trials is zero because successive trials are independent. The probability that the next toss is a head is not affected by whether the last was a head or a tail, although many people, imbued with a folk wisdom they call 'the law of averages', find this hard to believe.
What does it mean to say that the returns from investments are independent of each other? Suppose one company is drilling for oil in Kazakhstan and another has entered a new drug in clinical trials. These outcomes are independent – the dry hole in Kazakhstan does not influence the drug trial in the United States. The risks that a shareholder loses because an exploration does not find oil or that a drug trial fails are called specific risks, because their outcome is determined by factors that are specific to the company itself. However, the returns to shareholders in Exxon and in Novartis are probably correlated. There is no direct connection between oil companies and pharmaceutical companies. But both businesses depend on economic prospects in the same countries, and the share prices of both companies are influenced by the same vagaries of market sentiment. These common risks, which affect all businesses, are described as 'market risk'.
The relationship between the risk on an individual share and the market risk of a portfolio that follows the market as a whole is described by the parameter _beta_. Mathematically, _beta_ is computed from a regression equation that estimates the effect of market movements on the security in question. A security will have a _beta_ of one if a 1 per cent market movement will change its price by 1 per cent.
A share may have a low _beta_ for one or both of two distinct reasons. Shares will have low _betas_ if their risks are unrelated to the overall risk of the market. The prices of shares whose returns are derived from oil wells in Kazakhstan, or whose fortunes depend on the results of clinical trials, will have little correlation with the share prices of Apple and Exxon Mobil. A share will also have a low _beta_ if, although its returns are related to overall market risk, the company's activities are not very volatile.
Companies such as food retailers or utilities, whose sales will be little affected by economic recession, have low _betas_. Conversely, shares whose performance is very sensitive to the general economic cycle – producers of steel or basic chemicals – might have _betas_ that are greater than one. An index, which by definition yields the average return on the market, will have a _beta_ of one. For this reason, it is common to call the average return on that asset class 'the _beta_ ' of an asset class.
' _Alpha_ ' is then a measure of the extent to which a fund manager beats that average return, adjusted for the _beta_ of the portfolio – the outperformance after allowing for risk. Lovers of the Greek alphabet will find that derivatives markets extend the lexicon to _gamma_ , _delta_ and beyond. But _beta_ is key to the capital asset pricing model, which is the most influential economic theory in modern investment. It is therefore important to understand its structure, its effect on investment thinking and practice – and its limitations.
**The capital asset pricing model**
Risk-averse investors will need to earn a premium over the yield on a safe benchmark such as bunds or indexed government stock if they are to buy less secure assets. The greater the risk, the greater the premium. But what is the relevant meaning of risk?
At the beginning of this chapter I described the gulf of incomprehension that emerged when economists explained to business people their approach to risk. The latter group did not recognise any operational value to the distinction between specific and market risk that the economists believed was so important. Most people, including those business people, would describe an investment like Robb Caledon as extremely risky. They would say the same of drilling for oil in Kazakhstan, or a clinical trial for a new drug or accepting a fixed-price contract to develop new defence technology. They would think of a big holding in Nestlé or Exxon Mobil, by contrast, as safe but boring.
The 'mind your portfolio' principle says that this view gets matters the wrong way round. Robb Caledon and the clinical trial are not really speculative, because the risk can be spread and diversified. So long as the risk of any of these individual stocks is only a small part of your portfolio you can employ diversification to sleep soundly at night. Conversely, shares in Nestlé and Exxon Mobil are risky because investors cannot hedge or diversify them much – the returns from these investments are likely to be broadly in line with those of the market. Suppose everyone thought this way. Suppose everyone was a 'rational' investor, motivated by the principles of SEU. Then the general opinion would be that shares with specific risks were not very risky but that shares with market risk were. That general opinion, like all general opinions, would be 'in the price'. The implication is that what appear to be risky securities – shares in Robb Caledon or oil exploration in Kazakhstan – would offer lower yields than boring blue chips because the speculative shares have specific risks while the blue chips suffer market risks. This idea is the basis of the CAPM that won the Nobel Prize for its inventor, Bill Sharpe (of the Sharpe ratio). In the CAPM, risk and reward are determined by the _beta_ of a security. Robb Caledon had a _beta_ around zero, because the risk was entirely specific (the company may even have had a negative _beta_ , because its share price might have gone up when the market went down, and vice versa). Such an investment is almost as good as a risk-free asset and will therefore yield very little more than a risk-free asset. An asset with a _beta_ of one will have the same expected return as the market as a whole, and investors will demand a higher return still for an asset with a _beta_ above one. In principle, this approach can be applied to all assets, not just shares. You can calculate _betas_ for different kinds of property and for other asset classes. Commercial risk measurement services offer quantitatively minded investors estimates of _beta_ for the whole range of securities they might buy.
There is another customer base for these risk measurement services. The cost of equity capital to a firm is derived from the long-run expected rate of return on its shares. The expected return on equity is the combination of the _beta_ of the individual firm and the equity risk premium for the market. For adherents of the CAPM this expected return is the return required by the market and, therefore, the basis of the cost of capital estimates needed for the DCF appraisal of business investment.
Companies such as Robb Caledon, or a small oil exploration company, or a pharmaceutical business whose future depends on the outcome of a current clinical trial, should have a low cost of capital because the risk of investment in them, although large, is perfectly diversifiable. Companies such as Nestlé or Exxon Mobil, with returns likely to be more in line with the overall market, should have a higher cost of capital. This is the argument that the government economists were presenting to the defence contractors, who did not believe it.
And empirical evidence is not very favourable to the economists. The CAPM follows directly from the notion that people both should, and do, follow SEU. It predicts that the returns on different securities will be determined by two factors: the overall return on the market and the _beta_ appropriate to that security. There are problems with both parts of this claim. SEU explains the risk premium in terms of diminishing marginal utility. Investors do not judge risky ventures at their expected value because they value the second million of their wealth less highly than the first. But this factor doesn't seem to be nearly large enough to account for the size of the historic equity premium – the difference between the yield on shares and the yield on safe assets. Models based on risk aversion from diminishing marginal utility struggle to explain why the risks of equity investment would, in an SEU framework, justify a risk premium as high as 1 per cent.
The data in Chapter 2 suggested that over the last twenty years the difference in return between safe assets and shares has been around 5 per cent. The longer-term analyses published by Credit Suisse support figures of this magnitude with analysis from many markets over periods as long as a century. The inexplicably large difference between the predictions of SEU-based models and the observed equity premium is called the 'equity premium paradox'.
The second component of the CAPM is the use of _beta_ to explain relative returns on different assets. The most exhaustive empirical study of the issue, by Eugene Fama and Kenneth French, concluded that estimates of _beta_ contributed nothing to an explanation of the historic pattern of returns. These results have been disputed, but have not really been shaken.
Perhaps my business people, shaking their heads around the table, gave the answer. The CAPM describes what the world would be like if everyone behaved according to the principles of SEU. But its counterintuitive predictions suggest a world in which people don't behave that way. If the cost of capital were significantly lower for firms such as defence contractors, speculative oil drillers or small drugs companies, than for businesses with mainstream activities that followed closely the business cycle of major economies, then you might expect that people who had developed successful corporate careers would have noticed. I am sure those at that table were not dissimulating when they said that they had not. And nor had their bankers.
As SEU took hold in the 1950s, a French economist, Maurice Allais, developed a critique. Allais gave examples of common decisions that seemed to violate the principles of SEU (the Allais paradoxes). He wrote in French (no major economics journal would accept such an article today) and hinted that the whole exercise was an American plot. Although the power of his analysis created continuing nagging doubt, he failed to shake the consensus. Two decades later, two psychologists, Danny Kahneman and Amos Tversky, would undertake experiments in which they offered their subjects choices among risks – they found frequent examples of behaviour inconsistent with SEU. All these men would eventually receive the Nobel Prize for founding the subject now known as behavioural finance or behavioural economics. The motivation for behavioural finance is recognition that many people don't behave as SEU would seem to require.
Many real-life investors don't think probabilities, don't mind their portfolio and, above all, don't remain detached. They engage with the process, instead of focusing cold-bloodedly on the outcome. They look at their smartphones several times a day, fixated by screens and clients. They feel upset when shares they own go down (even if you are Warren Buffett, your shares will fall in value on almost half of all the days you own them). Some people need a premium to contemplate the risk of loss at all, even before they take specific account of what those losses might be.
SEU and CAPM suggest you need a premium much less than 5 per cent to compensate you for equity risks. The evidence suggests you will nevertheless earn that 5 per cent, because other investors, who don't think the SEU way, are wary of such investments. The CAPM says you have to pay a price for diversifying your portfolio. The empirical evidence says you probably don't.
SEU and CAPM are theories that are illuminating but, as explanations of actual behaviour in financial markets, are not true. They are illuminating, in part, because the investment strategies they suggest are more profitable if others don't adopt them. The equity premium paradox works in your favour if you think SEU and others don't. As with EMH, the intelligent investor can gain by understanding these theories, but will lose by believing them. That is why this chapter is the longest and most complicated, but also the most potentially rewarding, in this book.
8
**A World of Unknowns**
**Unknown unknowns**
There were no derivatives markets in the savannahs, where our ancestors developed modern brains. Casinos (in which the highest value counters were the blue chips) came into being only at a late stage of evolution, although an early stage of history. Hunters gambled in the evenings around the campfire and used animal bones as primitive dice. Human lives were always full of uncertainties, but pure risks – the well-defined problems found in games of chance that can be described using an approach to probabilities based on frequency of outcomes – were invented for popular amusement.
The claim made for SEU is that this calculus of probabilities derived from the gaming room can be applied to one-off events, such as the progress of shipbuilding nationalisation, as well as to repeated events, such as tossing a coin. Here is Milton Friedman, doyen of the Chicago economists who were founding the theory of finance, in 1976 (not entirely coincidentally, the year in which I bought those Robb Caledon shares):
> In his seminal work, Frank Knight drew a sharp distinction between risk, as referring to events subject to a known or knowable probability distribution, and uncertainty, as referring to events for which it was not possible to specify numerical probabilities. I have not referred to this distinction because I do not believe it is valid. I follow L. J. Savage in his view of personal probability, which denies any valid distinction along these lines. We may treat people as if they assigned numerical probabilities to every conceivable event.
The fate of Robb Caledon was a risk – a known unknown. Either Robb Caledon would be nationalised or it would not. Its shares would be worth either 100p or nothing. Some issues in financial markets are of this kind. Interest rate risk and credit risk seem to be known unknowns. But there are also many uncertainties, unknown unknowns – events that cannot be described sensibly with probabilities, because we do not know what these events might be.
SEU may work well for 'known unknowns'. If all potential states of the world are known (even if the one that will materialise is not), if the outcome of a gamble will be clearly and definitely recorded, if the processes that determine that outcome can be treated as random, then SEU is a guide to action. The gamble on Robb Caledon had some of these characteristics. There seemed only two possible outcomes; the time-scale was short and defined; the consequences of each appeared clear. But this case was exceptional. I have difficulty in thinking of another in my investment lifetime in which the issues were posed so sharply.
And even then there was an important difference between the fate of Robb Caledon and the toss of a coin or throw of a dice; the result of the vote on shipbuilding nationalisation was no random event. The outcome would be the result of a political process – the complexity of that process well illustrated in Graham's play – and any assessment of the likely result was properly a matter of expert judgement.
When President Obama held the crucial meeting to determine whether to go ahead with the raid that killed Osama bin Laden, he 'asked for confidence that bin Laden was in the Abbottabad compound'. 'The estimates ranged from 10 per cent to 95 per cent certainty.' Obama described his reaction: 'I'm accustomed to people offering me probabilities. In this situation, what you started getting was probabilities that disguised uncertainty as opposed to actually providing you with more useful information.' As discussion continued, the President interrupted. 'This is fifty-fifty. Look, guys, this is a flip of the coin. I can't base this decision on the notion that we have any greater certainty than that.'
When Obama said 'this is fifty-fifty', he did not mean that he thought the probability that bin Laden was in the compound was 0.5; still less was one of the most important decisions of his presidency based on 'a flip of the coin'. He was using these phrases, as people often do, to describe radical uncertainty. The mission was subject to known and unknown unknowns – the known unknown of whether bin Laden was or was not in the compound, the unknown unknowns which made the expedition by the SEALs so hazardous. Obama, and his advisers, simply did not know, and, as he recognised, the probabilistic statements of his advisers 'disguised uncertainty as opposed to actually providing you with more useful information'.
Obama recognised the limited value of probabilistic statements even in relation to 'known unknowns'. Keynes went further. Between completion of his work on probability and its publication the great economist had become famous for his polemical denunciation of the Versailles Treaty that ended the First World War. In a celebrated passage he described the confidence of the pre-war mood. The English upper-middle class viewed its comfortable, stable environment as a permanent condition. If that world was transformed by the war that followed, the world of the central European middle class was shattered. Wealthy families, whose lifestyle had seemed secure, lost everything in the default of Russian bonds, as a result of hyperinflation in Germany and central Europe, and from the expropriation of Jewish property. As parents waved goodbye to sons who went enthusiastically to war in 1914, they had no conception of what the future held in store. They fell victim to unknown unknowns – events that cannot be sensibly described with probabilities, because we do not know what these events might be.
The future has not become any more certain. The collapse of the Twin Towers was a major event for financial markets, as for international politics. No one, on 10 September 2001, could sensibly have framed or answered the question 'What is the probability that the World Trade Center will be destroyed tomorrow by a terrorist attack?' Keynes claimed that there could be no scientific basis for an assessment of probabilities in the face of such radical uncertainty. The right response to the question 'What will interest rates be in twenty years' time?' was and is 'We simply do not know.' A question about the level of interest rates appears one that might be answered probabilistically. But Keynes's confession of radical uncertainty even in that connection was oddly prescient. A holder of British government bonds in 1941 could not have been certain whether he would be repaid in sterling, in Reichsmarks or at all.1
Investors are vulnerable to defined, identifiable risks – interest rate risk, momentum and mean reversion in share prices. But investors are also vulnerable to fundamental uncertainty. Questions like 'What is the future of the Middle East?' 'What will be the economic consequences of China's rise?' or 'How will economic and political systems deal with climate change?' are open-ended. We cannot fully describe the range of outcomes, and decades from now there will still be disagreement over what the outcomes proved to be. And the question President Obama faced was not 'Is bin Laden in the compound?' but 'Should I order the raid, and what will the consequences be?'
We may attempt to transform these open-ended questions into more narrowly defined ones such as 'Will Bashar al-Assad still be in power in 2018?' or 'What will be China's GDP, or the average world temperature, in 2025?' But even if it were possible to make compelling, probabilistically expressed prognostications on these questions – and it is not – the numbers would not tell people what they really want to know. Imagine a time traveller from the nineteenth century asking us what the world we live in is like. They would not understand enough about that world to be able to frame sensible questions. We suffer not just from ignorance of the future but from a limited capacity to imagine what the future might be. People who are today concerned about conflict in the Middle East, China's rise or climate change would not have been worrying about these issues twenty years ago. They would have been worrying about the Cold War, Japan's economic pre-eminence and the effects of AIDS.
These earlier uncertainties have largely been resolved, and in ways that few people expected. But the key point is not that we mostly fail to anticipate the answers, but, rather, that we mostly fail to anticipate the relevant questions. No one predicted the catastrophes of the twentieth century – the stalemate of the First World War, the influenza pandemic, the murder of millions of people by deranged dictators.
The same was true of transforming political and economic developments – the rise and fall of Communism in Russia, decolonisation, the development of information technology, the changed role of women in society. Such failure of imagination is inevitable. If you could anticipate the functions and uses of the personal computer, you would already have taken the main steps towards inventing it. To describe a future political movement or economic theory or line of philosophical thought is to bring it into existence.
Many great geopolitical events are unknown unknowns – the First World War, the rise of Hitler, the attack on the Twin Towers. But unknown unknowns are also part of everyday life. Your journey home from work lends itself to description in frequentist terms. You have a rough mental probability distribution of outcomes in your mind, reflecting the schedule and reliability of trains, the variability of traffic conditions. But the outcome is also determined by events. You meet a long-lost acquaintance; you see an item in a shop window that attracts your attention; you slip on the pavement. You did not anticipate such events because you had not thought about them, and could not sensibly have attached a probability to them if you had.
More of these events will disrupt your journey than will accelerate it, which is why your estimate of journey length will be below its expected value. Most people think of risk as an adverse event rather than a distribution about a mean (this was another source of the gap of comprehension at my meeting between the business people and the economists). For good reasons, our thinking about an uncertain world is partly probabilistic, partly not.
In this light, consider again the risk appraisal tools described in Chapter 7, which looked at the outcomes associated with various percentiles of the frequency distribution of outcomes. Your routine experience of how long it takes you to get home from work is likely to be described well by a standard statistical distribution. But the same is not true of experiences that are non-routine: the occasional emergencies, the accidents, the days on which events at work make it impossible to go home at all.
These events are rare, but the extremes of the frequency distribution of outcomes are the product of these out-of-the-ordinary events. When the credit crunch began in August 2007, David Viniar, chief financial officer of Goldman Sachs, told news reporters that his firm had experienced a 25-sigma event several days in a row. A 25-sigma event is one with a probability so low that the noughts required to express the probability would occupy several lines of this page. Goldman Sachs could not expect to encounter a 25-sigma event on even one day if it had existed for the entire history of the universe. Of course, no 25-sigma outcome had happened. What had happened, and what Viniar perhaps intended to say, was that events had occurred which were not properly considered in Goldman Sachs's risk models.
If a coin comes up heads twenty times in a row, you should consider the possibility that the coin is biased before you conclude that the very low probability that this is the result of twenty tosses of a fair coin has materialised. The limits of probabilistic reasoning are set by the limitations of the models on which they are based, and any model is an abstraction from a more complex context. The actual distribution of outcomes is the product of a combination of the risks that the model describes and the uncertainties associated with the applicability of the model. That is the fundamental reason why a model can often be illuminating but will almost never be true. We use different tools in the face of uncertainty. Most often, we tell stories.
**Narratives and patterns**
If I ask people what they think is going to happen, they will usually respond with a story rather than a probability distribution. Thinking probabilities does not come easily to the human mind. Telling stories and constructing narratives do. The stockbrokers I called in 1976 could not separate in their minds the question of whether shipbuilding _would_ be nationalised from the question of whether shipbuilding _ought to_ be nationalised. They told a story about what they thought would happen, heavily influenced by what they thought should happen. They pontificated about the market and complained about the government, as old-fashioned stockbrokers were prone to do.
We weave narratives and fit events and expectations into them. There is evidence of halo effects – if we like something or someone, we interpret everything we know about them favourably – and of confirmation bias – people interpret new evidence in a manner consistent with the view they have already formed. The people who extended credit to subprime borrowers in the belief that house prices always went up, like the people who planned the Iraq war and thought it unnecessary to plan for the aftermath because troops would be greeted with garlands of flowers, were victims of halo effect and confirmation bias. Expectations had been confused with hopes and dreams.
Narratives and probabilistic thinking fit uncomfortably together because thinking probabilistically means thinking, at the same time, of two incompatible events – that Robb Caledon will go bankrupt, that Robb Caledon will be nationalised. It might have made sense to buy Robb Caledon shares even if you thought the bill was likely to fail. The issue for the probabilistic thinker was whether the expected value was above the market price. It may be appropriate to hold indexed bonds even if you do not think that inflation will rise, if you believe – and you should – that inflation _might_ rise.
Many people find it hard to reason in these ways. Scott Fitzgerald wrote, 'The test of a first-rate intelligence is the ability to hold two opposing ideas in mind at the same time and still retain the ability to function.' And thinking probabilistically requires that capacity: Robb Caledon will be nationalised, Robb Caledon will not be nationalised. In fact, thinking systematically about uncertainty requires that capacity.
The most systematic way of trying to handle incompatible futures is scenario planning, a technique pioneered by Shell and widely used in business. Scenario planning demands multiple narratives. Work out two, three or four internally consistent projections of the future, and plan for what you might do in each. But my experience is that if you try to talk about the future in scenarios you will always be approached by someone who asks, 'So what do you really think is going to happen?' They fail to understand that none of these scenarios will happen – the scenario is only a template for systematising our thought. For a highly intelligent decision-maker such as President Obama, or John F. Kennedy responding to the Cuban missile crisis, working through various scenarios with his advisers was an indispensable prelude to action.
Our ability to tell stories is a valuable asset, the means by which we make sense of disconnected information. But in the financial world this narrative skill often misleads. Bachelier and Samuelson discovered that, in securities markets, purposive and directed behaviour can produce outcomes with the appearance, and mathematical properties, of randomness. But we resist randomness. The search for patterns in randomly generated data warms the heart, as it hurts the wallet.
**Mirages, systems and hot hands**
Our abilities in pattern detection often lead us to mirages in the desert of data. We observe systematic relationships that do not exist, or attribute statistical noise to imagined underlying causes. Such confusions help chartists stay in business. Popular discussion, based on a poor understanding of probability, tells of a 'law of averages' – if there has been a run of heads, the next toss must be a tail. But successive trials are independent, and the probability of head or tail remains at 50 per cent. There is no law of averages, but there are many random walks. Chaos and randomness are rare in nature but common in financial markets.
Hopeful investors and tipsters trail through historical series to find systems that would, in the past, have yielded substantial profits. There will always be some, but that doesn't mean that these systems will be profitable in any, far less all, future periods. Many 'anomalies' in efficient markets disappear when they have been identified. This would be the case even if it were not also true that the very process of trading on the anomaly reduces or removes its profitability. The 'hot hand' – an outstanding series of scores by a player for whom it seems every ball ends in the net – is a well-reported phenomenon. Careful statistical analysis of sporting records suggests that there is not much evidence for 'the hot hand'. Sequences of extraordinary performance are no more frequent than would be expected from chance alone. But long sequences of brilliant scores (or mistakes) stick in the mind.
Are there investment managers with the 'hot hand'? Even if investment returns were the result of pure chance, some managers would seem to have a record of success. But if their returns are the result of chance, you cannot expect that these managers will perform any better in future than anyone else. And mostly they don't. Table 7 is a representative illustration of recent results, based on data compiled by S&P Dow Jones. If you are looking for a manager who will perform well over the next three years, any manager who has been in the top three quartiles is more or less equally likely to do well – or badly. Really badly performing funds, however, tend to remain really badly performing funds. Perhaps this consistently poor performance is the continuing effect of high charges and turnover; perhaps incompetence is more persistent than skill.
Table 7: **Percentage of US mutual funds in each performance quartile, 2009–12, which subsequently achieved above-median performance in 2012–15**
Source: S&P Dow Jones ('Persistence Scorecard', June 2015)
Be warned, however, that funds which perform in the top quartile in one three-year period are more than averagely likely to be in the bottom quartile in the succeeding three-year period. Probably this is because closet-indexed funds, which differ little in composition from their benchmarks, are not likely to be either top or bottom performers. The funds that shoot the lights out are specialist funds in what prove to be fashionable sectors, or idiosyncratic managers who hit it lucky. Even if there were no long-run mean reversion, such funds would stand a fair chance of being laggards (as well as leaders) in any three-year period.
In assessing all data of this kind, it is necessary to beware of 'survivor bias'. High-performing funds are more likely to continue in existence than poorly performing funds, which are often wound up or merged into more successful funds, so that the funds whose historic performance we can observe are better than average. But even without taking account of this effect, it is apparent that you will improve your chance of an above-average performance very little by selecting a fund that has done well in the past.
There are a very few fund managers, like Buffett, whose record of outperformance is so persistent, and continues long after it has been widely identified, that the evidence of competence over chance seems irrefutable. But the hot hand is frequently – and mistakenly – detected when what we see is the result of picking dimes in front of a steamroller. This exercise may be more formally described as an asymmetric distribution that produces frequent small gains punctuated by occasional very large losses. I call these Taleb distributions, or tailgating strategies, after a book in which Nassim Nicholas Taleb offers many illustrations (Taleb, 2001). As noted earlier, tailgating drivers save a few seconds on most trips. Occasionally, you see their smashed cars on the side of the road. Participants dissociate the infrequent failures from the frequent successes. The early arrival is the result of skilful driving; the crash is caused by bad luck. The drivers' judgement is vindicated by their success so far.
The 'carry trade' is a Taleb process in financial markets. Interest rates vary across classes of borrower and across currencies. Riskier borrowers pay higher interest rates. If markets price credit risk correctly, then the premium from the higher interest rate will equal the expected value of the losses from default. For example, if a certain type of borrower will fail to repay with a frequency of 2 per cent per year, then a bank with a portfolio of such loans will break even if it realises interest rates 2 per cent higher than those charged to prime borrowers.
But the distribution of gains and losses is not symmetric. Most borrowers meet the principal and interest on loans as they fall due, and the lender collects, year in, year out, the 2 per cent premium. But when the loan fails, the loss is usually the full amount outstanding. As with the tailgating driver, lenders win most of the time on risky loans, but when they lose, they lose big-time. And when the credit bubble burst in 2007–8, they did just that. In Mr Viniar's (erroneous) description, they experienced 25-standard-deviation events several days in a row. In 2008 the steamroller ran over them.
Still, like the tailgating driver, people who collect these regular profits congratulate themselves on their prowess, win the admiration of their bosses and appear to justify both their own bonuses and those of their superiors. Beneficiaries of Taleb distributions leave their more cautious competitors behind, so a process of rivalry forces others to follow, and to suppress any private doubts. From time to time accidents happen. The losses are large, described as unpredictable, unanticipated, exceptional. One-time heroes are fired. The process begins again. The same financial follies are repeated.
You may wonder, as you view the world of business and finance, why many positions in large corporations are filled by people of apparently modest talent; find it inexplicable that dealers with so little knowledge or skill are applauded for their trading success; be surprised at the stupidity of drivers who take large risks for seemingly inconsequential gains. The answer to all of these paradoxes is the same: the people you see are on the upside of their Taleb distribution (and may remain there until they retire).
Perhaps survivor bias is part of the resolution of the equity premium paradox. I'm writing this book about investing in shares and real estate, not about how to invest in Chinese bonds or South American utilities, both of which were popular choices a century ago. The markets from which we draw our experience are the markets that have performed well. There were no books written in the 1920s on investment in Russia, but, if there had been, they would have had to acknowledge that recent performance did not justify an investment.
Sceptics and heretics sometimes observed that paintings of the devout who had survived tempests and floods after praying for salvation were not persuasive, because there were no paintings of the devout who had drowned after praying for salvation. Many heroes of the financial world are immortalised for similar reasons, and the many books about their successful investment strategies resemble those paintings of the devout.
As with managers, so with asset categories. Markets show short-term positive serial price correlation but long-term negative serial price correlation. So don't be tempted by advertisements displaying how much money has been made recently in housing or commodities (by other people) – that is more likely to be a signal to sell than a signal to buy. Assets that have recently done badly deserve attention – but not indiscriminate attention. The objective is always to exploit negative market sentiment to buy cheaply. Detect regression towards the mean; correct for survivor bias; beware data mining.
**Models and their limits**
Models are an indispensable part of modern finance, and all banks and fund managers use them. The professionalism and sophistication of models and modellers have increased steadily. Anyone equipped with school mathematics can do a discounted cash flow (DCF) calculation. The Markowitz model of portfolio selection, the basis of the CAPM, needs more complex mathematics. But even if Milton Friedman worried that Markowitz's work was outside the scope of the economics of the time, the mathematical techniques are quite simple (and today well within that scope). The Black–Scholes model of option pricing, introduced in the 1970s, is both dynamic and stochastic (it involves differential equations and probability distributions).
These newer models are not rocket science (though they are often called such) but do require university-level training in maths or a maths-based subject. Today an understanding of the frontiers of finance theory demands an advanced degree in maths or physics. Because the financial services sector is very profitable, financial institutions offer large salaries to able mathematicians, but the executives to whom they ultimately report have little knowledge of what these employees do.
You don't have to understand what is under the bonnet of a car to drive it, and similarly you can be a competent manager of a pharmaceutical company or an electronics business without deep familiarity with the underlying chemical or physical characteristics of the product. But you cannot drive a car if you don't understand what a car does, and you cannot organise the marketing, manufacture or distribution of a product if you don't understand what the product can do – and what it can't do.
Senior people in the financial sector – bank executives or fund managers – are not themselves able to build the models on which these activities depend. That is not the problem. The problem is that the techniques are so far outside their comprehension that they do not understand the limitations of the models. While professing scepticism, they take too seriously the conclusions that models generate. I'll call their attitude cynical naïveté. Executives seek 99 per cent certainties from their risk management systems without appreciating that in an uncertain world there are never 99 per cent certainties. This was Mr Viniar's error. Ninety-nine per cent probabilities are derived from models whose applicability is never 100 per cent certain (and that applicability is rarely measurable). The first percentile in the distribution of outcomes is typically generated by events that were not incorporated into the structure of the model.
The attempt in Chapter 6 to apply DCF calculations to asset valuation illustrated a recurrent problem for all models in business and finance. Not only did we not know the numbers that the model required, but the outcome was very sensitive to the assumptions made in entering these numbers. This ignorance leads all too often to bogus quantification. People make up numbers they cannot know to derive supposedly scientific answers to questions whose answers are fundamentally unknowable.
If you have a well-trained financial adviser – one with a financial planning qualification or, better still, a CFA certification – then he or she will very likely build a model of your portfolio in the Markowitz tradition, emphasising the variance and covariances of different assets. That is what financial advisers are taught to do. Risk assessment models in large financial institutions have a similar structure.
In line with the principles of SEU and CAPM these models have three critical inputs:
* What is the expected return on each asset you hold?
* What will be the volatility of that expected return?
* What will be the correlation between the volatility of the return on a particular asset and the volatility of your portfolio as a whole?
But neither you nor your adviser knows, or can know, the answer to these questions.
The difficulties fall into two main groups:
* The assumptions made about the statistical distributions that link the data to the model.
* The reliance on history for that data.
I'll take each of these issues in turn. A common assumption is that short-term movements in securities prices can be described through a normal distribution. If we look at day-to-day movements in share prices, it appears that the normal distribution fits them quite well. But there is a problem of 'fat tails' – there are many more extreme events, especially downside events, than these statistical distributions would permit. If share price movements fitted the normal distribution, it is improbable that a market fall such as that of the US market on 19 October 1987 – a 20-sigma event – would occur even once in the entire history of the universe. But when several such 'once in a blue moon' incidents happen in the course of a few years – as indeed occurred in the last decade – it is time to rethink one's model.
Fat tails are a particularly troubling problem for people who are modelling risk in financial markets, because they suggest that these models fail in precisely the extreme situations in which they are most needed. Although the commonest reaction to the problem of fat tails is to note the problem and ignore it, there are two main strategies that attempt to deal with the issue. One approach involves _ad hoc_ modification to make the results of classical statistical distributions correspond better to reality. Examples are generalised autoregressive conditional heteroscedasticity (don't ask) and a technique more attractively labelled 'robust statistics'. These approaches continue to assume that there is some correct description of the world enabling the properties of its prospective behaviour to be deduced from historical information – that we can find models that are not just illuminating but true, that we deal with risks not uncertainties.
An alternative, perhaps more interesting, critique developed by the mathematician Benoit Mandelbrot, rejects the assumptions of classical statistics altogether. Mandelbrot looked at charts of security prices and saw not a random walk but a beautiful process known as 'fractals'. You encounter fractal geometry when you look at a snowflake through a microscope and discover that, whatever magnification you set, the picture you see is the same.
Distributions that follow fractal processes show altogether different properties from those of classical statistics, with fatter tails, and average outcomes in which extreme events play a much larger role. The physical analogies are not the diffuse movements of solids in solution described by Brownian motion, but earthquakes and avalanches – constant small slips, often imperceptible, punctuated by large shifts. Those analogies correspond to many people's experience of financial markets.
Whether you use the distributions of classical statistics, or the different mathematical structures of Mandelbrot's approach, the information you must use to build a model comes from data on what these returns have been, and how they varied, in the past. For good reasons – this historical record is the only objective information available. Such figures may not be a good guide to the future. Investment allocation models based on recent experience will probably suggest a high proportion in equities. That is because the returns from equities have in the immediate past been more than sufficient to compensate for their volatility. You may also be encouraged by asset allocation advisers to invest more in alternative assets. Alternative assets have similarly yielded high returns that were not strongly correlated, on a year-by-year basis, with other investment classes. But the model tells you these things only because someone has already told these things to the model.
No model can tell you what expected future returns on different asset classes, or the future correlation between these returns, will be. Robb Caledon was a hedge, whose price would move in the opposite direction to the main UK indices. But you would not have learned that by studying past covariances. Indeed, you would have found the opposite. Stocks subject to political risk tended to have high _betas_ , moving with the general market by often violent amounts.
Only the peculiar circumstances of 1976, in which the government proposed to nationalise the company for more than it was worth, gave that share its particular risk profile. If broad economic conditions had improved, lifting the price of most stocks, there would probably have been little impact, in either direction, on the share price of the shipyard. Neither in the past nor in the present was there any stable correlation, positive or negative, between Robb Caledon and the general market.
Such observations illustrate why historic measures of _beta_ have little value. Students of statistics are introduced to the alleged correlation between the stork population and the birth rate. The example illustrates that correlation does not imply causation. Births are a function of population, and dense populations attract storks. But almost all correlations between security prices have the property that gives rise to the stork fallacy. The correlation is the result of some common underlying factor that influences both variables rather than the product of a direct causal relationship between them. Since correlation and causation are not the same, only understanding the underlying causal mechanism can make it possible to know whether a correlation will, or will not, persist in any particular time period. The correlation of stork population and birth rate would continue to hold if new families moved into the neighbourhood, but would not hold if illness or a predator struck the storks. One relationship between Robb Caledon's share price and general market movements held under a Conservative government, and a different one held under a Labour government. Only information outside statistical analysis can tell us that.
The antidote to errors of data interpretation is general knowledge, what many people call common sense: the disparate facts about the world that we apprehend but do not articulate systematically, and which inform every decision we make. Keynes was a successful investor, as well as a man who achieved success in many other fields, in part because he brought to all his activities a range of experience and knowledge of both ideas and affairs probably unparalleled in the twentieth century.
The requirement for general knowledge drawn from outside the model does not make the model useless. A model that makes assumptions about correlations between variables – even arbitrary ones – can illustrate ways in which careful diversification can improve the relationship between risk and return. But the quantitative precision claimed for the results of these exercises is generally spurious. I am a better investor for having learned, and built, models of portfolio selection. I am a better investor partly because these models are illuminating. I understand the relationship between risk and return better as a result. But I am also a better investor because I understand the mistakes made by people who think these models are true.
**Forecasts and their limits**
If financial experts have limited ability to analyse the past, what of their ability to foresee the future? The commentators you hear on CNBC and other financial news channels make projections of general trends – the rise and fall of companies, industries and countries – and forecasts of specific economic variables – inflation, economic growth rate, exchange rates.
The record of the market pundits in anticipating events is poor. But this does not diminish the demand for their services. The power of conventional thinking strikes again. What is valued is not genuine knowledge of the future – to the very limited extent that such knowledge exists – but insight into the mind of the market. Keynes's analogy of the beauty contest is again relevant. The objective is not to predict what will happen, but to predict what others will themselves shortly predict.
The world in the 1990s experienced an unprecedented era of political and economic stability under American hegemony, following the end of the Cold War. The unthinking but contagious optimism of the era fuelled the conventional thinking of the New Economy bubble. Books and blogs proclaimed that the Dow Jones index, then around 6,000 would go to 36,000, or even 100,000, on the back of ever-soaring earnings and the disappearance of the risk premium. (It is currently around 18,000.) The beliefs were strikingly similar to those Keynes described for Britain exactly 100 years earlier. And in the noughties politicians announced the 'great moderation' until they were hit by the most serious financial crisis in eighty years.
Since we are ill equipped to handle inevitable uncertainty, the future is seen through the terms of the near present. The opening of Russia and Eastern Europe, China and India to the market economy is genuinely momentous. But the main beneficiaries of this opening will be the populations of the countries concerned. While it is certain that much money will be made in China, that does not necessarily imply that there is much money to be made there by you and me. And the rise of China and India has not exactly passed unnoticed.
**The China effect is 'in the price'**
People who pronounce on geopolitical developments are mostly talking not about tomorrow but about today. They tend to project current trends to an exaggerated extent and with exaggerated speed. As a result, they overestimate change in the short term while underestimating change in the long term. They treat current events as more momentous than they really are – hence 'It's different this time' or the frequency of portentous talk of 'new paradigms'. Simultaneously, such commentators fail to anticipate the transformational effect of events that have not yet happened and of which they can know nothing – the unknown unknowns that shape our lives. The safe course is to talk about current preoccupations, and that is what successful pundits mostly do. We can all then agree in being astonished by the pace of change.
Economic forecasts of recession or recovery, of interest rates and exchange rates, are part of every investment conversation. Every fund manager delivers an assessment of general economic prospects to clients. The EMH should make you suspicious when people who profess special insight into the future reveal that insight to large audiences – the value of such information would surely depend on its not being widely available. Professional forecasters have little, or nothing, to add to public information – information that is printed every day in the newspapers, which can be ascertained every hour of the day on the internet and which is available for free.
The economists who make such forecasts are valued, and rewarded by their employers, for their television manner rather than the accuracy of their predictions. It is a common perception that economists always disagree – two economists, three opinions. The facts are quite otherwise, at least as far as economic forecasters are concerned. There is typically far less divergence between different forecasts than there is between all forecasts and outcomes.
For economic forecasters, as for other financial pundits, it is better to be conventionally wrong than unconventionally right. To see correctly what others have failed to see, or chosen to ignore, damages careers more often than it enhances them. Such prescience undermines the claim – so important to those who disclaim responsibility for what went wrong on their watch – that what happened could not have been foreseen. Ask the people who tried to warn their superiors before 9/11 (or the credit crunch). The weaknesses of forecasting are compounded by the difficulty of specifying how exactly changes in GDP, or interest rate changes, will translate into market prices. Politicians, people in business and finance, and the public at large display cynical naïveté in their attitudes to economic forecasts, as in their attitudes to models – the same mixture of scepticism and credulousness. They express disdain for economic forecasts as they lap them up. Most professional economists regard this forecasting activity as embarrassing and ridiculous, but it continues because the demand is insatiable. Because so many people want answers to questions that cannot be answered, they refuse to accept the limits of knowledge in the face of uncertainty. They justifiably deride those who profess knowledge of the future they do not have. 'Economic forecasters are always wrong.' Less justifiably, they also deride those who disclaim such knowledge, and emphasise that the future can be described only through many divergent, yet plausible, scenarios – 'Give me a one-handed economist.'
The economic environment is like the weather. We cannot forecast it accurately, but we know quite a lot about its broad properties – it won't snow in England in July, and it might rain on any day of the year. General knowledge of this latter kind is already 'in the price', and people who claim more specific knowledge of the future are mostly charlatans.
The common mistake is to believe that the uncertainty described by Keynes and Knight can, through diligent research or analytic sophistication, be transformed into the well-defined, quantifiable risk that responds to the techniques developed by the successors of Ramsey and Savage. Keynes correctly observed that the only justified answer to many questions about the future is 'We simply do not know', but no one is rewarded for saying that. Many people in the financial services sector profess knowledge of the future they do not have, and cannot have. In contrast, legendary investors such as Buffett and Soros stand out for their readiness to acknowledge the limitations of their – and all – knowledge.
Here is Soros:
> My financial success stands in stark contrast with my ability to forecast events... With regard to events in the real world, my record is downright dismal. The outstanding feature of my predictions is that I keep on expecting developments that do not materialise.
>
> (Soros, 2003)
And here Buffett:
> In many industries, of course, Charlie [Munger] and I can't determine whether we are dealing with a 'pet rock' or a 'Barbie'. We wouldn't solve this problem, moreover, even if we were to spend years intensely studying these industries... Did we foresee thirty years ago what would transpire in the television manufacturing or computer industries? Of course not. (Nor did most of the investors and corporate managers who enthusiastically entered these industries.)
>
> (Buffett and Cunningham, 2002, p. 85)
Recall Keynes's 'There is no scientific basis on which to form any calculable probability whatever. We simply do not know.' The intelligent investor knows that he or she does not know and (like Buffett, Keynes and Soros) can enjoy the luxury of acknowledging that ignorance. Professional investment experts mostly cannot acknowledge such ignorance – they are hired to provide answers to unanswerable questions. Never mind, they are well paid for it. But not, if you are an intelligent investor, by you.
**Approaches to risk**
In this and the previous chapters I've described both the dominant SEU theory and the more eclectic and pragmatic approaches to risk and uncertainty favoured by Keynes and Knight. It is time to pull together the strands and spell out their implications for intelligent investors. In SEU theory, the term 'subjective' has a double significance. The probabilities are subjective, and so are the valuations of outcomes.
In that SEU world, probabilities and valuations differ between people but, once established for any individual, are held consistently. If the risks are the same, you use the same probabilities whatever the pay-offs; if the pay-offs are the same, you attach the same valuations, whatever the probabilities. There is a powerful underlying logic to this approach. The reasons the SEU school is today dominant in risk analysis in financial markets can be traced, directly and indirectly, to a clever argument constructed by Ramsey. If you don't adopt the SEU approach, people can devise schemes that will make money at your expense. The process is called a 'Dutch book'. (Offended residents of the Netherlands have attempted, without success, to track down the origins of the phrase.) In a Dutch book you make a series of choices, each of which you believe is to your advantage, whose net effect leaves you worse off. The only sure means of preventing inconsistency in your choices is to follow the precepts of subjective expected utility. The Allais paradoxes described compellingly attractive Dutch books.
While SEU may appear a good way of thinking about risk, it is certainly not the only way of thinking about risk. Ramsey's argument was never as decisive as it appeared. In the uncertain, constantly changing world that Keynes and Knight correctly perceived, it is impossible to be sure whether behaviour is consistent or simply stubborn. You are steadfast. I am pragmatic. There can be no objective basis for the claim that two similar situations are indeed identical.
The findings of behavioural economists are often interpreted as evidence of irrational behaviour. This interpretation is simplistic. The demonstration of supposedly irrational behaviour in the principal Allais paradox rests on the human tendency to value certainties much more highly than extremely probable events. There are good reasons for this trait. There is no such thing in everyday life as an event with a calculable 99 per cent probability. The first percentile of a distribution is generally encountered as a result of uncertainties outside the model. A common feature of these behavioural experiments is that the subject applies a rule that makes sense in everyday life – which is full of uncertainty – in an abstract experimental situation in which problems can be reduced to risks that are precisely defined and described.
There are choices in which rigorous application of the principles of SEU seems to make sense – Robb Caledon – but mostly calculations have to be tempered with the more qualitative judgement that enables us to handle uncertainty. It is not easy to draw a clear line between situations for which SEU is relevant and situations for which it is not. There is tension between people who are wired to 'think SEU' and those who are not. That was the situation President Obama faced in deciding whether to go ahead with the raid in Pakistan.
These conflicts are as old as history. Throwing dice is one of the oldest forms of gambling – bones shaped like cubes were used for this purpose thousands of years ago. Games of dice and other forms of gambling were invented precisely because the results emerge from a random process. Although those players millennia ago did not know it, they had created exactly the conditions in which a probability-based SEU approach is valuable. The modern casino similarly allows people who 'think SEU' – the operators – to make money at the expense of people who do not think SEU – the punters. The casino, like the insurance company, is concerned with expected values. The motivations of the customers of both groups of institution are more complex.
The ability to 'think probabilities', to apply the premises of SEU, is one that will save you from costly mistakes at the casino. But in an uncertain world in which risks cannot be clearly identified, we don't approach all questions probabilistically, because the probabilistic approach often isn't useful. If people turn out not to have consistent personal probabilities, the reason may be that a scheme of personal probabilities is incapable of fully describing the way they think about the First World War, the Twin Towers or even their daily journey home from work.
Detachment means that you focus single-mindedly on the outcome of a risky activity, and ignore those aspects of the process that you cannot control. This is also difficult. We worry more about risks we cannot control than about those we can, and with reason. This may be why so many people are nervous about flying, but far fewer are nervous about driving, even though it is more dangerous.
Sometimes involvement gives pleasure, as when we cheer the horses we have backed at the races. Some people experience a rush of adrenalin as lights flash on the fruit machine or as the roulette wheel spins. Sometimes involvement in events we cannot influence causes anxiety, as when I turned on the news to hear the progress of the shipbuilding nationalisation bill through Parliament.
Whether pleasurable or painful, the experience of engagement is costly. The thrill of the race and the excitement of the casino are paid for by the majority of punters who lose. Sleepless nights worrying about which way Members of Parliament will vote bring no benefit – this attention is not going to affect the outcome. But the fear of those sleepless nights dissuades most people from undertaking investments, like buying shares in Robb Caledon, even if the odds are attractive. If you are detached, you will pay no special attention to the prices of shares you have sold or decided not to buy. But you will!
In a world of unknowns, engagement is part of the process of managing uncertainty. To be detached about mountain-climbing or buying a lottery ticket is to miss the point of these activities. Regret for our mistakes is part of the process by which we learn from them. Compartmentalising our decisions and the information relevant to them – a process known as mental accounting – is the only way we can cope with a complex world.
But training yourself to think in line with the principles of SEU will give you a technique that will serve you well in investment decisions – so long as you do not make the mistake of thinking that this is the only way that you should think about risk and uncertainty, or the equally dangerous mistake of thinking that others behave according to its principles. We don't naturally or easily think probabilistically, achieve detachment, or always make decisions in the light of the totality of our circumstances. But if you want to make money in financial markets, there are substantial rewards to training yourself to think in that way. That is why this chapter and the preceding one, which are the most difficult in the book, may also be the most profitable.
If you don't practise SEU, you will be Dutch-booked – sold products that will be financially rewarding for the promoter but not for you. For years I taught that people would behave in line with SEU precepts since others would make money at their expense if they didn't. The logic was correct, but the conclusion was wrong. People often don't behave in line with SEU precepts, and others do make money at their expense, and that is a large part of what financial markets are about. Conversely, if you can learn to 'think SEU', then you will often make money at the expense of people who don't.
It isn't always a mistake to be Dutch-booked. The enjoyment of the daydream some people feel as the national lottery draw approaches, the excitement of watching the roulette wheel spin, the worry that is removed by buying an insurance policy – these feelings are real. But they are also expensive, and the other side of the Dutch book is the profits of the lottery and of insurance companies. If you can successfully suppress emotions of excitement and regret, you will be financially better off. You may prefer to enjoy these experiences, and I'm certainly not going to label you irrational in consequence. But this is a book about how to be a successful financial adviser to yourself. And one of the best pieces of financial advice you can give yourself is to familiarise yourself with the principles of SEU.
But people who think SEU are not necessarily more successful in life than people who do not think SEU, only more successful in not losing money at the casino. In casinos, the house wins and the punters lose. But perhaps the punters have the money to stake, and the attractive women on their arms, precisely because they are over-confident. They tend to view life in terms of lively stories and vivid narratives rather than the abstract world of Bayes's theorem inhabited by nerdy economists.
So the evolutionary argument, popular with these economists, that SEU will drive out other methods of thinking, is wrong. Around the campfires of those savannahs, _bon viveurs_ and entertainers brought together groups that contained many different personality types – the enthusiastic, energetic hunters who killed the game and the more analytic tribesmen who focused on dividing it out. All these groups cooperated and prospered, as they do in modern financial markets. The world accommodates different approaches to risk and uncertainty, and so must you.
**Footnote**
1The description of Obama's decision-making is based on the accounts of Mark Bowden described by John A. Gans, _The Atlantic_ (10 October 2012), and Mark Bowden, _The Finish_ (Atlantic Books, 2012).
9
**Modern Developments in Financial Markets**
**Derivatives**
In September 2000 Jim Chanos read the notes to the accounts of Enron. Enron's filings with the Securities and Exchange Commission ran to thousands of pages, including many notes printed in small type. The notes were ignored by the many Wall Street analysts who promoted Enron shares and didn't know, or didn't care, about the detail of the accounts. (The only Wall Street analyst to have been consistently critical of Enron, John Olson, left Merrill Lynch in 1998. His departure followed a complaint by Merrill's investment bankers, who feared being excluded from Enron transactions.) The notes described the complex relationships between Enron and associated companies. Chanos was not impressed by what he read.
If you like a security and you own it, you are 'long' of that security. If you don't like a security and you own it, you can sell it; if you don't own it, you can refrain from buying it. But Chanos and the fund he managed, Kynikos, took such a negative view of Enron that they wanted to sell it even though they didn't own it. Selling a share you don't own is called 'establishing a short position', and the market for derivatives enables such trades to be made. Derivatives are securities whose value is derived from the value of other securities. However complex the contract, almost all derivatives are based on shares, bonds, property, commodities or currencies.
Someone who is long of a security profits when its value rises. A short seller profits when the shares fall. When Enron shares plummeted, Chanos and other short sellers made substantial profits. The usual method of short selling today is to take out a 'contract for difference' (CFD). Suppose the share price today is $100. A contract for difference is an agreement that can be terminated at any time either to pay or to receive the difference between $100 and the market price. The value of the contract for difference varies from day to day in line with the market price. Such contracts are a means of speculating on price movements. Not much upfront cash is required. When Enron's market price fell to zero, the holder of a suitable contract for difference could close the transaction and receive what the market price of Enron shares had been on the day he made the agreement.
A contract for difference, like all derivatives contracts, is made with a counterparty – typically an investment bank. The bank may, but need not, hedge its side of the contract by itself buying, selling or 'borrowing' the shares. That decision will be part of the bank's management of its own portfolio of risks. In some countries a retail investor can engage in spread betting as an alternative to a contract for difference. This is a gambling contract in which you 'win' an amount for every upward movement of the price and lose a similar amount on every downward move (or vice versa).
Another common derivative contract is a 'swap'. Swaps were first widely used when exchange control made it difficult to buy foreign currency. You would 'swap' your domestic currency for an initially equivalent amount in the currency you wanted to hold, and agree to repay an increased or diminished amount depending on what happened to the exchange rate.
This market for currency swaps vanished when exchange controls did, but the underlying idea caught on, and today you can swap almost anything. Interest rate swaps are the most common. You exchange an interest rate that varies from day to day for an interest rate that is fixed for a period of years. Mortgage lenders use swaps to offer retail investors a choice of fixed and variable rate products. People who take out mortgages often trade in derivative markets even though they don't know it. When retail investors buy the structured products I describe below, they are often, even if they do not know or understand it, buying derivatives.
A futures contract is an agreement to buy or sell something in the future at a price that is fixed today. Futures have a long history. A merchant might have to wait three months till a ship reached port, and would be vulnerable to price changes while his cargo was at sea. A farmer might wish to establish, now, how much he will get for his crop. The earliest futures contracts brought such producers together with users concerned about how much they might have to pay for the raw materials they would need in future. Such contracts enabled both parties to satisfy their needs with greater certainty – to hedge. But you can buy a futures contract without any need, or intention, to hedge. You can buy it to speculate on what is going to happen to the price, and that is what most purchasers of futures contracts do.
The cost of buying a future is governed by the present price of the asset. If a commodity does not deteriorate physically, its anticipated price three months from now can't be more than the price today, plus the cost of storing it for three months. But if there is an immediate physical shortage, the current price could be higher than the futures price.
Storage costs influence the relationship between futures prices and current prices. There is a financing cost. Money that has been borrowed, or which could otherwise be invested, is tied up in the purchase. A physical commodity needs to be warehoused and insured. For securities there is only the financing cost, and this cost will be reduced if there is a dividend on the share, or interest on a bond. Interest rates differ in different currencies – generally there will be one interest rate at which banks borrow and lend dollars, and different interest rates at which they borrow and lend euros or pounds. The cost of owning a currency is the difference between the interest rate on that currency and the interest rate in your home country.
A futures contract is a firm agreement to buy and sell at a pre-agreed price at a future date. An option confers the right, but not the obligation, to buy or sell at a specified price, known as the 'strike price'. Some options have a fixed exercise date, others can be exercised at any time until they expire, but since you usually won't want to exercise an option before expiry the difference isn't great. Since the option confers right but not obligation, an option seems preferable to a futures contract, which entails both right and expectation. If the option were free, it would be preferable. But while you can usually buy a futures contract in a commodity or a security for something close to the current price, you have to pay a premium for an option. The right to buy is known as a 'call option', while the right to sell is called a 'put'. What are these options worth?
Two influences on value are obvious. The longer the life of an option, the greater its value. The value of a call option diminishes, and the value of a put option increases, the higher is the strike price relative to the current price. There is another, less obvious, principle of option pricing. The value of an option increases with the volatility of the underlying asset. The riskier a security, the more valuable are options to buy or sell it. This may seem paradoxical, but note that the riskier the asset, the more likely it is that you will be glad you own an option rather than the security itself.
Until the 1970s, assessments of option value were guesses. Then the Black–Scholes model provided an analysis of the relationship between an option price and the value of the underlying security. That apparently scientific basis for appraising derivatives was essential to the subsequent explosion of derivatives markets. The value of the option depends on its exercise price, its expiry date and the volatility of the price of the underlying security. Given these parameters, the standard model calculates an option value on the basis that successive price changes are independent of each other (follow a random walk) and observe the normal distribution. That is fine if these assumptions of independence and normality hold. But there are strong reasons for querying both.
Deviations from a random walk arise from momentum or positive short-term serial correlation. The 'fat tail problem' – extreme events happen more often than the normal distribution allows – is especially troubling. These valuation models seem to work well in relatively placid times, but it's not so clear that they work well when markets are unsettled. That's a major problem, because it is in unsettled markets that options are most useful, most profitable and most dangerous, and the need for measures of what they are worth is most urgent.
'Mind your portfolio' – look at your investment portfolio as a whole. Most derivatives are, in themselves, risky investments – the return on an individual transaction is very uncertain. But adding derivatives to an existing portfolio can reduce the risk, or improve the expected return on the portfolio without additional risk. A futures contract can hedge a portfolio against a market fall; a put option will limit the maximum loss. Derivatives can lock in profits or anticipate future needs, or allow you to benefit from a market rise while limiting risks. But hedging and insurance aren't the main reasons why people buy derivatives and related products based on them. They buy them because they think they are a good bet.
The normal experience of gamblers is that only the house makes money in the long run. This is as true of speculation in derivative markets as it is elsewhere. Valuing derivatives is complex, and there are no correct answers. But the financial institutions that sell derivative products have access to sophisticated computer programs, extensive databases, Black–Scholes and other models, and roomfuls of maths and physics PhDs. They are better at derivative valuation than the fund managers and treasurers of big companies and public authorities who buy them. Buying derivatives because you think they are cheap, rather than as hedges in a programme of portfolio diversification, is almost certainly a mug's game. The risks are in the price, and if you think the risks are not in the price, the most likely reason is that you are wrong. Why do I want to buy what they want to sell?
Trade in derivative markets has grown steadily, and the products have become more opaque. Some people – notably Alan Greenspan, former chairman of the Federal Reserve Board – explained this growth in trade in terms of an ever more sophisticated risk allocation. Perhaps the German regional banks that invested in asset-backed securities related to the value of subprime mortgages in American inner cities did so because they were particularly knowledgeable about these loans, or believed they hedged or diversified other assets in their portfolios. More likely, victims of information asymmetry, they did not understand what they were doing.
When assets are difficult to value, they will be owned by people who overestimate that value. Most derivatives are bought by people who are making a mistake, and the proliferation of ever more complex financial instruments, hard to assess even with sophisticated models, encourages these mistakes.
There is a history of spectacular crises in derivatives markets. Sometimes credulous individuals in public agencies or large corporations fall for a sales pitch. In the 1990s many embarked on programmes of trading in derivatives or structured products whose nature they at best dimly understood. The municipality of Orange County went bust after trades made by the appropriately named Robert Citron came unstuck. The London borough of Hammersmith and Fulham was spared similar embarrassment when the courts ruled that the council had no authority to gamble with local residents' money and the banks could get lost.
But the most spectacular of the many failures of that decade was certainly not the product of naïveté – or not that kind of naïveté; the demise of Long-Term Capital Management in 1998. This collapse received particular attention – and caused particular joy in some quarters – because of the personalities involved. Founded by John Meriwether, a legendary Salomon trader, the management team included two Nobel Prize-winners, Bob Merton and Myron Scholes (of the Black–Scholes model). LTCM relied on arbitrage trades based on fundamental value – reversion to the mean in relationships between similar but different securities. But momentum may drive prices sufficiently far away from fundamental value for sufficiently long to exhaust the patience, and the resources, of investors – 'markets can be wrong for longer than you can be solvent'. Apparently unrelated bets proved to be correlated in the widespread panic that followed the Asian financial crisis and Russian debt repudiation of 1997–8. The scale of LTCM's borrowings – the magic of leverage meant that this relatively small fund had liabilities of trillions of dollars – was such that the Federal Reserve orchestrated a rescue operation.
Many derivatives in modern securities markets are complex. However elaborate, they are mostly based on combinations of the four basic mechanisms I have described above: shorts, swaps, futures and options. Most private investors will not trade in derivatives directly. But they may do so through hedge funds and structured products, or through funds that invest in hedge funds or structured products.
**When bonds were no longer boring**
Bonds had traditionally been boring. Nick Carraway, the colourless narrator of Scott Fitzgerald's 1925 novel _The Great Gatsby_ , was a bond trader. But in the 1970s and 1980s several distinct innovations turned the bond market into an exciting place. The introduction of an interest rate futures contract on the Chicago Mercantile Exchange, almost simultaneous with the work of Fischer Black and Myron Scholes at the neighbouring university, was the beginning of the development of derivative contracts related to bond markets.
The 'junk bond' was invented by Michael Milken, whose graduate school thesis demonstrated that the premium rate attached to risky bonds was more than sufficient to affect the default rate. Most of the bonds Milken studied were once sound securities whose issuers had fallen on hard times. Milken's innovation was to issue bonds that were obviously highly risky from inception.
Banks created asset-backed securities, bundling together packages of loans and dividing them into safer and riskier tranches. 'Mortgages are math,' declared Lew Ranieri, who ran fixed interest at Salomon Brothers and promoted the mortgage-backed security. Ranieri would be a central figure in Michael Lewis's coruscating 1989 account of the trading floor at Salomon. Two years earlier Tom Wolfe had described the New York of the era in his novel _The Bonfire of the Vanities._ His central figure, Sherman McCoy, Master of the Universe, was, like Nick Carraway, a bond salesman. But in a very different environment.
These innovations did not end well. The first era of junk bond issues ended with the collapse of Milken's firm, Drexel Burnham Lambert, which had more or less monopolised their issue. The 'big short' – the bet on the collapse of US house prices – was undertaken through the medium of a credit default swap based on collateralised debt obligations – a many times repackaged bundle of subprime mortgages. The failure of many of these supposedly asset-backed securities was central to the 2008 global financial crisis.
The fundamental mistake was to believe that by bundling risky loans it was possible to create relatively safe securities. The 'mind your portfolio' principle tells us that this belief is not as implausible as it may sound, but relative safety depends on the underlying risks being uncorrelated with each other. The risks of mortgage defaults were not uncorrelated. If you seek guarantees, you will have to be content with the bonds of the governments of countries such as Germany and Switzerland – and the paltry returns that go with true guarantees.
**Structured products**
Few retail investors will trade directly in derivative markets, and few should. But you may be offered opportunities to do so indirectly, through structured products or hedge funds. A collateralised debt obligation, the instrument at the heart of the global financial crisis, is a structured product, a security whose risk and return characteristics have been created by financial engineering. Such engineering can take the form of packaging one security with others: splitting the risk and return from an asset into several components; combining a security with a derivative based on the same or another security; or, frequently, a combination of all these methods. A synthetic investment resembles a fund, but the issuer does not hold the underlying assets to which its value is related; the risks may be (but need not be) hedged through derivatives.
Financial engineering sounds complicated, and the legal and economic structure of these instruments is indeed complicated. But good financial engineering is like watchmaking. Its objective is to create products whose properties appear simple, even if the underlying mechanisms are not. Much financial engineering, however, is more like the skills of the illusionist: what the audience sees is not the reality.
The aim of financial engineering is either to increase the yield on bonds with little addition to risk or to reduce the risk in equity investment with little loss of return. Avoid these offers. The issuer of these bonds writes a complex derivative contract with an investment bank. The bank makes a payment but is entitled to receive a much larger sum if the complex combination of events described in the small print materialises (the 'punt'). The premium pays for the higher interest rate you are guaranteed (and for the marketing expenses of the bond, which will be substantial).
The investment bank has done its own modelling and considers that the punt is a good deal. But if the punt is a good deal for the bank, it is a bad deal for you. It is very unlikely that you would buy this kind of complex gamble with negative expected value as a stand-alone product. Once again, you can apply the 'mind your portfolio' principle to the structured product by analysing its components. You will get to the same answer quickly and simply through two other principles: 'Why do I want to buy what they want to sell?' and 'Keep it simple'. Both these maxims are good guides in the world of structured products, and they tell you not to go there.
If you adhere to the 'mind your portfolio' principle, you will not want to buy structured products. The structured product alters the risk/return characteristics of individual investments. What matters to you is the risk/return characteristics of your overall portfolio. You can and should achieve the overall objectives – the balance of risk and return – by balancing your overall mix of investments. Mind your portfolio! You do not need funds or companies to gear or diversify on your behalf, because you can do these things yourself. The best and cheapest way of limiting the risk in shares is to hold a lower proportion of your assets in shares.
But rigorous application of the 'mind your portfolio' principle is hard. Many intelligent investors will feel better if they limit the risk on the individual components of their portfolio as well as on the aggregate. Others like the risk and return combination offered by particular structured products. One of the very first structured products, and still the one with most appeal to retail investors, is an enhanced bond from the British government. The premium bond was introduced in 1956 with a showman's style by the future Prime Minister Harold Macmillan. The bond offers no income, only the chance to participate in a monthly prize draw. The premium bond is a combination of a deposit account and a lottery, and since the deposit rate is competitive and the lottery is fair, the combination is an attractive proposition for people who are interested in both holding bonds and entering lotteries.
This method of analysing a structured product unravels the financial engineering of the product's designers. The technique can also be applied to any structured product, and should be applied by anyone who adheres to the 'mind your portfolio' principle. Split the package into its component parts, and ask whether each component is something you would want to add to your profits.
A bond issued by a large company will offer a higher return than a government stock, but at a higher risk. There is a greater possibility (though still not a very large one) that the company will default. Deutsche Telekom's 2023 bond has a gross redemption yield of around 0.6 per cent, as against around zero for the equivalent German government bond. Neither looks an attractive opportunity. If you buy the Telekom bond, you are in effect insuring the – admittedly small – credit risk associated with Deutsche Telekom in return for a very modest premium.
You do not have the information needed to assess the risk on these policies. There are companies that sell credit insurance (in 2008 they were rapidly disappearing). Credit insurance is an activity whose returns are strongly correlated with your overall portfolio (a situation in which Deutsche Telekom defaults on its bonds is likely to be one in which there is mayhem in other financial markets). The additional risk and return that the corporate bond implies relative to the government bond is probably not a risk you would add to your portfolio if it were offered on a stand-alone basis.
Many markets offer bundled products. Buy one, get the second half-price. Television programmes are sold as packages of channels. Buy coffee and croissant together. Eat a _prix fixe_ menu. The cost of the menu is typically less than the cost of the separate components. Not so in financial markets, where the cost of the structured product is generally higher than the cost of the components, and the premium is the source of the financial engineer's profit. No one would buy a _table d'hôte_ menu at a price higher than the sum of the items on the _à la carte_ menu. The premium also indicates that structured products are often bought (in my view, mostly bought) by people who misunderstand them. The analogy of the illusionist is more relevant than the analogy of the watchmaker.
The market for structured products illustrates all the issues of information asymmetry. Asking 'Would I choose to buy each of the separate components of this package?' is an important discipline in both the restaurant and in financial markets, but especially in financial markets. Because structured products are often bought by people who are making mistakes, they are usually better bought second-hand. If you are going to hold bonds, hold bonds. Complex financial engineering is expensive, and the expense is likely to be yours.
**Hedge funds**
Keynes was an active speculator, famously trading from bed in the mornings, managing his own money and that of King's College, Cambridge, shorting securities and dealing in a wide range of assets. He seems to have been successful in the long run, though not extraordinarily so. He died in 1946 with assets of around £½ million, a wealthy man by the standards of the time but not rich on the scale of Soros or Buffett. Benjamin Graham, the legendary value investor, would establish short positions in businesses he did not like as well as having long positions in those he did.
These styles of investing were the precursors of the modern hedge fund. The first investor to attract that label seems to have been Alfred Jones, who traded aggressively through the 1950s. Hedge funds attracted wide popular attention with the famous bet against sterling by the Quantum Fund of George Soros in 1991 and came into the spotlight again with the failure of Long-Term Capital Management in 1998.
Until 2000 hedge funds were used only by rich individuals and a few sophisticated institutions. Academic endowments, notably those of Harvard and Yale, were early and successful supporters. After the end of the New Economy bubble, investors searched for new ways of generating the high returns that had seemed to come so easily in the 1990s. They turned to hedge funds. Exceptional individuals such as Soros had been able to command high fee levels for their services, and newer funds set their charges at the same level. The prospect of such fees proved irresistible to many people in investment banks. As newcomers rushed through the door, Soros made for the exit. So did another legend, Julian Robertson. While the long-term performance of his Tiger Fund rivalled that of Soros, Robertson may have lost more money in his last year than he had made throughout his career.
Different hedge funds pursue different strategies. A long/short fund, like Graham's, will deploy the stock-picking skills of its managers to select losers as well as winners. Jim Chanos, who helped expose Enron, specialised in taking short positions. A market-neutral fund will hedge its exposure to general market movements. It will focus on _alpha_ and eliminate _beta_. A macroeconomic event fund in the style of Keynes or Soros aspires to predict major financial and political developments in the world economy – such as Britain's EMS failure – and trades on the basis of these forecasts.
Arbitrage strategies, such as those deployed at LTCM, rely on stable relationships between the prices of related, but different, securities. Such pairs might be currencies within a region, or the debt and equity of the same company, or the shares of an acquiring company and its potential target. As at LTCM, arbitrage strategies are often based on complex modelling. Relying on history in this way is dangerous, as it proved for the LTCM partners. Distressed debt funds buy bonds of businesses or countries in financial difficulty. These funds may hope to profit from buying below fundamental value or to increase that fundamental value through litigation or the threat to block proposals for financial reconstruction.
Although most hedge funds are based in London or the United States, for legal and regulatory purposes they are normally registered offshore. While the City of London (and its increasingly important annexe at Canary Wharf) remains the centre of London's financial services business, hedge fund managers emphasise differentiation and exclusivity by clustering in St James's. Charging typical fees of 2 per cent of funds under management and 20 per cent of profits, the managers can afford West End rents. US hedge funds have similarly chosen to distance themselves from Wall Street, and Greenwich, Connecticut, is now generally regarded as the centre of the hedge fund world.
The fee structure, in effect, shares profits but not losses. Investors look for reliable month-by-month appreciation in the value of their funds, and hedge fund managers are closely monitored month by month. Both these incentives and this supervision encourage the adoption of strategies with Taleb characteristics – frequent small profits, occasional large losses. Hedge funds make extensive use of the 'carry trade', which relies on interest rate differentials between good and bad credits, or strong and weak currencies.
Retail investors cannot easily access major hedge funds, because of regulatory restrictions and high minimum investments; moreover, many of the most highly regarded funds are closed to new investors. Retail investors have the opportunity to invest in more conventionally open- or closed-ended funds which themselves invest in hedge funds. Since many hedge fund managers reveal only limited information about what they do, intermediaries may be helpful. These funds of funds can diversify across funds with different strategies. The returns on, say, a long/short fund should not be correlated with the returns on, say, a distressed debt fund. Historical analysis supports this claim.
But do not count on the stability of historical correlations in times of extreme financial strain. Funds learnt this lesson in 2007–8. The shift from mainstream to alternative investments also favoured private equity. Private equity is a means by which investment institutions, either directly or through managed funds, invest in unquoted businesses. Before the late 1990s private equity used to be more or less synonymous with venture capital – the provision of funds for start-up and early-stage businesses.
Not all private equity deals were of this kind. As large corporations restructured, unwanted 'non-core' divisions were often sold to consortia led by their managers and funded by private equity. The newly established business was often refinanced or the subject of a fresh IPO within a few years. Many individual managers became very rich through these transactions, a development that helped to raise the pay aspirations of all executives.
The development of junk bond financing in the 1980s made it possible to use debt to buy even very large companies. That boom ended with the relatively unsuccessful outcome of the massive takeover of RJR Nabisco by private equity house KKR and the failure of a number of other transactions, such as the hopeless acquisition of many of America's leading department stores by the ambitious Canadian Robert Campeau. Still, the search for new investment avenues after 2000 led to explosive growth in the funds available for private equity, and the scale of the typical transaction grew rapidly. Private equity funds would buy an established business from the existing owners. These owners might be other companies, family and founding shareholders or, increasingly, the shareholders of public companies quoted on the stock exchange.
The RJR Nabisco transaction is still the largest (inflation-adjusted) private equity deal in history, though in nominal terms it has been overtaken by the $45 billion acquisition of TXU by a consortium of KKR, Goldman Sachs and a third private equity group, Texas Pacific – a transaction that ended in the bankruptcy of the largest power company in Texas. In these transactions private equity has come a long way from its roots in the provision of funds to small growing businesses. Private equity investment became an indirect and expensive method of buying shares in medium and large companies similar to those in the main index. The returns from this kind of investment will therefore be correlated with the returns from other equity investments, but, given the leverage involved, the _beta_ associated with such equity is well above one.
The concentrated share ownership of private equity deals allows more effective monitoring of company management. Such concentrated ownership has been common for many years in continental Europe. Public companies find their ability to focus on fundamental value inhibited by the management of quarterly earnings forecasts and reports. Some patient private equity investors are willing to hold their financial investments for many years, encouraging more long-term strategic thinking and productive investment. But the major private equity firms, and the investors who back them, are generally in search of a rapid exit. The objective is normally to sell the company or float it on the stock market within three to five years, returning the money obtained to the investors in that particular fund.
I described in Chapter 6 the numerous ways in which businesses can enhance earnings in the short term. Private equity, and the threat of takeover by private equity, has increased pressure to use these strategies. During a recent stay in a hotel, looking at the frayed carpet and overpriced extras, I thought, 'This hotel has been bought out in a private equity transaction.' It had been. It is easy to enhance the earnings of most businesses if your time horizon is three years.
Both the citizen and the investor should be concerned about the difficulty modern management confronts in taking a long-term view of the growth and development of the business. The public company encounters the blight of quarterly earnings reporting; the private equity owned business must accommodate the time-scale of investors seeking a quick profit. This short- termism is reinforced by the incentive schemes offered to managers – even, or perhaps especially, the so-called long-term incentive schemes that now constitute a large part of executive pay. Even if you are concerned with the fundamental value of the business in which you own shares, the managers may not be.
It is difficult for the investor to judge the overall historic success of private equity, far less its prospective future returns. Some private equity houses may have been able to supervise management teams more effectively than public markets. Some may possess skills that add value through financial engineering. Private equity houses have recently (but no longer) had access to debt financing whose pricing did not reflect the risk involved. The reporting of historical performance is confounded by survivor bias – it is the more successful investors who are raising new funds – and by the capacity of leverage to produce very high returns to equity investors in a period of rising share prices (with the opposite effect when share prices are falling).
Against these potential advantages must be set the certainty of charges. 'Two and twenty' is a common charging basis for both hedge funds and private equity investments. The private equity house takes a management fee of 2 per cent of the fund per year, and 20 per cent of the profits, known in private equity as the 'carried interest'. Like other investment managers, they may also arrange for many of their own costs to be directly or indirectly charged to their funds. When these percentages are applied to large transactions – such as those of TXU – the numbers become eye-watering. Many hedge funds and private equity managers have become very rich very quickly. But what of the customers' yachts? 'Two and twenty' reduces a 10 per cent underlying return to 6 per cent. A retail investor will normally make an investment in a hedge fund or private equity fund through a fund of funds or a feeder fund, for which a charge of 1.5 per cent of assets for management and 10 per cent of profit is common. The addition of these charges means that a 10 per cent return on the underlying investment might be no more than 3.5 per cent in the hands of the investors. Even if the fund secured a 20 per cent yield, the net return to the investor might be only half of that, with 6 per cent going to the underlying fund and 3 per cent to the fund of funds. Tax and inflation take a further bite of the return.
Such figures make no sense. Returns on financial investment are ultimately governed by returns on productive investment. While some investors in hedge funds and private equity will make excellent returns from a mixture of skill and luck on the part of their managers, most stand no chance of earning profits commensurate with the risks.
**Other alternative assets**
Hedge funds and private equity are not the only alternative assets. Other options include commodities such as oil, gold and other metals; undeveloped land such as farms, forests and building plots; and collectables such as wine, art, jewels and furniture.
Commodity exchanges, large and active, meet the needs of people who produce or use the physical commodities. Commodities have always been a fertile field for speculation by those who believe they can anticipate 'the mind of the market'. If this is you, then good luck and good fortune; you will need good luck to preserve good fortune. If you simply want to invest, there are funds whose value is linked to commodity prices, which I will describe in the next chapter. The growth of such funds has fuelled the remarkable instability of commodity prices over the last decade, an instability that is disrupting the supply of such commodities.
But over the long run commodity investment has not been very profitable. Despite continuing worries about forthcoming resource scarcity, new technology and new discoveries have more than offset the exploitation of existing reserves. It is cheaper to own commodities in the ground than in a warehouse, and mining companies are generally better investments than the commodities themselves, and are good portfolio diversifiers. The aphorism that a small minerals company is a hole in the ground with a crook at the top is, however, often true.
Other alternative assets are marketed to private investors either as hot tips or as additions to a diversified portfolio. The general rule is to avoid these areas unless you have specialist knowledge. Buy wine, art, jewels or furniture if you like wine, art, jewels or furniture. While it is certainly possible to make money in these investments, the differences between buying and selling prices are wide, and there is none of the regulatory apparatus that protects small investors in securities markets. If you are expert, and even more if you are not, buy only collectable items that you like.
10
**The Conventional Investor**
**Establish your portfolio**
Every financial adviser is trained to begin by asking you to define your investment objectives and your attitude to risk. As your own financial adviser, you need to pose these questions to yourself. The professional adviser must tick boxes to complete the 'fact find' so he can get on and sell you products. You really want to know the answers because they will determine your investment strategy.
Are you a net saver or a net spender? Are you building up assets or managing existing assets to support your standard of living? Most people before retirement are, or should be, net savers. Many people, after retirement, will be net spenders. People before retirement need to consider how much saving they should do. People after retirement need to consider how much they can afford to spend.
If you gathered nuts for twenty-five years, and planned to retire for twenty-five years, you would need to set aside half your nuts for your old age. If you can earn a return on your investment, you can reduce significantly the amount you need to set aside. Table 8 shows how great a reduction you can make. If you meet the 8 per cent target rate of return, 12.5 per cent of income will grow to a cache large enough to meet your needs. You can now eat almost 90 per cent of the nuts you collect.
Table 8: **Save now, spend later**
Every 2 per cent that you gain or lose in investment return will make a big difference. You can sacrifice more than 2 per cent through poor investment performance, bad timing of sales and purchases or unsuitable allocation across investment categories. You might lose 2 per cent in tax, and another 2 per cent by paying more than you need in fees and charges. And you can lose 2 per cent, or more, by making large allocations to assets that advisers describe as low-risk investments which offer lower but more predictable returns.
In this and the following chapters I will describe how you can minimise the leakage of return under each of these headings. I'll suggest how to turn the timing of investments to your advantage, and how to approach asset allocation and stock selection. I'll stress the importance of keeping down costs and charges, and explain how to do it. And I will offer practical guidance on how to think about risk and uncertainty based on the principles outlined in Chapters 7 and 8. The lesson of that discussion is that you don't necessarily have to accept low returns to achieve low risk.
Chapter 2 described the essential preliminaries – listing assets and liabilities, reviewing your banking arrangements and your mortgage. Now you need a stockbroker. Most people still have an image of the stockbroker derived from the days before players replaced gentlemen: grand but seedy; with good social skills and connections; streetwise but not clever; well off but not hardworking. These people still exist but are a dying breed. They will levy annual management charges, or substantial commissions on purchases and sales, or both. They are expensive. So are the wealth management services of banks, which offer more modern versions of these services.
If you still lack confidence to build even a conventional investment portfolio yourself, the robo-adviser offers an inexpensive substitute, replacing by a computer the personalised service of the individual broker, private banker or wealth manager. These online services began in the United States and have now spread to several other countries. Betterment, the US market leader, charges a maximum fee of 0.35 per cent and requires no minimum account balance. However, this chapter will explain how you can do it yourself more effectively, more cheaply – and almost as easily.
Your stockbroker should be an online share-dealing service. You need an execution-only service, in which you make decisions and give instructions yourself. These services are very cheap, because computers do most of the work – so cheap, in fact, that price should not be a major factor in your choice.
While many brokers will allow you to trade foreign stocks, you will need to be careful about additional charges and poor foreign exchange rates. It is often best to buy securities from a broker in the country where those securities are listed. This is not an issue if you buy ETFs, which almost all brokers will allow you to do. If you are an EU resident, and often even if you are not, you can use a broker in any EU member state. You can use a single account for all eurozone securities. (Luxembourg may be a good choice.)
When you set up this account, you will want to take advantage of the tax concessions available to retail investors. Almost all countries offer substantial tax concessions for pension savings, and there are often other schemes to favour saving, although you may find that much or all of the benefit is absorbed by intermediaries. The price of accessing the relief, through explicit charges, trading costs and management fees within funds, and inflexibility and poor-quality asset management, may exceed the tax.
While securities will be the main component of your investment portfolio, the 'mind your portfolio' principle implies that you should consider insurance at the same time as you contemplate investment. You should think probabilities when you take out insurance. But so does the insurance company, and the company knows probabilities well. That is its business.
You should be detached. While it is vexing to have your television stolen, or to lose your bag on holiday, insurance will not bring back either your television or your bag. The financial loss from a stolen television or lost bag is probably less than you will incur on a bad day on the stock market. Much of the premium on policies that insure against these minor contingencies goes not to pay policyholders but in the administrative costs of small claims.
So anyone who is truly minding their portfolio, and recognises that the insurance company has calculated the expected value of their policy, will insure only things they cannot afford to lose. You need insurance against your house burning down, but not for replacing the bedroom carpet; you need insurance against being hospitalised in the United States, but not for the cost of an extra night's hotel accommodation because your plane is delayed.
Many people find this advice difficult to accept. Insurers observe that their customers have little appetite for policies that rarely pay out, even though low-probability risks are precisely those that are appropriate for insurance. Policyholders like the reassurance of occasional small cheques even if these cheques add up to much less than their premiums. Regret is a powerful human emotion, even if often an unproductive one.
You will be financially better off if you can be objective and learn to control these emotions, but it is difficult, even for people educated in intelligent investment, to exercise such control. If, when an uninsured television set is stolen, you can't refrain from kicking yourself, or your spouse can't refrain from kicking you, then you should take out the policy. This strategy is, however, likely to cost you money – even after taking account of the insurance companies' cheques.
**Pay less**
Over the fifty years that Warren Buffett has been in charge of Berkshire Hathaway, the company has earned an average compound rate of return of 20 per cent per year. For Buffett himself. But also for his investors. The lucky people who have been his fellow shareholders through all that time have enjoyed just the same rate of return as he has. The fortune he has accumulated is the result of the rise in the value of his share of the collective fund.
Suppose that Buffett had deducted from the returns on his own investment – his own, not that of his fellow shareholders – a notional investment management fee, based on the standard 2 per cent annual charge and 20 per cent of gains formula of the hedge fund and private equity business. There would then be two pots: one created by reinvestment of the fees Buffett was charging himself, and one created by the growth in the value of Buffett's own original investment. Call the first pot the wealth of Buffett Investment Management, the second pot the wealth of the Buffett Foundation.
How much of Buffett's $70 billion would be the property of Buffett Investment Management and how much the property of the Buffett Foundation? The – completely astonishing – answer is that Buffett Investment Management would have $64 billion and the Buffett Foundation $6 billion. The cumulative effect of 'two and twenty' over fifty years is so large that the earnings of the investment manager completely overshadow the earnings of the investor. That sum tells you why it was the giants of the financial services industry, not the customers, who owned the yachts.
The least risky way to increase the returns from your financial investments is to minimise agency costs – to ensure that the return on the underlying investments goes into your own pocket rather than someone else's. The effect of charges on investments is so large that it is as important to understand the structure of charges as it is to understand the principles of investment analysis. The first charge that hits you when you invest is the spread between the buying price and the selling price. For government bonds and for blue chip shares, such as Exxon Mobil, this spread will be extremely small, and you can deal at any time in any quantity you like. The only other dealing costs are commission (you should expect to pay less than €20 for an internet deal) and any government or exchange levies on transactions.
Small companies are different. The spread between buying and selling prices may be 3 or 4 per cent. Even a purchase of €10,000 may require a telephone call rather than a mouse click, and telephone orders will usually incur higher commission. The difference between what it will cost you to buy and what you will get if you sell could be as much as 5 per cent.
The effect of these costs on returns depends on the frequency with which you deal. Online trading is so inexpensive and easy that you may be tempted to trade often. Only one thing eats up investment returns faster than fees and commissions, and that is frequent trading. Do not succumb. The total costs of running your own portfolio should be well under 1 per cent per year.
Investing in actively managed funds will cost you more. The choice of funds available, both open- and closed-ended, is unbelievably wide. There are more funds investing in shares than there are shares to invest in. This situation doesn't make sense, and is both cause and effect of the high charges. Costs need to be high to recover the expenses of running so many different, mainly small, funds which all do much the same thing. At the same time the high level of charges encourages financial services companies to set up even more funds.
The proliferation of funds means that choosing a fund may be no easier than choosing individual investments. You can hire an adviser to recommend funds. There are also many advisers. The problem seems to multiply itself, as do the fees. The fees attract more advisers, and so on. The underlying problem is one of information asymmetry. The marketing of financial services emphasises quality, not price, and for good reasons. It would be worth paying more – a lot more – to get a good fund manager. But since it is hard to identify a good fund manager, good and bad managers all charge high fees, with the consequences described above. It is hard to escape the dilemma posed by this market inefficiency.
If you own an actively managed equity fund for five years, it is possible that the direct and indirect costs and charges you incur in buying, holding and selling that investment will total 3 per cent a year. Other investment funds may cost you more. The total charges on a fund of hedge funds are such that it might yield less than a government bond even if the underlying investments returned more than 10 per cent per year.
Costs and charges for funds fall into three broad categories:
* annual costs of managing and administering the investment;
* costs and charges incurred in buying and selling the investment, which include initial fees, commission charges and any difference between the quoted buying and selling price (the effect of these charges on your annual return depends on the length of time for which you hold the investment);
* costs incurred by the fund manager and charged against the value of the fund (e.g. commission on transactions and the difference between the buying and selling prices of the underlying securities).
Annual management charges range from 0.1 or 0.2 per cent for some indexed funds to 2 per cent or more for an insurance-linked fund or a hedge fund. Other administrative costs will usually be between 0.2 and 0.4 per cent. If you buy a fund of funds that invests in other funds, you may have to pay more than one level of these charges. Some funds also charge a performance fee, which gives the managers a share of any profit, or of outperformance relative to a benchmark. If you buy one of the funds recommended to neophyte investors, you will probably end up in an open-ended fund linked to a selection of mainstream shares.
Most such funds are closet-indexed – although they are actively managed, their composition is very close to the make-up of the index against which they are benchmarked. That minimises the risk – to the manager – that the fund will significantly underperform the market. If the largest holdings are the largest companies in the country, the fund is closet-indexed. You can achieve a similar result much more cheaply yourself, and you should.
Today most of the largest providers of retail funds globally are specialist asset managers such as Vanguard, BlackRock and Fidelity. These businesses do not suffer the conflicts of interest inherent in the cross-selling of retail products or the provision of investment banking services to companies in which the funds invest. In the United States, Vanguard Group, a not-for-profit company with a messianic founder, John Bogle, pioneered low-cost index funds for retail investors, and competitors have followed suit. In many other countries banks and life assurance companies still have a large share of the market, typically with high management fees. However, the worst US fund managers are greedier, and probably more corrupt, than their counterparts elsewhere. And, as Table 7 showed, their performance is dismal.
However Vanguard, Fidelity and BlackRock all now operate internationally. All provide open-ended index funds with very low charges. However, you may find that you have to use a 'platform' – an agent – to access these open-ended funds, and with some platforms charging 0.5 per cent or more per annum this can dramatically increase the cost of these options. You may be able to find a platform with a low fee, or one that charges a flat fee which will be almost irrelevant if your portfolio is large enough. The best option available for an open-ended fund varies from time to time and country to country. Although I would recommend an open-ended tracker if you can access it cheaply, an ETF – both BlackRock and Vanguard offer ETFs – will always be available cheaply through an execution-only stockbroker.
**The conventional investor's portfolio**
Where to begin? Many people will sensibly start the job of being their own investment manager as conventional investors. The conventional investor is hesitant to trust his or her own judgement and prefers to rely on the consensus of professional opinion. The conventional investor is probably hesitant to 'think SEU' and is still not quite convinced that a share such as Robb Caledon is part of a low-risk investment strategy. I shall describe a strategy for the conventional investor and then suggest how that strategy might be developed by those who are more ready to use their own judgement.
Even twenty or thirty years ago the commonest form of long-term saving was through the investment funds of life insurance companies, which managed a complete range of investments – shares, bonds and property. In this way the conventional investor could hand over investment allocation decisions to a fund manager. But recent returns on with-profits policies have been poor and the bases on which returns are determined opaque, and these are no longer attractive investments (except to those who sell them). Even conventional investors must now make their own asset allocation decisions.
Large institutions that are conventional investors get advice on asset allocation from consultants, such as Mercer or Towers Watson. Most of these firms originated as actuarial practices (actuaries measure the liabilities of insurance companies or pension funds) and have broadened their business activities. Consultants will usually base their advice on portfolio models and on their knowledge of what other similarly placed investors are doing. The activities of consultants are a means of disseminating conventional thinking.
The conventional investor can, simply and cheaply, replicate the portfolio of these institutions with a sum as small as €20,000. Let us begin by matching the average asset allocation of the three giant investors discussed in Chapter 2 – the Norwegian Oil Fund, the principal Dutch pension fund, and CalPERS. We will put €10,000 into global equities, €7,000 into bonds, €1,500 into real estate and keep €1,500 in cash – or for fun, in other investments that take our fancy.
You can invest €10,000 in the Vanguard Total World Stock ETF. You now own a share of Apple, Exxon Mobil and Microsoft. The largest non-US companies you hold are Nestlé and another Swiss-headquartered company, the pharmaceutical business Novartis. You also own some Sony, and BMW, and Apple, and HSBC. You are invested in over 7,000 companies overall, about the same number as the Norwegian fund. With only €10,000! And for an annual charge of 0.17 per cent.
Seven thousand euros is earmarked for bonds. We'll give €5,000 to the BlackRock Global Index Bond Fund. About 40 per cent of this is lent to the US Treasury, and 16 per cent to the government of Japan. Only about 6 per cent is in Germany, because the prudent Germans don't have much debt. Your investment is roughly equally divided across short- and long-dated securities. I am going to put €2,000 of your €7,000 bond allocation in inflation-linked securities – again either BlackRock or Vanguard will let me do this for you through a fund diversified across the countries (principally the US and UK) which have issued such bonds.
You may have a choice of leaving the currency of your ETF unhedged or hedging these funds into your own or another currency. For a long-run investor I don't think the choice matters much. Microsoft is an American company and Nestlé a Swiss one, but they sell to customers around the world in local currency, not in US dollars or Swiss francs. And while your future liabilities will be in your own currency, you give yourself a degree of protection by holding assets whose value is determined by the global economy rather than your own.
Finally, there is €1,500 for property. You are not going to buy a property of your own for €1,500, so your choice is between funds. The best option – which is sometimes available – is to buy a closed-end property fund with modest gearing at a discount to its asset value.
And that is it. With four clicks of a mouse you have a diversified portfolio similar to that adopted by a well-advised large investment fund with long-term aspirations to total return. And you have constructed it at low initial and ongoing cost. You can safely forget about this portfolio for a long time – years if you like. Your investments may or may not do well, but their performance will most likely be similar to, but better than, the balanced portfolio a consultant or adviser would construct on your behalf. Similar to, because it reproduces the balanced portfolio of the conventional institutional investor; better than, because you have greatly reduced the costs of establishment and management.
You should have aspirations to do better still. The conventional investor should consider separating a part of his or her portfolio in order to test his or her own judgement. As experience and confidence grow, the plan is to devote more and more funds to intelligent investment.
11
**The Intelligent Investor's Strategy**
**Developing a strategy**
Conventional investors – often tacitly, sometimes knowingly – employ a conventional approach to risk, based on SEU. In that approach, risk has the same meaning for everyone: i.e. short-term volatility. The well-advised conventional investor finds a balance between risk and return by selecting a point on the 'efficient frontier' described by Harry Markowitz. The appropriate choice of the point on the frontier depends on your 'risk appetite' or 'risk tolerance'. Risk has a market price, determined by the average risk appetite of all investors. A financial adviser will quiz you to establish your 'risk tolerance'; this interrogation is an essential part of the 'know your client' exercise which all financial advisers and investment consultants must undertake.
Actually, most conventional investors do not really think this way: they mostly follow conventional wisdom and do what other people in loosely similar situations do. When Markowitz was asked how he had invested his own retirement savings, he replied:
> I should have compared the historical advantages of the asset classes and drawn an efficient frontier. Instead, I visualised my grief if the stock market went way up and I wasn't in it, or if it went down and I was completely in it. My intention was to minimise my future regret. So I split my contributions 50/50 between bonds and equities.1
For the intelligent investor, risk is _not_ short-term volatility of asset prices. Risk is failure to meet realistic aspirations. So the intelligent investor needs to begin by defining his or her investment goals. Every individual has different objectives, but the purposes of investment generally fall into a few broad categories. Most people need to save for retirement. You may want to save for some other specific purpose – to make a deposit on a house, or to buy a secondary property, or to fund your children through college or help them buy their own apartment. You may have a lump sum to invest, perhaps from inheritance or from the sale of a property or business. A cautious person will not want to splurge it right away but hope to earn returns in order to enjoy a more comfortable standard of living in future. Or you may have no very clear objective in mind – you are saving for a rainy day, or perhaps for a sunny one.
Whatever his or her objectives, the intelligent investor thinks total return. In Chapter 2 I suggested that 8 per cent was an appropriate target for that total return – a figure that includes both income and capital gains, but is measured before allowing for tax and inflation. But the intelligent investor should plan around another, lower, rate of return figure – the prudent spending rate. How much can you draw from your investments without impairing their long-term capacity to allow you to maintain or increase the level of expenditure to which you have become accustomed?
The prudent spending rate is a key concept for the trustees of endowments: the managers of the Norwegian oil fund, the overseers of the Harvard endowment, the trustees of the Gates Foundation. Because sophisticated investors like these think total return, they are no longer bound, either legally or in their own planning, by the traditional convention that you should match your spending to the income you receive – the total of the dividends, interest, rents etc., arising year by year. But faced with considerable volatility of total return as a result of gyrations in market prices, they have to make decisions as to how much it is provident to spend. And so will you. If you have built up a fund for retirement, or benefited from a windfall, you will need to construct a portfolio of assets and decide how much you can safely withdraw each year.
The definition of income given by the economist John Hicks provides the starting point. Hicks defined 'a man's income as the maximum value which he can consume during a week, and still expect to be as well off at the end of the period as he was at the beginning' (Hicks, 1939, p. 172) (The sexism is remarkable since Hicks's formidable wife was a much better practical manager than her husband.) So if you plan for an 8 per cent total return, and achieve it, you could spend 8 per cent of the value of your portfolio each year and still meet Hicks's test.
Well, not quite. First, 8 per cent is a target, not money in the bank. Prudence suggests you should plan on the basis of a lower return, and therefore spend less. Second, taxes will eat into your total return. I've suggested you might allow 2 per cent for that, giving a realistic target return of 6 per cent. Third, inflation will erode the value of your assets. Even if prices rise at only 2 per cent per annum, you will need to earn a 2 per cent total return simply to stand still. Taking these factors of prudence, tax and inflation together, you might conclude that spending 3 to 4 per cent of assets each year would be consistent with Hicks's test. This is the sort of prudent spending rate widely used by sophisticated charitable funds such as university endowments. US foundations are required to spend 5 per cent of the value of their assets each year. This is to prevent the establishment of trusts such as the Netherlands foundation of Ingvar Kamprad, the fabulously wealthy Swedish founder of IKEA, which spends very little and grows indefinitely in value. The prudent spending rate should be a little less than your target rate, converted into a real post-tax rate of return, and should be revised from time to time in line with experience.
As an individual, you might choose a prudent spending rate a little higher than 3 to 4 per cent. Trustees must be more conservative than individuals. You, however, are mortal. Ben Franklin famously remarked that only two things were inevitable: death and taxes. For individuals, both are inevitable; for charitable institutions, neither applies. In setting a target rate of return, you might plausibly conclude that one roughly offsets the other. You can spend more than a fund with an indefinite time horizon, because you are mortal and there is no point in being the richest person in the graveyard: but you are probably subject to heavier taxation than the endowment. Older people, therefore, might be a little less conservative than younger ones – conventional wisdom tends to have it the other way round. Enjoy those round-the-world cruises!
Buying an annuity was a traditional method of providing for security in old age by converting a lump sum into a fixed lifetime income. In the modern world these investments are not attractive. Annuity rates are poor, because of high charges and investment strategies that focus on individually low-risk assets. There is also the problem of moral hazard created by information asymmetry. Annuities are bought by people in good health, not people with low life expectancies. Purchasers of annuities are more long-lived than the population as a whole. So a man aged sixty-five can expect to live another seventeen years, but an annuity for him will today yield, before tax, less than 6 per cent and, if linked to inflation, less than 4 per cent. These are rates available in dollars or sterling; an annuity in euros may offer 1 per cent less. All these figures are well below a realistic target rate of return.
**Reviewing your goals**
Turn now to these various categories of investment purpose: relevant planning, saving for a specific goal, prudent management of a lump sum and 'rainy day' saving.
Retirement planning is critical for almost every household, and even young ones must give it thought. At the beginning of the last chapter I offered an illustrative calculation of what you might need to save. You might want to make a similar calculation based on your own circumstances. You will want to tweak the figures in many ways. You will probably need less in retirement than in your working lifetime. You are likely to spend less, you will pay less tax and you will have some other sources of pension income.
Readers of this book are, on average, healthier than average, so your plan should cover at least twenty-five years of retirement.2 But – if you can – you should also plan to save for more than twenty-five years before retirement, to benefit from the extraordinary power of compounding returns. Your capacity to save may increase over time, along with your income. You may already have some assets which can be invested more effectively. Your calculation needs to be tailored to your particular circumstances.
This is the sort of problem that a model helps to solve. Remember that no such calculation will be true; like any model, it gives you the 'right' answer to the arithmetic problem, but not necessarily to the substantive problem – there are too many unknown unknowns. But the exercise can be illuminating and will help to illustrate whether your hopes and plans are feasible. You may want to use one of many useful programs on the internet. But with simple arithmetic you – or your children or grandchildren – should be able to construct a spreadsheet to calculate what you should aim to save for your retirement.
If your investment goal is a specific purchase, the lowest-risk method of achieving that objective may be to buy it now, even if you have to borrow. If you accept that risk is failure to achieve your realistic investment objectives, then both the financial difficulty that may follow excessive ambition and the inability to achieve your financial goals that may result from overly modest aspirations are risks. You will have to balance the risk that servicing the borrowing will prove difficult against the risk that you will not be able to afford to make the purchase in future. We are taught to regard the former, but not the latter, as a risk – to shun failure, but to accept disappointment. But both are sources of unhappiness and regret. The 'mind your portfolio' principle tells you that leverage is not always risky.
If you cannot, or do not want to, achieve your goal immediately, then you may be able to hedge it in whole or part. Buying a property – a secondary residence for yourself or an apartment for your children – is a common investment objective. The clearer you are about the characteristics of such a property, the better placed you are to hedge. Buy-to-let property may be relevant. If you are determined to buy a house in France, then you can bias your portfolio towards property and assets denominated in euros. Another common specific investment objective is to fund the education of your family. There are many schemes to help you do this. Take advantage of any tax breaks offered, but do not pay initial commissions or high ongoing charges to access these schemes. DIY will often be the best route.
The cautious investor of a lump sum will build a diversified portfolio of the kind I will describe below, but should not aim to do so immediately. Gradual investment, drip-feeding your funds into your portfolio, will allow you to benefit from 'dollar cost averaging', a potent 'get rich slow' scheme which I will also describe below.
For the lump sum investor looking to supplement earnings from investment income, determination of the prudent spending rate is critical. Of course, you may not want to spread the benefit over the whole of your life, but the calculation of a prudent spending rate provides a key benchmark. You may well find that annual interest and dividends are less than your prudent spending rate. If so, you will need to sell assets from time to time if your expenditure is in line with your prudent spending rate.
You may find this hard. The notion that a cautious person spends no more than his or her income is firmly ingrained in our culture, and for good reasons. A wise farmer would divide the crop into the amount that should feed his family and provide for their clothes and education – the income – and the amount – the capital – that should be planted for next year's crop. But principles that worked well on the land do not necessarily serve us well in a world of swaps and derivatives. Habits honed by evolution can be costly. It requires discipline to resist mental accounting. One mechanism for challenging this mental accounting is to set up an automatic transfer from your broker's account to your bank.
And then there is the 'rainy day' saver. The most attractive financial goal for most people is to know that no likely contingency will cause financial embarrassment. While your own circumstances will be particular to you, the meaning of financial security for most readers of this book is confidence about their future standard of living. There are several threats to the achievement of such security through an investment portfolio. The portfolio may decline in monetary value through market risk or credit risk. The monetary value of the portfolio may lose its purchasing power through inflation risk. I will discuss below how to control these risks through diversification, emphasis on long time horizons and a focus on real assets.
But beyond these known unknowns, the world is also uncertain. There are many historical instances of people who thought they had secured their financial future but who were proved wrong. Wealthy central Europeans, who had never expected to work for a living, discovered in the economic turbulence of the 1920s that they had lost all they had, because banks failed or because hyperinflation made their savings worthless. Prosperous Jews thought that owning their own houses gave them security but discovered that they would have been better off with assets overseas when Nazi persecution made it impossible for them to enjoy their German property.
Many people pay a high price for the appearance of certainty – docile citizens of East Germany, or the people who accepted dull jobs in large corporations in the belief that their jobs were for life. Both groups discovered that the certainty they craved was illusory. To take control of one's own life or financial affairs may seem to create uncertainty, but taking control is a prerequisite of security in a world that is inescapably uncertain.
It is impossible to eliminate financial worries altogether, as those twentieth-century European households discovered. More recently, Mikhail Khodorkovsky and Boris Berezovsky became billionaires in the 1990s as oligarchs prospered from the collapse of the Soviet state. But both these individuals fell out with the Putin regime. Khodorkovsky spent ten years in prison. Berezovsky, who had fled to London, was engaged for years in ruinous litigation with another oligarch, Roman Abramovich, until, bankrupt, Berezovsky was found hanged in unexplained circumstances in 2013. Such vicissitudes of fortune have been the experience of many oligarchs throughout history.
A few investment gurus, harbingers of the apocalypse, recommend gold bars, or the equivalent, which can be cashed in the event of financial collapse. Barton Biggs, Morgan Stanley's long-term naysayer, urged everyone to have 'a farm or a ranch somewhat far off the beaten track but which you can get to quickly and easily'. This is not a realistic recommendation for most people. But there are episodes in history in which coins or jewellery have literally been lifesavers.
**Diversification is the key to managing risk**
The conventional investor believes that there is a trade-off between risk and return. This is another aphorism that is sometimes illuminating, but often untrue. The assumption of trade-off gives insufficient attention to the 'mind your portfolio' principle. Even if high risk and high expected returns were indissolubly associated at the level of the individual security, it does not follow that the same relationship holds in a portfolio. You cannot measure the risk of a portfolio by simply aggregating the risks of individual investments.
A portfolio that consists of a collection of idiosyncratic but individually risky investments, such as Robb Caledon, can be a low-risk portfolio. Such a portfolio may carry a lower risk, in fact, than a collection of blue chips – stocks that are individually safe, but whose returns are likely to be strongly correlated with each other. For the intelligent investor, risk is a characteristic of a portfolio, rather than of the individual security. This is the illuminating insight of the capital asset pricing model, and that insight remains valid independently of reservations about the truth of the CAPM.
In fact, the insight is profitable for investors precisely because the capital asset pricing model is not true. The pattern of returns on different assets reflects other people's views of risk, uncertainty and security. The risk premium they offer is the product of the average of other people's attitudes to risk. But other people's attitudes to risk are different from yours. They avoid mistakes for which they might be criticised. They are wedded to benchmarks. They focus on the riskiness of individual assets, rather than the risk of their overall portfolio. Their mistakes, or at least their different approaches to risk, provide your opportunities.
Conventional investors regard risk as short-term volatility of return, but the intelligent investor has a different conception of risk. _Your risk is your failure to meet your realistic aspirations._ Your strategy is the product of your long-term investment goals, and the longer your time horizon, the better the opportunities for you to achieve satisfying combinations of risk and return.
Suppose the premium that volatile assets command over what are conventionally described as risk-free assets is 4 per cent, a common assumption for the size of the equity premium. The magnitude of this figure reflects these conventional investors' fear of short-term fluctuations in the value of their assets. Suppose also the standard deviation of annual equity returns is 15 per cent (approximately the historical average). Then over the next week the expected return on an equity investment is less than 0.1 per cent more than the risk-free asset.
And the spread of outcomes is much wider; that follows from the way risk has been defined. Even Warren Buffett and George Soros lose money most days. If the distribution of returns is normal – a reasonable assumption over one week – then there is almost a 50 per cent chance that the 'risk-free' asset will do better. Over a year, however, the expected value of the additional return from volatile assets is 4 per cent, and the probability that there will be _some_ additional return to compensate for not, has risen to around 60 per cent.3
Stretch the time horizon to twenty-five years, and the divergence becomes much greater. The expected value of the 'risky' portfolio is 167 per cent higher. If the distribution of returns is normal, this 167 per cent advantage is also the median and the modal outcome. Only the very lowest percentiles of the distribution of volatile returns will fall short of the returns on the 'safe' asset. In fact, you will do better in the risky asset on almost 99 per cent of occasions. And the longer the time horizon, the more certain it is that volatile assets will outperform (on the assumptions above about risk premium and volatility). The notion that cash and short-dated bonds are less risky than more volatile assets is the product of the confusion between certainty and security identified in Chapter 7.
Moreover, we have seen that there is no such thing as a risk-free asset. Extreme percentiles of the distribution of outcomes are typically generated not by risks but from uncertainties. These extreme events are sometimes the result of very low-probability risks incorporated in models: the risk that a fair coin comes up heads twenty times in a row. But disastrous outcomes are most often a consequence of the large-scale breakdown of social, political and economic institutions; the Russian Revolution, the hyperinflation of the Weimar republic and the lawlessness of the Nazi regime which followed and, as I am writing this, the breakdown of the Venezuelan state. It is events such as these, rather than runs of bad luck, which are most likely to destroy the value of an investment portfolio.
Even in Britain such institutional breakdown was within sight on three occasions in the last century: in 1931, when the Great Depression threatened political stability; in 1940, when Britain faced the real prospect of hostile invasion; and in 1974, when securities markets collapsed as a weak government presided over inflation and labour unrest that seemed out of control. The United States weathered the political and economic crisis that had closed the nation's banks as Franklin Roosevelt was inaugurated President in 1933; but the survival of the US market economy was no foregone conclusion. In these latter cases breakdown was averted. But in none was the margin so wide as to suggest that collapse was inconceivable.
Since distributions of investment outcomes have fat tails, especially at the lower ends of the distribution, even the most conventional portfolio can suffer from events that fall far short of these apocalyptic outcomes. The financial failures with which we are more recently familiar – the 1987 crash, the credit crisis of 2007–8, defaults of Greek debt – represent bad, but not extreme, percentiles of potentially predictable risk distributions (although, as Mr Viniar revealed, even these events were not within the range of possibilities contemplated by Goldman Sachs's risk models4). Big upsets come from even milder crises – the Asian collapse of 1997–8 or the bursting of the New Economy bubble. All these events, however, are sufficiently abnormal to have fallen outside the scope of conventional modelling.
You do not know what investments will pay off in these extreme circumstances, and nor does anyone else. You can build an investment portfolio for a rainy day, but what of a monsoon? The only certainty you can have is that a well-diversified portfolio gives you the best chance that at least some assets will return their value – as it did for those refugees who fled with their jewellery.
**The illusory security of cash and bonds**
Over the years since 1980, intelligent investors have been able to make a lot of money in bonds. As worldwide inflation subsided, interest rates steadily fell from the peaks they achieved in the 1970s. These declines in short- and long-term interest rates provided capital gains to bondholders. Even Queen Victoria's Consols proved a highly rewarding investment.
But even after inflation had more or less ended in developed economies, bond yields continued to fall and their prices to rise. The policy known as 'quantitative easing' was adopted in response to the 2008 global financial crisis. Central banks – initially those of Japan, the UK and US, and more recently the European Central Bank – have bought bonds in enormous quantities. The Federal Reserve Board holds US Treasury securities worth trillions of dollars. The resulting distortion of the bond market has made the sector unattractive to intelligent investors.
Safe bonds from stable countries with interest rates that are around zero or even negative do not sound like attractive investments, and they are not. If you were thinking of buying the 2046 bund described in Chapter 3, with its 1 per cent redemption yield, you might instead consider an apartment in Berlin or Frankfurt. That investment will yield between 3 and 5 per cent. Even if the rent never rose and the property was worthless, by 2046 you would have earned more from the property than from the bund. Instead of the thirty-year US Treasury bond, you might consider the stock of Johnson & Johnson, the diversified healthcare company. These shares yield 3 per cent, more than the 2.6 per cent GRY on the US Treasury bond. But while the Treasury will make a fixed pay-out till 2046, Johnson & Johnson has increased its dividend every year for the last fifty years, and that dividend today is well covered by the company's earnings.
Since safe government bonds today are so unlikely to pay more than reasonable alternative investments over the long run, who holds them, and why? As a result of quantitative easing, governments themselves are, through their central banks, by far the largest owners of their own securities. But even now, most government bonds are in private hands. Some long-dated bonds are held speculatively, by people who hope to sell at a profit next week and have no interest in what happens in 2046. Others are held by institutions which are only interested in what happens in 2046. These are typically pension funds and insurance companies, which have fixed monetary liabilities stretching far into the future, and which are under regulatory or other pressure to find assets that more or less exactly match these liabilities.
But you do not fall into either category. You are not trying to anticipate the changing 'mind of the market', nor attempting to meet a fixed money claim falling due in 2046. With bonds so unattractive that you do not want to hold them directly, you also do not want to hold them indirectly, by participating in insurance or pension funds which hold them in significant quantity.
Cash and short-dated bonds are often described as low-risk investments. But they are not low-risk investments for those who want to protect their future real living standards. Many people find that statement difficult. Part of the problem is confusion between the minimisation of risk and the search for certainty. With cash and short-dated bonds you know – in a sense – what is going to happen; it is highly probable that you will get back the cash you put in. But if security is confidence in your long-term standard of living – the ability to be relaxed about retirement, the confidence that you can provide for your children or one day buy the property you have dreamed about – then cash does not provide that security.
Few if any of your long-term future liabilities are fixed in money terms. Your plans for the future are defined in real terms – the cash needed to fulfil your objectives is dependent on the purchasing power of the money of that future time. The inflation protection provided by index-linked bonds potentially offers greater certainty. Even then it is difficult to match the bonds exactly to your personal objectives. You are exposed to interest rate risk unless the pattern of payments on the bond precisely corresponds to the time path of your planned expenditures. You are also exposed to personal inflation risk – the price of the goods and services you buy is not the same as the representative basket chosen by official statisticians. And the return is extremely low; currently most maturities of index-linked bonds offer a negligible or even negative real yield.
Even in more normal circumstances cash and bonds have little to offer the intelligent investor. The greater certainty that cash appears to provide is perhaps only the certainty that you will not do well, since nothing can provide certainty that exceptionally unfavourable outcomes are unlikely. Even in a relatively stable political and economic environment, a portfolio of bonds offers little security. Investors in British Consols lost more or less their entire wealth in the course of the twentieth century. A portfolio of indexed bonds would perform better, although this is a modern asset category that has never been tested in extreme conditions. Risks can never be eliminated, and uncertainties are inescapable. The key issue for the intelligent investor is to understand that achieving security lies not in avoiding all risk – which is impossible – but in diversification. Do not be too vulnerable to any particular contingency, however improbable. Hold as many options as possible against a necessarily unpredictable future.
The relevance of bonds to an investment portfolio is not that they are risk-free, but that they may do well in circumstances in which other investments will not. British government bonds served their holders well in the Great Depression of the early 1930s, and were the only major asset category to do so. In Britain the oil crisis of 1973–4 became a political crisis leading to rapid inflation and a collapse of shares and property values. If indexed bonds had been available, they would have performed strongly, and would almost certainly have been the only asset class to do so.
I don't think cash or conventional fixed interest bonds normally have a role in an intelligent investor's portfolio. You will want to keep enough cash on hand for impulse and emergency purchases. Many, probably most, of the investments in the intelligent investor's portfolio are as good as cash for these purposes. You can turn an investment in a blue chip share into money in the bank within a week. But we employ mental accounting. We don't think of our liquid securities as cash. Like me, you will probably keep more money in a deposit account than you need. You relinquish return, and gain only superficial security, from these habits.
These conclusions represent a very different approach to risk from that of the conventional investor. Most financial advisers will recommend a mix of safe and risky assets, assuming that the meaning of safety is both obvious and universal, and that investors need to be paid a premium to relinquish that safety. The robo-adviser, constrained by regulation to follow the conventional wisdom, will make similar proposals. But the most rewarding strategy for the intelligent investor is to construct a low-risk portfolio from a collection of assets that the conventional investor perceives as risky.
**Contrary thinking**
The conventional investor avoids judgements wherever possible. The intelligent investor is willing to make judgements. But most retail investors show bad judgement. The average investor does much worse than the market average mutual fund. The US investment research house Dalbar has for many years measured the average return earned by investors in mutual funds. Over twenty years the annual return to mutual fund savers who invested in stocks has been 5.2 per cent. Not too bad, you may think – until you are told that the total return on the S & P 500 index, the principal US stock market index, has been 9.9 per cent. The average return to holders of bond funds has been a miserable 0.8 per cent, against 6.2 per cent for the equivalent bond index. This underperformance – averaging 5 per cent per year – has persisted, year in year out, in the Dalbar analysis.
Why? The average mutual fund investor, with encouragement from the financial services industry, gets timing badly wrong. Momentum effects are picked up most quickly by proprietary traders, next by professional fund managers, and finally by the marketing departments of financial services businesses. Their sales people, naturally, promote the funds that have recently done best. So money floods into fashionable sectors. Retail investors bought technology in 1999–2000 and were encouraged to buy bond funds after equity markets had collapsed. Subsequently, property funds were strongly promoted. In each case, investors bought at the top and quickly incurred losses. Doubtless they will have the same experience in bonds when interest rates return to more normal levels.
Retail investors, towards the back of the crowd, tend to suffer from mean reversion rather than to benefit from momentum. They enter the market when short-term positive serial correlation turns into long-term negative serial correlation. Jaded investors later sell the funds that promised much but delivered little. They get it wrong both when they buy and when they sell. I suspect that more than a few readers of this book recognise their own experience.
Timing peaks and troughs in the market is as hopeless as looking for hot tips. Professional doomsters continually predict market crashes and are, like stopped clocks, occasionally right. Chartists detect patterns or investment indicators that provide buy and sell signals in randomly generated data. There is little evidence of anyone, amateur or professional, generating superior returns from systematic identification of the tops and bottoms of market cycles. George Soros and Warren Buffett understood well that the 1999–2000 New Economy bubble would burst and that markets would crash. But they didn't know the date, and so they didn't profit. Even if you know that long-term mean reversion will eventually lead prices back towards fundamental values, you can't tell when. If Soros and Buffett can't – and no longer try – nor can you.
Once you have accepted that you can't time the market, there is a great deal you can still do. The dismal record of mutual fund investors suggests a place to start. Watch what these people do, and do the opposite. Their underperformance will be your outperformance. Look at the sectors promoted in advertisements. These are the sectors to avoid. When the promotions favour technology and emerging markets, think infrastructure and property. When they show photographs of roads and offices, think technology and emerging markets. Be contrarian. Your aim is to avoid following in the rear of the crowd, and to be travelling in a different – even the opposite – direction.
Modern financial markets are dominated by the power of conventional thinking. People in the financial sector share superficial views with each other and project current trends too far and too fast. Social and commercial pressures, reinforced by ubiquitous benchmarking, encourage professional fund managers to act on these widely shared opinions regardless of private reservations.
There is, almost always, an underlying element of truth in conventional thinking. The internet was an important commercial and social development. The economic progress of China and India does have a major impact on the global economy. This information is 'in the price'. More than that – the herd behaviour of professional investors means that these conventional opinions are more than fully in the price. That is part of the reason why prices display positive serial correlation in the short run and negative serial correlation in the long run. In the short run, momentum drives prices as fashion spreads; in the long run, mean reversion drives prices as fashion fades.
There are two possible ways of exploiting this behaviour. One is to ride momentum. But this is like riding a tiger. The experience will certainly be stressful and exciting, and it may be rewarding for a time, but the difficulty of timing accurately when you get on and when you get off is so great that, sooner or later, it is probable that you will be mauled (as happened to Julian Robertson). There is not much evidence that anyone can do it successfully over a long period. If anyone can do it successfully, it will almost certainly be someone very close to the market, who can detect shifts in sentiment in that brief interval when it is still possible to act but before these shifts are incorporated in prices. It is very unlikely that you will be that person.
In the second strategy – defying the fashion – the retail investor potentially has an advantage over the professional. Not being close to the market means that you are not subject to the pressures the professionals experience. You don't have to talk to trustees or clients who want to act on the news they have read in the newspaper that morning (and which everyone else has also read) – that China is growing rapidly or that the housing market is dull.
The converse of the power of conventional thinking is the power of contrarian investment. There is something paradoxical in the idea that the best way to use the expertise of the financial services industry is to do the opposite of what it recommends. This is not because that expertise is no good. It is because that expertise is 'in the price' and characteristically more 'in the price' than is objectively justified.
Some of the biggest mistakes in my own investment history have been following the crowd when the crowd was going in the right direction – for example, buying European property in 2006, when the mind of the market identified it as an undervalued asset class. Even if the crowd is right about fundamental value (and in that case I think it was), the fashion is 'in the price'. When the fashion fades, so will some of the money you have paid (and within two years, it had).
Being contrarian should not be interpreted as perversity, or counter-suggestibility. Begin by limiting your contrarian strategy to broad asset categories and to funds. Be much more hesitant about taking a contrarian view of individual shares. There is a real likelihood that a share price has fallen because someone knows something you don't. It is a common mistake to believe that if a share once sold for €50 it will reach that value again. A stock that has fallen by 95 per cent from its high has halved in value since the date at which it had fallen from its high by 90 per cent.
A contrarian investor will, however, see cases where a stock has fallen for a specific reason but by an amount far larger than that specific reason would justify, as Buffett did at American Express. The scandal was soon forgotten by both customer and market, and the stock price rose steadily. If, however, the fall had affected all house-builders, or most Japanese stocks, then it is more likely that the explanation of the price fall is to be found in general knowledge exaggerated in conventional thinking than that it is the result of inside knowledge not yet available to the markets.
Contrarianism is equally relevant to your purchase of funds. Buy the funds that aren't promoted rather than the ones that are. Closed-end funds move to large discounts to asset value when the sector in which they specialise falls out of favour, and this is an opportunity to buy.
**Market timing**
Here is a scheme for beating the market that really works. Imagine a volatile share that sells for 50c in odd years and €1 in even years. If you invest €100 every year in this share, over a ten-year period you will have accumulated 1,500 shares at an average price of 66.7c, well below the average market price, which is 75c.
This system will, on average, outperform the market, and the more volatile the markets, the greater the gains. The method is known as dollar cost averaging. It works through its built-in mechanism for buying more when prices are relatively low and less when prices are relatively high. No judgement is required by the investor that prices are relatively low or relatively high.
Many individuals do, and more should, make regular savings to build up an investment portfolio. Some investment managers use the benefits of dollar cost averaging and extol the benefits of a regular savings scheme in their marketing literature. On this occasion, believe them. The benefits are real. But keep an eye, as usual, on their charges. Regular saving is a practice you can organise for yourself through an execution-only stockbroker. You do not need to pay someone a commission to take direct debits from your account.
The effectiveness of dollar cost averaging illustrates how low share prices are an opportunity rather than a problem for the intelligent investor. Conventional investors are usually excited when they hear that the market is going up, disappointed when they learn that it has fallen. As Buffett has occasionally pointed out, this is odd. We are pleased when we hear that a favourite shop has a sale, or car dealers have reduced their prices. We want high prices when we are sellers, but low prices when we are buyers. We are usually only buyers of clothes or durable goods. However, for shares – as perhaps for cars and houses – we may be both buyers and sellers at different times.
When should the intelligent investor sell? The simple answer is 'not very often'. Frequent trading endangers returns. But sometimes the mind of the market will take a security to a price well in excess of any reasonable estimate of its fundamental value. Money is a means to an end, not an end in itself, and you will sometimes sell investments in order to spend, especially if most of your total return is capital gains. In recent years the rate at which small companies have been acquired has ensured that portfolios have a regular inflow of cash from acquisitions. This will not be true if you invest mainly in funds rather than individual securities. But if you invest in well-chosen funds, you have even less reason to wish to trade.
Dollar cost averaging makes sense for both conventional and intelligent investors. Conventional investors can feel relieved that their approach yields profits without requiring judgement. Intelligent investors know that market timing is unlikely to make money and can feel happy with an approach that is inherently contrarian. The reasoning that makes dollar cost averaging attractive applies to decisions about asset classes as well as to decisions about market timing. Let's see how.
**Asset allocation**
Investment consultants and managed funds follow each other closely. Trustees or managers are required to set benchmarks for asset allocations, and deviations from the norm – the average of what others do – require considered justification. The best way for them to minimise risk is to do much the same as everyone else. This is, of course, not a course of action that minimises risk for their beneficiaries.
When an asset category rises in value, the proportion of that asset class in the benchmark increases. Individual investors may choose to follow the market allocation; institutional investors feel obliged to follow it. Such benchmarking has led to a steady increase in the share of equities, the best-performing asset class, in conventional portfolios.
The consequences can be perverse. At the peak of the Japanese stock market bubble of the 1980s, Japanese shares accounted for almost half of the value of all shares – American, European, Brazilian, Australian – in the world. American and European investors, who had thought they were brave if 10 per cent of their assets were in Japan, felt under pressure to acquire more securities in Japan. Fifteen years later, Japanese share prices had fallen, while those in other countries had risen. Today Japan accounts for less than 10 per cent of the market value of world indices.
Common sense suggests that an allocation of half your portfolio to Japan in the late 1980s was far too much, given that Japan never accounted for more than 10 per cent of world output. Most investors did, at least intuitively, understand this. While few people fell comprehensively into this Japanese trap, many investors have been victims of milder versions of the error, buying technology shares, bonds, infrastructure and property at the wrong times. Looking at market weightings when deciding asset allocation leads institutions and fund managers to buy high and sell low. This strategy leads them to purchase overpriced assets in order to achieve desired portfolio weightings.
The search for 'fundamental indices' is an attempt to resist these perverse behaviours. The idea behind a fundamental index is that asset-class weightings should be based not on market value but on some underlying measure of economic contribution. This insight leads intelligent investors to look behind financial assets and judge the value of the productive assets that underpin them. A benchmark for allocation to Japan might, therefore, be Japanese national income as a percentage of world national income, implying a range of 5–10 per cent that would remain unchanged by the vagaries of the Japanese stock markets.
From this perspective the model pension fund portfolio contains a heavy concentration of oil companies, pharmaceutical businesses and banks relative to the economic importance of these activities. The investor who has thought in terms of fundamental indices is bound to wonder why a conventional institutional portfolio allocates ten times as much to stocks as to property. Very large economic sectors, such as agriculture, education and healthcare, and legal and accounting services, are not represented at all in the model pension fund portfolio, or only in very limited ways.
You can change that. There are few quoted securities in agriculture, education or health, but there are some. There are (as yet) only a few stock market prices for law firms or accountancy practices (the first and largest quoted law firm in the UK was called Quindell, and many people pronounced its name with emphasis on the first syllable). There are other businesses, such as recruitment agencies and public relations firms, whose fortunes are closely related. The conventional investor's assets are much less diversified than they might be.
The numbers that emerge from analysis based on fundamental indices may be illuminating but not true – certainly no one should regard them as more than qualitative guides. The illuminating insight is that investors should be wary of allowing fluctuations in market prices to influence their target allocations to different asset classes. If the price of property rises relative to the price of your other assets, consider reducing the proportion of your assets you hold in property, and vice versa. Many people find this paradoxical. Should we really sell securities just because they have done well? In a world characterised by momentum and mean reversion, you should. That way you can realise the benefits of dollar cost averaging in asset allocation as well as market timing.
But what is the right asset allocation? How should an intelligent investor's portfolio be divided between shares, bonds and property? I'm not sure that this is the right way to pose the asset allocation question. To do so supposes that all shares are much more like each other than they are like any of the assets in the other categories, such as property and bonds, and that the same is true in turn of these asset categories.
For the assets that institutional funds typically hold, it may well be true that all the securities within a given asset class are much the same. Their share portfolio will mainly contain large, global companies, whose prices tend to go up and down together; the bonds will mainly be medium- and long-dated government securities; the properties will mainly be well-located city centre office blocks with similar prospective growth in both rental levels and capital values. These funds will have followed the conventional investor's strategy of minimising the risks on individual investments rather than the intelligent investor's strategy of looking at the risk of their portfolio as a whole. So when they buy property, they buy whichever category of property their advisers currently tout as the best mixture of return and reward, and they buy several such properties. Following this conventional strategy, they fail to diversify effectively.
The 'mind your portfolio' principle implies that you should look at the expected return from individual assets and their correlation with your overall portfolio. Your objective is that every single investment you buy should significantly diversify your portfolio. You don't need, and probably shouldn't have, an intermediate process of asset allocation. Asset allocation percentages will be the result, not the preliminary, of diversifying investment decisions about individual securities. In the next chapter I'll discuss how these securities should be selected.
**Footnotes**
1Interview reported by Jason Zweig, 2008.
2Seriously. Life-expectancy increases with education and income level.
3This calculation assumes that the distribution of returns is normal and that successive period returns are independent. As I have explained, there are good reasons for scepticism about both these assumptions. These illustrative figures should be treated as illuminating rather than accurate.
4Perhaps significantly, Goldman had done more than its competitors to protect itself against the events of 2007–8, suggesting that its qualitative analysis was more reliable than its modelling.
12
**Intelligent Investment Choices**
**Diversify**
Diversification within an investment portfolio is the key to risk management. The conventional investor's portfolio is unnecessarily risky even by reference to its own measure of risk – short-term volatility. Between 2007 and 2009 the conventional investor's portfolio would have fallen in value by around 30 per cent. Many pension funds, insurance companies and charities suffered badly during that period. Nor was that fall unprecedented. There was a similar setback in 2000–2. The loss in 1973–4 would have been greater, and there would also have been other significant dips in value.
But in practice short-term volatility is not the only risk that concerns many conventional investors. They interpret low risk as meaning avoidance of investments that involve the possibility of significant loss, and aim to ensure that each individual stock in their portfolio is, in this sense, low risk. There is evidence that the distress people experience from losses is significantly greater than the joy they experience from gains of similar magnitude. If they react in this way to losses on individual securities as well as those on their portfolio as a whole, the discomfort they will experience on reviewing a pool of assets with any price volatility at all is likely to be considerable. Of course, they can minimise their regret by 'minding their portfolio' and refraining from looking constantly at the value of their assets; but that requires mental discipline.
And from the perspective of the pension fund trustees and their investment managers such aversion to loss is understandable. As I have emphasised, the major risk a financial adviser runs is not the risk that his clients do badly but the risk that his clients do worse than other people. Every committee and large organisation is crowded with people who will say 'I told you so' when things go wrong. The conventional institutional investor will be reluctant ever to hold an asset that the institution, or the responsible manager, can be criticised for having bought. I can take more individual stock risks with my personal investment portfolio because I don't have to justify decisions, before or after the event. I can do that even though my personal style is to maintain a low-risk portfolio. I am free to 'mind my portfolio', and so are you. The typical institutional investor is not. Robb Caledon would be a risky purchase for anyone managing someone else's funds, because there is a substantial probability of total loss.
Conventional investors focus on large, widely held stocks. There was for many years a maxim that 'no one ever got fired for buying IBM', and that held for IBM stock as well as IBM computers. Conventional investors then claim to diversify by including many of these popular stocks with big market capitalisations in their portfolio. But holding many stocks is not the same as diversification. Look behind the index and the broad allocations to asset categories and see the disposition of the conventional investor's investment in a global index. There are large holdings in the oil majors – Exxon, Shell and BP, and in General Electric, Pfizer and Nestlé, Apple and Microsoft. The portfolio will, even after the recent collapse of the banking sector, have a large exposure to finance. All these companies sell mainly to the advanced economies of the United States and Western Europe. Their fortunes wax and wane with the world economy, and so does market sentiment towards them. Returns from all these stocks are strongly correlated with each other. Neither the market indices nor the conventional investor's portfolio are well diversified.
So how should the intelligent investor approach diversification? You do not need to hold many different securities if the returns from those you do hold are uncorrelated with each other. Recall how the volatility of outcomes of the coin-tossing game diminished rapidly as the number of tosses of the coin increased. Understanding correlation is the key to successful diversification.
In the last decade many conventional investors who emphasise asset allocation between broad categories – quoted equities, real estate, private equity etc. – have been concerned that all these asset classes appear strongly correlated with each other. During the global financial crisis all these real assets performed badly, and even companies whose business was little affected by the financial crisis experienced stock price falls as conventional investors, struggling for liquidity, sold whatever could be sold. Then the loose monetary policies of low interest rates and quantitative easing pushed up all asset prices almost indiscriminately.
But while short-term correlations are governed by the mind of the market, long-term correlations are the product of fundamental value. An apartment in Berlin, a field of land with development potential within reach of London and an office block in Hong Kong are all classed as real estate; the stock of Apple, shares in a supermarket chain in Taiwan and a Silicon Valley start-up are all categorised as global equities. And in the short run, market liquidity and herd behaviour have similar effects on them all. But over twenty years the factors that determine the success of each of these investments are not only different but specific to each individual property or stock.
While the covariance matrices that investment consultants use help to focus attention on diversification, they are of little use to intelligent investors. The asset categories they use are insufficiently granular – domestic equities, real estate – and the historical data used to compile them are often too short-term and are always likely to be undermined by technological or institutional change. The only way of understanding the correlation between Robb Caledon shares and the wider market in 1976 was to have knowledge of the peculiar politics of that time – knowledge that could never be expressed in a covariance matrix.
Almost everyone thinking about investment opportunities begins by asking the question 'What is likely to go up?' But general knowledge about companies, industries and countries is knowledge that you share with everyone else; it is already 'in the price'. The record of most investment professionals in correctly identifying what will go up is poor. You don't know what is going to go up. That is why you may have been tempted to take advice. But those who offer to advise you have little or no more valuable knowledge than you, and they have different interests from yours.
The intelligent investor gives as much attention to correlation as to expected return and looks at both by reference to fundamental value. There are two groups of question to ask in considering any particular proposed investment. First, is the potential return on this investment consistent with my overall investment objective? Can I reasonably anticipate that this holding might meet my target total return of 8 per cent per annum over a five-year period? And second, what are the factors specific to this asset that will determine its success or failure over that five-year period, or longer? Are the characteristics of this investment significantly different from those of other securities I already own?
This approach means rejecting investments for which there is a plausible case because the returns from these investments are correlated with investments you already have. This will happen often, since the reasons that made the earlier purchase seem attractive may also apply to the new one. An indexed portfolio is more diversified than a portfolio in which you keep using similar arguments to buy more assets of the same kind.
An indexed portfolio is neither diversified nor contrarian, because the weights in the index are driven by the power of conventional thinking. A more diversified portfolio should offer you expected returns similar to those of an indexed portfolio with lower volatility. (I need to enter again the health warning that if you accept the conclusions of the CAPM, as most finance academics – though few practical people – do, then the diversified strategy will be so popular that it will force down the average return from adopting it. Perhaps that proposition should be true, but I don't think it is.)
Increasing diversification means less in pharmaceuticals and oil companies, and more in sectors whose performance is less strongly correlated with the investments you already hold. General insurance companies, for example, experience cycles of their own that are not necessarily in sync with the general economic cycle.
The performance of gold mines or other commodity producers is likely to be very different from that of the general stock market. Diversify internationally. But there is no point in reducing your holding in Glaxo in order to increase your holding in Pfizer – these companies have very similar businesses. A small pharmaceutical company – whatever its nationality – whose fortunes will probably rise or fall with the success of individual drugs rather than the sector as a whole will offer returns with lower correlation. Look to Japan, or Russia or Taiwan, but real diversification from these investments will come from companies oriented to their domestic economies, rather than businesses with which you are more familiar. Sony, Gazprom and Lenovo sell to the same people as Microsoft, Shell and Apple.
**Fund selection**
Most investors will, and should, begin with funds rather than individual stocks. Funds may be distinguished by investment style or by sectoral or geographic specialisation. Choose funds whose returns are not strongly correlated with each other – be diversified. Choose funds which are out of favour rather than fashionable – be contrarian.
A web site such as Morningstar will give you a wide – far too wide – selection. In every category there are many similar funds doing similar things. I am not going to recommend specific funds, but the criteria I will describe will narrow the range very considerably.
You might consider exchange-traded funds here, since the range of simple index funds is limited, and this sector offers a variety of country options. You can invest not just in the indices of the major developed markets but also in Chile or Taiwan and other small and emerging economies. In the larger markets you can choose to focus on smaller or larger companies, or on a particular sector, such as utilities or real estate. Plan to build up a diversified portfolio of such funds, emphasising sectors that are unfashionable.
Then consider an allocation to actively managed funds. Pick two or three funds with widely different styles and approaches. This gives a better balance of risk and return. But keep a close eye on charges. A company that charges 1 per cent or more for a closet index fund is ripping you off. Buying a closed-end fund with low charges and a large discount to asset value gives you a chance of earning the return generated by the underlying productive investments. Most closed-end funds go to a discount of 15 per cent or more from time to time if you are content to wait. The widening of the discount is a good contrarian signal when it is the result of the unfashionability of the sector rather than the poor performance of the fund. If management costs are not substantially less than 1 per cent, the fund must justify these charges by the proven success of its distinctive style (including low correlation with market indices).
How to distinguish proven success from the 'hot hand'? Favour funds that are strongly identified with an individual. Good fund managers are people rather than corporations, although on occasion good processes are the creation of innovative founders. Idiosyncratic funds with a strong record are typically provided by boutiques, but there are exceptions – legendary fund managers Anthony Bolton and Peter Lynch ran funds for Fidelity.
Buy funds that have a history of good performance, but look for a long history and, above all, ensure that the track record was good both before and after the fund manager became famous. Many good track records are the product of lucky streaks, and are then aggressively marketed, only to disappear. There is some evidence of investment skill, but it is rare.
**Property**
While most equity funds are open-ended, most property funds are closed-end. Many countries, including the US and UK, have established a special real estate investment trust (REIT) tax status, under which shareholders are taxed broadly as if they owned a share of the properties of the fund. There are also closed-end property funds that do not qualify as REITs, typically those with large development programmes and extensive borrowings.
Closed-end property funds generally have substantial leverage – much more so than closed-end funds that invest in shares. You need to be careful about this leverage. Your €1,000 investment in a fund might correspond to €3,000 of property and €2,000 of borrowings. If you hold leveraged assets, you should remember the 'mind your portfolio' principle, and understand that the leverage of the fund is also your leverage. Your €1,000 in the fund corresponds to a €3,000 commitment to real estate.
Open-ended property funds mostly do not use leverage. Some invest directly in property, and sell and redeem units at a price that reflects the value of the underlying assets. When the property market fell in 2007, some open-ended funds suspended redemption. Other open-ended property funds hold REITs and other property shares. While some property funds cover a wide spectrum of properties, most have a bias towards particular geographical areas or sectors. As in other areas of investment, closed-end funds are attractive when you can buy them at significant discounts to their asset value; again, do not buy closed-end funds at a premium. You should expect to pay higher management charges for a property fund than for an equity fund, because the costs of looking after buildings are greater than the costs of looking after paper.
With such a range of possibilities it is easy with two or three purchases to construct a portfolio of property assets diversified across different kinds of property – offices, shops, industrial, residential – and across geographical locations.
**Stock picking**
As your experience of intelligent investment grows, you will wish to consider individual stocks. Even if you have a large, diversified portfolio, you will probably not own more than 1 per cent of the stocks you could potentially buy. That means you can, and should, be extremely selective. It is as though you were speed-dating thousands of potential partners. You are fortunate in being spoilt for choice, and one good reason for rejection is enough.
The basic principles of stock picking follow from Chapters 5 to 8. You are concerned with fundamental value, not momentum in the share price – with the characteristics of the company, not the stock price history. Conventional investors are unduly averse to specific risk which affects one company alone and which can and should be diversified. You do not mind that an individual stock is risky, so long as it does not add greatly to the risk of your portfolio as a whole. That proposition implies, of course, that you can understand what these risks are.
If conventional investors are sometimes too prone to avoid risk – the known unknowns – they are also too prone to accept uncertainties – the unknown unknowns. Use probabilities to assess risk, be detached and minimise uncertainties by knowing when you don't know.
Everywhere and always in financial markets, you are vulnerable to information asymmetry. The company and its executives know more – or are at least capable of knowing more – than you do. The greater the information asymmetry, the more likely it is that you are being offered a lemon. If you don't understand something, or the affairs of the company or the strategy or construction of the fund appear unnecessarily difficult to understand, do not buy. If the share price is falling for inexplicable reasons, do not buy.
Institutional investors will have access to the research on companies that banks circulate to their clients. You will find it difficult to obtain much of this, although you can see similar material in newsletters (tip sheets) or even on bulletin boards. Research reports are more useful for the information they contain than for their recommendations (though you probably will have readier access to the latter). You should understand a little of what financial analysts do, not in order to reproduce their conclusions but because their approach and conclusions are 'in the price'.
An analyst will provide 'coverage' of a group of companies with similar activities. If these businesses are very large, the analyst will be responsible for only a very few companies, though an analyst of smaller companies may cover twenty or more stocks. The bank that employs the analyst frequently hopes to obtain corporate finance business from the companies on which it undertakes (or has already undertaken) research; for small companies, the bank that managed the IPO will often provide the only coverage. This limits objectivity.
An analyst is rewarded for 'good calls'. (ThomsonReutersExtel ratings, based on opinion polls of institutions, have a major influence on the reputation and remuneration of analysts.) A good call is a successful prediction of a sharp upward or downward movement in a share price. But the time-scale on which the call is judged will rarely exceed six months. Although many research reports appear to be concerned with fundamental value, a good call will often anticipate a news announcement relevant to the company (generally either merger and acquisition activity or a substantial revision to earnings expectations). Or, in the manner of Keynes's beauty contest, an analyst may make good calls by being quick to ascertain the changing mood of other analysts.
The favoured analyst will be close to the companies he follows, able to infer from nudges and winks whether announcements will be behind, or ahead of, market expectations. The less favoured analyst – one whose reporting has a critical tone – will be shut out. Analysts who take a negative view of a company frequently receive abuse from chief executives, and the senior management of banks that employ critical analysts frequently receive complaints from the offended companies.
As I described in Chapter 5, the effect of the rise of financial conglomerates was to transform analysts into a sales force for investment banks. Following the New Economy bubble, the damage to the reputations of the financial conglomerates concerned, and pressure from regulators, forced investment bankers to concede greater independence to analysts. Recommendations still have a strongly positive bias, but experienced investors know this and are able, at least in part, to discount this bias. Analysts' research is focused on short-term earnings announcements and projections.
Most analysts simply project current earnings in a mechanical fashion. Even if they are trying to assess fundamental value, they typically do the calculation by making estimates of earnings and translating them into a DCF model. These analysts' valuations are what is 'in the price'. Estimates of fundamental value play a relatively minor role.
**Stock selection**
A basic checklist for any company you consider investing in should contain the following points: the company should be clear and transparent in the presentation of its affairs and accounts; and the background and remuneration of its senior executives should suggest that they are experienced in and interested in the business, rather than that they are people whose primary concerns are self-aggrandisement or self-enrichment. These are necessary precautions against information asymmetry, and necessary preliminaries to judging risks. If you wouldn't buy a used car from the directors, then you shouldn't buy a share in the company, for essentially the same reasons.
There should also be a clear, comprehensible business model, based on a distinctive capability that has the potential to create sustainable and appropriable competitive advantage. The only sources of fundamental value are tangible assets and competitive advantage. If you cannot identify tangible assets from the accounts that support the valuation (i.e. the market capitalisation) or don't understand the source of the competitive advantage of the business, then you cannot judge either its fundamental value or the risks associated with it. Avoid unnecessary uncertainty.
You will normally look for companies that are soundly financed. Borrowings should be less than the value of readily realisable assets, and the business should generate cash. Operational risk in a company is inescapable. But when an established business compounds operational risk with financial risk, and becomes dependent on the indulgence of its bank or the vagaries of interest rates, and vulnerable to modest economic or business setbacks, it adds uncertainty to risk. Usually this vulnerability in profit and loss accounts or balance sheets is the product of debt-financed acquisitions or of financial engineering. Both large-scale acquisition and financial engineering generally send negative signals about the preoccupations of the company's management.
Where the weak balance sheet is the result of recent poor trading, the position may be more interesting. Most players in financial markets tell stories rather than think probabilities. They believe that the company is going bust, or that it is not – they rarely express the view that there is a probability of 0.4 that it might, or might not, go bust. Analysts, brokers and fund managers are not often detached. They may still be in love with the purchase they made at a much higher price, or worried by the criticism they might receive from bosses or trustees who possess the benefit of hindsight. As at Robb Caledon, the market price of a risky investment may be a poor guide to the expected value.
Be wary of companies with growing profits and negative cash flow. A new, rapidly expanding business or a firm with a large investment programme may have transparent reasons for these. But the difficulty of explaining the discrepancy between profits and cash should have steered intelligent investors away from Enron.
You can reduce uncertainties by focusing on stable businesses. There are fewer uncertainties in the markets for water – a monopoly whose basic technology was developed millennia ago – than in more competitive and exciting activities such as computers or biotechnology. Even companies in relatively stable markets can create uncertainty by programmes of acquisitions and disposals. Most of the fundamental value of any company derives from the cash it will generate five and more years into the future. If you don't know what the company will be doing five years from now, how can you assess its value? Such strategic uncertainty is unnecessary and avoidable.
Look for clarity and transparency in the company's account of its affairs: a simple business model based on distinctive capability that offers sustainable competitive advantage, and a strong balance sheet with cash flow generation and a stable business. A stock that meets all of these criteria is unlikely to be a ten or twenty bagger, but you are looking for a reliable 8 per cent annual target return, not ten or twenty baggers.
Most companies will fail to meet at least one of these criteria. But there are many more fish in the sea. Your aim is to buy good companies on a contrarian basis, probably when their sector is out of fashion. If investors generally applied these criteria in selecting stocks, their decisions would transform the behaviour of corporations. And overwhelmingly for the better. There would be less financial engineering, fewer acquisitions, simpler and more modest executive remuneration. And, above all, less emphasis on investor relations activities to influence the mind of the market and more emphasis on the creation of fundamental value through the identification and development of competitive advantage in operating businesses.
The preoccupations of most professionals lie elsewhere. They are focused on earnings guidance and their quarterly performance figures. They are fixated on 'corporate activity' – a term that is often simply a euphemism for mergers and acquisitions, and more generally means an announcement that may have a material effect on share prices. Such activity is the basis of market gossip and keeps market turnover high.
The intelligent investor isn't, and doesn't want to be, party to that market gossip. The intelligent investor isn't seeking votes in the ThomsonReutersExtel survey. Your reward comes from good investment returns, not from the bonuses you receive for the business you attract.
13
**The Customers' Yachts**
'Why do I want to buy what they want to sell?' This question is fundamental to investment. Both buyers and sellers own securities in the hope of income and capital gains. But the returns the buyer will obtain are exactly the returns the seller could have obtained. Why should you buy what they want to sell?
In any market in which there is wide and unresolvable uncertainty, in any market where participants have different information and beliefs, many trades will be the result of mistakes. In financial markets, uncertainty and differential information are endemic. When you trade, you need to be confident that it is not you who is making the mistake. Bear in mind the old gambler's maxim: if you don't know who is the patsy in the room, it is probably you.
Keep it simple. If you don't understand a financial product, don't buy it. We purchase cars and computers and many other things without understanding how they work – it is enough to understand their purpose. We rely on the reputation of Mercedes or Microsoft for our belief that their products will actually deliver what they promise. Many people dislike thinking about money, and would like to hand over their financial affairs to someone they can trust. Unfortunately, though, simply placing your trust in financial services businesses is likely to be a mistake and isn't enough to deal with product complexity in modern financial markets. The nature of trade in financial services and the ethics and behaviour of financial services businesses are different from those that pertain in other sectors. In the financial services sector, good reputations seem to survive bad behaviour. Many of the suckers who were ripped off in the New Economy bubble came back to be victims of the credit bubble.
The robo-adviser now offers a possible route if you simply want to hand over all responsibility to someone else and are willing to let it be a computer. But if you have reached this point in this book, I hope you have enough information and enthusiasm to do it better yourself.
Scepticism is appropriate. If it sounds too good to be true, it usually is. In most markets, you tend to get what you pay for. But that is not true of the investment world. The greater the extent of differential information, the less likely it is that costs and charges represent value. The most certain way of increasing returns on your investments is to pay less in fees and commissions. You are on your own, but today the internet gives cheap and easy access to financial products for everyone, and immediate access to information that once required diligent and time-consuming research. Innovations in financial markets – such as the growth of exchange-traded funds and property vehicles – have transformed the opportunities available to retail investors to build for themselves a portfolio that achieves low risk through wide diversification. The intelligent investor today is in greater danger of trading too much because it is easy, than of being deprived of opportunities because access is too difficult.
The notion that market prices are the result of a plebiscite on competing estimates of fundamental value is far removed from the reality of frenzied trading in modern financial markets. Most participants are preoccupied not with long-term economic trends and the competitive advantage of companies but with evolving market opinion and the ephemeral news that passes across the Bloomberg screen.
In an efficient market what is 'in the price' is an average of divergent views. Yet often what is in the price is the manifestation of a common view, generally held within the financial community, but not necessarily well founded. The New Economy bubble was an extreme instance of widely held perceptions that were also widely at variance with reality. Prices that are going up tend to continue going up – markets are characterised in the short term by short-term serial price correlation, or momentum. In the long run, prices that have gone up by a lot tend to go down – there is mean reversion to fundamental value. If only we knew when the short run becomes the long! The efficient market hypothesis is illuminating but not true. There are investment lessons and profit opportunities both in what is illuminating and in what is not true. Across the market as a whole, the profits on financial investment must ultimately depend on the profits on productive investment. For all the intricacies of modern finance, the yield of securities depends on the ability of trading businesses to find profitable ways of serving their customers, on the willingness of tenants to pay rents on shops, offices, houses and factories, and on the ability of companies, consumers and governments to service and repay loans that finance their recent investment or current consumption.
The value of companies depends on their ability to establish competitive advantages over their rivals; sustainable competitive advantage is the only enduring source of superior returns. The value of a property depends on the rents derived from it. The value of a bond depends on the interest it yields and the principal it repays. These determinants of fundamental value necessarily drive securities prices in the long run.
It is possible, though unusual, to make money out of observing and anticipating market momentum. Some people, though not many, are close to the market and have an intuitive feel for, or perhaps an analytic understanding of, its determinants. Whatever the books on trading will tell you, the chances are very small that the person who makes money out of observing and anticipating these fluctuations will be you, the retail investor.
The only basis on which the intelligent investor can hope to keep up with, far less beat, the market, is by attention to fundamental value. Through attention to fundamental value, the intelligent investor gains a big advantage over investment professionals who are judged on their performance quarterly, or even more frequently. There is no way in which an investor, amateur or professional, who is focused on fundamental value can reliably outperform the market on a monthly or quarterly basis – the noise is far too loud. But, over the long run, mean reversion works in your favour.
Not being part of the conventional wisdom of investment professionals gives you an opportunity to be contrarian. Contrarianism requires scepticism but is not perversity. The opportunity for contrarianism arises when the question 'Why do I want to buy what they want to sell?' can attract the answer 'Because they have bad reasons for wanting to sell it.' The investment community cannot afford to ignore the noise, but you can. Patience pays. Assets are worth more to a long-term investor than to a trader.
The principal risk investment professionals run is the risk of underperformance relative to their peers. Most investment institutions benchmark their performance against their competitors and market averages. There are good reasons for this practice, but also malign consequences. In any event, what interests the intelligent investor is how much money he makes, not how much money he makes relative to other people. Relative performance doesn't pay the bills.
Relative performance has no spending power, but money has equal spending power whether it comes as income or capital gain. Think 'total return'. Treat both income and capital gain as part of the yield on your investment. Set a prudent spending rate by reference to your expected total return over a period of years. If your underlying investment objective is capital growth, then reinvestment of dividends will contribute substantially to capital appreciation in the long run. If your underlying investment objective is to obtain a steady income from investments, then an 'income' can be generated, if necessary, through occasional sales from a growing portfolio. A 'think total return' strategy is equally relevant to investors who are accumulating wealth and to investors who are using their assets to support their standard of living.
The return on productive investments provides the basis for realistic expectations of what an investment portfolio can yield. If you earn higher returns than the yield on productive investments, these returns can only be obtained at the expense of other people who are obtaining lower returns than the yield on productive investment. The intelligent investor can do this through a better understanding of risk and a long-term perspective which emphasises fundamental value.
Your objectives colour your strategy. One justification for the substantial fees charged by the financial services industry is that retail investors need individually tailored advice. While there are some honest and conscientious independent financial advisers, the majority of people who make contact with small savers lack the capacity or knowledge to give competent, individually tailored advice, even if they have the inclination and the financial motivation to do so. If you still doubt this, test them with some questions about the meanings of terms defined in the Glossary.
I am not convinced that much tailoring of investment strategy to particular needs is required anyway. Most investors fall within one of the four broad categories described in Chapter 11 – retirement saver, purposive investor, lump sum investor and rainy day saver. It is far more important to understand the meaning of risk by reference to these objectives than to explore 'risk appetite' or 'risk tolerance'. Most people have low risk tolerance, when risk is properly defined and understood. Those who don't should be kept well away from the casino or the stock market.
I've given a lot of space in this book to different ways of thinking about risk, and described at length in Chapter 7 the theory I have called SEU. Treated both prescriptively and descriptively, this theory completely dominates quantitative approaches to investment today. Intelligent investors should learn to 'think SEU' – to understand its principles but not to follow them blindly. That means learning to think probabilities and expected values. Such an approach recognises that many different outcomes are possible, and assesses, even quantifies, the likelihood of attractive and unattractive outcomes.
SEU implies detachment in investment decisions. If you are a gambler by temperament, you may enjoy the search for a ten bagger, but the expected value of the search is likely to be negative. If you are concerned principally to minimise the regret that you will feel (or that your bosses will make you feel) with hindsight, then the expected value of your strategy is also likely to be negative. These losses of expected value are the costs of emotional involvement in your investments.
The most important implication of SEU is the 'mind your portfolio' principle. Every investment decision should be judged not just by the character of the individual investment but also by the contribution it makes to your overall portfolio. An incidental, but large, benefit of being your own investment manager is that you are the only person who knows everything you are doing.
Like the efficient market hypothesis, SEU is an illuminating theory so long as you do not make the mistake of believing that it is true: It is not true either as a prescriptive theory – What should I do? – or as a descriptive theory – How do market participants respond to risks and uncertainties? The conventional view is that you, the typical dumb investor, don't think SEU but that markets do. This view is mistaken. Markets don't generally think this way, but you often, though by no means always, should. SEU is a powerful tool for thinking about well-defined risks, the things we know we don't know. SEU can describe the risks you encounter when you play roulette, or wonder what the European Central Bank will decide about interest rates at its next meeting. But SEU helps little when we contemplate uncertainties. The things we don't know we don't know.
The prevalence of uncertainty explains why SEU thinking doesn't come naturally. We didn't experience the complexities of modern financial markets in the long process of evolution that hardwired our brains, or in the schools and playgrounds where we were taught to cope with life. Many, though by no means all, situations we face in financial markets are ones for which SEU methods are relevant. We make mistakes. Sometimes the mistake is not to use SEU principles when they might help us – as with the stockbrokers who could not take a detached view of Robb Caledon. More sophisticated investors sometimes make the error of using SEU principles where they don't help us – as in the banks whose risk modelling failed to describe the events that brought them to their knees.
The sophisticated world of modern finance relies on models and forecasts in its assessment of both risks and fundamental values. Most people who are not professionally engaged in building models or producing forecasts respond to models and forecasts with cynical naïveté – a curious mixture of disdain and credulousness. At a visceral level these people know that no model can capture the complexity of the world of finance, business and politics, yet they treat the output of models with the utmost seriousness. They believe that economic forecasts convey little useful information, and yet they pay almost obsessive attention to them.
Financial analysis without models is impossible. But investment managers and executives of financial services firms who rely on the output of models they don't understand have relinquished decision-making to a black box. Such individuals profess adherence to science, but their behaviour is identical to that of earlier generations of decision-makers who consulted the oracles or employed soothsayers to read the entrails. You don't have to be able to build models yourself, but you do have to understand a little of what the models can do to understand and make best use of the limited but real insights they can provide. Numbers can illuminate, but they can also blind.
You need analysis and models more than you need forecasts. If you were able to see the future, profitable investment would be easy, but you can't. Nor can anyone else. The only useful information that people who predict the level of the stock market a year from now give you when they speak is that you should pay very little attention to anything they say. General knowledge about the future is 'in the price'. Specific knowledge about the future is vouchsafed to very few, and the people who have it are mostly prohibited from trading on it.
Instead of devoting time to speculation about what the future holds, recognise that the future is inescapably uncertain. Equip yourself with what will be a portfolio that will be robust to many different contingencies. Such diversification is the most effective means of coping with a complex world. Knowing what you don't know gives you a considerable advantage over those, amateurs and professionals alike, who don't know what they don't know, or do know what ain't so. A broadly diversified portfolio based on real assets will give you considerable protection against risk and uncertainty with little compromise on return. In a 100 per cent efficient market there would be a trade-off between risk and return, but in a partly efficient market there is not. New investment technology using internet dealing means that the retail investor can achieve a diversified portfolio with relatively modest sums.
Three simple rules – pay less, diversify more and be contrarian – will serve almost everyone well. If you have an established investment portfolio based on advice, you will almost certainly find that a substantial part of your total assets is in open-ended managed funds that are closet-indexed. This costs you too much and is inadequately diversified. You can transfer this money into a group of exchange-traded funds, closed-end funds and real estate investment trusts on substantial discounts. That strategy will give you exposure to a range of countries, types of security and styles of management. It will also give you lower charges, and less risk in your overall portfolio. An emphasis on market sectors at a discount will automatically imply a contrarian stance. You will lose what you have already paid in initial charges to set up your investments, but that money has gone anyway. If you are at an earlier stage of your investment career, you can structure your portfolio along these lines from the beginning.
Modern financial markets are complex, but much of the complexity is for the benefit of providers, rather than consumers, of financial services. If you don't understand it, don't do it. That simple maxim would have saved both amateurs and professionals billions of pounds over the years, and the more recent the years, the larger the savings.
Whether you hire a financial adviser or not, you are on your own. I hope this book can be your companion.
**Glossary**
**Absolute return** The total return on a portfolio. Absolute return is measured without benchmarking or reference to market movements. ****
**Acquisition** The takeover of one company by another. ****
**Acquisition accounting** Accounting treatment designed to flatter the accounts of the acquiring company. ****
**Active management (portfolio)** Professional asset selection. Contrast with passive management, which simply follows an index. ****
**Agency costs** Costs of intermediation, which reduce the returns on financial investment relative to those of the underlying productive investment. ****
**Allais paradox** A problem in behavioural economics devised by the French economist Maurice Allais which demonstrates a 'Dutch book'. **, , **
**_Alpha_** Outperformance of an investment or investment fund, relative to a benchmark, adjusted for risk. (The _beta_ of the portfolio is calculated to measure what part of excess return is accounted for by return to above average risk.) ****
**Alternative investment** Asset outside mainstream categories of shares, bonds and property. Traditionally alternative assets were mostly physical objects (such as paintings and commodities), but more recently the term is mainly used to refer to private equity and hedge funds. **, **
**Amortisation** _See_ DEPRECIATION.
**Anomalies** Deviations from the efficient market hypothesis. **, **
**Arbitrage** A strategy to profit from divergences in the prices of similar or related securities. ****
**Arbitrageurs** Traders using arbitrage strategies, especially those that trade between an acquirer's stock and that of the prospective acquisition. ****
**Asset-backed securities (ABS)** Tradable securities based on a pool of loans. An asset-backed security can have several tranches of security relating to the same asset pool; if the value of the pool deteriorates, the senior tranches will be paid first. ****
**Asset stripping** Breaking up a company to realise the value of its underlying assets. ****
**Asset value** The value of the assets underlying a security (which may differ from the market price of the security). **, **
**Bagger (as in 'ten or twenty bagger')** Speculative stock which increases greatly in value. ****
**Basis point** Each percentage point of yield is 100 basis points.
**Bayesian** The view that degrees of belief can be expressed as probabilities updated with new information. ****
**Behavioural finance, behavioural economics** Analysis based on empirical or experimental studies of behaviour (especially towards risk) rather than models of rational choice. **, , **
**Benchmark** The index specified to judge an investment manager's performance. ****
**Best execution** The obligation to place a trade at the best price available in the market.
**_Beta_** The predicted effect on the price of a security of a 1 per cent movement in the benchmark index. ****
**Black–Scholes model** The most widely used model for pricing options. **, **
**Blue chips** Shares in large companies. **, **
**Bonds** Securities with fixed interest rates and redemption dates. ****
**BRIC** Principal emerging markets – Brazil, Russia, India, China. South Africa has sometimes also been included.
**Call option** The right to buy a security at a fixed price at a future date ( _see also_ PUT OPTION). ****
**Capital asset pricing model (CAPM)** A model that predicts that expected rates of return will be determined by a combination of the risk-free rate and a risk premium determined by the asset category and the asset _beta_. ****
**Carry trade** Arbitrage strategies that yield repeated small profits, usually from interest rate differentials ( _see also_ TALEB DISTRIBUTIONS). ****
**Chartist** Someone who predicts security prices by identifying trends from charts. ****
**Chinese walls** Measures (including rules and physical separation) to prevent different divisions (principally buy side and sell side) of a financial institution from exchanging information. ****
**Closed-ended** An investment fund whose shares can be acquired or disposed of only by selling to another investor. Such a fund may therefore sell at a premium or discount to its asset value. **, **
**Closet indexation** Active management which closely follows the content (and performance) of the benchmark index. ****
**Collateral** Provision against a liability.
**Collateralised debt obligation (CDO)** An asset-backed security whose value is based on a (generally) variable pool of debt instruments of specified quality. ****
**Commission bias** Investment recommendation influenced by the adviser's remuneration. ****
**Competitive advantage** The ability of a business to produce at lower cost, or to command higher prices, than other businesses serving the same market. ****
**Complex systems** Social or natural systems whose mathematical description demands dynamic, non-linear relationships. ****
**Compound interest** Calculation in which returns are increased by interest on reinvested interest. ****
**Confirmation bias** A tendency to select evidence that supports views that are already held. ****
**Consensus forecast** The median (or perhaps mean or modal) forecast, a product of the tendency of all economic forecasters to cluster together. ****
**Consumer prices index** A measure of inflation calculated by measuring the changing price of a fixed basket of goods.
**Contract for difference (CFD)** An agreement to pay (or receive) the difference between the present and future prices of a security. ****
**Contrarian** Someone who buys assets not currently favoured by the mind of the market. ****
**Conventional investor** An investor who applies closet indexation. **, **
**Correlation** Assets are correlated if movements in their prices tend to move in the same direction ( _see also_ BETA). ****
**Cost of capital** The cost to a firm of raising capital (typically an average of the cost of corporate debt and the total return required by equity shareholders). ****
**Counterparty** The person on the other side of a transaction. Market counterparties are institutions trading securities. ****
**Credit default swap** A derivative contract that pays in the event of the default of a borrower. ****
**Credit rating** An assessment of credit risk, often by a rating agency.
**Credit risk** The risk that a borrower will fail to meet the interest or principal on a loan. **, **
**Day traders** Small speculators who buy and sell frequently, perhaps within a single day. ****
**Depreciation** An allowance against profits for the fall in value of assets through time or use. ****
**Derivatives** Securities whose value is based on the value of other securities. ****
**Diminishing marginal utility** A tendency for additions to income or wealth to be valued less by the beneficiary as that income or wealth increases. ****
**Discount rate, discounted cash flow (DCF)** The process of discounting future revenues to compute a present value. ****
**Distressed debt** Bonds or loans on which the borrower may fail to make payment. ****
**Diversification** The process of reducing risk by buying assets whose returns are uncorrelated. **, **
**Dividend** The dividend on a share is the (usually quarterly or biannual) payment declared by the company. ****
**Dividend yield** The annual dividend per share as a percentage of the share price. ****
**Dutch book** A sequence of apparently attractive gambles which, if accepted, will leave the taker worse off. ****
**Earnings before interest, tax, depreciation, amortisation (EBITDA)** A profit figure reported by companies seeking to deliver flattering accounts of their affairs ( _see also_ PRO FORMA EARNINGS). ****
**Earnings guidance** The process by which companies influence analysts' forecasts of their earnings. ****
**Earnings per share** Profits after tax, divided by the total number of shares in issue. ****
**Earnings yield** ( _see also_ PRICE:EARNINGS RATIO) Earnings per share, as a percentage of the share price. ****
**Eat what you kill** The practice in professional firms of rewarding partners by reference to the revenues the individual partners themselves generate. ****
**Efficient market hypothesis (EMH)** The proposition that all relevant information about the value of a security is reflected in its price. **, **
**Emerging market** Country outside the principal developed economies (e.g. BRIC countries). ****
**Enhanced bonds** A structured product that uses derivatives to provide a higher yield (and risk) than the underlying bond. ****
**Equity** An equity interest is the right to receive what is left after prior claims have been met: e.g. ordinary shareholders receive the residual after debts and preferred shareholders have been paid. The owner's equity in a house is its value less the value of any outstanding mortgage. ****
**Equity premium, equity risk premium** The expected difference between the return on shares and the return on a safe asset. **, , **
**Exchange-traded fund (ETF)** A fund traded on a stock exchange whose value is linked to an index. ****
**Execution only** The service offered by a broker who will deal but not advise. ****
**Exercise price** The price at which you can buy or sell a share under option ( _see also_ STRIKE PRICE). ****
**Expected value** The average outcome of a gamble, calculated by multiplying potential pay-offs by their probabilities. ****
**Expiry date** The last date at which an option can be exercised. ****
**Fat tails** A phenomenon in which extreme events happen more frequently than a standard statistical distribution (e.g. binominal or normal distribution) allows. ****
**Financial engineering** The attempt to create value by repackaging investment products. ****
**Financial investment, productive investment** Financial investment is investment in securities; productive investment is investment in the activities that give these securities their value. ****
**Fractal** A geometric shape, each part of which resembles the whole. ****
**Free cash flow** Net cash from operations after deducting necessary investment. ****
**Fundamental indices** Market indices in which the weights of different securities are derived from economic indicators other than their market values. ****
**Fundamental value** The expected value of the cash flows generated by an asset. ****
**Futures** The right to buy or sell a security or commodity at a fixed price on a future date. ****
**General knowledge** Publicly available information. **, **
**Generally accepted accounting principles** The principal US accounting standard. ****
**Going concern basis** The assumption (for valuation purposes) that a company will continue in business.
**Gross redemption yield (GRY)** The total return from a bond, including any decline or increase in capital value between the present and maturity. ****
**Halo effects** The tendency to give high ratings to all characteristics of a favoured item or individual. ****
**Hedge** An asset whose price is expected to move in the opposite direction from the price of the asset it hedges. ****
**Hedge fund** An adventurous, open-ended investment company charging high fees. ****
**High-net-worth** Rich. ****
**Hot hand** Persistence of strong performance ( _see also_ SERIAL CORRELATION). ****
**Independent** Two variables are independent if their movements are unrelated. ****
**Index fund** Fund whose value follows that of a benchmark. ****
**Index-linked stocks** Securities on which interest and principal increase in line with inflation. ****
**Information asymmetry** Characteristic of markets in which the seller knows more about the goods than the buyer. ****
**Initial public offering (IPO)** The first offering of stock in a company to the public. **, **
**Insider trading** Dealing by corporate executives or their advisers (or their agents) on the basis of information about the company not yet publicly disclosed. ****
**Insolvent** A company or individual is insolvent when liabilities exceed assets.
**Intangible assets** Assets other than property, plant and investments: e.g., brands, patents, reputation. Sometimes called 'goodwill'. Figures for intangible assets reported in balance sheets mainly refer to overpayment for acquisitions and mean little. ****
**Intelligent investor** Someone who defines their own investment strategy by reference to fundamental value, thinks total return and minds their portfolio. ****
**Interest rate risk** The risk that a bond will fall in value owing to interest rate rises. ****
**International financial reporting standards** Accounting standards applied in most countries outside the United States. ****
**Investment banking** Traditionally, banking focused on securities issuance and corporate advice (principally IPOs and acquisitions). Modern investment banks also undertake (principally undertake) market-making, trading and asset management. Most are now part of financial conglomerates with corporate and retail banking operations. **, **
**Junk bonds** Bonds below investment grade. ****
**Know your customer** The regulatory obligation on a financial adviser to base recommendations on specific knowledge of the investor's circumstances. ****
**Lemons** Goods that the seller, but not the buyer, knows to be of inferior quality. ****
**Leverage** The process of increasing risk and potential return on an asset through borrowing. ****
**Life insurance** Insurance policy which pays on the death of the insured. Often packaged with savings products – such a package is called life assurance. ****
**Likelihood** The degree of belief in a narrative. ****
**Long** The opposite of short: a positive position in a security. ****
**Long/short fund** A fund that takes short positions as well as long. (A 130/30 fund will own shares worth 130 per cent of its value and have short positions equivalent to 30 per cent.) ****
**Mark-to-market** The practice of revaluing assets in line with current market values. ****
**Market capitalisation** The total value of the shares of a company. ****
**Market risk** The risk profile created by general market movements. ****
**Mean reversion** A tendency for variables to return to their long-term average. ****
**Mental accounting** An inclination to compartmentalise a portfolio rather than to mind a portfolio as an aggregate. ****
**Merger** The combination of two firms.
**Momentum** The tendency for price movements to continue in the same direction ( _see also_ SERIAL CORRELATION). ****
**Monoliners** Financial services firms that specialise in a single product or product group. ****
**Moral hazard** The tendency for average product quality to deteriorate when there is information asymmetry. ****
**Mutual fund** The US term for a European OEIC ( _see_ OPEN-ENDED INVESTMENT COMPANY). ****
**New issues** The sale of new shares for cash by a company ( _see_ IPO). ****
**Noise** Short-term price movements not justified by changes in fundamental value. ****
**Noise traders** Frequent traders who act on noise ( _see also_ DAY TRADERS). ****
**Non-recurring items** Costs that a company does not want to charge against profits. ****
**Normal distribution** The most common – bell-shaped – statistical distribution. ****
**Open-ended investment company (OEIC)** A European fund that creates or redeems shares or units in line with investor demand (cf. MUTUAL FUND). ****
**Option** The right to buy or sell securities at a price agreed in advance. ****
**Overweight** Holding a larger proportion of one's portfolio in a particular security than the share of that security in the relevant index. ****
**Passive fund, passive investment** A fund or investment strategy that tracks an index. ****
**Payment-in-kind securities** Bonds on which the interest rate increases as the creditworthiness of the borrower declines. ****
**Payment protection insurance** Retail product which may meet loan repayments during sickness or unemployment. **, **
**PE ratio** See PRICE:EARNINGS. ****
**PEG ratio** The ratio of the price:earnings ratio and the growth rate of earnings. ****
**Penny shares** Shares whose value is only a few cents per share (even though there may be many millions of shares). Often issued to appeal to noise traders. ****
**Personal probability (subjective probability)** An estimate of the probability of a particularly outcome of a single event. ****
**Price:earnings ratio** The ratio of the market price of a security to earnings per share after tax. ****
**Principal** The amount outstanding on a loan. ****
**Private equity fund** A fund that takes large stakes in unquoted companies. **, **
**Pro forma earnings** The profits companies would like you to think they have earned. ****
**Productive investment** _See_ FINANCIAL INVESTMENT. ****
**Proprietary trading ('prop desks')** The internal hedge funds of banks. ****
**Put option** An option to sell a security at a price agreed in advance ( _see also_ CALL OPTION). ****
**Quant** Someone who specialises in mathematical modelling of investment questions. ****
**Quantitative easing** The purchase of debt securities by a central bank. ****
**Random walk** A process in which each change is independent of all previous changes. **, **
**Rational expectations** The assumption that individual expectations reflect all potentially available information. ****
**Real estate investment trusts (REITs)** Closed-ended funds which invest in property and enjoy a special tax regime. **, **
**Real return** Investment return after allowing for inflation. ****
**Redemption date** The date on which a loan will be repaid.
**Reflexivity** A characteristic of a process in which expected outcomes of a process affect the outcomes themselves. ****
**Relative performance** Investment performance relative to a benchmark. ****
**Retained profits** Profits that are not distributed as dividends. ****
**Revenue recognition** Anticipation of revenues not yet received in cash. ****
**Risk-averse** Disposed to value risks or gambles at less than their expected value. ****
**Risk premium, equity risk premium** The difference between the return on a risky asset and the yield on a safe one. **, **
**Running yield** The income yield on a bond whose market price differs from its redemption value ( _see_ GROSS REDEMPTION YIELD). ****
**Scenario planning** A mechanism for reviewing plans by positing several different scenarios of the future. ****
**Secured loans** Loans secured against specific assets. ****
**Serial correlation** The relationship between price movements in successive time periods. Short-term positive serial correlation is when price movements continue in the same direction (momentum). Long-term negative serial correlation is the tendency for long-term movements to reverse (mean reversion). ****
**Sharpe ratio** The ratio of standard deviation (variability) to the return from an asset. ****
**Short seller** A negative position established to benefit from the fall in value of a share. **, **
**Short-term positive serial correlation** The expectation that short-term price movements will be followed by other moves in a similar direction ( _see also_ MOMENTUM). ****
**Specific risks** Risks that are specific to the investment concerned and unrelated to movements in the general level of share prices. ****
**Spread betting** Betting on the rise and fall of a security or index ( _see also_ CONTRACT FOR DIFFERENCE). ****
**Statistical arbitrage** Computerised trading systems for exploiting short-term discrepancies between prices of similar securities. ****
**Stock options** The right (often given to senior executives) to buy the company's shares at a future date at a favourable price. ****
**Strike price** The price at which an option is granted. ****
**Structured product** Security with complex relationship to the underlying assets and liabilities. The result of financial engineering. ****
**Subjective expected utility (SEU)** A theory that individuals should or do approach risk by maximising the expected value of pay-offs using subjective (personal) probabilities. ****
**Subprime borrower** Subprime borrowers have poor credit ratings and the loans they take out in turn attract the adjective subprime. ****
**Survivor bias** Misleading generalisation about performance based on analysis of a population from which poor performers have been removed. ****
**Sustainable competitive advantage** A competitive advantage that a business can successfully maintain over time. ****
**Swap, swap rates** (generally of interest rates). A swap rate defines the terms on which fixed and variable rate obligations can be exchanged. The transaction is a swap. ****
**Synergies** Alleged benefits from mergers. ****
**Taleb distributions** Processes in which frequent small gains are punctuated by occasional large losses. ****
**Technical analysis** What chartists do. ****
**Time preference** The 'exchange rate' between present and future consumption. ****
**Total return** Investment return as a percentage of assets, including both income and capital gain. **, **
**Tracking error** A measure of how closely an index fund follows the benchmark index. ****
**Trading floor** Traders in investment banks typically work at desks and screens in one trading room. ****
**Underweight** _See_ OVERWEIGHT. ****
**Value at risk** In risk modelling, the anticipated loss at some low percentile of the risk distribution. ****
**Variability, volatility** Measures of how much and how rapidly a security price goes up and down. ****
**Winner's curse** The tendency for assets to be bought by the person who makes the highest assessment of their value. ****
**Yield curve** The relationship between interest rates and bond maturities. ****
**Bookshelves and Bookmarks**
_The Economist, Financial Times_ and _Wall Street Journal_ all offer guidebooks that cover most investment topics. If you plan to buy individual stocks, you should have a book on how to read accounting statements, such as Bob Parker's _Understanding Company Financial Statements_. And you should learn basics of corporate strategy. There are many books written on corporate strategy and for investment analysts, but not many target readers of this book would want to finish them. Dick Rumelt's _Good Strategy, Bad Strategy_ is witty and perceptive and has lessons for investors as well as corporate executives. I'm bound to recommend my own The _Hare and the Tortoise_ , a distillation of _Foundations of Corporate Success_. The world of business and finance is notorious for writing that luxuriates in jargon and complexity. John Lanchester's _How to Speak Money_ is an excellent antidote and guide. Peter Bernstein's _Against the Gods_ , on risk, is an exemplar of how to make complex concepts appear simple.
Do you want a book that contains the secrets of how to be rich? I hope not. But you might want advice from Soros and Buffett. The most relevant of George Soros's several books is _The Alchemy of Finance_ , while Lawrence Cunningham offers an entertaining collection of extracts from the essays Buffett writes in his annual reports to Berkshire Hathaway shareholders (which are available on the company's web site). _Snowball_ , by Alice Schroeder, is an exhaustive biography of the sage of Omaha.
If you want a small shelf of books about the industry that are fun to read, you might consider: from the 1980s, Burrough and Helyar's _Barbarians at the Gate_ , an account of the last days of RJR Nabisco, and _Liar's Poker_ , by Michael Lewis, set in Salomon in its heyday; and John Cassidy's account of the New Economy bubble, _dot.com,_ best read in parallel with J. K. Galbraith's account of _The Great Crash, 1929,_ which is beautifully written; its lessons are all still relevant.
The best narrative account of the global financial crisis is Andrew Ross Sorkin's _Too Big to Fail;_ Michael Lewis's _The Big Short_ is a highly entertaining account of one aspect of its origins, and the same author's more recent _Flash Boys_ is an introduction to some of the issues raised by the absurd world of the high-frequency trader. I do not recommend the self-congratulatory memoirs of those who claim to have saved the world from the crisis for which they themselves were largely responsible; but Mervyn King's _The End of Alchemy_ instead tries to draw lessons for finance and economics in relatively detached manner.
A comparison of Tom Wolfe's _The Bonfire of the Vanities_ and Anthony Trollope's _The Way We Live Now_ reveals that, while the actions and the instruments change, the fundamentals of financial follies do not.
The inspiration for a book like this one, which aims to provide practical advice with a rigorous intellectual basis, is Burton Malkiel's _A Random Walk down Wall Street_ , which has been for years what I recommended to the encouragingly numerous people who sought such a book. _Random Walk_ has been through eleven editions since it was first published in 1973, but its age shows, and the institutions and products it considers are American. So are the jokes, which are not to European taste. Still, it remains a classic. The permanent classic is Benjamin Graham's _The Intelligent Investor_.
Other academics who have written popular books on investment are Jeremy Siegel, whose _Stocks for the Long Run_ is an unashamed paean to the virtues of equity investment, and Robert Shiller, who had the good fortune to publish _Irrational Exuberance_ just as the New Economy bubble was bursting. Similar books you might consider are William Bernstein's _Seven Pillars of Investing Wisdom_ and either of David Swenson's volumes, _Pioneering Portfolio Management_ and _Unconventional Success_. Swenson, Yale's much-fêted investment manager, adopts a rather similar approach to the one described here, but Swenson invests better than he writes.
Do you want to pursue some of the more philosophical issues that have arisen in the course of this book? Quants and risk managers in the financial world make constant case of probabilities without much thought about what the numbers they readily deploy mean. Ian Hacking's _An Introduction to Probability and Inductive Logic_ provides a good introduction to the underlying complexities. Questions about the nature of explanation – What does it mean to say a theory is illuminating but not true? – are at the heart of this book. People of a philosophical bent will appreciate the influence of Richard Rorty, whose related ideas are best set out in _Philosophy and the Mirror of Nature_. I found Wade Hands's _Reflections without Rules_ an invaluable introduction to the central relevance of this sort of approach in modern economics.
Whatever books you put on your shelves, you will want to add some bookmarks to your computer. The internet gives you easy access to both products and information. All financial services organisations use web sites to market and give access to their products, and competitive pressures ensure that most are easy to locate and use. But good web sites cost money to build and maintain, and, one way or another, he who pays the piper at least influences the tune. While comparison sites listing financial products are invaluable, bear in mind that comparison sites are financed through commission.
The internet also makes access to information much easier. Most companies have an investors relations section on their web site, from which you can obtain annual reports and other financial information. Your stockbroker's site will probably have a research section that summarises these data and gives you other data about the company. Bulletin boards, which are a forum for title tattle about individual stocks, are amusing, but both the information and opinions on them should be treated with great caution.
**Bibliography**
Allais, M., 1953, Le Comportement de l'Homme Rationnel devant le Risque: Critique des Postulats et Axiomes de l'Ecole Americaine. _Econometrica_ , v. 21, no. 4, p. 503, doi:10.2307/1907921.
Bachelier, L., trans. M. H. A. Davis and A. Etheridge, 2006, _Louis Bachelier's Theory of Speculation: The Origins of Modern Finance._ Princeton, NJ: Princeton University Press.
Bayes, M., and M. Price, 1763, An Essay towards Solving a Problem in the Doctrine of Chances. By the Late Rev. Mr. Bayes, F.R.S. Communicated by Mr. Price, in a Letter to John Canton, A.M.F.R.S. _Philosophical Transactions of the Royal Society of London_ , v. 53, pp. 370–418, doi: 10.1098/rstl.1763.0053.
Bernoulli, D., 1954, Exposition of a New Theory on the Measurement of Risk. _Econometrica_ , v. 22, no. 1, p. 23, doi: 10.2307/1909829.
Biggs, B., 2008, _Wealth, War, and Wisdom_. Hoboken, NJ: John Wiley & Sons.
Black, F., and M. Scholes, 1973, The Pricing of Options and Corporate Liabilities. _Journal of Political Economy_ , v. 81, no. 3, pp. 637–54, doi: 10.1086/260062.
Bowden, M., 2012, _The Finish: The Killing of Osama bin Laden_. New York: Atlantic Monthly Press.
Cunningham, L., 2002, _How to Think Like Benjamin Graham and Invest Like Warren Buffett_. London: McGraw-Hill Education.
Dimson, E., P. Marsh, and M. Staunton, 2002, _Triumph of the Optimists: 101 Years of Global Investment Returns_. Princeton, NJ: Princeton University Press.
Fama, E. F., and K. R. French, 1992, _The Cross-Section of Expected Stock Returns_. Chicago, IL: Center for Research in Security Prices, Graduate School of Business, the University of Chicago.
Friedman, M., 1976, _Price Theory: A Provisional Text_. Chicago, IL: Aldine.
Friedman, M., and L. J. Savage, 1948, The Utility Analysis of Choices Involving Risk. _Journal of Political Economy_ , v. 56, no. 4, pp. 279–304, doi: 10.1086/256692.
'H. W. M.', 1933, Review of J. M. Keynes, _Essays in Biography_. _Journal of the Royal Statistical Society_ , v. 96, no. 3, p. 512, doi: 10.2307/2342137.
Harrod, R. F., and J. R. Hicks, 1939, Value and Capital. _The Economic Journal_ , v. 49, no. 194, p. 294, doi: 10.2307/2225091.
Hicks, J., 1939, _Value and Capital; An Inquiry into Some Fundamental Principles of Economic Theory_. Oxford: Clarendon Press.
Kahnemann, D., P. Slovic, and A. Tversky, 1982, _Judgment under Uncertainty: Heuristics and Biases_. Cambridge: Cambridge University Press.
Kay, J., 2009, _The Long and the Short of It: A Guide to Finance and Investment for Normally Intelligent People Who Aren't in the Industry_. London: Erasmus.
Keynes, J. M., 1920, _The Economic Consequences of the Peace_. New York: Harcourt, Brace and Howe.
Knight, F. H., 1921, _Risk, Uncertainty and Profit_. Boston, MA: Houghton Mifflin.
Lo, A. and MacKinlay, A., 2002, _A Non-Random Walk Down Wall Street_. NJ: Princeton University Press.
Malkiel, B., 1973, _A Random Walk down Wall Street_. New York: W.W. Norton.
Markowitz, H., 1952, Portfolio Selection. _Journal of Finance_ , v. 7, no. 1, p. 77, doi: 10.2307/2975974.
Pigou, A. C., and J. M. Keynes, 1921, A Treatise on Probability. _Economic Journal_ , v. 31, no. 124, p. 507, doi: 10.2307/2223083.
Sharpe, W. F., 1964, Capital Asset Prices: A Theory of Market Equilibrium under Conditions of Risk. _The Journal of Finance_ , v. 19, no. 3, p. 425, doi: 10.2307/2977928.
Shiller, R. J., 2000, _Irrational Exuberance_. Princeton, NJ: Princeton University Press.
Soros, G., 2003, _The Alchemy of Finance_ (2nd edn). London: John Wiley and Sons Ltd.
Taleb, N. N., 2001, _Fooled by Randomness: The Hidden Role of Chance in the Markets and in Life_. New York: Texere.
Zweig, J., 2008, _Your Money and Your Brain_. New York: Simon & Schuster.
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\section{introduction}
The study of counting the number of the possible representation of a positive integer as sum of primes, prime powers and, in general,
sum of elements that belong to some fixed subset of $\mathbb{N}$ is classical in number theory and it has been highly developed in
recent years. Probably, the most popular problem in this context is the ternary and binary Goldbach conjecture, which states that every
odd number greater than $3$ is sum of three primes and every even number greater than $2$ is the sum of two primes, respectively. While
the ternary conjecture has been partially solved by Vinogradov \cite{Vin} and then definitively solved by Helfgott in a series of papers \cite{Helf3,Helf1,Helf2}, the binary conjecture is still open; only partial results were obtained
and, often, only conditional on the Riemann hypothesis (see, e.g.
the historical account about the Goldbach binary problem \cite{BhoHal}).
Given the difficulty of the problem, it was decided to tackle simplified
versions of it, such as, for example, considering the number
of representations on average and with suitable weights. In this context
are inserted the works related to the study of averages, with Ces\`aro
weight, of functions that count the number representations of an integer
as the sum of elements that are in some fixed subset of $\mathbb{N}$. Similar averages of arithmetical
functions are common in the literature: see, e.g., \cite{Ber}.
This approach was used in \cite{LanZac} for the binary Goldbach
problem: see Languasco's paper \cite{Languasco2016} for a thorough
introduction. The presence of a smooth weight allowed to obtain an asymptotic
formula with terms of decreasing orders of magnitude and depending
on the non-trivial zeros of the Riemann Zeta function; furthermore,
the weights allowed to obtain results independent of the Riemann hypothesis.
Since the Ces\`aro weights depend on a non-negative real parameter
$k$ and are equal to $1$ if $k=0$ (that is, for $k=0$ we have
a simple average without weights) it is important to obtain results
with the smallest possible $k$. During the last few years there have
been some improvements regarding the optimal $k$ in the case of the
Goldbach's problem: in \cite{LanZac} results hold for $k>1$, in
\cite{GolYan} (assuming Riemann hypothesis) and in \cite{Can1} (unconditionally)
for $k=1$ and in \cite{BruKacPer} for $k>0$. Unfortunately, for
case $k=0$, that is, without the Ces\`aro weights, it is not yet
possible to obtain in the same form or with the same quantity of terms
as in the other cases (see, for example, \cite{LanZac2},\cite{Can2}
and \cite{Pin}). Given the flexibility of the technique introduced
in \cite{LanZac}, the latter has been applied to other types of additive
problems (see \cite{Can3}\cite{Can5}\cite{LanZac3}\cite{LanZac4}\cite{LanZac5}).
In this paper, we prove a result which incorporates all the previous
results in the case of Ces\`aro averages and we show how the technique,
although very general and applicable to many problems, makes the lower
bound of $k$ worse as the number of primes and squares involved increases,
so as to confirm what has already been suggested by the lower
bound for $k$ obtained in \cite{Can3} and \cite{Can5}.
\section{preliminary definitions and main theorem}
Let $d,h,N\in\mathbb{N},\,d>0$, $N\geq2$, $\mathbf{m}=\left(m_{1},\dots,m_{d}\right)\in\mathbb{N}^{d},\,\mathbf{r}:=\left(r_{1},\dots,r_{d}\right)\in\left(\mathbb{\mathbb{N}}^{+}\right)^{d},$ where $1\le r_1\le r_2\le \cdots\le r_d$, $\mathbf{\mathbf{t}}:=\left(t_{1},\dots,t_{h}\right)\in\mathbb{N}^{h}$
and, in general, with bold letters, for example $\mathbf{f}$, we
will indicate some vector that belongs to $\mathbb{N}^{\alpha}$ or
$\left(\mathbb{N}^{+}\right)^{\alpha},$ for some positive integer
$\alpha.$ With the symbol $\left\Vert \cdot\right\Vert $ we will
indicate the usual Euclidean norm, with the symbol $\rho$, with or without subscripts, we will always indicate the non-trivial zeros of the Riemann zeta function and the series $\sum_{\rho}$ will always indicates the sum over all non trivial zeros of $\zeta(s)$, with or without subscripts. With $\bm{\rho}:=(\rho_{s_{1}},\dots,\rho_{s_{v}})$, where $s_{j},\, j=1,\dots,v$ belong to some subset of $\mathbb{N}^{+}$.
For every $\mathfrak{J}\subseteq\mathfrak{D}$ we will define the scalar product
\[
\tau\left(\bm{\Psi},\mathbf{r},\mathfrak{J}\right):=\sum_{j\in\mathfrak{J}}\frac{\Psi_{j}}{r_{j}}
\]
where $\bm{\Psi}=(\Psi_1,\ldots,\Psi_d)$. In most cases along the paper we will use $\bm{\Psi}=\bm{\rho}$; in addition, we will use the short definition $\tau\left(\bm{1},\mathbf{r},\mathfrak{J}\right)=\tau\left(\mathbf{r},\mathfrak{J}\right):=\sum_{j\in\mathfrak{J}}\frac{1}{r_{j}}$.
We will also indicate by $\sum_{\mathfrak{J}\subseteq\mathfrak{D}}$
the sums over all the possible subsets of of $\mathfrak{D}$. Taking
$n\in\mathbb{N}$, we set
\[
R_{d,h,\mathbf{r}}\left(n\right):=\sum_{m_{1}^{r_{1}}+\dots+m_{d}^{r_{d}}+t_{1}^{2}+\dots+t_{h}^{2}=n}\Lambda\left(m_{1}\right)\cdots\Lambda\left(m_{h}\right)
\]
where $\Lambda\left(m\right)$ is the usual von Mangoldt function.
We want to find an asymptotic formula, as $N\rightarrow+\infty$,
for
\[
\sum_{n\leq N}R_{d,h,\mathbf{r}}\left(n\right)\frac{\left(N-n\right)^{k}}{\Gamma\left(k+1\right)}
\]
where $k>0$ is a real parameter and $\Gamma\left(x\right)$ is the
Euler Gamma function.
Let $Z:=\left\{ s\in\mathbb{C},\,0\leq\mathrm{Re}\left(s\right)\leq1:\,\zeta\left(s\right)=0\right\}$ be the set of the non-trivial zeros of the Riemann Zeta function and let $\mathfrak{J}\subseteq\mathfrak{D}$. We will use the symbols
$$\sum_{\bm{\rho}\in Z^{\left|\mathfrak{J}\right|}}=\sum_{\rho_{j_{1}}}\cdots\sum_{\rho_{j_{\left|\mathfrak{J}\right|}}}$$
and
$$ \frac{1}{\textbf{r}}\Gamma\left(\frac{\bm{\rho}}{\textbf{r}}\right):=
\frac1{r_{j_1}}\Gamma\left(\frac{\rho_{j_{1}}}{r_{j_{1}}}\right)\cdots\frac1{r_{j_{_{\left|\mathfrak{J}\right|}}}}\Gamma\left(\frac{\rho_{j_{_{\left|\mathfrak{J}\right|}}}}{r_{j_{\left|\mathfrak{J}\right|}}}\right)$$
where $j_{\alpha}\in\mathfrak{J},\,\alpha=1,\dots,\left|\mathfrak{J}\right|$, every $\rho_{j_{\alpha}}\in Z$ and $r_{j_{\alpha}}$ is the $j_{\alpha}$-th coordinate of the fixed vector $\textbf{r}=\left(r_{1},\dots,r_{d}\right)$.
In analogy to the previous definition, we will use the following symbol
$$ \frac{1}{\textbf{r}}\Gamma\left(\frac{1}{\textbf{r}}\right):=
\frac1{r_1}\Gamma\left(\frac{1}{r_{1}}\right)\cdots\frac1{r_{d}}\Gamma\left(\frac{1}{r_{d}}\right).$$
We introduce the following abbreviation for
the terms of the development:
\begin{align*}
M_{1}\left(N,k,d,h,\mathbf{r}\right) :=&
\frac{1}{2^{h}}\sum_{\ell=0}^{h}\dbinom{h}{\ell}\frac{\pi^{\frac{\ell}{2}}\left(-1\right)^{h-\ell}N^{k+\tau\left(\mathbf{r},\mathfrak{D}\right)+\frac{\ell}{2}}}{\Gamma\left(k+1+\tau\left(\mathbf{r},\mathfrak{D}\right)+\frac{\ell}{2}\right)}
\frac{1}{\textbf{r}}\Gamma\left(\frac{1}{\textbf{r}}\right),
\\
M_2\left(N,k,d,h,\mathbf{r}\right) :=&
\frac{N^{\frac{k+\tau\left(\mathbf{r},\mathfrak{D}\right)}{2}}}{\pi^{k+\tau\left(\mathbf{r},\mathfrak{D}\right)}}\sum_{\eta=0}^{h-1}\frac{\dbinom{h}{\eta}}{2^{\eta}}\sum_{\ell=0}^{\eta}\dbinom{\eta}{\ell}\left(-1\right)^{\eta-\ell}
\mathfrak{B}\big(\tau\left(\mathbf{r},\mathfrak{D}\right)\big)
\frac{1}{\textbf{r}}\Gamma\left(\frac{1}{\textbf{r}}\right),
\\
M_3\left(N,k,d,h,\mathbf{r}\right) :=&
\frac{N^{k}\left(-1\right)^{d}}{2^{h}}\sum_{\ell=0}^{h}\dbinom{h}{\ell}\left(N\pi\right)^{\frac{\ell}{2}}\left(-1\right)^{h-\ell}\sum_{\bm{\rho}\in Z^{d}}
\frac1{\textbf{r}}\Gamma\left(\frac{\bm{\rho}}{\textbf{r}}\right)
\frac{N^{\tau\left(\bm{\rho},\mathbf{r},\mathfrak{D}\right)}}{\Gamma\left(k+1+\frac{\ell}{2}+\tau\left(\bm{\rho},\mathbf{r},\mathfrak{D}\right)\right)}.
\\
M_4\left(N,k,d,h,\mathbf{r}\right) :=&
\frac{N^{k/2}\left(-1\right)^{d}}{\pi^{k}}\sum_{\eta=0}^{h-1}\frac{\dbinom{h}{\eta}}{2^{\eta}}\sum_{\ell=0}^{\eta}\dbinom{\eta}{\ell}\left(-1\right)^{\eta-\ell}
\sum_{\bm{\rho}\in Z^{d}}
\frac1{\textbf{r}}\Gamma\left(\frac{\bm{\rho}}{\textbf{r}}\right)
\frac{N^{\tau\left(\bm{\rho},\mathbf{r},\mathfrak{D}\right)/2}}{\pi^{\tau\left(\bm{\rho},\mathbf{r},\mathfrak{D}\right)}}
\mathfrak{B}\big(\tau\left(\bm{\rho},\mathbf{r},\mathfrak{D}\right)\big),
\\
M_5\left(N,k,d,h,\mathbf{r}\right) :=&
\frac{N^{k/2}}{\pi^{k}}\sum_{\underset{{\scriptstyle \left|I\right|\geq1}}{I\subseteq\mathfrak{D}}}N^{\frac{\tau\left(\mathbf{r},I\right)}{2}}\left(-1\right)^{\left|\mathfrak{D}\setminus I\right|}\sum_{\eta=0}^{h}\frac{\dbinom{h}{\eta}}{2^{\eta}}\sum_{\ell=0}^{\eta}\dbinom{\eta}{\ell}\left(-1\right)^{\eta-\ell}
\sum_{\bm{\rho}\in Z^{|\mathfrak{D}\setminus I|}}
\frac1{\textbf{r}}\Gamma\left(\frac{\bm{\rho}}{\textbf{r}}\right)
\frac{N^{\tau\left(\bm{\rho},\mathbf{r},\mathfrak{D}\setminus I\right)/2}}{\pi^{\tau\left(\bm{\rho},\mathbf{r},\mathfrak{D}\setminus I\right)}}
\\
&\times
\sum_{\bm{\rho}\in Z^{|\mathfrak{D}\setminus I|}}
\frac1{\textbf{r}}\Gamma\left(\frac{\bm{\rho}}{\textbf{r}}\right)
\frac{N^{\tau\left(\bm{\rho},\mathbf{r},\mathfrak{D}\setminus I\right)/2}}{\pi^{\tau\left(\bm{\rho},\mathbf{r},\mathfrak{D}\setminus I\right)}}
\mathfrak{B}\big(\tau\left(\mathbf{r},I\right)+\tau\left(\bm{\rho},\mathbf{r},\mathfrak{D}\setminus I\right)\big),
\end{align*}
where $\rho_{j}$ runs over the non-trivial zeros of the Riemann Zeta
function and $J_{v}\left(u\right)$ are the Bessel $J$ function of
real argument $u$ and complex order $v$ and
\begin{align*}
\mathfrak{B}(x)
=
\mathfrak{B}_{k,h,\eta,\ell,N}(x)
=
N^{\frac{h-\eta+\ell}{4}}
\sum_{\mathbf{f}\in\left(\mathbb{N}^{+}\right)^{h-\eta}}\frac{J_{x+k+(h-\eta+\ell)/2}\left(2\pi\sqrt{N}\left\Vert \mathbf{f}\right\Vert \right)}{\left\Vert \mathbf{f}\right\Vert ^{x+k+(h-\eta+\ell)/2}},
\end{align*}
with
\begin{align*}
\sum_{\mathbf{f}\in\left(\mathbb{N}^{+}\right)^{c}}:=\sum_{f_{1}\geq1}\cdots\sum_{f_{c}\geq1}.
\end{align*}
The convergence of the mentioned series will be proved in the section \S \ref{conv}. The main result of this
article is the following theorem:
\begin{thm}\label{main_thm}
Let $d,h\in\mathbb{N},\,d>0,$ let $N$ be a sufficiently large integer.
Let $\mathfrak{D}:=\left\{ 1,\dots,d\right\} $ and, for every $\mathfrak{J}\subseteq\mathfrak{D}$
(or with the notation $I\subseteq\mathfrak{D})$ let $\tau\left(\mathbf{r},\mathfrak{J}\right):=\sum_{j\in\mathfrak{J}}\frac{1}{r_{j}}$, where $1\le r_1\le r_2\le \cdots\le r_d$.
Then, for $k>\frac{d+h}{2}$, we have that
\[
\sum_{n\leq N}R_{d,h,\mathbf{r}}\left(n\right)\frac{\left(N-n\right)^{k}}{\Gamma\left(k+1\right)}=
\sum_{j=1}^{5}M_{j}\left(N,k,d,h,\mathbf{r}\right)+
O_{\mathbf{r},d,h}\left(N^{k+h/2+\tau\left(\mathbf{r},\mathfrak{D}\right)-1/r_d}\right).
\]
\end{thm}
It is important to underline that in some particular configurations of the parameters some terms of the asymptotic (but not the dominant term) could be incorporated in the error. Despite the apparently complicated form of the terms, it is not difficult to recognize
the results obtained in the previous work on this topic, for example
setting $d=2$, $h=0$ and $\mathbf{r}=\left(1,1\right)$ (the Goldbach
numbers case \cite{LanZac}) or $\mathbf{r}=\left(\ell_{1},\ell_{2}\right),\,1\leq\ell_{1}\leq\ell_{2}$
integers (the generalized Goldbach numbers case \cite{LanZac4}).
Furthermore, it is quite natural to conjecture that at least the main
term of this asymptotic is valid for $k\geq0$ instead of $k>\frac{d+h}{2}$
as suggested by similar studies but with other techniques (see, e.g.,
\cite{CanGamZac2}\cite{CanGamLangZac}). Now, let's briefly explain
the major ideas behind this theorem; one of the main tools of this
technique is the formula, due to Laplace \cite{Lap}, namely
\begin{equation}
\frac{1}{2\pi i}\int_{\left(a\right)}v^{-s}e^{v}\,\mathrm{d} v=\frac{1}{\Gamma\left(s\right)}\label{eq:onemaintool}
\end{equation}
for $\mathrm{Re}\left(s\right)>0$ and $a>0$ (see formula $5.4(1)$
on page $238$ of \cite{ErdMagObeTric}), where
\[
\int_{\left(a\right)}:=\int_{a-i\infty}^{a+i\infty}.
\]
From (\ref{eq:onemaintool}) and suitable hypotheses, which we will
explain in detail in the next sections, we are able to write
\begin{equation}
\sum_{n\leq N}R_{d,h,\mathbf{r}}\left(n\right)\frac{\left(N-n\right)^{k}}{\Gamma\left(k+1\right)}=\frac{1}{2\pi i}\int_{\left(a\right)}e^{Nz}z^{-k-1}\widetilde{S}_{r_{1}}\left(z\right)\cdots\widetilde{S}_{r_{d}}\left(z\right)\omega_{2}\left(z\right)^{h}\,\mathrm{d} z\label{eq:main}
\end{equation}
where $z=a+iy$, $a>0$, $y\in\mathbb{R}$, where
\begin{equation}
\widetilde{S}_{r}\left(z\right):=\sum_{m\geq1}\Lambda\left(m\right)e^{-m^{r}z},\qquad\omega_{2}\left(z\right):=\sum_{m\geq1}e^{-m^{2}z},\label{eq:Stildaeomega}
\end{equation}
are the series that embody the prime powers and the squares, respectively.
Since, as we will see, it is possible to develop $\widetilde{S}_{r}\left(z\right)$
as an asymptotic formula, the idea is to substitute this formula for
$\widetilde{S}_{r}\left(z\right)$, exchange the integral with all
the terms which are obtained from the various products and and finally
calculate the error. Another important aspect to emphasize is that
we work with squares, and so with $\omega_{2}\left(z\right)$, because
this function is linked to the well-known Jacobi theta $3$ function
\[
\theta_{3}\left(z\right):=\sum_{m\in\mathbb{Z}}e^{-m^{2}z}=1+2\omega_{2}\left(z\right)
\]
and $\theta_{3}\left(z\right)$ satisfies the functional equation
\[
\theta_{3}\left(z\right)=\left(\frac{\pi}{z}\right)^{1/2}\theta_{3}\left(\frac{\pi^{2}}{z}\right),\,\mathrm{Re}\left(z\right)>0
\]
(see, for example, Proposition VI.$4.3$, page $340$, of \cite{FreBus})
which implies a functional equation for $\omega_{2}\left(z\right)$
\begin{equation}
\omega_{2}\left(z\right)=\frac{1}{2}\left(\frac{\pi}{z}\right)^{1/2}-\frac{1}{2}+\left(\frac{\pi}{z}\right)^{1/2}\omega_{2}\left(\frac{\pi^{2}}{z}\right).\label{eq:funceqomega}
\end{equation}
This is fundamental for the present technique, because this functional equation
allows us to find the terms involving the Bessel $J$ function and,
since we do not have a functional equation of this type for other
powers than squares, we can only deal with this particular case.
\section{Settings}
For our purposes, we need a general version of the formula (\ref{eq:onemaintool}),
so we recall the following relations:
\begin{equation}
\frac{1}{2\pi}\int_{\mathbb{R}}\frac{e^{iDu}}{\left(a+iu\right)^{s}}\,\mathrm{d} u=\begin{cases}
\frac{D^{s-1}e^{-aD}}{\Gamma\left(s\right)}, & D>0\\
0, & D<0
\end{cases}\label{eq:gen1}
\end{equation}
with $\mathrm{Re}\left(s\right)>0$, $\mathrm{Re}\left(a\right)>0$
and
\begin{equation}
\frac{1}{2\pi}\int_{\mathbb{R}}\frac{1}{\left(a+iu\right)^{s}}\,\mathrm{d} u=\begin{cases}
0, & \mathrm{Re}\left(s\right)>1\\
1/2, & \mathrm{Re}\left(s\right)=1
\end{cases}\label{eq:gen2}
\end{equation}
with $\mathrm{Re}\left(a\right)>0$ (see formulas $\left(8\right)$
and $\left(9\right)$ of \cite{Aze}). We also need an integral representation
of the Bessel $J$ function with real argument $u$ and complex order
$v$:
\begin{equation}
J_{v}\left(u\right):=\frac{\left(u/2\right)}{2\pi i}\int_{\left(a\right)}s^{-v-1}e^{s}e^{-u^{2}/\left(4s\right)}\mathrm{d} s\label{eq:bessel}
\end{equation}
for $a>0$, $u,v\in\mathbb{C}$ with $\mathrm{Re}\left(v\right)>-1$
(see, e.g., equation $\left(8\right)$ on page $177$ of \cite{Wat}).
Assume that $k>0$. From the definition of $\widetilde{S}_{r}\left(z\right)$
and $\omega_{2}\left(z\right)$ (\ref{eq:Stildaeomega}), it is not
difficult to note that
\[
\widetilde{S}_{r_{1}}\left(z\right)\cdots\widetilde{S}_{r_{d}}\left(z\right)\omega_{2}\left(z\right)^{h}=\sum_{n\geq1}R_{d,h,\mathbf{r}}\left(n\right)e^{-nz}.
\]
Furthermore, from (\ref{eq:gen1}) and (\ref{eq:gen2}), we have
that
\begin{equation}
\sum_{n\leq N}R_{d,h,\mathbf{r}}\left(n\right)\frac{\left(N-n\right)^{k}}{\Gamma\left(k+1\right)}=\sum_{n\geq1}R_{d,h,\mathbf{r}}\left(n\right)\left(\frac{1}{2\pi i}\int_{\left(a\right)}e^{\left(N-n\right)z}z^{-k-1}\,\mathrm{d} z\right).\label{eq:applicazgen}
\end{equation}
Now we want to show that it is possible to exchange the integral with
the series in the right side of (\ref{eq:applicazgen}).
By the Prime Number Theorem, we have that
\begin{equation}
\widetilde{S}_{r_{j}}\left(a\right)\sim\frac{\Gamma\left(\frac{1}{r_{j}}\right)}{r_{j}a^{1/r_{j}}}\label{eq:PNT}
\end{equation}
as $a\rightarrow0^{+}$(see \cite{LanZac5}) and
\begin{equation}
\left|\omega_{2}\left(z\right)\right|\leq\omega_{2}\left(a\right)\leq\int_{0}^{+\infty}e^{-au^{2}}\,\mathrm{d} u\leq a^{-1/2}\int_{0}^{+\infty}e^{-v^{2}}\,\mathrm{d} v\ll a^{-1/2}\label{eq:omegaest}
\end{equation}
and so
\[
\sum_{n\geq1}\left|R_{d,h,\mathbf{r}}\left(n\right)e^{-nz}\right|=\sum_{n\geq1}R_{d,h,\mathbf{r}}\left(n\right)e^{-na}=\widetilde{S}_{r_{1}}\left(a\right)\cdots\widetilde{S}_{r_{d}}\left(a\right)\omega_{2}\left(a\right)^{h}
\]
\[
\ll_{\mathbf{r},d,h}a^{-\tau\left(\mathbf{r},\mathfrak{D}\right)-h/2}.
\]
From the trivial estimate
\begin{equation}
\left|e^{Nz}\right|\left|z^{-k-1}\right|\asymp e^{Na}\begin{cases}
a^{-k-1}, & \left|y\right|\leq a\\
\left|y\right|^{-k-1}, & \left|y\right|>a
\end{cases}\label{eq:trivial estimate}
\end{equation}
where $f\asymp g$ means $g\ll f\text{\ensuremath{\ll g}, we have}$
\begin{align*}
\frac{1}{2\pi i}\int_{\left(a\right)}e^{Nz}z^{-k-1}\widetilde{S}_{r_{1}}\left(z\right)\cdots\widetilde{S}_{r_{d}}\left(z\right)\omega_{2}\left(z\right)^{h}\,\mathrm{d} z & \ll_{\mathbf{r},d,h}e^{Na}a^{-\tau\left(\mathbf{r},\mathfrak{D}\right)-h/2}\left(\int_{-a}^{a}a^{-k-1}\,\mathrm{d} y+\int_{a}^{+\infty}y^{-k-1}\,\mathrm{d} y\right)\\
& \ll_{\mathbf{r},d,h}e^{Na}a^{-\tau\left(\mathbf{r},\mathfrak{D}\right)-h/2-k}
\end{align*}
for $k>0$. Then, we can exchange the integral with the series and
so we obtain the main formula \eqref{eq:main}.
\section{Lemmas}\label{conv}
In this section we present some technical lemmas that will be useful
later and some basic facts in complex analysis. First, we recall that
if $z=a+iy,\,a>0$ and $w\in\mathbb{C}$, we have that
\[
z^{-w}=\left|z\right|^{-\mathrm{Re}\left(w\right)-i\mathrm{Im}\left(w\right)}\exp\left(\left(-i\mathrm{Re}\left(w\right)+\mathrm{Im}\left(w\right)\right)\arctan\left(\frac{y}{a}\right)\right)
\]
and so
\begin{equation}
\left|z^{-w}\right|=\left|z\right|^{-\mathrm{Re}\left(w\right)}\exp\left(\mathrm{Im}\left(w\right)\arctan\left(\frac{y}{a}\right)\right).\label{eq:complex power}
\end{equation}
We also recall the Stirling formula
\begin{equation}
\left|\Gamma\left(x+iy\right)\right|\sim\sqrt{2\pi}e^{-\pi\left|y\right|/2}\left|y\right|^{x-1/2}\label{eq:Stirling}
\end{equation}
which holds uniformly for $x\in\left[x_{1},x_{2}\right]$, $x_{1},\,x_{2}$
fixed and $\left|y\right|\rightarrow+\infty$ (see, e.g., \cite{Tit},
section $4.42$).
Now we introduce the ``explicit formula'' of $\widetilde{S}_{r}\left(z\right),\,r\in\mathbb{N}^{+}$.
\begin{lem}
(Lemma $1$ of \cite{LanZac4}) Let $r\geq1$ be an integer, let $z=a+iy,\,a>0,\,y\in\mathbb{R}$.
Let
\begin{equation}
T\left(z,r\right):=\frac{\Gamma\left(\frac{1}{r}\right)}{rz^{1/r}}-\frac{1}{r}\sum_{\rho}z^{-\rho/r}\Gamma\left(\frac{\rho}{r}\right).\label{eq:asmipt2}
\end{equation}
Then
\begin{equation}
\widetilde{S}_{r}\left(z\right)=T\left(z,r\right)+E\left(a,y,r\right).\label{eq:asimpt}
\end{equation}
where
\begin{equation}
\left|E\left(a,y,r\right)\right|\ll_{r}1+\left|z\right|^{1/2}\begin{cases}
1, & \left|y\right|\leq a\\
1+\log^{2}\left(\frac{\left|y\right|}{a}\right), & \left|y\right|>a.
\end{cases}\label{eq:error estimate}
\end{equation}
\end{lem}
Note that in Lemma $1$ of \cite{LanZac4} $T\left(z,r\right)$ is
defined as
\[
T\left(z,r\right):=\frac{\Gamma\left(\frac{1}{r}\right)}{rz^{1/r}}-\frac{1}{r}\sum_{\rho}z^{-\rho/r}\Gamma\left(\frac{\rho}{r}\right)-\log\left(2\pi\right)
\]
but in our context, to make the main term combinatorically more tractable,
it is better to insert $\log\left(2\pi\right)$ in the error term
$E\left(a,y,r\right)$. Furthermore, from (\ref{eq:PNT}) and (\ref{eq:error estimate})
we immediately get the important estimate
\begin{equation}
\left|\sum_{\rho}z^{-\rho/r}\Gamma\left(\frac{\rho}{r}\right)\right|\ll_{r}a^{-1/r}+1+\left|z\right|^{1/2}\begin{cases}
1, & \left|y\right|\leq a\\
1+\log^{2}\left(\frac{\left|y\right|}{a}\right), & \left|y\right|>a
\end{cases}\label{eq:estserieszeros0}
\end{equation}
which can be rewritten, if $0<a<1$ and $r\ge1$, in the more compact
form
\begin{equation}
\left|\sum_{\rho}z^{-\rho/r}\Gamma\left(\frac{\rho}{r}\right)\right|\ll_{r}\begin{cases}
a^{-1/r}, & \left|y\right|\leq a\\
a^{-1/r}+\left|z\right|^{1/2}\log^{2}\left(\frac{2\left|y\right|}{a}\right), & \left|y\right|>a.
\end{cases}\label{eq:estserieszeros}
\end{equation}
\begin{lem}
\label{lem:generalized1}Let $\lambda\in\mathbb{N}^{+}$, $r_{1},\dots,r_{\lambda}\in\mathbb{N}^{+}$
and $\mathbf{r}:=\left(r_{1},\dots,r_{\lambda}\right)\in\left(\mathbb{N}^{+}\right)^{\lambda}$.
Let $\rho_{j}=\beta_{j}+i\gamma_{j},\,j\in\left\{ 1,\dots,\lambda\right\} $,
run over the non trivial zeros of Riemann Zeta function and $\alpha>1$
be a parameter. Then, for any fixed $b>1$ and $c\geq0$, the series
\[
\sum_{\rho_{1}:\,\gamma_{1}>0}\left(\frac{\gamma_{1}}{r_{1}}\right)^{\beta_{1}/r_{1}-1/2}\cdots\sum_{\rho_{\lambda}:\,\gamma_{\lambda}>0}\left(\frac{\gamma_{\lambda}}{r_{\lambda}}\right)^{\beta_{\lambda}/r_{\lambda}-1/2}\int_{1}^{+\infty}\log^{c}\left(bu\right)\exp\left(-\arctan\left(\frac{1}{u}\right)\tau\left(\bm{\gamma},\mathbf{r},\mathfrak{J}_{\lambda}\right)\right)\frac{\,\mathrm{d} u}{u^{\alpha+\tau\left(\bm{\beta},\mathbf{r},\mathfrak{J}_{\lambda}\right)}}
\]
converges if $\alpha>\frac{\lambda}{2}+1$.
\end{lem}
\begin{proof}
Following the proof of Lemma $2$ of \cite{LanZac4}, we can see that
\[
\int_{1}^{+\infty}\exp\left(-\arctan\left(\frac{1}{u}\right)\tau\left(\bm{\gamma},\mathbf{r},\mathfrak{J}_{\lambda}\right)\right)\frac{\,\mathrm{d} u}{u^{\alpha+\tau\left(\bm{\beta},\mathbf{r},\mathfrak{J}_{\lambda}\right)}}
\ll_{\alpha,\mathbf{r}}\tau\left(\bm{\gamma},\mathbf{r},\mathfrak{J}_{\lambda}\right)^{1-\alpha-\tau\left(\bm{\beta},\mathbf{r},\mathfrak{J}_{\lambda}\right)}\int_{0}^{+\infty}e^{-w}w^{\alpha+\tau\left(\bm{\beta},\mathbf{r},\mathfrak{J}_{\lambda}\right)-2}\,\mathrm{d} w
\]
and the integral converges since $0<\beta_{j}<1,\,j=1,\dots,\lambda$
and $\alpha>1.$ Hence, we have to consider
\[
\sum_{\rho_{1}:\,\gamma_{1}>0}\cdots\sum_{\rho_{\lambda}:\,\gamma_{\lambda}>0}\frac{\left(\frac{\gamma_{1}}{r_{1}}\right)^{\beta_{1}/r_{1}-1/2}\cdots\left(\frac{\gamma_{\lambda}}{r_{\lambda}}\right)^{\beta_{\lambda}/r_{\lambda}-1/2}}{\tau\left(\bm{\gamma},\mathbf{r},\mathfrak{J}_{\lambda}\right)^{\alpha+\tau\left(\bm{\beta},\mathbf{r},\mathfrak{J}_{\lambda}\right)-1}}.
\]
Now, it is not difficult to note that
\begin{equation}
\frac{\left(\frac{\gamma_{1}}{r_{1}}\right)^{\beta_{1}/r_{1}}\cdots\left(\frac{\gamma_{\lambda}}{r_{\lambda}}\right)^{\beta_{\lambda}/r_{\lambda}}}
{\tau\left(\bm{\gamma},\mathbf{r},\mathfrak{J}_{\lambda}\right)^{\tau\left(\bm{\beta},\mathbf{r},\mathfrak{J}_{\lambda}\right)}}\leq1\label{eq:stima semplice}
\end{equation}
so we analyze
\[
\sum_{\rho_{1}:\,\gamma_{1}>0}\cdots\sum_{\rho_{\lambda}:\,\gamma_{\lambda}>0}\frac{\left(\frac{\gamma_{1}}{r_{1}}\right)^{-1/2}\cdots\left(\frac{\gamma_{\lambda}}{r_{\lambda}}\right)^{-1/2}}
{\tau\left(\bm{\gamma},\mathbf{r},\mathfrak{J}_{\lambda}\right)^{\alpha-1}}
\le \sum_{\rho_{1}:\,\gamma_{1}>0}\left(\frac{\gamma_{1}}{r_{1}}\right)^{-\frac12-\frac{\alpha-1}{\lambda}}\cdots\sum_{\rho_{\lambda}:\,\gamma_{\lambda}>0}\left(\frac{\gamma_{\lambda}}{r_{\lambda}}\right)^{-\frac12-\frac{\alpha-1}{\lambda}}
\]
by the inequality of arithmetic and geometric means.
From the asymptotic formula of $N\left(T\right)$, where $N\left(T\right)$
is the number of non-trivial zeros of the Riemann zeta function with
imaginary part $0\leq\gamma\leq T$, it is not difficult to prove,
putting $\gamma\left(k\right)$ the imaginary part of the
$k$-th non-trivial zeros of $\zeta\left(s\right)$, that
\[
\gamma\left(k\right)\sim\frac{2\pi k}{\log\left(k\right)}
\]
as $k\rightarrow+\infty$. So the series converges if $\alpha>\frac{\lambda}{2}+1.$
The treatment is similar for the case $c>0$.
\end{proof}
\begin{lem}
\label{lem:generalizlemnew}Let $N,\lambda,\alpha$ be positive integers,
let $h\in\mathbb{Q}^{+},$ let $\rho_{j}=\beta_{j}+i\gamma_{j},\,j\in\left\{ 1,\dots,\lambda\right\} $,
run over the non-trivial zeros of the Riemann Zeta function, $\left\Vert \cdot\right\Vert $
the Euclidean norm in $\mathbb{R}^{d},\,d\in\mathbb{N}^{+}$ and $k>0$
a real number. For sake of simplicity we define $\delta:=\sum_{j=1}^{\lambda}\gamma_{j}$.
Then, for every fixed integer $b>1$ and $c>0$,
\[
\sum_{\rho_{1}:\,\gamma_{1}>0}\cdots\sum_{\rho_{\lambda}:\,\gamma_{\lambda}>0}\frac{\gamma_{1}^{-\frac{1}{2}}\cdots\gamma_{\lambda}^{-\frac{1}{2}}}{\delta^{k+h+\alpha}}
\sum_{\mathbf{f}\in\left(\mathbb{N}^{+}\right)^{\alpha}}
\int_{0}^{\delta}v^{k-1+h+\alpha+\tau\left(\bm{\beta},\mathbf{r},\mathfrak{J}_{\lambda}\right)}
e^{-\left\Vert \mathbf{f}\right\Vert ^{2}Nv^{2}/\delta^{2}-v}
\log^{2c}\left(\frac{b\,\delta}{v}\right)\,\mathrm{d} v
\]
converges if $k>\frac{\lambda}{2}-h.$
\end{lem}
\begin{proof}
We consider the integral
\begin{equation}
\sum_{\rho_{1}:\,\gamma_{1}>0}\cdots\sum_{\rho_{\lambda}:\,\gamma_{\lambda}>0}\frac{\gamma_{1}^{-\frac{1}{2}}\cdots\gamma_{\lambda}^{-\frac{1}{2}}}{{\delta}^{k+h+\alpha}}
\sum_{\mathbf{f}\in\left(\mathbb{N}^{+}\right)^{\alpha}}
\int_{0}^{\delta}v^{k-1+h+\alpha+\tau\left(\bm{\beta},\mathbf{r},\mathfrak{J}_{\lambda}\right)}e^{-\left\Vert \mathbf{f}\right\Vert ^{2}Nv^{2}{\delta}^{-2}}\exp\left(-v\right)\,\mathrm{d} v.\label{int lemma}
\end{equation}
Now we claim that we can exchange the integral with the multiple series
$\sum_{\mathbf{f}\in\left(\mathbb{N}^{+}\right)^{\alpha}}$. To show
this we consider
\[
\int_{0}^{\delta} v^{k-1+h+\alpha+\tau\left(\bm{\beta},\mathbf{r},\mathfrak{J}_{\lambda}\right)}
\sum_{f_{1}\geq1}e^{-f_{1}Nv^{2}{\delta}^{-2}}\omega_{2}^{\alpha-1}(Nv^{2}\delta^{-2})\exp\left(-v\right)\,\mathrm{d} v.
\]
Now, since for every $M\geq1$ we have
\[
\sum_{f_{1}\leq M}e^{-f_{1}Nv^{2}\delta^{-2}}\leq\sum_{f_{1}\geq1}e^{-f_{1}Nv^{2}\delta^{-2}}
=
\omega_{2}(Nv^{2}\delta^{-2})
\ll_{N}
\frac{\delta}{v}
\]
from (\ref{eq:omegaest}) and so we have to deal with
\[
\int_{0}^{\delta}v^{k-2+h+\alpha+\tau\left(\bm{\beta},\mathbf{r},\mathfrak{J}_{\lambda}\right)}\omega_{2}^{\alpha-1}(Nv^{2}\delta^{-2})\exp\left(-v\right)\,\mathrm{d} v\ll_{N,\alpha}
\delta^{\alpha-1}\int_{0}^{\delta}v^{k+h-1+\tau\left(\bm{\beta},\mathbf{r},\mathfrak{J}_{\lambda}\right)}\exp\left(-v\right)\,\mathrm{d} v
\]
which is convergent since $k>0$, then we obtain
\[
\sum_{f_{1}\geq1}\int_{0}^{\delta}v^{k-1+h+\alpha+\frac{\beta_{1}}{r_{1}}+\dots+\frac{\beta_{\lambda}}{r_{\lambda}}}
e^{-f_{1}Nv^{2}{\delta}^{-2}}
\omega_{2}^{\alpha-1}(Nv^{2}\delta^{-2})\exp\left(-v\right)\,\mathrm{d} v
\]
by the Dominated Convergence Theorem. Clearly, we can repeat the same
argument for every factor in the product $\omega_{2}^{\alpha-1}(Nv^{2}\delta^{-2})$
and so we can write (\ref{int lemma}) as
\[
\sum_{\rho_{1}:\,\gamma_{1}>0}\cdots\sum_{\rho_{\lambda}:\,\gamma_{\lambda}>0}\frac{\gamma_{1}^{-\frac{1}{2}}\cdots\gamma_{\lambda}^{-\frac{1}{2}}}{\delta^{k+h+\alpha}}
\int_{0}^{\delta}v^{k-1+h+\alpha+\tau\left(\bm{\beta},\mathbf{r},\mathfrak{J}_{\lambda}\right)}
\omega_{2}^{\alpha}(Nv^{2}\delta^{-2})\exp(-v)\,\mathrm{d} v
\]
and again using (\ref{eq:omegaest})
we have to deal with
\[
\sum_{\rho_{1}:\,\gamma_{1}>0}\cdots\sum_{\rho_{\lambda}:\,\gamma_{\lambda}>0}\frac{\gamma_{1}^{-\frac{1}{2}}\cdots\gamma_{\lambda}^{-\frac{1}{2}}}{{\delta}^{k+h+\alpha}}
\int_{0}^{\delta}v^{k-1+h+\tau\left(\bm{\beta},\mathbf{r},\mathfrak{J}_{\lambda}\right)}\exp\left(-v\right)\,\mathrm{d} v.
\]
Now, since $k+h+\tau\left(\bm{\beta},\mathbf{r},\mathfrak{J}_{\lambda}\right)>0$, then
\[
\int_{0}^{\delta}v^{k-1+h+\tau\left(\bm{\beta},\mathbf{r},\mathfrak{J}_{\lambda}\right)}\exp\left(-v\right)\,\mathrm{d} v\ll\int_{0}^{+\infty}v^{k-1+h+\tau\left(\bm{\beta},\mathbf{r},\mathfrak{J}_{\lambda}\right)}\exp\left(-v\right)\,\mathrm{d} v<+\infty.
\]
Then, from arithmetic mean - geometric mean inequality, we get
\begin{equation}
\sum_{\rho_{1}:\,\gamma_{1}>0}\cdots\sum_{\rho_{\lambda}:\,\gamma_{\lambda}>0}\frac{\gamma_{1}^{-\frac{1}{2}}\cdots\gamma_{\lambda}^{-\frac{1}{2}}}{\delta^{k+h+\alpha}}
\ll
\sum_{\rho_{1}:\,\gamma_{1}>0}\gamma_{1}^{-\frac{k}{\lambda}-\frac{1}{2}-\frac{h}{\lambda}}\cdots\sum_{\rho_{\lambda}:\,\gamma_{\lambda}>0}\gamma_{\lambda}^{-\frac{k}{\lambda}-\frac{1}{2}-\frac{h}{\lambda}}\label{eq:sumgammageneralized}
\end{equation}
and the series converges if $k>\frac{\lambda}{2}-h.$ Clearly, if
we have a log factor into the integral the bound for $k$ is the same.
Indeed, we note that
\[
\log^{2c}\left(b\frac{\delta}{v}\right)
\ll
\log^{2c}(\delta)+\log^{2c}\left(bv\right)
\]
and
\[
\log^{2c}(\delta)\leq\log^{2c}\left(\lambda\max_{\gamma_{j},\,j=1,\dots,\lambda}\gamma_{j}\right):=\log^{2c}\left(\lambda\gamma_{\star}\right)
\]
so we have in (\ref{eq:sumgammageneralized}) one series such that
\[
\sum_{\rho_{\star}:\,\gamma_{\star}>0}\gamma_{\star}^{-\frac{k}{\lambda}-\frac{1}{2}-\frac{h}{\lambda}}\log^{2c}\left(\lambda\gamma_{\star}\right)
\]
and clearly the log factor does not affect the bound for $k$; if
we have
\[
\int_{0}^{+\infty}v^{k-1+h+\tau\left(\bm{\beta},\mathbf{r},\mathfrak{J}_{\lambda}\right)}\exp\left(-v\right)\log^{2c}\left(bv\right)\,\mathrm{d} v
\]
again, we have the same bounds for $k$ and so the Lemma is proved.
\end{proof}
\section{proof of the main theorem}
In this section we prove the main theorem. We first show that the
error bound in the main formula is ``small'', then we prove that
all the exchange of symbols is justified and finally we evaluate
the integrals.
\subsection{Error term}
From (\ref{eq:asmipt2}), (\ref{eq:asimpt}) and following the subdivision
in \cite{CanGamZac}, formula $(2)$, we can write
\[
\widetilde{S}_{r_{1}}\left(z\right)\cdots\widetilde{S}_{r_{d}}\left(z\right)=T\left(z,r_{1}\right)\cdots T\left(z,r_{d}\right)
\]
\[
+\sum_{j=1}^{d}E\left(a,y,r_{j}\right)\left(\prod_{i\neq j}\widetilde{S}_{r_{i}}\left(z\right)\right)+\sum_{\underset{{\scriptstyle \left|I\right|\geq2}}{I\subseteq\mathfrak{D}}}c_{d}\left(I\right)\left(\prod_{i\in\mathfrak{D}\setminus I}T\left(z,r_{i}\right)\right)\left(\prod_{\ell\in I}E\left(a,y,r_{\ell}\right)\right)
\]
for some suitable coefficients $c_{d}\left(I\right)$, so we get
\begin{align*}
\frac{1}{2\pi i}\int_{\left(a\right)}e^{Nz}z^{-k-1}&\widetilde{S}_{r_{1}}\left(z\right)\cdots\widetilde{S}_{r_{d}}\left(z\right)\omega_{2}\left(z\right)^{h}\,\mathrm{d} z =\frac{1}{2\pi i}\int_{\left(a\right)}e^{Nz}z^{-k-1}T\left(z,r_{1}\right)\cdots T\left(z,r_{d}\right)\omega_{2}\left(z\right)^{h}\,\mathrm{d} z\\
& +\frac{1}{2\pi i}\sum_{j=1}^{d}\int_{\left(a\right)}e^{Nz}z^{-k-1}E\left(a,y,r_{j}\right)\left(\prod_{i\neq j}\widetilde{S}_{r_{i}}\left(z\right)\right)\omega_{2}\left(z\right)^{h}\,\mathrm{d} z\\
& +\sum_{\underset{{\scriptstyle \left|I\right|\geq2}}{I\subseteq\mathfrak{D}}}c_{d}\left(I\right)\int_{\left(a\right)}e^{Nz}z^{-k-1}\left(\prod_{i\in\mathfrak{D}\setminus I}T\left(z,r_{i}\right)\right)\left(\prod_{\ell\in I}E\left(a,y,r_{\ell}\right)\right)\omega_{2}\left(z\right)^{h}\,\mathrm{d} z\\
& =:A_{1}+A_{2}+A_{3}.
\end{align*}
Now we have to estimate the error term. From (\ref{eq:PNT}), (\ref{eq:omegaest})
and (\ref{eq:error estimate}) we obtain
\[
\left|A_{2}\right|\ll\sum_{j=1}^{d}\int_{\left(a\right)}\left|e^{Nz}\right|\left|z^{-k-1}\right|\left|E\left(a,y,r_{j}\right)\right|\prod_{i\neq j}\left|\widetilde{S}_{r_{i}}\left(z\right)\right|\left|\omega_{2}\left(z\right)^{h}\right|\,\mathrm{d} y
\]
\[
\ll_{\mathbf{r},d,h}e^{Na}a^{-h/2}\sum_{j=1}^{d}a^{-\tau\left(\mathbf{r},\mathfrak{D}\right)+\frac{1}{r_{j}}}\left(\int_{0}^{a}a^{-k-1}\left(1+a^{1/2}\right)\,\mathrm{d} y+\int_{a}^{+\infty}y^{-k-1}\left(1+y^{1/2}\left(1+\log^{2}\left(\frac{y}{a}\right)\right)\right)\,\mathrm{d} y\right)
\]
\begin{equation}
\ll_{\mathbf{r},d,h}e^{Na}a^{-k-h/2}\sum_{j=1}^{d}a^{-\tau\left(\mathbf{r},\mathfrak{D}\right)+\frac{1}{r_{j}}}\label{eq:main error term}
\end{equation}
for $k>0$.
For the estimation of $A_{3}$ we fix $I\subseteq\mathfrak{D}$
and we consider
\[
\left|A_{3.I}\right|:=\int_{\left(a\right)}\left|e^{Nz}\right|\left|z^{-k-1}\right|\prod_{i\in\mathfrak{D}\setminus I}\left|T\left(z,r_{i}\right)\right|\prod_{\ell\in I}\left|E\left(a,y,r_{\ell}\right)\right|\left|\omega_{2}\left(z\right)^{h}\right|\,\mathrm{d} y.
\]
We know from (\ref{eq:PNT}) and (\ref{eq:asimpt}) that
\[
\left|T\left(z,r\right)\right|\ll_{r}a^{-1/r}+\left|E\left(a,y,r\right)\right|
\]
hence, using formula \eqref{eq:omegaest} it is enough to work with
\[
a^{-h/2}\int_{\left(a\right)}\left|e^{Nz}\right|\left|z^{-k-1}\right|\prod_{i\in\mathfrak{D}\setminus I}\left(a^{-1/r_{i}}+\left|E\left(a,y,r_{i}\right)\right|\right)\prod_{\ell\in I}\left|E\left(a,y,r_{\ell}\right)\right|\,\mathrm{d} y.
\]
Now, observing that
\[
\prod_{i\in\mathfrak{D}\setminus I}\left(a^{-1/r_{i}}+\left|E\left(a,y,r_{i}\right)\right|\right)=\sum_{\mathfrak{J}\subseteq\mathfrak{D}\setminus I}a^{-\tau\left(\mathbf{r},\mathfrak{J}\right)}\prod_{i\in\mathfrak{D}\setminus\left(I\cup\mathfrak{J}\right)}\left|E\left(a,y,r_{i}\right)\right|
\]
we have by \eqref{eq:error estimate},
\begin{align*}
\left|A_{3.I}\right|\ll_{\mathbf{r},d,h}&e^{Na}a^{-h/2}\sum_{\mathfrak{J}\subseteq\mathfrak{D}\setminus I}\int_{\left(a\right)}a^{-\tau\left(\mathbf{r},\mathfrak{J}\right)}\prod_{i\in\mathfrak{D}\setminus\left(I\cup\mathfrak{J}\right)}\left|E\left(a,y,r_{i}\right)\right|\prod_{\ell\in I}\left|E\left(a,y,r_{\ell}\right)\right|\left|z^{-k-1}\right|\,\mathrm{d} y
\\
\ll_{\mathbf{r},d,h}&e^{Na}a^{-h/2}\sum_{\mathfrak{J}\subseteq\mathfrak{D}\setminus I}a^{-\tau\left(\mathbf{r},\mathfrak{J}\right)}\int_{0}^{a}a^{-k-1}\left(1+a^{1/2}\right)^{\left|\mathfrak{D}\setminus\mathfrak{J}\right|}\,\mathrm{d} y
\\
&+e^{Na}a^{-h/2}\sum_{\mathfrak{J}\subseteq\mathfrak{D}\setminus I}a^{-\tau\left(\mathbf{r},\mathfrak{J}\right)}\int_{a}^{+\infty}y^{-k-1}\left(1+y^{1/2}\left(1+\log^{2}\left(\frac{y}{a}\right)\right)\right)^{\left|\mathfrak{D}\setminus\mathfrak{J}\right|}\,\mathrm{d} y
\\
\ll_{\mathbf{r},d,h}&e^{Na}a^{-k-h/2}\sum_{\mathfrak{J}\subseteq\mathfrak{D}\setminus I}a^{-\tau\left(\mathbf{r},\mathfrak{J}\right)}
\end{align*}
for $k>\frac{\left|\mathfrak{D}\setminus\mathfrak{J}\right|}{2}$
and since this inequality must holds for all subsets $\mathfrak{J}\subseteq\mathfrak{D},$
we have to assume $k>\frac{d}{2}$. Hence
\[
\left|A_{3}\right|\ll_{\mathbf{r},d,h}e^{Na}a^{-k-h/2}\sum_{\underset{{\scriptstyle \left|I\right|\geq2}}{I\subseteq\mathfrak{D}}}\sum_{\mathfrak{J}\subseteq\mathfrak{D}\setminus I}a^{-\tau\left(\mathbf{r},\mathfrak{J}\right)}.
\]
Now we take $a=1/N$ and we observe that
\[
\sum_{\underset{{\scriptstyle \left|I\right|\geq2}}{I\subseteq\mathfrak{D}}}\sum_{\mathfrak{J}\subseteq\mathfrak{D}\setminus I}N^{\tau\left(\mathbf{r},\mathfrak{J}\right)}\ll_{d}\max_{\underset{{\scriptstyle \left|I\right|\geq2}}{I\subseteq\mathfrak{D}}}\max_{\mathfrak{J}\subseteq\mathfrak{D}\setminus I}N^{\tau\left(\mathbf{r},\mathfrak{J}\right)}\ll_{d}N^{\tau\left(\mathbf{r},\mathfrak{D}\right)-\frac{1}{r_{j_{1}}}-\frac{1}{r_{j_{2}}}}
\ll_{d}
N^{k+h/2+\tau\left(\mathbf{r},\mathfrak{D}\right)-1/r_d}
\]
remembering that $1\le r_1\le r_2\le \ldots\le r_d$. This error term is compatible with that of Theorem \ref{main_thm}.
\subsection{Evaluation of the main term}
According to (14) we rewrite $A_{1}$ in the following form
\begin{align*}
A_{1} & =\frac{1}{2\pi i}\int_{\left(1/N\right)}e^{Nz}z^{-k-1}T\left(z,r_{1}\right)\cdots T\left(z,r_{d}\right)\omega_{2}\left(z\right)^{h}\,\mathrm{d} z\\
& =\frac{1}{2\pi i}
\frac1{\textbf{r}}\Gamma\left(\frac{1}{\textbf{r}}\right)
\int_{\left(1/N\right)}e^{Nz}z^{-k-1-\tau\left(\mathbf{r},\mathfrak{D}\right)}\omega_{2}\left(z\right)^{h}\,\mathrm{d} z\\
& +\frac{\left(-1\right)^{d}}{2\pi i}\int_{\left(1/N\right)}e^{Nz}z^{-k-1}\left(
\sum_{\bm{\rho}\in Z^{d}}\frac1{\textbf{r}}\Gamma\left(\frac{\bm{\rho}}{\textbf{r}}\right)
z^{-\tau(\bm{\rho},\textbf{r},\mathfrak{D})}
\right)\omega_{2}\left(z\right)^{h}\,\mathrm{d} z
\\
& +\frac{1}{2\pi i}\sum_{\underset{{\scriptstyle \left|I\right|\geq1}}{I\subseteq\mathfrak{D}}}\left(-1\right)^{\left|\mathfrak{D}\setminus I\right|}\int_{\left(1/N\right)}e^{Nz}z^{-k-1-\tau\left(\mathbf{r},I\right)}
\left(
\sum_{\bm{\rho}\in Z^{|\mathfrak{D}\setminus I|}}
\frac1{\textbf{r}}\Gamma\left(\frac{\bm{\rho}}{\textbf{r}}\right)
z^{-\tau\left(\bm{\rho},\mathbf{r},\mathfrak{D}\setminus I\right)}
\right)\omega_{2}\left(z\right)^{h}\,\mathrm{d} z\\
& =:I_{1}+I_{2}+I_{3}.
\end{align*}
$I_1$ corresponds to the terms $M_1$ and $M_2$ of Theorem \ref{main_thm}, $I_2$ corresponds to the terms $M_3$ and $M_4$ and finally $I_3$ corresponds to $M_3$.
\subsubsection{Evaluation of $I_{1}$}
We study $I_{1}$. By (\ref{eq:funceqomega}) and the binomial theorem,
we get
\begin{align*}
\omega_{2}\left(z\right)^{h}=&\left(\frac{1}{2}\left(\frac{\pi}{z}\right)^{1/2}-\frac{1}{2}+\left(\frac{\pi}{z}\right)^{1/2}\omega_{2}\left(\frac{\pi^{2}}{z}\right)\right)^{h}
\\
=&\sum_{\eta=0}^{h}\frac{\dbinom{h}{\eta}}{2^{\eta}}\left(\left(\frac{\pi}{z}\right)^{1/2}-1\right)^{\eta}\omega_{2}\left(\frac{\pi^{2}}{z}\right)^{h-\eta}\left(\frac{\pi}{z}\right)^{\frac{h-\eta}{2}}
\\
=&\sum_{\eta=0}^{h}\frac{\dbinom{h}{\eta}}{2^{\eta}}\sum_{\ell=0}^{\eta}\dbinom{\eta}{\ell}\left(-1\right)^{\eta-\ell}\left(\frac{\pi}{z}\right)^{\frac{h-\eta+\ell}{2}}\omega_{2}\left(\frac{\pi^{2}}{z}\right)^{h-\eta}
\end{align*}
and so
\begin{align*}
I_{1} & =\frac{1}{2\pi i}
\frac1{\textbf{r}}\Gamma\left(\frac{1}{\textbf{r}}\right)
\int_{\left(1/N\right)}e^{Nz}z^{-k-1-\tau\left(\mathbf{r},\mathfrak{D}\right)}\omega_{2}\left(z\right)^{h}\,\mathrm{d} z\\
& =\frac{1}{2\pi i}\sum_{\eta=0}^{h}\frac{\dbinom{h}{\eta}}{2^{\eta}}\sum_{\ell=0}^{\eta}\dbinom{\eta}{\ell}\pi^{\frac{h-\eta+\ell}{2}}\left(-1\right)^{\eta-\ell}
\frac1{\textbf{r}}\Gamma\left(\frac{1}{\textbf{r}}\right)
\int_{\left(1/N\right)}e^{Nz}z^{-k-1-\tau\left(\mathbf{r},\mathfrak{D}\right)-\frac{h-\eta+\ell}{2}}\omega_{2}\left(\frac{\pi^{2}}{z}\right)^{h-\eta}\,\mathrm{d} z.
\end{align*}
Our main goal is to show that, for a suitable $k$, we can exchange
the integral with the involved series; in this case, with the series
related to $\omega_{2}$. We consider two cases: if $\eta=h$ we get
\[
I_{1,1}:=\frac{1}{2^{h+1}\pi i}\sum_{\ell=0}^{h}\dbinom{h}{\ell}\pi^{\frac{\ell}{2}}\left(-1\right)^{h-\ell}
\frac1{\textbf{r}}\Gamma\left(\frac{1}{\textbf{r}}\right)
\int_{\left(1/N\right)}e^{Nz}z^{-k-1-\tau\left(\mathbf{r},\mathfrak{D}\right)-\frac{\ell}{2}}\,\mathrm{d} z
\]
which corresponds to the term $M_1$ in Theorem \ref{main_thm} and, from the substitution $Nz=u$ and (\ref{eq:onemaintool}), we
get
\begin{align*}
I_{1,1}=&\frac{1}{2^{h+1}\pi i}\sum_{\ell=0}^{h}\dbinom{h}{\ell}\pi^{\frac{\ell}{2}}\left(-1\right)^{h-\ell}N^{k+\tau\left(\mathbf{r},d\right)+\frac{\ell}{2}}
\frac1{\textbf{r}}\Gamma\left(\frac{1}{\textbf{r}}\right)
\int_{\left(1\right)}e^{u}u^{-k-1-\tau\left(\mathbf{r},\mathfrak{D}\right)-\frac{\ell}{2}}\,\mathrm{d} u
\\
=&\frac{1}{2^{h}}\sum_{\ell=0}^{h}\dbinom{h}{\ell}\frac{\pi^{\frac{\ell}{2}}\left(-1\right)^{h-\ell}N^{k+\tau\left(\mathbf{r},\mathfrak{D}\right)+\frac{\ell}{2}}}{\Gamma\left(k+1+\tau\left(\mathbf{r},\mathfrak{D}\right)+\frac{\ell}{2}\right)}
\frac1{\textbf{r}}\Gamma\left(\frac{1}{\textbf{r}}\right)
\end{align*}
for $k+1+\tau\left(\mathbf{r},\mathfrak{D}\right)+\frac{\ell}{2}>0$,
which is trivially true if $k>0$. Now, fix $1\leq\lambda\leq h-\eta$.
We consider the general case
\begin{equation}
I_{1,2,\lambda}:=\sum_{f_{1}\geq1}\cdots\sum_{f_{\lambda}\geq1}\int_{\left(1/N\right)}\left|e^{Nz}\right|\left|z\right|^{-k-1-\tau\left(\mathbf{r},\mathfrak{D}\right)-\frac{h-\eta+\ell}{2}}e^{-\pi^{2}\mathrm{Re}\left(1/z\right)\left(f_{1}^{2}+\dots+f_{\lambda}^{2}\right)}\left|\omega_{2}\left(\frac{\pi^{2}}{z}\right)\right|^{h-\eta-\lambda}\left|\,\mathrm{d} z\right|.\label{eq:I12 general case}
\end{equation}
Note that if $I_{1,2,\lambda}$ converges for all $\lambda,$ the
exchange between series and integral is justified. By the trivial
estimate
\[
\mathrm{Re}\left(\frac{1}{z}\right)=\frac{N}{1+y^{2}N^{2}}\gg\begin{cases}
N, & \left|y\right|\leq1/N\\
1/\left(Ny^{2}\right), & \left|y\right|>1/N,
\end{cases}
\]
by (\ref{eq:trivial estimate}) and by (\ref{eq:omegaest}), we obtain
\begin{align*}
I_{1,2,\lambda}\ll_{h,\eta,\lambda}&\sum_{\mathbf{f}\in\left(\mathbb{N}^{+}\right)^{\lambda}}\int_{0}^{1/N}N^{k+1+\tau\left(\mathbf{r},\mathfrak{D}\right)+\frac{\lambda+\ell}{2}}e^{-\pi^{2}N\left(\left\Vert \mathbf{f}\right\Vert ^{2}\right)}\,\mathrm{d} y
\\
&+N^{h-\eta-\lambda}\sum_{\mathbf{f}\in\left(\mathbb{N}^{+}\right)^{\lambda}}\int_{1/N}^{+\infty}y^{-k-1-\tau\left(\mathbf{r},\mathfrak{D}\right)+\frac{h-\eta-\ell}{2}-\lambda}e^{-\frac{\pi^{2}\left(\left\Vert \mathbf{f}\right\Vert ^{2}\right)}{Ny^{2}}}\,\mathrm{d} y.
\end{align*}
The first integral and the series trivially converge since $N$ is
positive, then we can consider only the second integral. Making the
substitution $v=\frac{\pi^{2}\left(\left\Vert \mathbf{f}\right\Vert ^{2}\right)}{Ny^{2}},$ we
get
\begin{align*}
\sum_{\mathbf{f}\in\left(\mathbb{N}^{+}\right)^{\lambda}}&\int_{1/N}^{+\infty}y^{-k-1-\tau\left(\mathbf{r},\mathfrak{D}\right)+\frac{h-\eta-\ell}{2}-\lambda}e^{-\frac{\pi^{2}\left(\left\Vert \mathbf{f}\right\Vert ^{2}\right)}{Ny^{2}}}\,\mathrm{d} y
\\
&\ll_{N,h,\eta,\lambda}
\sum_{\mathbf{f}\in\left(\mathbb{N}^{+}\right)^{\lambda}}
\left\Vert \mathbf{f}\right\Vert ^{-\big(k+\tau\left(\mathbf{r},\mathfrak{D}\right)-\frac{h-\eta-\ell}{2}+\lambda\big)}
\int_{0}^{+\infty}v^{\frac12 \big(k+\tau\left(\mathbf{r},\mathfrak{D}\right)-\frac{h-\eta-\ell}{2}+\lambda-1\big)-1}e^{-v}\,\mathrm{d} v.
\end{align*}
Now, the integral is convergent if $k+\tau\left(\mathbf{r},\mathfrak{D}\right)-\frac{h-\eta-\ell}{2}+\lambda-1>0$,
which means $k>-\tau\left(\mathbf{r},\mathfrak{D}\right)+\frac{h-\eta-\ell}{2}-\lambda+1$
and the series is convergent if $k+\tau\left(\mathbf{r},\mathfrak{D}\right)-\frac{h-\eta-\ell}{2}+\lambda>\lambda$,
from the inequality of arithmetic and geometric means, and so $k>-\tau\left(\mathbf{r},\mathfrak{D}\right)+\frac{h-\eta-\ell}{2}$.
Since the inequalities must holds for all $1\leq\lambda\leq h-\eta$,
for all $0\leq\ell\leq\eta$ and for all $0\leq\eta\leq h-1$, we
can conclude that we can exchange all the series with the integral
if $k>-\tau\left(\mathbf{r},\mathfrak{D}\right)+\frac{h}{2}.$ Hence,
using (\ref{eq:bessel}) we can finally write
\begin{align*}
I_{1,2} & =\frac{1}{2\pi i}\sum_{\eta=0}^{h-1}\frac{\dbinom{h}{\eta}}{2^{\eta}}\sum_{\ell=0}^{\eta}\dbinom{\eta}{\ell}\pi^{\frac{h-\eta+\ell}{2}}\left(-1\right)^{\eta-\ell}
\frac1{\textbf{r}}\Gamma\left(\frac{1}{\textbf{r}}\right)
\sum_{\mathbf{f}\in\left(\mathbb{N}^{+}\right)^{h-\eta}}\int_{\left(1/N\right)}e^{Nz}z^{-k-1-\tau\left(\mathbf{r},\mathfrak{D}\right)-\frac{h-\eta+\ell}{2}}e^{-\frac{\pi^{2}\left\Vert \mathbf{f}\right\Vert ^{2}}{z}}\,\mathrm{d} z\\
& =\frac{N^{k+\tau\left(\mathbf{r},\mathfrak{D}\right)}}{2\pi i}\sum_{\eta=0}^{h-1}\frac{\dbinom{h}{\eta}}{2^{\eta}}\sum_{\ell=0}^{\eta}\dbinom{\eta}{\ell}\left(N\pi\right)^{\frac{h-\eta+\ell}{2}}\left(-1\right)^{\eta-\ell}
\frac1{\textbf{r}}\Gamma\left(\frac{1}{\textbf{r}}\right)
\sum_{\mathbf{f}\in\left(\mathbb{N}^{+}\right)^{h-\eta}}\int_{\left(1\right)}e^{u}u^{-k-1-\tau\left(\mathbf{r},\mathfrak{D}\right)-\frac{h-\eta+\ell}{2}}e^{-\frac{\pi^{2}\left\Vert \mathbf{f}\right\Vert ^{2}N}{u}}\,\mathrm{d} u\\
& =\frac{N^{\frac{k+\tau\left(\mathbf{r},\mathfrak{D}\right)}{2}}}{\pi^{k+\tau\left(\mathbf{r},\mathfrak{D}\right)}}\sum_{\eta=0}^{h-1}\frac{\dbinom{h}{\eta}}{2^{\eta}}\sum_{\ell=0}^{\eta}\dbinom{\eta}{\ell}N^{\frac{h-\eta+\ell}{4}}\left(-1\right)^{\eta-\ell}
\frac1{\textbf{r}}\Gamma\left(\frac{1}{\textbf{r}}\right)
\sum_{\mathbf{f}\in\left(\mathbb{N}^{+}\right)^{h-\eta}}\frac{J_{k+\tau\left(\mathbf{r},\mathfrak{D}\right)+\frac{h-\eta+\ell}{2}}\left(2\pi\sqrt{N}\left\Vert \mathbf{f}\right\Vert \right)}{\left\Vert \mathbf{f}\right\Vert ^{k+\tau\left(\mathbf{r},\mathfrak{D}\right)+\frac{h-\eta+\ell}{2}}}
\end{align*}
for $k>-\tau\left(\mathbf{r},\mathfrak{D}\right)+\frac{h}{2}.$ This term corresponds to $M_2$ in Theorem \ref{main_thm}.
\subsubsection{Evaluation of $I_{2}$}
As in the previous case, we split the integral into two pieces
\begin{align*}
I_{2} & =\frac{\left(-1\right)^{d}}{2\pi i}\int_{\left(1/N\right)}e^{Nz}z^{-k-1} \left(
\sum_{\bm{\rho}\in Z^{d}}\frac1{\textbf{r}}\Gamma\left(\frac{\bm{\rho}}{\textbf{r}}\right)
z^{-\tau(\bm{\rho},\textbf{r},\mathfrak{D})}
\right)\omega_{2}\left(z\right)^{h}\,\mathrm{d} z
\\
& =\frac{\left(-1\right)^{d}}{2\pi i}\sum_{\eta=0}^{h}\frac{\dbinom{h}{\eta}}{2^{\eta}}\sum_{\ell=0}^{\eta}\dbinom{\eta}{\ell}\pi^{\frac{h-\eta+\ell}{2}}\left(-1\right)^{\eta-\ell}\int_{\left(1/N\right)}e^{Nz}z^{-k-1-\frac{h-\eta+\ell}{2}}\left(
\sum_{\bm{\rho}\in Z^{d}}\frac1{\textbf{r}}\Gamma\left(\frac{\bm{\rho}}{\textbf{r}}\right)
z^{-\tau(\bm{\rho},\textbf{r},\mathfrak{D})}
\right)\omega_{2}\left(\frac{\pi^{2}}{z}\right)^{h-\eta}\,\mathrm{d} z
\\
& =\frac{\left(-1\right)^{d}}{2^{h+1}\pi i}\sum_{\ell=0}^{h}\dbinom{h}{\ell}\pi^{\frac{\ell}{2}}\left(-1\right)^{h-\ell}\int_{\left(1/N\right)}e^{Nz}z^{-k-1-\frac{\ell}{2}}\left(
\sum_{\bm{\rho}\in Z^{d}}\frac1{\textbf{r}}\Gamma\left(\frac{\bm{\rho}}{\textbf{r}}\right)
z^{-\tau(\bm{\rho},\textbf{r},\mathfrak{D})}
\right)\,\mathrm{d} z
\\
& +\frac{\left(-1\right)^{d}}{2\pi i}\sum_{\eta=0}^{h-1}\frac{\dbinom{h}{\eta}}{2^{\eta}}\sum_{\ell=0}^{\eta}\dbinom{\eta}{\ell}\pi^{\frac{h-\eta+\ell}{2}}\left(-1\right)^{\eta-\ell}\int_{\left(1/N\right)}e^{Nz}z^{-k-1-\frac{h-\eta+\ell}{2}}\left(
\sum_{\bm{\rho}\in Z^{d}}\frac1{\textbf{r}}\Gamma\left(\frac{\bm{\rho}}{\textbf{r}}\right)
z^{-\tau(\bm{\rho},\textbf{r},\mathfrak{D})}
\right)\omega_{2}\left(\frac{\pi^{2}}{z}\right)^{h-\eta}\,\mathrm{d} z\\
& =:I_{2,1}+I_{2,2}.
\end{align*}
Let us consider $I_{2,1}$ which corresponds to $M_3$ in Theorem \ref{main_thm}. We want to show that it is possible to
exchange the integral with the product of the series involving the
non-trivial zeros if the Riemann Zeta function. To prove this, we
fix an arbitrary $1\leq\lambda\leq d$ and we analyze
\[
\sum_{\rho_{1}}\frac{\left|\Gamma\left(\frac{\rho_{1}}{r_{1}}\right)\right|}{r_{1}}\cdots\sum_{\rho_{\lambda}}\frac{\left|\Gamma\left(\frac{\rho_{\lambda}}{r_{\lambda}}\right)\right|}{r_{\lambda}}\int_{\left(1/N\right)}\left|e^{Nz}\right|\left|z\right|^{-k-1-\frac{\ell}{2}}\left|z^{-\tau\left(\bm{\rho},\mathbf{r},\mathfrak{J}_{\lambda}\right)}\right|\prod_{s=\lambda+1}^{d}\left|\sum_{\rho_{s}}\frac{\Gamma\left(\frac{\rho_{s}}{r_{s}}\right)}{r_{s}}z^{-\frac{\rho_{s}}{r_{s}}}\right|\left|\,\mathrm{d} z\right|
\]
with the convention that, if $\lambda=d$, then $\prod_{s=\lambda+1}^{d}\left|\sum_{\rho_{s}}\frac{\Gamma\left(\frac{\rho_{s}}{r_{s}}\right)}{r_{s}}z^{-\frac{\rho_{s}}{r_{s}}}\right|=1$.
From Stirling formula (\ref{eq:Stirling}) and (\ref{eq:estserieszeros0})
we have that it is enough to study the convergence of
\begin{align*}
\sum_{\rho_{1}}\left|\gamma_{1}\right|^{\frac{\beta_{1}}{r_{1}}-\frac{1}{2}}\cdots\sum_{\rho_{\lambda}}\left|\gamma_{\lambda}\right|^{\frac{\beta_{\lambda}}{r_{\lambda}}-\frac{1}{2}}&\int_{\mathbb{R}}\left|z\right|^{-k-1-\frac{\ell}{2}-\tau\left(\bm{\beta},\mathbf{r},\mathfrak{J}_{\lambda}\right)}
\\
&\times\exp\left(\sum_{j=1}^{\lambda}\left(\frac{\gamma_{j}}{r_{j}}\arctan\left(Ny\right)-\frac{\pi\left|\gamma_{j}\right|}{2r_{j}}\right)\right)\prod_{s=\lambda+1}^{d}\left|\sum_{\rho_{s}}\frac{\Gamma\left(\frac{\rho_{s}}{r_{s}}\right)}{r_{s}}z^{-\frac{\rho_{s}}{r_{s}}}\right|\,\mathrm{d} y.
\end{align*}
We split the integral in $\left|y\right|\leq1/N$ and $\left|y\right|>1/N$.
Assume that $\left|y\right|\leq1/N$, then, by (\ref{eq:estserieszeros0}),
we have
\begin{align*}
\sum_{\rho_{1}}\left|\gamma_{1}\right|^{\frac{\beta_{1}}{r_{1}}-\frac{1}{2}}\cdots\sum_{\rho_{\lambda}}\left|\gamma_{\lambda}\right|^{\frac{\beta_{\lambda}}{r_{\lambda}}-\frac{1}{2}}
&\int_{-1/N}^{1/N}\left|z\right|^{-k-1-\frac{\ell}{2}}\left|z\right|^{-\tau\left(\bm{\beta},\mathbf{r},\mathfrak{J}_{\lambda}\right)}
\exp\left(\sum_{j=1}^{\lambda}\left(\frac{\gamma_{j}}{r_{j}}\arctan\left(Ny\right)-\frac{\pi\left|\gamma_{j}\right|}{2r_{j}}\right)\right)N^{d-\lambda}\,\mathrm{d} y
\\
&\ll_{N,\lambda,d}\sum_{\rho_{1}}\left|\gamma_{1}\right|^{\frac{\beta_{1}}{r_{1}}-\frac{1}{2}}\exp\left(-\frac{\pi\left|\gamma_{1}\right|}{4r_{1}}\right)\cdots\sum_{\rho_{\lambda}}\left|\gamma_{\lambda}\right|^{\frac{\beta_{\lambda}}{r_{\lambda}}-\frac{1}{2}}\exp\left(-\frac{\pi\left|\gamma_{\lambda}\right|}{4r_{\lambda}}\right)
\end{align*}
and the series trivially converges, so assume that $\left|y\right|>1/N.$
It is enough considering the case
\begin{align*}
\sum_{\rho_{1}}\left|\gamma_{1}\right|^{\frac{\beta_{1}}{r_{1}}-\frac{1}{2}}\cdots\sum_{\rho_{\lambda}}\left|\gamma_{\lambda}\right|^{\frac{\beta_{\lambda}}{r_{\lambda}}-\frac{1}{2}}
&\int_{\left|y\right|>1/N}\left|z\right|^{-k-1-\frac{\ell}{2}-\tau\left(\bm{\beta},\mathbf{r},\mathfrak{J}_{\lambda}\right)+\frac{\alpha}{2}}
\\
&\times\exp\left(\sum_{j=1}^{\lambda}\left(\frac{\gamma_{j}}{r_{j}}\arctan\left(Ny\right)-\frac{\pi\left|\gamma_{j}\right|}{2r_{j}}\right)\right)\log^{2\alpha}\left(2N\left|y\right|\right)\,\mathrm{d} y
\end{align*}
for $1\leq\alpha\leq d-\lambda$, since the powers of $N$ do not
affect the study of the convergence and so can be omitted. Assume
$y>1/N$ and $\gamma_{j}>0,\,j=1,\dots,\lambda$. Putting $Ny=u$
and using the well-known identity $\arctan(x)-\frac{\pi}{2}=-\arctan\left(\frac{1}{x}\right)$
we get
\[
\sum_{\rho_{1}:\,\gamma_{1}>0}\gamma_{1}^{\frac{\beta_{1}}{r_{1}}-\frac{1}{2}}\cdots\sum_{\rho_{\lambda}:\,\gamma_{\lambda}>0}\gamma_{\lambda}^{\frac{\beta_{\lambda}}{r_{\lambda}}-\frac{1}{2}}
\int_{1}^{+\infty}u^{-k-1-\frac{\ell}{2}-\tau\left(\bm{\beta},\mathbf{r},\mathfrak{J}_{\lambda}\right)+\frac{\alpha}{2}}\exp\left(-\arctan\left(\frac{1}{u}\right)\tau\left(\bm{\gamma},\mathbf{r},\mathfrak{J}_{\lambda}\right)\right)\log^{2\alpha}\left(2u\right)\,\mathrm{d} u
\]
and, by Lemma \ref{lem:generalized1}, we have the convergence if
$k>\frac{\lambda+\alpha-\ell}{2}$, and since this inequality must
be holds for all $0\leq\ell\leq d$ and all $1\leq\alpha\leq d-\lambda$
we can conclude that $k>\frac{d}{2}.$ Now, fix $1\leq\eta\leq\lambda$
and assume that $\gamma_{1},\dots,\gamma_{\eta}>0$ and $\gamma_{\eta+1},\dots,\gamma_{\lambda}<0$.
In this case, recalling that $y>1/N$ and so $\frac{\gamma_{j}}{r_{j}}\arctan\left(Ny\right)-\frac{\pi\left|\gamma_{j}\right|}{2r_{j}}\leq-\frac{\pi\left|\gamma_{j}\right|}{2r_{j}}$
for $j>\eta$, we have to work with
\begin{align*}
\sum_{\rho_{1}:\,\gamma_{1}>0}\gamma_{1}^{\frac{\beta_{1}}{r_{1}}-\frac{1}{2}}\cdots&\sum_{\rho_{\eta}:\,\gamma_{\eta}>0}\gamma_{\eta}^{\frac{\beta_{\eta}}{r_{\eta}}-\frac{1}{2}}\sum_{\rho_{\eta+1}:\,\gamma_{\eta+1}<0}\left|\gamma_{\eta+1}\right|^{\frac{\beta_{\eta+1}}{r_{\eta+1}}-\frac{1}{2}}\exp\left(-\frac{\pi\left|\gamma_{\eta+1}\right|}{2r_{\eta+1}}\right)\cdots\sum_{\rho_{\lambda}:\,\gamma_{\lambda}<0}\left|\gamma_{\lambda}\right|^{\frac{\beta_{\lambda}}{r_{\lambda}}-\frac{1}{2}}\exp\left(-\frac{\pi\left|\gamma_{\lambda}\right|}{2r_{\lambda}}\right)
\\
&\times\int_{y>1/N}y^{-k-1-\frac{\ell}{2}-\tau\left(\bm{\beta},\mathbf{r},\mathfrak{J}_{\lambda}\right)+\frac{\alpha}{2}}\exp\left(\sum_{j=1}^{\eta}\left(\frac{\gamma_{j}}{r_{j}}\arctan\left(Ny\right)-\frac{\pi\gamma_{j}}{2r_{j}}\right)\right)\log^{2\alpha}\left(2Ny\right)\,\mathrm{d} y
\end{align*}
with $1\leq\alpha\leq d-\lambda$. Letting $Ny=u$, we note that we
have to deal with
\begin{align*}
\sum_{\rho_{1}:\,\gamma_{1}>0}\gamma_{1}^{\frac{\beta_{1}}{r_{1}}-\frac{1}{2}}\cdots&\sum_{\rho_{\eta}:\,\gamma_{\eta}>0}\gamma_{\eta}^{\frac{\beta_{\eta}}{r_{\eta}}-\frac{1}{2}}\sum_{\rho_{\eta+1}:\,\gamma_{\eta+1}<0}\left|\gamma_{\eta+1}\right|^{\frac{\beta_{\eta+1}}{r_{\eta+1}}-\frac{1}{2}}\exp\left(-\frac{\pi\left|\gamma_{\eta+1}\right|}{2r_{\eta+1}}\right)\cdots\sum_{\rho_{\lambda}:\,\gamma_{\lambda}<0}\left|\gamma_{\lambda}\right|^{\frac{\beta_{\lambda}}{r_{\lambda}}-\frac{1}{2}}\exp\left(-\frac{\pi\left|\gamma_{\lambda}\right|}{2r_{\lambda}}\right)
\\
&\times\int_{1}^{+\infty}u^{-k-1-\frac{\ell}{2}-\tau\left(\bm{\beta},\mathbf{r},\mathfrak{J}_{\lambda}\right)+\frac{\alpha}{2}}
\exp\left(-\arctan\left(\frac{1}{u}\right)\tau\left(\bm{\gamma},\mathbf{r},\mathfrak{J}_{\eta}\right)\right)\log^{2\alpha}\left(2u\right)\,\mathrm{d} u
\end{align*}
(also in this case we omit the powers of $N$ because they do not
affect the convergence) and, since
\[
\left(\frac{1}{u}\right)^{\frac{\beta_{j}}{r_{j}}}<1,\,u>1,\,j=1,\dots,\lambda,
\]
it is enough to consider
\[
\sum_{\rho_{1}:\,\gamma_{1}>0}\gamma_{1}^{\frac{\beta_{1}}{r_{1}}-\frac{1}{2}}\cdots\sum_{\rho_{\eta}:\,\gamma_{\eta}>0}\gamma_{\eta}^{\frac{\beta_{\eta}}{r_{\eta}}-\frac{1}{2}}\int_{1}^{+\infty}u^{-k-1-\frac{\ell}{2}-\tau\left(\bm{\beta},\mathbf{r},\mathfrak{J}_{\eta}\right)+\frac{\alpha}{2}}\exp\left(-\arctan\left(\frac{1}{u}\right)\tau\left(\bm{\gamma},\mathbf{r},\mathfrak{J}_{\eta}\right)\right)\log^{2\alpha}\left(2u\right)\,\mathrm{d} u
\]
and so, arguing as in the previous case, the convergence if $k>\frac{\eta+d-\lambda}{2}$
and so, since $\eta\leq\lambda$, a complete convergence in the case
$k>\frac{d}{2}$. If $y<-1/N$ we get the same bounds for $k$, by
symmetry.
Now, since $\left|\mathfrak{D}\right|=d$, we have
\begin{align*}
I_{2,1}=&\frac{\left(-1\right)^{d}}{2^{h+1}\pi i}\sum_{\ell=0}^{h}\dbinom{h}{\ell}\pi^{\frac{\ell}{2}}\left(-1\right)^{h-\ell}\sum_{\bm{\rho}\in Z^{d}}\frac1{\textbf{r}}\Gamma\left(\frac{\bm{\rho}}{\textbf{r}}\right)\int_{\left(1/N\right)}e^{Nz}z^{-k-1-\frac{\ell}{2}-\tau\left(\bm{\rho},\mathbf{r},\mathfrak{D}\right)}\,\mathrm{d} z
\\
=&\frac{N^{k}\left(-1\right)^{d}}{2^{h}}\sum_{\ell=0}^{h}\dbinom{h}{\ell}\left(N\pi\right)^{\frac{\ell}{2}}\left(-1\right)^{h-\ell}\sum_{\bm{\rho}\in Z^{d}}
\frac1{\textbf{r}}\Gamma\left(\frac{\bm{\rho}}{\textbf{r}}\right)
\frac{N^{\tau\left(\bm{\rho},\mathbf{r},\mathfrak{D}\right)}}{\Gamma\left(k+1+\frac{\ell}{2}+\tau\left(\bm{\rho},\mathbf{r},\mathfrak{D}\right)\right)},
\end{align*}
from \eqref{eq:onemaintool}.
Now, we analyze $I_{2,2}$ (which corresponds to $M_4$ in Theorem \ref{main_thm}) and we prove that we can switch the integral
with the series involving the non-trivial zeros of the Riemann Zeta
function and with the powers of $\omega_{2}\left(\frac{\pi^{2}}{z}\right)$.
As the previous calculations, we fix $1\leq\lambda\leq d$ and $1\leq\alpha\leq h-\eta$.
So we have to consider
\begin{align*}
\sum_{\rho_{1}}\frac{\left|\Gamma\left(\frac{\rho_{1}}{r_{1}}\right)\right|}{r_{1}}\cdots\sum_{\rho_{\lambda}}\frac{\left|\Gamma\left(\frac{\rho_{\lambda}}{r_{\lambda}}\right)\right|}{r_{\lambda}}
&\sum_{\mathbf{f}\in\left(\mathbb{N}^{+}\right)^{\alpha}}\int_{\left(1/N\right)}\left|e^{Nz}\right|e^{-\mathrm{Re}\left(\frac{\pi^{2}}{z}\right)\left\Vert \mathbf{f}\right\Vert ^{2}}\left|z\right|^{-k-1-\frac{h-\eta+\ell}{2}}\left|z^{-\tau\left(\bm{\rho},\mathbf{r},\mathfrak{J}_{\lambda}\right)}\right|
\\
&\times\prod_{s=\lambda+1}^{d}\left|\sum_{\rho_{s}}\frac{\Gamma\left(\frac{\rho_{s}}{r_{s}}\right)}{r_{s}}z^{-\frac{\rho_{s}}{r_{s}}}\right|\left|\omega_{2}\left(\frac{\pi^{2}}{z}\right)\right|^{h-\eta-\alpha}\left|\,\mathrm{d} z\right|
\end{align*}
again with the convention $\prod_{s=\lambda+1}^{d}\left|\sum_{\rho_{s}}\frac{\Gamma\left(\frac{\rho_{s}}{r_{s}}\right)}{r_{s}}z^{-\frac{\rho_{s}}{r_{s}}}\right|=1$
if $\lambda=d$. From (\ref{eq:Stirling}) and recalling that the powers of $N$ do not affect the convergence, it is enough to study the convergence of
\begin{align*}
\sum_{\rho_{1}}\left|\gamma_{1}\right|^{\frac{\beta_{1}}{r_{1}}-\frac{1}{2}}\cdots&\sum_{\rho_{\lambda}}\left|\gamma_{\lambda}\right|^{\frac{\beta_{\lambda}}{r_{\lambda}}-\frac{1}{2}}
\sum_{\mathbf{f}\in\left(\mathbb{N}^{+}\right)^{\alpha}}\int_{\mathbb{R}}\left|z\right|^{-k-1-\frac{h-\eta+\ell}{2}}\left|z\right|^{-\tau\left(\bm{\beta},\mathbf{r},\mathfrak{J}_{\lambda}\right)}e^{-\frac{\left\Vert \mathbf{f}\right\Vert ^{2}N}{1+N^{2}y^{2}}}
\\
&\times\exp\left(\sum_{j=1}^{\lambda}\left(\frac{\gamma_{j}}{r_{j}}\arctan\left(Ny\right)-\frac{\pi\left|\gamma_{j}\right|}{2r_{j}}\right)\right)\prod_{s=\lambda+1}^{d}\left|\sum_{\rho_{s}}\frac{\Gamma\left(\frac{\rho_{s}}{r_{s}}\right)}{r_{s}}z^{-\frac{\rho_{s}}{r_{s}}}\right|\left(\frac{1+y^{2}N^{2}}{N}\right)^{\frac{h-\eta-\alpha}{2}}\,\mathrm{d} y.
\end{align*}
If $\left|y\right|\leq1/N$ we have
\begin{align*}
\sum_{\rho_{1}}\left|\gamma_{1}\right|^{\frac{\beta_{1}}{r_{1}}-\frac{1}{2}}\cdots&\sum_{\rho_{\lambda}}\left|\gamma_{\lambda}\right|^{\frac{\beta_{\lambda}}{r_{\lambda}}-\frac{1}{2}}\sum_{\mathbf{f}\in\left(\mathbb{N}^{+}\right)^{\alpha}}\int_{\mathbb{-}1/N}^{1/N}\left|z\right|^{-k-1-\frac{h-\eta+\ell}{2}}\left|z\right|^{-\tau\left(\bm{\beta},\mathbf{r},\mathfrak{J}_{\lambda}\right)}e^{-\frac{\left\Vert \mathbf{f}\right\Vert ^{2}N}{1+N^{2}y^{2}}}
\\
&\times\exp\left(\sum_{j=1}^{\lambda}\left(\frac{\gamma_{j}}{r_{j}}\arctan\left(Ny\right)-\frac{\pi\left|\gamma_{j}\right|}{2r_{j}}\right)\right)\prod_{s=\lambda+1}^{d}\left|\sum_{\rho_{s}}\frac{\Gamma\left(\frac{\rho_{s}}{r_{s}}\right)}{r_{s}}z^{-\frac{\rho_{s}}{r_{s}}}\right|\left(\frac{1+y^{2}N^{2}}{N}\right)^{\frac{h-\eta-\alpha}{2}}\,\mathrm{d} y
\\
\ll_{N,k,\alpha,h,\eta,\ell,\mathbf{r}}&\sum_{\rho_{1}}\left|\gamma_{1}\right|^{\frac{\beta_{1}}{r_{1}}-\frac{1}{2}}\exp\left(-\frac{\pi\left|\gamma_{1}\right|}{4r_{1}}\right)\cdots\sum_{\rho_{\lambda}}\left|\gamma_{\lambda}\right|^{\frac{\beta_{\lambda}}{r_{\lambda}}-\frac{1}{2}}\exp\left(-\frac{\pi\left|\gamma_{1}\right|}{4r_{1}}\right)\sum_{\mathbf{f}\in\left(\mathbb{N}^{+}\right)^{\alpha}}e^{-\left\Vert \mathbf{f}\right\Vert ^{2}N}
\end{align*}
and trivially the convergence, so we consider now $\left|y\right|>1/N$.
If we fix $1\leq\mu\leq d-\lambda$, from (\ref{eq:estserieszeros0}),
it is enough to work with
\begin{align*}
\sum_{\rho_{1}}\left|\gamma_{1}\right|^{\frac{\beta_{1}}{r_{1}}-\frac{1}{2}}\cdots&\sum_{\rho_{\lambda}}\left|\gamma_{\lambda}\right|^{\frac{\beta_{\lambda}}{r_{\lambda}}-\frac{1}{2}}
\sum_{\mathbf{f}\in\left(\mathbb{N}^{+}\right)^{\alpha}}
\int_{\left|y\right|>1/N}\left|z\right|^{-k-1-\frac{h-\eta+\ell}{2}+\frac{\mu}{2}}\left|z\right|^{-\tau\left(\bm{\beta},\mathbf{r},\mathfrak{J}_{\lambda}\right)}e^{-\frac{\left\Vert \mathbf{f}\right\Vert ^{2}}{Ny^{2}}}
\\
&\times\exp\left(\sum_{j=1}^{\lambda}\left(\frac{\gamma_{j}}{r_{j}}\arctan\left(Ny\right)-\frac{\pi\left|\gamma_{j}\right|}{2r_{j}}\right)\right)\log^{2\mu}\left(2N\left|y\right|\right)\left|y\right|^{h-\eta-\alpha}\,\mathrm{d} y.
\end{align*}
Assume $y>1/N$ and $\gamma_{j}>0,\,j=1,\dots,\lambda$. Since $\arctan\left(\frac{1}{Ny}\right)\gg\frac{1}{Ny}$,
We have
\begin{align*}
\sum_{\rho_{1}:\,\gamma_{1}>0}\gamma_{1}^{\frac{\beta_{1}}{r_{1}}-\frac{1}{2}}\cdots&\sum_{\rho_{\lambda}:\,\gamma_{\lambda}>0}\gamma_{\lambda}^{\frac{\beta_{\lambda}}{r_{\lambda}}-\frac{1}{2}}
\sum_{\mathbf{f}\in\left(\mathbb{N}^{+}\right)^{\alpha}}
\int_{1/N}^{+\infty}y^{-k-1-\frac{h-\eta+\ell}{2}-\tau\left(\bm{\beta},\mathbf{r},\mathfrak{J}_{\lambda}\right)+\frac{\mu}{2}}e^{-\frac{\left\Vert \mathbf{f}\right\Vert ^{2}}{Ny^{2}}}
\\
&\times\exp\left(-\frac{\tau\left(\bm{\gamma},\mathbf{r},\mathfrak{J}_{\lambda}\right)}{Ny}\right)\log^{2\mu}\left(2Ny\right)y^{h-\eta-\alpha}\,\mathrm{d} y.
\end{align*}
Putting $v=\frac{\sum_{j=1}^{\lambda}\frac{\gamma_{j}}{r_{j}}}{Ny},$
we obtain
\begin{align*}
\frac{\sum_{\rho_{1}:\,\gamma_{1}>0}\gamma_{1}^{\frac{\beta_{1}}{r_{1}}-\frac{1}{2}}\cdots\sum_{\rho_{\lambda}:\,\gamma_{\lambda}>0}\gamma_{\lambda}^{\frac{\beta_{\lambda}}{r_{\lambda}}-\frac{1}{2}}}{\left(\tau\left(\bm{\gamma},\mathbf{r},\mathfrak{J}_{\lambda}\right)\right)^{k+\frac{h-\eta+\ell}{2}+\tau\left(\bm{\beta},\mathbf{r},\mathfrak{J}_{\lambda}\right)-h+\eta+\alpha-\frac{\mu}{2}}}
&\sum_{\mathbf{f}\in\left(\mathbb{N}^{+}\right)^{\alpha}}
\int_{0}^{\tau\left(\bm{\gamma},\mathbf{r},\mathfrak{J}_{\lambda}\right)}v^{k-1+\frac{h-\eta+\ell}{2}+\tau\left(\bm{\beta},\mathbf{r},\mathfrak{J}_{\lambda}\right)-\frac{\mu}{2}}
\\
&\times e^{-\frac{\left\Vert \mathbf{f}\right\Vert ^{2}Nv^{2}}{\tau\left(\bm{\beta},\mathbf{r},\mathfrak{J}_{\gamma}\right)^{2}}}\exp\left(-v\right)\log^{2\mu}\left(2\frac{\tau\left(\bm{\gamma},\mathbf{r},\mathfrak{J}_{\lambda}\right)}{v}\right)v^{-h+\eta+\alpha}\,\mathrm{d} v.
\end{align*}
Now, from (\ref{eq:stima semplice}) and by elementary manipulations
we can study
\begin{align*}
\frac{\sum_{\rho_{1}:\,\gamma_{1}>0}\gamma_{1}^{-\frac{1}{2}}\cdots\sum_{\rho_{\lambda}:\,\gamma_{\lambda}>0}\gamma_{\lambda}^{-\frac{1}{2}}}{\left(\tau\left(\bm{\gamma},\mathbf{r},\mathfrak{J}_{\lambda}\right)\right)^{k-\frac{h-\eta}{2}+\frac{\ell}{2}+\alpha-\frac{\mu}{2}}}
&\sum_{\mathbf{f}\in\left(\mathbb{N}^{+}\right)^{\alpha}}
\int_{0}^{\tau\left(\bm{\gamma},\mathbf{r},\mathfrak{J}_{\lambda}\right)}v^{k-1-\frac{h-\eta}{2}+\frac{\ell}{2}+\tau\left(\bm{\beta},\mathbf{r},\mathfrak{J}_{\lambda}\right)+\alpha-\frac{\mu}{2}}
\\
&\times e^{-\frac{\left\Vert \mathbf{f}\right\Vert ^{2}Nv^{2}}{\tau\left(\bm{\gamma},\mathbf{r},\mathfrak{J}_{\lambda}\right)^{2}}}\exp\left(-v\right)\log^{2\nu}\left(2\frac{\tau\left(\bm{\gamma},\mathbf{r},\mathfrak{J}_{\lambda}\right)}{v}\right)\,\mathrm{d} v
\end{align*}
and so, by Lemma \ref{lem:generalizlemnew}, we have the convergence
if $k>\frac{\lambda}{2}+\frac{h-\eta-\ell+\mu}{2}.$ Since $\mu\leq d-\lambda$
and $0\leq\ell\leq\eta$ we have the complete convergence for all
possible cases if $k>\frac{d+h}{2}.$
Now fix $1\leq\xi\leq\lambda$ and assume that $\gamma_{1},\dots,\gamma_{\xi}>0$
and $\gamma_{\xi+1},\dots,\gamma_{\lambda}<0$. In this case, recalling
that $y>1/N,$we have to work with
\begin{align*}
&\sum_{\rho_{1}:\,\gamma_{1}>0}\gamma_{1}^{\frac{\beta_{1}}{r_{1}}-\frac{1}{2}}\cdots\sum_{\rho_{\xi}:\,\gamma_{\xi}>0}\gamma_{\xi}^{\frac{\beta_{\xi}}{r_{\xi}}-\frac{1}{2}}\sum_{\rho_{\xi+1}:\,\gamma_{\xi+1}<0}\left|\gamma_{\xi+1}\right|^{\frac{\beta_{\xi+1}}{r_{\xi+1}}-\frac{1}{2}}\cdots\sum_{\rho_{\lambda}:\,\gamma_{\lambda}<0}\left|\gamma_{\lambda}\right|^{\frac{\beta_{\lambda}}{r_{\lambda}}-\frac{1}{2}}
\\
&\times\sum_{\mathbf{m}\in\left(\mathbb{N}^{+}\right)^{\alpha}}
\int_{1/N}^{+\infty}y^{-k-1-\frac{h-\eta+\ell}{2}-\tau\left(\bm{\beta},\mathbf{r},\mathfrak{J}_{\lambda}\right)+\frac{\mu}{2}}e^{-\frac{\left\Vert \mathbf{m}\right\Vert ^{2}}{Ny^{2}}}
\exp\left(\sum_{j=1}^{\lambda}\left(\frac{\gamma_{j}}{r_{j}}\arctan\left(Ny\right)-\frac{\pi\left|\gamma_{j}\right|}{2r_{j}}\right)\right)\log^{2\mu}\left(2Ny\right)y^{h-\eta-\alpha}\,\mathrm{d} y.
\end{align*}
Now, since $y>1/N$, if $\gamma_{j}<0$, we observe that $\frac{\gamma_{j}}{r_{j}}\arctan\left(Ny\right)-\frac{\pi\left|\gamma_{j}\right|}{2r_{j}}\leq-\frac{\pi\left|\gamma_{j}\right|}{2r_{j}}$,
$y^{-\beta_{j}/r_{j}}\leq N^{\beta_{j}/r_{j}}\leq N^{1/r_{j}}$ and
\[
\sum_{\rho_{j}:\,\gamma_{j}<0}\left|\gamma_{j}\right|^{\frac{\beta_{j}}{r_{j}}-\frac{1}{2}}\exp\left(-\frac{\pi\left|\gamma_{j}\right|}{2r_{j}}\right)
\]
trivially converges, so it is enough to consider
\begin{align*}
\sum_{\rho_{1}:\,\gamma_{1}>0}\gamma_{1}^{\frac{\beta_{1}}{r_{1}}-\frac{1}{2}}&\cdots\sum_{\rho_{\xi}:\,\gamma_{\xi}>0}\gamma_{\xi}^{\frac{\beta_{\xi}}{r_{\xi}}-\frac{1}{2}}
\sum_{\mathbf{f}\in\left(\mathbb{N}^{+}\right)^{\alpha}}
\int_{1/N}^{+\infty}y^{-k-1-\frac{h-\eta+\ell}{2}-\tau\left(\bm{\beta},\mathbf{r},\mathfrak{J}_{\xi}\right)+\frac{\mu}{2}}e^{-\frac{\left\Vert \mathbf{f}\right\Vert ^{2}}{Ny^{2}}}
\\
&\times\exp\left(\sum_{j=1}^{\xi}\left(\frac{\gamma_{j}}{r_{j}}\arctan\left(Ny\right)-\frac{\pi\gamma_{j}}{2r_{j}}\right)\right)\log^{2\mu}\left(2Ny\right)y^{h-\eta-\alpha}\,\mathrm{d} y
\end{align*}
and so, following the previous case, we have the convergence if $k>\frac{h+d-\lambda+\xi}{2}$
and since $\xi\leq\lambda$, we have the complete convergence if $k>\frac{d+h}{2}$.
If $y<-1/N$ we have the same bounds, by the symmetry of the non-trivial
zeros of the Riemann Zeta function, so we can finally exchange the
integral with the series and get
\[
I_{2,2}=\frac{\left(-1\right)^{d}}{2\pi i}\sum_{\eta=0}^{h-1}\frac{\dbinom{h}{\eta}}{2^{\eta}}\sum_{\ell=0}^{\eta}\dbinom{\eta}{\ell}\pi^{\frac{h-\eta+\ell}{2}}\left(-1\right)^{\eta-\ell}
\sum_{\bm{\rho}\in Z^{d}}\frac1{\textbf{r}}\Gamma\left(\frac{\bm{\rho}}{\textbf{r}}\right)
\sum_{\mathbf{f}\in\left(\mathbb{N}^{+}\right)^{h-\eta}}
\int_{\left(1/N\right)}e^{Nz}z^{-k-1-\frac{h-\eta+\ell}{2}-\tau\left(\bm{\rho},\mathbf{r},\mathfrak{D}\right)}e^{-\frac{\pi^{2}\left\Vert \mathbf{f}\right\Vert ^{2}}{z}}\,\mathrm{d} z
\]
which is, taking $Nz=v$,
\[
\frac{\left(-1\right)^{d}}{2\pi i}\sum_{\eta=0}^{h-1}\frac{\dbinom{h}{\eta}}{2^{\eta}}\sum_{\ell=0}^{\eta}\dbinom{\eta}{\ell}\pi^{\frac{h-\eta+\ell}{2}}\left(-1\right)^{\eta-\ell}
\sum_{\bm{\rho}\in Z^{d}}\frac1{\textbf{r}}\Gamma\left(\frac{\bm{\rho}}{\textbf{r}}\right)
N^{k+\frac{h-\eta+\ell}{2}+\tau\left(\bm{\rho},\mathbf{r},\mathfrak{D}\right)}\sum_{\mathbf{f}\in\left(\mathbb{N}^{+}\right)^{h-\eta}}\int_{\left(1\right)}z^{-k-1-\frac{h-\eta+\ell}{2}-\tau\left(\bm{\rho},\mathbf{r},\mathfrak{D}\right)}e^{v-\frac{\pi^{2}N\left\Vert \mathbf{f}\right\Vert }{v}}\,\mathrm{d} v
\]
\[
=\frac{N^{k/2}\left(-1\right)^{d}}{\pi^{k}}\sum_{\eta=0}^{h-1}\frac{\dbinom{h}{\eta}}{2^{\eta}}\sum_{\ell=0}^{\eta}\dbinom{\eta}{\ell}\left(-1\right)^{\eta-\ell}
\sum_{\bm{\rho}\in Z^{d}}\frac1{\textbf{r}}\Gamma\left(\frac{\bm{\rho}}{\textbf{r}}\right)
\frac{N^{\frac{h-\eta+\ell}{4}+\tau\left(\bm{\rho},\mathbf{r},\mathfrak{D}\right)/2}}{\pi^{\tau\left(\bm{\rho},\mathbf{r},\mathfrak{D}\right)}}
\sum_{\mathbf{f}\in\left(\mathbb{N}^{+}\right)^{h-\eta}}\frac{J_{k+\frac{h-\eta+\ell}{2}+\tau\left(\bm{\rho},\mathbf{r},\mathfrak{D}\right)}\left(2\pi\sqrt{N}\left\Vert \mathbf{f}\right\Vert \right)}{\left\Vert \mathbf{f}\right\Vert ^{k+\frac{h-\eta+\ell}{2}+\tau\left(\bm{\rho},\mathbf{r},\mathfrak{D}\right)}}.
\]
\section{Evaluation of $I_{3}$}
In this section we evaluate
\[
I_{3}=\frac{1}{2\pi i}\sum_{\underset{{\scriptstyle \left|I\right|\geq1}}{I\subseteq\mathfrak{D}}}\left(-1\right)^{\left|\mathfrak{D}\setminus I\right|}\int_{\left(1/N\right)}e^{Nz}z^{-k-1-\tau\left(\mathbf{r},I\right)} \left(
\sum_{\bm{\rho}\in Z^{|\mathfrak{D}\setminus I|}}
\frac1{\textbf{r}}\Gamma\left(\frac{\bm{\rho}}{\textbf{r}}\right)
z^{-\tau\left(\bm{\rho},\mathbf{r},\mathfrak{D}\setminus I\right)}
\right)\omega_{2}\left(z\right)^{h}\,\mathrm{d} z
\]
which corresponds to $M_5$ in Theorem \ref{main_thm} and has a similar structure to $I_{2}$. If we fix $I\subseteq\mathfrak{D}$
we can repeat the previous argument to justify the exchange the integral
with the series only considering $k+\tau\left(\mathbf{r},I\right)$
instead of $k$ and $\left|\mathfrak{D}\setminus I\right|$ instead
of $d=\left|\mathfrak{D}\right|$. So, we can conclude that all exchanges
are justified if $k>\frac{\left|\mathfrak{D}\setminus I\right|+h}{2}-\tau\left(\mathbf{r},I\right)$,
and so we have
\begin{align*}
I_{3}=&\frac{1}{2\pi i}\sum_{\underset{{\scriptstyle \left|I\right|\geq1}}{I\subseteq\mathfrak{D}}}\left(-1\right)^{\left|\mathfrak{D}\setminus I\right|}\sum_{\eta=0}^{h}\frac{\dbinom{h}{\eta}}{2^{\eta}}\sum_{\ell=0}^{\eta}\dbinom{\eta}{\ell}\left(-1\right)^{\eta-\ell}\pi^{\frac{h-\eta+\ell}{2}}
\\
&\times
\sum_{\bm{\rho}\in Z^{|\mathfrak{D}\setminus I|}}
\frac1{\textbf{r}}\Gamma\left(\frac{\bm{\rho}}{\textbf{r}}\right)
\int_{\left(1/N\right)}e^{Nz}z^{-k-1-\tau\left(\mathbf{r},I\right)-\frac{h-\eta+\ell}{2}-\tau\left(\bm{\rho},\mathbf{r},\mathfrak{D}\right)}\omega_{2}\left(\frac{\pi^{2}}{z}\right)^{h-\eta}\,\mathrm{d} z
\\
=&\frac{1}{2\pi i}\sum_{\underset{{\scriptstyle \left|I\right|\geq1}}{I\subseteq\mathfrak{D}}}\left(-1\right)^{\left|\mathfrak{D}\setminus I\right|}\sum_{\eta=0}^{h}\frac{\dbinom{h}{\eta}}{2^{\eta}}\sum_{\ell=0}^{\eta}\dbinom{\eta}{\ell}\left(-1\right)^{\eta-\ell}\pi^{\frac{h-\eta+\ell}{2}}
\\
&\times
\sum_{\bm{\rho}\in Z^{|\mathfrak{D}\setminus I|}}
\frac1{\textbf{r}}\Gamma\left(\frac{\bm{\rho}}{\textbf{r}}\right)
\sum_{\mathbf{f}\in\left(\mathbb{N}^{+}\right)^{h-\eta}}\int_{\left(1/N\right)}e^{Nz}z^{-k-1-\tau\left(\mathbf{r},I\right)-\frac{h-\eta+\ell}{2}-\tau\left(\bm{\rho},\mathbf{r},\mathfrak{D}\right)}e^{-\frac{\pi^{2}\left\Vert \mathbf{f}\right\Vert ^{2}}{z}}\,\mathrm{d} z
\end{align*}
and so, taking $Nz=v$ and using (\ref{eq:bessel}), we have that
\begin{align*}
I_{3}=&\frac{N^{k}}{2\pi i}\sum_{\underset{{\scriptstyle \left|I\right|\geq1}}{I\subseteq\mathfrak{D}}}N^{\tau\left(\mathbf{r},I\right)}\left(-1\right)^{\left|\mathfrak{D}\setminus I\right|}\sum_{\eta=0}^{h}\frac{\dbinom{h}{\eta}}{2^{\eta}}\sum_{\ell=0}^{\eta}\dbinom{\eta}{\ell}\left(-1\right)^{\eta-\ell}\pi^{\frac{h-\eta+\ell}{2}}
\\
&\times
\sum_{\bm{\rho}\in Z^{|\mathfrak{D}\setminus I|}}
\frac1{\textbf{r}}\Gamma\left(\frac{\bm{\rho}}{\textbf{r}}\right)
N^{\frac{h-\eta+\ell}{2}-\tau\left(\bm{\rho},\mathbf{r},\mathfrak{D}\setminus I\right)}\sum_{\mathbf{f}\in\left(\mathbb{N}^{+}\right)^{h-\eta}}\int_{\left(1\right)}e^{v}v^{-k-1-\tau\left(\mathbf{r},I\right)-\frac{h-\eta+\ell}{2}-\tau\left(\bm{\rho},\mathbf{r},\mathfrak{D}\right)}e^{-\frac{\pi^{2}N\left\Vert \mathbf{f}\right\Vert ^{2}}{v}}\,\mathrm{d} v
\\
=&\frac{N^{k/2}}{\pi^{k}}\sum_{\underset{{\scriptstyle \left|I\right|\geq1}}{I\subseteq\mathfrak{D}}}N^{\tau\left(\mathbf{r},I\right)/2}\left(-1\right)^{\left|\mathfrak{D}\setminus I\right|}\sum_{\eta=0}^{h}\frac{\dbinom{h}{\eta}}{2^{\eta}}\sum_{\ell=0}^{\eta}\dbinom{\eta}{\ell}\left(-1\right)^{\eta-\ell}
\\
&\times
\sum_{\bm{\rho}\in Z^{|\mathfrak{D}\setminus I|}}
\frac1{\textbf{r}}\Gamma\left(\frac{\bm{\rho}}{\textbf{r}}\right)
\frac{N^{\frac{h-\eta+\ell}{4}+\tau\left(\bm{\rho},\mathbf{r},\mathfrak{D}\setminus I\right)/2}}{\pi^{\tau\left(\bm{\rho},\mathbf{r},\mathfrak{D}\setminus I\right)}}
\sum_{\mathbf{f}\in\left(\mathbb{N}^{+}\right)^{h-\eta}}\frac{J_{k+\tau\left(\mathbf{r},I\right)+\frac{h-\eta+\ell}{2}+\tau\left(\bm{\rho},\mathbf{r},\mathfrak{D}\setminus I\right)}\left(2\pi\sqrt{N}\left\Vert \mathbf{f}\right\Vert \right)}{\left\Vert \mathbf{f}\right\Vert ^{k+\tau\left(\mathbf{r},I\right)+\frac{h-\eta+\ell}{2}+\tau\left(\bm{\rho},\mathbf{r},\mathfrak{D}\setminus I\right)}}
\end{align*}
and this completes the proof.
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 39
|
Martin Austin Fido (18 October 1939 – 2 April 2019) was a university professor, true crime writer and broadcaster. His many books include The Crimes, Detection and Death of Jack the Ripper, The Official Encyclopedia of Scotland Yard, Serial Killers, and The Murder Guide to London. He is also one of the authors of The complete Jack the Ripper A to Z.
After leaving Balliol College, Oxford, in 1966, where he had been a junior research fellow in English, he went to the University of Leeds where he lectured in English until 1973. In 1971 he went to Michigan State University in the USA where he was a visiting associate professor for one year, and in 1973 he became a reader in English Literature and head of the English department at the University of the West Indies, Cave Hill Campus, Barbados. In the West Indies he was active in theatre and educational broadcasting.
In 1983 he returned to England and became a freelance writer and broadcaster, specialising in true crime. He broadcast a weekly radio programme called Murder After Midnight on London's LBC Radio from 1987 to 2001, some of which were produced and released commercially on cassette and CD by his friend (and fellow LBC broadcaster) Paul Savory. Aside from his many true crime books he has also written illustrated biographies of Charles Dickens, William Shakespeare, Rudyard Kipling and Oscar Wilde, and books on Agatha Christie, and Sherlock Holmes. He translated Louis Cazamian's Le Roman Social en Angleterre, and his play Let's Go Bajan! was performed successfully in Barbados and London.
A number of Murder After Midnight stories are available on iTunes with additional titles planned for early 2013 release.
In 2000 he settled in Cape Cod, Massachusetts, USA, to help his third wife (Karen, née Sandel, died October 2013) nurse her parents through their terminal illnesses, and from 2001 until his death he taught writing and research at Boston University.
Martin Fido died on 2 April 2019 of complications resulting from having suffered a fall.
References
1939 births
2019 deaths
People from Penzance
Writers from Cornwall
Alumni of Balliol College, Oxford
University of the West Indies academics
British non-fiction crime writers
|
{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 5,438
|
This page is available in the following languages:
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Last updated: 2/12-2022
Bengtsfors municipality is best known for its beautiful countryside. The municipality has five populations centres: Bengtsfors (central town), Billingsfors, Skåpafors, Bäckefors and Dals Långed.
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Comprehensive schools for years 1–5 are located in Bengtsfors, Dals Långed and Bäckefors. Bengtsfors also has a comprehensive school for years 6–9 and upper secondary school with a broad range of courses. Bengtsfors also has a comprehensive special school for children with various disabilities.
Strömkullegymnasiet
Bengtsfors offers opportunities for adult education. Adult education offers a broad range of courses, both our own and in collaboration with universities and other actors. Here you can also study Swedish for Immigrants, SfI. Dals Långed also has a craft and design college – HDK Steneby.
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Bengtsfors has a medical centre with ante-natal clinic and child healthcare clinic. It also has a number of dentists. Bäckefors has a hospital with an emergency centre.
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The municipality uses authorised interpreter from the interpreter centres in Karlstad and Region West.
Getting to and from Bengtsfors is easiest by bus, it is Västtrafik that operates buses in the area. If you want to travel by train, you first have to get to one of our neighbouring municipalities, Dals-Ed or Åmål.
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|
{
"redpajama_set_name": "RedPajamaCommonCrawl"
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| 7,226
|
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This meeting should take place at the clients home as that is where the care is to be provided. Our Senior Care Manager will come to meet you at the agreed time. The Needs Assessment meeting is our opportunity to collectively talk about you and your needs, discuss the care service options available and formulate a care package that is best for you.
This is also our opportunity to get to know each other as well as assessing any potential risks within your home environment either for yourself or our care staff.
Once we are fully aware of your needs and preferences and have received any relevant information, if required, from any other parties or agencies involved in caring for you, we then produce a proposal document of the care package that is most appropriate for you.
This proposal document outlines the full extent of the service we can offer, how and when we will deliver the care, and a full breakdown of the costs and contractual terms involved.
Once you are fully satisfied with our proposal and are happy to proceed with working with River Garden Care, we all sign and agree the contract which details our obligations to each other. We both keep a copy for our records.
Our Care Plan details what to do if you want to change any aspect of the care service, the procedures to follow if there are any issues as well as full details of every aspect of the care service we will be delivering to you.
During the first few care visits we will always accompany our care worker with a senior care manager. This allows us the opportunity to assess how our assigned care worker is bonding with you and if there are any concerns you can easily discuss it with the senior manager.
Once you, our senior care manager and the actual care worker are satisfied with all aspects of working directly with each other, only then do we allow the care worker to work alone with the client.
As part of our commitment to quality assurance and to ensure 100% client satisfaction, we continuously assess and review the service we offer our clients. From quarterly satisfaction surveys, to our continuous care worker appraisals, we ensure every aspect of our service is consistently delivered to the highest standard.
We underpin every part of our service with our quality guarantee, so you can always trust in us to do the right thing for your benefit.
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{"url":"http:\/\/gate-exam.in\/ME\/Syllabus\/Fluid-Mechanics-Thermal-Sciences\/FM-Fluid-Mechanics","text":"# GATE Questions & Answers of Fluid Mechanics Mechanical Engineering\n\n#### Fluid Mechanics 81 Question(s) | Weightage 09 (Marks)\n\nFor a two-dimensional incompressible flow field given by $\\overset\\rightharpoonup u=A\\left(x\\widehat i-y\\widehat j\\right)$, where $A>0$ , which one of the following statements is FALSE?\n\nA tank open at the top with a water level of 1 m, as shown in the figure, has a hole at a height of 0.5 m. A free jet leaves horizontally from the smooth hole. The distance X (in m) where the jet strikes the floor is\n\nIn a Lagrangian system, the position of a fluid particle in a flow is described as $x=x_\\circ e^{-kt}$ and $y=y_\\circ e^{-kt}$ where t is the time while $x_\\circ$, $y_\\circ$, and k are constants. The flow is\n\nA solid block of 2.0 kg mass slides steadily at a velocity V along a vertical wall as shown in the figure below. A thin oil film of thickness h = 0.15 mm provides lubrication between the block and the wall. The surface area of the face of the block in contact with the oil film is 0.04 $m^2$ . The velocity distribution within the oil film gap is linear as shown in the figure. Take dynamic viscosity of oil as $7\\times10^{-3}$ Pa-s and acceleration due to gravity as 10 $m\/s^2$ . Neglect weight of the oil. The terminal velocity V (in m\/s) of the block is _________ (correct to one decimal place).\n\nThe viscous laminar flow of air over a flat plate results in the formation of a boundary layer. The boundary layer thickness at the end of the plate of length L is $\\delta_L$. When the plate length is increased to twice its original length, the percentage change in laminar boundary layer thickness at the end of the plate (with respect to $\\delta_L$) is ________ (correct to two decimal places).\n\nAir flows at the rate of 1.5 m3\/s through a horizontal pipe with a gradually reducing crosssection as shown in the figure. The two cross-sections of the pipe have diameters of 400 mm and 200 mm. Take the air density as 1.2 kg\/m3 and assume inviscid incompressible flow. The change in pressure $\\left(p_2-p_1\\right)$ (in kPa) between sections 1 and 2 is\n\nFor steady flow of a viscos incompressible fluid through a circular pipe of constant diameter, the averavge velocity in the fully developed region is constant. Which one of the following statements about the average velocity in the developing region is TRUE?\n\nConsider the two-dimensional velocity filed given by $\\style{font-family:'Times New Roman'}{\\overrightarrow V=(5+a_1x+b_1y)\\widehat i+(4+a_2x+b_2y)\\widehat j,}$ where a1, b1, a2 and b2 are constants. Which one of the following conditions needs to be satisfied for the flow to be incompressible?\n\nConsider steady flow of an incomressible fluid through two long and straight pipes of diameters $d_1$ and $d_2$\u00a0arranged in series. Both pipes are of equal length and the flow is turbulent in both pipes. The friction factor for turbulent flow though pipes is of the form, $f=K(Re)^{-n}$, where $K$\u00a0and $n$\u00a0are known positive constants and Re is the Reynolds number. Neglecting minir losses, the ratio of the frictional pressure drop in pipe 1 to that \u00a0in pipe 2, $\\style{font-family:'Times New Roman'}{\\left(\\frac{\\triangle P_1}{\\triangle P_2}\\right)}$, is given by\n\nFor a steady flow, the velocity field is $\\style{font-family:'Times New Roman'}{\\overrightarrow V=(-x^2+3y)\\widehat i+(2xy)\\widehat j}$. The magnitude of the acceleration of a particle at (1,-1) is\n\nFor the stability of a floating body the\n\nConsider a laminar flow at zero incidence over a flat plate. The shear stress at the wall is denoted by ${\\tau }_{\\omega }$ . The axial position x1 and x2 on the plate are measured from the leading edge in the direction of flow. If x2>x1 then\n\nThe arrangement shown in the figure measures the velocity V of a gas of density 1 kg\/m2 flowing through a pipe. The acceleration due to gravity is 9.81 m\/s2. If the manometric fluid is water (density 1000 kg\/m3) and the velocity V is 20 m\/s, the differential head h (in mm) between the two arms of the manometer is__________\n\nA 60 mm-diameter water jet strikes a plate containing a hole of 40 mm diameter as shown in the figure. Part of the jet passes through the hole horizontally, and the remaining is deflected vertically. The density of water is 1000 kg\/m3. If velocities are as indicates in the figure, the magnitude of horizontal force (in N) required to hold the plate is_______\n\nFor a laminar flow of water over a sphere, the drag coefficient CF is defined as ${C}_{F}=F\/\\left(\\rho {U}^{\\mathit{2}}{D}^{\\mathit{2}}\\right)$ , where F is the drag force, $\\rho$ is the fluid density, U is the fluid velocity and D is the diameter of the sphere. The density of water is 1000 kg\/m3. The diameter of the sphere is 100 mm and the velocity is 2 m\/s, the drag coefficient is 0.5. If water now flows over another sphere of diameter 200 mm under dynamically similar conditions, the drag force (in N) on the sphere is___________\n\nThe instantaneous stream-wise velocity of a turbulent flow is given as follows:\n$u(x,y,z,t)$=$\\overset\\_u(x,y,z)$+$u'(x,y,z,t)$\nThe time-average of the fluctuating velocity $u'(x,y,z,t)$ is\n\nFor a floating body, buoyant force acts at the\n\nConsider two hydraulic turbines having identical specific speed and effective head at the inlet. If the speed ratio $\\left(N_1\/N_2\\right)$ of the two turbines is 2, then the respective power ratio $\\left(P_1\/P_2\\right)$ is _____________\n\nAn inverted U-tube manometer is used to measure the pressure difference between two pipes A and B, as shown in the figure. Pipe A is carrying oil (specifc gravity=0.8) and pipe B is carrying water. The densities of air and water are 1.16 $\\mathrm{kg}\/\\mathrm m^3$ and 1000 $\\mathrm{kg}\/\\mathrm m^3$, respectivly. The pressure difference between pipes A and B is _________ kPa.\n\nOil (kinematic viscosity,$v_{oil}=1.0\\times10^{-5}\\;\\mathrm m^2\/\\mathrm s$) flows through a pipe of 0.5 m diameter with a velocity of 10 m\/s. Water (kinematic viscosity, $v_{oil}=0.89\\times10^{-6}\\;\\mathrm m^2\/\\mathrm s$) is flowing through a model pipe of diameter 20 mm. For satisfying the dynamic similarity, the velocity of water (in m\/s) is __________\n\nA steady laminar boundary layer is formed over a flat plate as shown in the figure. The free stream velocity of the fluid is U0. The velocity profile at the inlet a-b is uniform, while that at a downstream location c-d is given by $\\mathrm u=U_0\\left[2\\left(\\frac y\\delta\\right)-\\left(\\frac y\\delta\\right)^2\\right]$\nThe ratio of the mass flow rate,$\\overset.{M_{bd}}$, leaving through the horizontal section $b-d$ to that entering through the vertical section $a-b$\u00a0is ___________\n\nThe volumetric flow rate (per unit depth) between two streamlines having stream functions $\\psi_1$ and $\\psi_2$ is\n\nThe large vessel shown in the figure contains oil and water. A body is submerged at the interface of oil and water such that 45 percent of its volume is in oil while the rest is in water. The density of the body is _________ $\\mathrm{kg}\/\\mathrm m^3$.\n\nThe specific gravity of oil is 0.7 and density of water is 1000 $\\mathrm{kg}\/\\mathrm m^3$.\n\nAcceleration due to gravity\u00a0$\\mathbf g\\boldsymbol=\\mathbf{10}\\boldsymbol\\;\\mathbf m\\boldsymbol\/\\mathbf s^\\mathbf2$.\n\nConsider fluid flow between two infinite horizontal plates which are parallel (the gap between them being 50 mm). The top plate is sliding parallel to the stationary bottom plate at a speed of 3 m\/s. The flow between the plates is solely due to the motion of the top plate. The force per unit area (magnitude) required to maintain the bottom plate stationary is _________ $\\mathrm N\/\\mathrm m^2$.\n\nViscosity of the fluid $\\mu=0.44\\;\\mathrm{kg}\/\\mathrm m-\\mathrm s$ and density $\\rho=888\\;\\mathrm{kg}\/\\mathrm m^3$.\n\nConsider a frictionless, massless and leak-proof plug blocking a rectangular hole of dimensions $2R\\times L$\u00a0at the bottom of an open tank as shown in the figure. The head of the plug has the shape of a semi-cylinder of radius $R$. The tank is filled with a liquid of density $\\rho$ up to the tip of the plug. The gravitational acceleration is $g$. Neglect the effect of the atmospheric pressure.\n\nThe force F required to hold the plug in its position is\n\nA channel of width 450 mm branches into two sub-channels having width 300 mm and 200 mm as shown in figure. If the volumetric flow rate (taking unit depth) of an incompressible flow through the main channel is $0.9\\;\\mathrm m^3\/\\mathrm s$ and the velocity in the sub-channel of width 200 mm is 3 m\/s, the velocity in the sub-channel of width 300 mm is _____________ m\/s.\n\nAssume both inlet and outlet to be at the same elevation.\n\nFor a certain two-dimensional incompressible flow, velocity field is given by $2xy\\;\\widehat i-y^2\\widehat j$ . The streamlines for this flow are given by the family of curves\n\nGrashof number signifies the ratio of\n\nConsider a fully developed steady laminar flow of an incompressible fluid with viscosity $\\mu$\u00a0through a circular pipe of radius R. Given that the velocity at a radial location of $R\/2$\u00a0from the centerline of the pipe is\u00a0$U_1$ , the shear stress at the wall is\u00a0\u00a0$K\\mu U_1\/R$, where K is __________\n\nThe water jet exiting from a stationary tank through a circular opening of diameter 300 mm impinges on a rigid wall as shown in the figure. Neglect all minor losses and assume the water level in the tank to remain constant. The net horizontal force experienced by the wall is ___________ kN.\n\nDensity of water is 1000 kg\/m3.\n\nAcceleration due to gravity g = 10 m\/s2.\n\nFor a two-dimensional flow, the velocity field is are the basis vectors in the x-y Cartesian coordinate system. Identify the CORRECT statements from below\n\n(1) The flow is incompressible.\n(3) $y-$\u00a0component of acceleration, ${a}_{y}=\\frac{-y}{{\\left({x}^{2}+{y}^{2}\\right)}^{2}}$\n(4) $x-$\u00a0component of acceleration ${a}_{x}=\\frac{-\\left(x+y\\right)}{{\\left({x}^{2}+{y}^{2}\\right)}^{2}}$\n\nConsider fully developed flow in a circular pipe with negligible entrance length effects. Assuming the mass flow rate , density and friction factor to be constant, if the length of the pipe is doubled and the diameter is halved, the head loss due to friction will increase by a factor of\n\nThe Blasius equation related to boundary layer theory is a\n\nMatch the following pairs:\n\n Equation Physical Interpretation P $\\nabla\\times\\overrightarrow V=0$ I Incompressible continuity equation Q $\\nabla\\cdot\\overrightarrow V=0$ II Steady flow R $\\frac{D\\overrightarrow V}{Dt}=0$ III Irrotational flow S $\\frac{\\partial\\overrightarrow V}{\\partial t}=0$ IV Zero acceleration of fluid particle\n\nThe velocity field on an incompressible flow is given by $V=(a_1x+a_2y+a_3z)\\boldsymbol i+(b_1x+b_2y+b_3z)\\boldsymbol j+(c_1x+c_2y+c_3z)\\boldsymbol k$, where $a_1=2$\u00a0and $c_3=-4$\u00a0. The value of $b_2$\u00a0is________.\n\nAir $(p=1.2\\;kg\/m^3$\u00a0and kinematic viscosity , $v=2\\times10^{-5}m^2\/s)$\u00a0with a velocity of 2 m\/s flows over the top surface of a flat plate of length 2.5 m . If the average value of friction coefficient is ${C}_{f}=\\frac{1.328}{\\sqrt{R{e}_{x}}}$, the total drag force (in N) per unit width of the plate is _____.\n\nWater ($\\rho$ = 1000kg\/m3) flows through a venturimeter with inlet diameter 80 mm and throat diameter 40 mm.The inlet and throat gauge pressures are measured to be 400 kPa and 130 kPa respectively . Assuming the venturimeter to be horizontal and neglecting friction , the inlet velocity (in m\/s) is_______.\n\nIf the fluid velocity for a potential flow is given by $\\mathbf V(x,y)=u(x,y)\\mathbf i+v(x,y)\\mathbf j$\u00a0with usual notations, then the slope of potential line at $(x,y)$\u00a0is\n\nWithin a boundary layer for a steady incompressible flow, the Bernoulli equation\n\nThe head loss for a laminar incompressible flow through a horizontal circular pipe is $h_1$\u00a0Pipe length and fluid remaining the same, if the average flow velocity doubles and the pipe diameter reduces to half its previous value, the head loss is $h_2$. The ratio $h_2\/h_1$\u00a0is\n\nFor a fully developed laminar flow of water (dynamic viscosity 0.001 Pa-s) through a pipe of radius 5 cm, the axial pressure gradient is - 10 Pa\/m The magnitude of axial velocity (in m\/s) at a radial location of 0.2 cm is________.\n\nCouette flow is characterized by\n\nThree parallel pipes connected at the two ends have flow-rates Q1, Q2 and Q3 respectively, and the corresponding frictional head losses are hL1, hL2 and hL3 respectively. The correct expressions for total flow rate (Q) and frictional head loss across the two ends (hL) are\n\nA Prandtl tube (Pitot-static tube with $C=1$) is used to measure the velocity of water. The differential manometer reading is 10 mm of liquid column with a relative density of 10. Assuming g = 9.8 m\/s2, the velocity of water (in m\/s) is _______.\n\nWhich of the following statements are TRUE, when the cavitation parameter \u03c3 = 0?\n(i) the local pressure is reduced to vapor pressure\n(ii) cavitation starts\n(iii) boiling of liquid starts\n(iv) cavitation stops\n\nFor a completely submerged body with centre of gravity \u2018G\u2019 and centre of buoyancy \u2018B\u2019, the condition of stability will be\n\nFor a fully developed flow of water in a pipe having diameter 10 cm, velocity 0.1 m\/s and kinematic viscosity 10\u22125 m2\/s, the value of Darcy friction factor is _______\n\nIn a simple concentric shaft-bearing arrangement, the lubricant flows in the 2 mm gap between the shaft and the bearing. The flow may be assumed to be a plane Couette flow with zero pressure gradient. The diameter of the shaft is 100 mm and its tangential speed is 10 m\/s. The dynamic viscosity of the lubricant is 0.1 kg\/m.s. The frictional resisting force (in newton) per 100 mm length of the bearing is _______\n\nThe difference in pressure (in N\/m2) across an air bubble of diameter 0.001 m immersed in water (surface tension = 0.072 N\/m) is _______\n\nA spherical balloon with a diameter of 10 m, shown in the figure below is used for advertisements. The balloon is filled with helium (RHe = 2.08 kJ\/kg.K) at ambient conditions of 15\u00b0C and 100 kPa. Assuming no disturbances due to wind, the maximum allowable weight (in newton) of balloon material and rope required to avoid the fall of the balloon (Rair = 0.289 kJ\/kg.K) is _______\n\nConsider laminar flow of water over a flat plate of length 1 m. If the boundary layer thickness at a distance of 0.25 m from the leading edge of the plate is 8 mm, the boundary layer thickness (in mm), at a distance of 0.75 m, is _______\n\nConsider the turbulent flow of a fluid through a circular pipe of diameter, D. Identify the correct pair of statements.\n\n I. The fluid is well-mixed II. The fluid is unmixed III. ReD < 2300 IV. ReD > 2300\n\nA siphon is used to drain water from a large tank as shown in the figure below. Assume that the level of water is maintained constant. Ignore frictional effect due to viscosity and losses at entry and exit. At the exit of the siphon, the velocity of water is\n\nA fluid of dynamic viscosity 2 \u00d7 10\u22125 kg\/m.s and density 1 kg\/m3 flows with an average velocity of 1 m\/s through a long duct of rectangular (25 mm \u00d7 15 mm) cross-section. Assuming laminar flow, the pressure drop (in Pa) in the fully developed region per meter length of the duct is _______\n\nA flow field which has only convective acceleration is\n\nConsider the following statements regarding streamline(s):\n\n (i) It is a continuous line such that the tangent at any point on it shows the velocity vector at that point (ii) There is no flow across streamlines (iii) $\\frac{\\mathrm{dx}}{u}=\\frac{\\mathrm{dy}}{v}=\\frac{\\mathrm{dz}}{w}$ is the differential equation of a streamline, where u, v and w are velocities in directions x, y and z, respectively (iv) In an unsteady flow, the path of a particle is a streamline\n\nWhich one of the following combinations of the statements is true?\n\nConsider a velocity field $\\overrightarrow V=K\\left(y\\widehat i+x\\widehat K\\right)$ where K is a constant. The vorticity, \u03a9Z , is\n\nFor steady, fully developed flow inside a straight pipe of diameter D, neglecting gravity effects, the pressure drop over a length L and the wall shear stress ${\\tau }_{\\omega }$ are related by\n\nA hinged gate of length 5 m, inclined at 30\u00b0 with the horizontal and with water mass on its left, is shown in the figure below. Density of water is 1000 kg\/m3. The minimum mass of the gate in kg per unit width (perpendicular to the plane of paper), required to keep it closed is\n\nOil flows through a 200 mm diameter horizontal cast iron pipe (friction factor, $f=0.0225$) of length 500 m. The volumetric flow rate is 0.2 m3\/s. The head loss (in m) due to friction is (assume g = 9.81 m\/s2)\n\nAn incompressible fluid flows over a flat plate with zero pressure gradient. The boundary layer thickness is 1 mm at a location where the Reynolds number is 1000. If the velocity of the fluid alone is increased by a factor of 4, then the boundary layer thickness at the same location, in mm will be\n\nA large tank with a nozzle attached contains three immiscible, inviscid fluids as shown. Assuming that the changes in h1, h2 and h3 are negligible, the instantaneous discharge velocity is\n\nA streamline and an equipotential line in a flow field\n\nFigure shows the schematic for the measurement of velocity of air (density = 1.2kg \/m3) through a constant-area duct using a pitot tube and a water-tube manometer. The differential head of water (density = 1000 kg \/m3) in the two columns of the manometer is 10mm. Take acceleration due to gravity as\u00a0 9.8m\/ s2. The velocity of air in m\/s is\n\nFor the stability of a floating body, under the influence of gravity alone, which of the following is TRUE?\n\nThe maximum velocity of a one-dimensional incompressible fully developed viscous flow, between two fixed parallel plates, is 6ms-1. The mean velocity (in ms-1) of the flow is\n\nA phenomenon is modeled using n dimensional variables with k primary dimensions. The number of non-dimensional variables is\n\nVelocity vector of a flow field is given as$\\stackrel{\\to }{v}=2xy\\stackrel{^}{i}-{x}^{2}z\\stackrel{^}{j}$.the velocity vector at (1,1,1)is\n\nA lightly loaded full journal bearing has a journal of 50mm, bush bore of 50.05mm and bush length of 20mm. if rotational speed of journal is 1200rpm and average viscosity of liquid lubricant is 0.03 Pa s, the power loss (in W) will be\n\nConsider steady, incompressible and irrotational flow through a reducer in a horizontal pipe where the diameter is reduced from 20 cm to 10 cm. The pressure in the 20 cm pipe just upstream of the reducer is 150 kPa. The fluid has a vapour pressure of 50 kPa and a specific weight of 5 kN\/m3. Neglecting frictional effects, the maximum discharge (in m3\/s) that can pass through the reducer without causing cavitation is\n\nYou are asked to evaluate assorted fluid flows for their suitability in a given laboratory application. The following three flow choices, expressed in terms of the two-dimensional velocity fields in the xy-plane, are made available.\n\nP. \u00a0 u = 2y, v = -3x\nQu = 3xy, v = 0\nR.\u00a0 u = -2x, v = 2y\n\nWhich flow(s) should be recommended when the application requires the flow to be incompressible and irrotational?\n\nWater at 25 \u00b0C is flowing through a 1.0 km long G.I pipe of 200 mm diameter at the rate of 0.07 m3\/s. If value of Darcy friction factor for this pipe is 0.02 and density of water is 1000 kg\/m3, the pumping power (in kW) required to maintain the flow is\n\nThe velocity profile of a fully developed laminar flow in a straight circular pipe, as shown in the figure, is given by the expression $u\\left(r\\right)=-\\frac{{R}^{2}}{4\\mu }\\left(\\frac{dp}{dx}\\right)\\left(1-\\frac{{r}^{2}}{{R}^{2}}\\right)$ where $\\frac{dp}{dx}$ is a constant.\n\nThe average velocity of fluid in the pipe is\n\nFor the continuity equation given by $\\overrightarrow\\nabla\\bullet\\overrightarrow V=0$ to be valid, where $\\overrightarrow V$ is the velocity vector, which one of the following is a necessary condition?\n\nA journal bearing has shaft diameter of 40mm and a length of 40mm. The shaft is rotating at 20 rad\/s and the viscosity of the lubricant is 20 mPa.s. The clearance is 0.020mm .The loss of torque due to the viscosity of the lubricant is approximately\n\nThe gap between a moving circular plate and a stationary surface is being continuously reduced, as the circular plate comes down at a uniform speed V towards the stationary bottom surface, as shown in the figure. In the process, the fluid contained between the two plates flows out radially. The fluid is assumed to be incompressible and inviscid.\n\nThe radial velocity vr at any radium r, when the gap width is h, is\n\nThe gap between a moving circular plate and a stationary surface is being continuously reduced, as the circular plate comes down at a uniform speed V towards the stationary bottom surface, as shown in the figure. In the process, the fluid contained between the two plates flows out radially. The fluid is assumed to be incompressible and inviscid.\n\nThe radial component of the fluid acceleration at r = R is\n\nConsider an incompressible laminar boundary layer flow over a flat plate of length L, aligned with the direction of an oncoming uniform free stream. If F is the ratio of the drag force on the front half of the plate to the drag force on the rear half, then\n\nIn a steady flow through a nozzle, the flow velocity on the nozzle axis is given by v = uo(1+3x\/L)i, where x is the distance along the axis of the nozzle from its inlet plane and L is the length of the nozzle. The time required for a fluid particle on the axis to travel from the inlet to the exit lane of the nozzle is\n\nConsider a steady incompressible flow through a channel as shown below.\n\nThe velocity profile is uniform with a value of uo at the inlet section A. The velocity profile at section B downstream is\n\n$\\mathrm{u}=\\left\\{\\begin{array}{lc}{\\mathrm{V}}_{\\mathrm{m}}\\frac{\\mathrm{y}}{\\delta },& 0\\le \\mathrm{y}\\le \\delta \\\\ {\\mathrm{V}}_{\\mathrm{m}},& \\delta \\le \\mathrm{y}\\le H-\\delta \\\\ {\\mathrm{V}}_{\\mathrm{m}}\\frac{H-\\mathrm{y}}{\\delta },& H-\\delta \\le \\mathrm{y}\\le H\\end{array}\\right\\$\n\nThe ratio Vm\/uo is\n\n$\\mathrm{u}=\\left\\{\\begin{array}{lc}{\\mathrm{V}}_{\\mathrm{m}}\\frac{\\mathrm{y}}{\\delta },& 0\\le \\mathrm{y}\\le \\delta \\\\ {\\mathrm{V}}_{\\mathrm{m}},& \\delta \\le \\mathrm{y}\\le H-\\delta \\\\ {\\mathrm{V}}_{\\mathrm{m}}\\frac{H-\\mathrm{y}}{\\delta },& H-\\delta \\le \\mathrm{y}\\le H\\end{array}\\right\\$\nThe ratio $\\frac{{\\mathrm{p}}_{\\mathrm{A}}-{\\mathrm{p}}_{\\mathrm{B}}}{\\frac{1}{2}{{\\mathrm{\\rho u}}_{\\mathrm{o}}}^{2}}$(where pA and pB are the pressures at section A and B, respectively, and \u03c1 is the density of the fluid) is","date":"2020-12-02 10:28:19","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 72, \"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\": 0.7471382021903992, \"perplexity\": 573.9659789934716}, \"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-2020-50\/segments\/1606141706569.64\/warc\/CC-MAIN-20201202083021-20201202113021-00432.warc.gz\"}"}
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{"url":"https:\/\/www.aimsciences.org\/article\/doi\/10.3934\/jgm.2018012","text":"# American Institute of Mathematical Sciences\n\n\u2022 Previous Article\nThe Euler-Poisson equations: An elementary approach to integrability conditions\n\u2022 JGM\u00a0Home\n\u2022 This Issue\n\u2022 Next Article\nAlternative angle-based approach to the $\\mathcal{KS}$-Map. An interpretation through symmetry and reduction\nSeptember\u00a0 2018,\u00a010(3):\u00a0331-357. doi:\u00a010.3934\/jgm.2018012\n\n## A family of compact semitoric systems with two focus-focus singularities\n\n 1 University of Antwerp, Department of Mathematics and Computer Science, Middelheimlaan 1, B-2020 Antwerpen, Belgium 2 Rutgers University, Department of Mathematics, Hill Center - Busch Campus, 110 Frelinghuysen Road, Piscataway, NJ 08854, USA\n\n* Corresponding author: Sonja Hohloch\n\nReceived\u00a0 October 2017 Revised\u00a0 June 2018 Published\u00a0 August 2018\n\nFund Project: The first author was partially supported by the Research Fund of the University of Antwerp and the second author is partially supported by an AMS-Simons travel grant\n\nAbout 6 years ago, semitoric systems were classified by Pelayo & V\u0169 Ng\u1ecdc by means of five invariants. Standard examples are the coupled spin oscillator on $\\mathbb{S}^2 \\times \\mathbb{R}^2$ and coupled angular momenta on $\\mathbb{S}^2 \\times \\mathbb{S}^2$, both having exactly one focus-focus singularity. But so far there were no explicit examples of systems with more than one focus-focus singularity which are semitoric in the sense of that classification. This paper introduces a $6$-parameter family of integrable systems on $\\mathbb{S}^2 \\times \\mathbb{S}^2$ and proves that, for certain ranges of the parameters, it is a compact semitoric system with precisely two focus-focus singularities. Since the twisting index (one of the semitoric invariants) is related to the relationship between different focus-focus points, this paper provides systems for the future study of the twisting index.\n\nCitation: Sonja Hohloch, Joseph Palmer. A family of compact semitoric systems with two focus-focus singularities. Journal of Geometric Mechanics, 2018, 10 (3) : 331-357. doi: 10.3934\/jgm.2018012\n##### References:\n [1] J. Alonso, H. Dullin and S. Hohloch, Taylor series and twisting-index invariants of coupled spin-oscillators, arXiv: 1712.06402 (to appear in the Journal of Geometry and Physics).Google Scholar [2] J. Alonso, H. Dullin and S. Hohloch, Symplectic classification of coupled angular momenta, arXiv: 1808.05849.Google Scholar [3] M. F. Atiyah, Convexity and commuting Hamiltonians, Bull. London Math. Soc., 14 (1982), 1-15. doi:\u00a010.1112\/blms\/14.1.1. Google Scholar [4] A. V. Bolsinov and A. T. Fomenko, Integrable Hamiltonian Systems, Chapman & Hall\/CRC, Boca Raton, FL, 2004, Geometry, topology, classification, Translated from the 1999 Russian original. doi:\u00a010.1201\/9780203643426. Google Scholar [5] P.-L. Buono, F. Laurent-Polz and J. Montaldi, Symmetric Hamiltonian bifurcations, in Geometric Mechanics and Symmetry, vol. 306 of London Math. Soc. Lecture Note Ser., Cambridge Univ. Press, Cambridge, 2005, 357-402, Based on lectures by Montaldi. doi:\u00a010.1017\/CBO9780511526367.007. Google Scholar [6] R. Cushman and L. Bates, Global Aspects of Classical Integrable Systems, 2nd edition, Birkh\u00e4user\/Springer, Basel, 2015, URL https:\/\/doi.org\/10.1007\/978-3-0348-0918-4. doi:\u00a010.1007\/978-3-0348-0918-4. Google Scholar [7] R. Cushman\u00a0and\u00a0V. N. San, Sign of the monodromy for Liouville integrable systems, Ann. Henri Poincar\u00e9, 3 (2002), 883-894. doi:\u00a010.1007\/s00023-002-8640-7. Google Scholar [8] T. Delzant, Hamiltoniens p\u00e9riodiques et images convexes de l'application moment, Bull. Soc. Math. France, 116 (1988), 315-339. doi:\u00a010.24033\/bsmf.2100. Google Scholar [9] H. Dullin\u00a0and\u00a0\u00c1. Pelayo, Generating hyperbolic singularities in semitoric systems via Hopf bifurcations, J. Nonlinear Sci., 26 (2016), 787-811. doi:\u00a010.1007\/s00332-016-9290-0. Google Scholar [10] K. Efstathiou\u00a0and\u00a0N. Martynchuk, Monodromy of Hamiltonian systems with complexity 1 torus actions, J. Geom. Phys., 115 (2017), 104-115. doi:\u00a010.1016\/j.geomphys.2016.05.014. Google Scholar [11] L. H. Eliasson, Hamiltonian Systems with Poisson Commuting Integrals, PhD thesis, University of Stockholm, 1984.Google Scholar [12] M. Gaudin, Diagonalisation d'une classe d'hamiltoniens de spin, J. Phys. France, 37 (1976), 1087-1098. doi:\u00a010.1051\/jphys:0197600370100108700. Google Scholar [13] V. Guillemin\u00a0and\u00a0S. Sternberg, Convexity properties of the moment mapping, Invent. Math., 67 (1982), 491-513. doi:\u00a010.1007\/BF01398933. Google Scholar [14] S. Hohloch,\u00a0S. Sabatini\u00a0and\u00a0D. Sepe, From compact semi-toric systems to Hamiltonian $S^1$-spaces, Discrete Contin. Dyn. Syst., 35 (2015), 247-281. Google Scholar [15] S. Hohloch, S. Sabatini, D. Sepe and M. Symington, Faithful semitoric systems, SIGMA, 14 (2018), 084, 66 pages.Google Scholar [16] D. M. Kane, J. Palmer and \u00c1. Pelayo, Classifying toric and semitoric fans by lifting equations from $\\rm{SL}_2(\\mathbb{Z})$, SIGMA Symmetry Integrability Geom. Methods Appl., 14 (2018), Paper No. 016, 43 pp. doi:\u00a010.3842\/SIGMA.2018.016. Google Scholar [17] D. M. Kane,\u00a0J. Palmer\u00a0and\u00a0\u00c1. Pelayo, Minimal models of compact symplectic semitoric manifolds, J. Geom. Phys., 125 (2018), 49-74. doi:\u00a010.1016\/j.geomphys.2017.12.005. Google Scholar [18] Y. Karshon, Periodic Hamiltonian flows on four-dimensional manifolds, Mem. Amer. Math. Soc., 141 (1999), \u2177+71pp. doi:\u00a010.1090\/memo\/0672. Google Scholar [19] Y. Le Floch and \u00c1. Pelayo, Symplectic geometry and spectral properties of classical and quantum coupled angular momenta, arXiv: 1607.05419.Google Scholar [20] Y. Le Floch,\u00a0\u00c1. Pelayo\u00a0and\u00a0S. V\u0169 Ng\u1ecdc, Inverse spectral theory for semiclassical Jaynes-Cummings systems, Math. Ann., 364 (2016), 1393-1413. doi:\u00a010.1007\/s00208-015-1259-z. Google Scholar [21] J. Marsden and T. Ratiu, Introduction to Mechanics and Symmetry, vol. 17 of Texts in Applied Mathematics, 2nd edition, Springer-Verlag, New York, 1999, https:\/\/doi.org\/10.1007\/978-0-387-21792-5, A basic exposition of classical mechanical systems. doi:\u00a010.1007\/978-0-387-21792-5. Google Scholar [22] E. Miranda\u00a0and\u00a0N. T. Zung, Equivariant normal form for nondegenerate singular orbits of integrable Hamiltonian systems, Ann. Sci. \u00c9cole Norm. Sup. (4), 37 (2004), 819-839. doi:\u00a010.1016\/j.ansens.2004.10.001. Google Scholar [23] J. Palmer, Moduli spaces of semitoric systems, J. Geom. Phys., 115 (2017), 191-217. doi:\u00a010.1016\/j.geomphys.2017.02.008. Google Scholar [24] \u00c1. Pelayo, Hamiltonian and symplectic symmetries: An introduction, Bull. Amer. Math. Soc. (N.S.), 54 (2017), 383-436. doi:\u00a010.1090\/bull\/1572. Google Scholar [25] \u00c1. Pelayo,\u00a0T. Ratiu\u00a0and\u00a0S. V\u0169 Ng\u1ecdc, The affine invariant of proper semitoric integrable systems, Nonlinearity, 30 (2017), 3993-4028. doi:\u00a010.1088\/1361-6544\/aa8aec. Google Scholar [26] \u00c1. Pelayo\u00a0and\u00a0S. V\u0169 Ng\u1ecdc, Semitoric integrable systems on symplectic 4-manifolds, Invent. Math., 177 (2009), 571-597. doi:\u00a010.1007\/s00222-009-0190-x. Google Scholar [27] \u00c1. Pelayo\u00a0and\u00a0S. V\u0169 Ng\u1ecdc, Constructing integrable systems of semitoric type, Acta Math., 206 (2011), 93-125. doi:\u00a010.1007\/s11511-011-0060-4. Google Scholar [28] \u00c1. Pelayo and S. V\u0169 Ng\u1ecdc, Symplectic theory of completely integrable Hamiltonian systems, Bull. Amer. Math. Soc. (N. S.), 48 (2011), 409-455. doi:\u00a010.1090\/S0273-0979-2011-01338-6. Google Scholar [29] \u00c1. Pelayo\u00a0and\u00a0S. V\u0169 Ng\u1ecdc, Hamiltonian dynamics and spectral theory for spin-oscillators, Comm. Math. Phys., 309 (2012), 123-154. doi:\u00a010.1007\/s00220-011-1360-4. Google Scholar [30] M. Petrera, Integrable Extensions and Discretizations of Classical Gaudin Models, PhD thesis, Dipartimento di Fisica, Universit\u00e1 degli Studi di Roma Tre, 2007.Google Scholar [31] T. Ratiu, C. Wacheux and N. T. Zung, Convexity of singular affine structures and toric-focus integrable hamiltonian systems, arXiv: 1706.01093.Google Scholar [32] D. A. Sadovski\u00ed\u00a0and\u00a0B. I. Z\u0125ilinski\u00ed, Monodromy, diabolic points, and angular momentum coupling, Phys. Lett. A, 256 (1999), 235-244. doi:\u00a010.1016\/S0375-9601(99)00229-7. Google Scholar [33] M. Symington, Four dimensions from two in symplectic topology, in Topology and geometry of manifolds (Athens, GA, 2001), vol. 71 of Proc. Sympos. Pure Math., Amer. Math. Soc., Providence, RI, 2003,153-208. doi:\u00a010.1090\/pspum\/071\/2024634. Google Scholar [34] S. V\u0169 Ng\u1ecdc, On semi-global invariants for focus-focus singularities, Topology, 42 (2003), 365-380. doi:\u00a010.1016\/S0040-9383(01)00026-X. Google Scholar [35] S. V\u0169 Ng\u1ecdc, Syst\u00e9mes Int\u00e9grables Semi-Classiques: Du Local au Global, vol. 22 of Panoramas et Synth\u00e9ses [Panoramas and Syntheses], Soci\u00e9t\u00e9 Math\u00e9matique de France, Paris, 2006. Google Scholar [36] S. V\u0169 Ng\u1ecdc, Moment polytopes for symplectic manifolds with monodromy, Adv. Math., 208 (2007), 909-934. doi:\u00a010.1016\/j.aim.2006.04.004. Google Scholar [37] C. Wacheux, Syst\u00e9mes Int\u00e9grables Semi-toriques et Polytopes Moment, PhD thesis, Universit\u00e9 de Rennes 1, 2013.Google Scholar [38] J. Williamson, On the algebraic problem concerning the normal form of linear dynamical systems, Amer. J. Math., 58 (1936), 141-163. doi:\u00a010.2307\/2371062. Google Scholar\n\nshow all references\n\n##### References:\n [1] J. Alonso, H. Dullin and S. Hohloch, Taylor series and twisting-index invariants of coupled spin-oscillators, arXiv: 1712.06402 (to appear in the Journal of Geometry and Physics).Google Scholar [2] J. Alonso, H. Dullin and S. Hohloch, Symplectic classification of coupled angular momenta, arXiv: 1808.05849.Google Scholar [3] M. F. Atiyah, Convexity and commuting Hamiltonians, Bull. London Math. Soc., 14 (1982), 1-15. doi:\u00a010.1112\/blms\/14.1.1. Google Scholar [4] A. V. Bolsinov and A. T. Fomenko, Integrable Hamiltonian Systems, Chapman & Hall\/CRC, Boca Raton, FL, 2004, Geometry, topology, classification, Translated from the 1999 Russian original. doi:\u00a010.1201\/9780203643426. Google Scholar [5] P.-L. Buono, F. Laurent-Polz and J. Montaldi, Symmetric Hamiltonian bifurcations, in Geometric Mechanics and Symmetry, vol. 306 of London Math. Soc. Lecture Note Ser., Cambridge Univ. Press, Cambridge, 2005, 357-402, Based on lectures by Montaldi. doi:\u00a010.1017\/CBO9780511526367.007. Google Scholar [6] R. Cushman and L. Bates, Global Aspects of Classical Integrable Systems, 2nd edition, Birkh\u00e4user\/Springer, Basel, 2015, URL https:\/\/doi.org\/10.1007\/978-3-0348-0918-4. doi:\u00a010.1007\/978-3-0348-0918-4. Google Scholar [7] R. Cushman\u00a0and\u00a0V. N. San, Sign of the monodromy for Liouville integrable systems, Ann. Henri Poincar\u00e9, 3 (2002), 883-894. doi:\u00a010.1007\/s00023-002-8640-7. Google Scholar [8] T. Delzant, Hamiltoniens p\u00e9riodiques et images convexes de l'application moment, Bull. Soc. Math. France, 116 (1988), 315-339. doi:\u00a010.24033\/bsmf.2100. Google Scholar [9] H. Dullin\u00a0and\u00a0\u00c1. Pelayo, Generating hyperbolic singularities in semitoric systems via Hopf bifurcations, J. Nonlinear Sci., 26 (2016), 787-811. doi:\u00a010.1007\/s00332-016-9290-0. Google Scholar [10] K. Efstathiou\u00a0and\u00a0N. Martynchuk, Monodromy of Hamiltonian systems with complexity 1 torus actions, J. Geom. Phys., 115 (2017), 104-115. doi:\u00a010.1016\/j.geomphys.2016.05.014. Google Scholar [11] L. H. Eliasson, Hamiltonian Systems with Poisson Commuting Integrals, PhD thesis, University of Stockholm, 1984.Google Scholar [12] M. Gaudin, Diagonalisation d'une classe d'hamiltoniens de spin, J. Phys. France, 37 (1976), 1087-1098. doi:\u00a010.1051\/jphys:0197600370100108700. Google Scholar [13] V. Guillemin\u00a0and\u00a0S. Sternberg, Convexity properties of the moment mapping, Invent. Math., 67 (1982), 491-513. doi:\u00a010.1007\/BF01398933. Google Scholar [14] S. Hohloch,\u00a0S. Sabatini\u00a0and\u00a0D. Sepe, From compact semi-toric systems to Hamiltonian $S^1$-spaces, Discrete Contin. Dyn. Syst., 35 (2015), 247-281. Google Scholar [15] S. Hohloch, S. Sabatini, D. Sepe and M. Symington, Faithful semitoric systems, SIGMA, 14 (2018), 084, 66 pages.Google Scholar [16] D. M. Kane, J. Palmer and \u00c1. Pelayo, Classifying toric and semitoric fans by lifting equations from $\\rm{SL}_2(\\mathbb{Z})$, SIGMA Symmetry Integrability Geom. Methods Appl., 14 (2018), Paper No. 016, 43 pp. doi:\u00a010.3842\/SIGMA.2018.016. Google Scholar [17] D. M. Kane,\u00a0J. Palmer\u00a0and\u00a0\u00c1. Pelayo, Minimal models of compact symplectic semitoric manifolds, J. Geom. Phys., 125 (2018), 49-74. doi:\u00a010.1016\/j.geomphys.2017.12.005. Google Scholar [18] Y. Karshon, Periodic Hamiltonian flows on four-dimensional manifolds, Mem. Amer. Math. Soc., 141 (1999), \u2177+71pp. doi:\u00a010.1090\/memo\/0672. Google Scholar [19] Y. Le Floch and \u00c1. Pelayo, Symplectic geometry and spectral properties of classical and quantum coupled angular momenta, arXiv: 1607.05419.Google Scholar [20] Y. Le Floch,\u00a0\u00c1. Pelayo\u00a0and\u00a0S. V\u0169 Ng\u1ecdc, Inverse spectral theory for semiclassical Jaynes-Cummings systems, Math. Ann., 364 (2016), 1393-1413. doi:\u00a010.1007\/s00208-015-1259-z. Google Scholar [21] J. Marsden and T. Ratiu, Introduction to Mechanics and Symmetry, vol. 17 of Texts in Applied Mathematics, 2nd edition, Springer-Verlag, New York, 1999, https:\/\/doi.org\/10.1007\/978-0-387-21792-5, A basic exposition of classical mechanical systems. doi:\u00a010.1007\/978-0-387-21792-5. Google Scholar [22] E. Miranda\u00a0and\u00a0N. T. Zung, Equivariant normal form for nondegenerate singular orbits of integrable Hamiltonian systems, Ann. Sci. \u00c9cole Norm. Sup. (4), 37 (2004), 819-839. doi:\u00a010.1016\/j.ansens.2004.10.001. Google Scholar [23] J. Palmer, Moduli spaces of semitoric systems, J. Geom. Phys., 115 (2017), 191-217. doi:\u00a010.1016\/j.geomphys.2017.02.008. Google Scholar [24] \u00c1. Pelayo, Hamiltonian and symplectic symmetries: An introduction, Bull. Amer. Math. Soc. (N.S.), 54 (2017), 383-436. doi:\u00a010.1090\/bull\/1572. Google Scholar [25] \u00c1. Pelayo,\u00a0T. Ratiu\u00a0and\u00a0S. V\u0169 Ng\u1ecdc, The affine invariant of proper semitoric integrable systems, Nonlinearity, 30 (2017), 3993-4028. doi:\u00a010.1088\/1361-6544\/aa8aec. Google Scholar [26] \u00c1. Pelayo\u00a0and\u00a0S. V\u0169 Ng\u1ecdc, Semitoric integrable systems on symplectic 4-manifolds, Invent. Math., 177 (2009), 571-597. doi:\u00a010.1007\/s00222-009-0190-x. Google Scholar [27] \u00c1. Pelayo\u00a0and\u00a0S. V\u0169 Ng\u1ecdc, Constructing integrable systems of semitoric type, Acta Math., 206 (2011), 93-125. doi:\u00a010.1007\/s11511-011-0060-4. Google Scholar [28] \u00c1. Pelayo and S. V\u0169 Ng\u1ecdc, Symplectic theory of completely integrable Hamiltonian systems, Bull. Amer. Math. Soc. (N. S.), 48 (2011), 409-455. doi:\u00a010.1090\/S0273-0979-2011-01338-6. Google Scholar [29] \u00c1. Pelayo\u00a0and\u00a0S. V\u0169 Ng\u1ecdc, Hamiltonian dynamics and spectral theory for spin-oscillators, Comm. Math. Phys., 309 (2012), 123-154. doi:\u00a010.1007\/s00220-011-1360-4. Google Scholar [30] M. Petrera, Integrable Extensions and Discretizations of Classical Gaudin Models, PhD thesis, Dipartimento di Fisica, Universit\u00e1 degli Studi di Roma Tre, 2007.Google Scholar [31] T. Ratiu, C. Wacheux and N. T. Zung, Convexity of singular affine structures and toric-focus integrable hamiltonian systems, arXiv: 1706.01093.Google Scholar [32] D. A. Sadovski\u00ed\u00a0and\u00a0B. I. Z\u0125ilinski\u00ed, Monodromy, diabolic points, and angular momentum coupling, Phys. Lett. A, 256 (1999), 235-244. doi:\u00a010.1016\/S0375-9601(99)00229-7. Google Scholar [33] M. Symington, Four dimensions from two in symplectic topology, in Topology and geometry of manifolds (Athens, GA, 2001), vol. 71 of Proc. Sympos. Pure Math., Amer. Math. Soc., Providence, RI, 2003,153-208. doi:\u00a010.1090\/pspum\/071\/2024634. Google Scholar [34] S. V\u0169 Ng\u1ecdc, On semi-global invariants for focus-focus singularities, Topology, 42 (2003), 365-380. doi:\u00a010.1016\/S0040-9383(01)00026-X. Google Scholar [35] S. V\u0169 Ng\u1ecdc, Syst\u00e9mes Int\u00e9grables Semi-Classiques: Du Local au Global, vol. 22 of Panoramas et Synth\u00e9ses [Panoramas and Syntheses], Soci\u00e9t\u00e9 Math\u00e9matique de France, Paris, 2006. Google Scholar [36] S. V\u0169 Ng\u1ecdc, Moment polytopes for symplectic manifolds with monodromy, Adv. Math., 208 (2007), 909-934. doi:\u00a010.1016\/j.aim.2006.04.004. Google Scholar [37] C. Wacheux, Syst\u00e9mes Int\u00e9grables Semi-toriques et Polytopes Moment, PhD thesis, Universit\u00e9 de Rennes 1, 2013.Google Scholar [38] J. Williamson, On the algebraic problem concerning the normal form of linear dynamical systems, Amer. J. Math., 58 (1936), 141-163. doi:\u00a010.2307\/2371062. Google Scholar\nAn image of the momentum map $(J_{(1, 2)}, H_{(s_1, s_2)})$ with the rank 0 points marked for varying values of $s_1, s_2\\in [0, 1]$. Notice that the coupled angular momenta system shown in Figure 3 is the bottom row of the system shown in this figure since the coupled angular momenta is the special case for which $s_2 = 0$\nLeft: a plot of the set $\\gamma$, which is the union of $\\gamma_{(S, N)}$ (blue) and $\\gamma_{(N, S)}$ (orange), see Equation (23). Right: Values of $(s_1, s_2)$ for which the system $(J_{(1, 2)}, H_{(s_1, s_2)})$ has focus-focus values at: only the point $(S, N)$ (blue), only the point $(N, S)$ (orange), or at both points (green). The system is degenerate on the black curves. Compare with Figure 1\nFour semitoric polygons associated to the system (1). The slanted edges all have slope $\\pm 1$. For each polygon the $x$-coordinates of the vertices, from left to right, are $-R_1-R_2$, $R_1-R_2$, $-R_1+R_2$, and $R_1+R_2$ (since we assume $R_1<R_2$)\nThe momentum map image for the coupled angular momenta system with the rank zero points marked. As the coupling parameter $t$ changes one of the rank zero points transitions from being elliptic-elliptic to being focus-focus and then back to elliptic-elliptic\nThis figure analyses the right hand side of Equation (22): The plot on the left shows the graph of the right hand side of Equation (22) which is always below $-1.06066$. The contour plot on the right displays the associated level sets\nCase $(e_1, e_2) = (1, 1)$: on the left, the graph of $(s_1, s_2) \\mapsto \\Delta _{((s_1, s_2), (1, 1))}$ (orange) and a plane through zero (blue) are displayed. On the right, the associated level sets of $(s_1, s_2) \\mapsto \\Delta _{((s_1, s_2), (1, 1))}$ are shown\nCase $(e_1, e_2) = (1, -1)$: on the left, the graph of $(s_1, s_2) \\mapsto \\Delta _{((s_1, s_2), (1, -1))}$ (orange) and a plane through zero (blue) are displayed. On the right, the associated level sets of $(s_1, s_2) \\mapsto \\Delta _{((s_1, s_2), (1, -1))}$ are shown\n [1] Lingling Liu, Bo Gao, Dongmei Xiao, Weinian Zhang. Identification of focus and center in a 3-dimensional system. Discrete & Continuous Dynamical Systems - B, 2014, 19 (2) : 485-522. doi: 10.3934\/dcdsb.2014.19.485 [2] Dmitry Treschev. A locally integrable multi-dimensional billiard system. Discrete & Continuous Dynamical Systems - A, 2017, 37 (10) : 5271-5284. doi: 10.3934\/dcds.2017228 [3] R\u0103zvan M. Tudoran, Anania G\u00eerban. On the Hamiltonian dynamics and geometry of the Rabinovich system. Discrete & Continuous Dynamical Systems - B, 2011, 15 (3) : 789-823. doi: 10.3934\/dcdsb.2011.15.789 [4] Misha Bialy, Andrey E. Mironov. Rich quasi-linear system for integrable geodesic flows on 2-torus. Discrete & Continuous Dynamical Systems - A, 2011, 29 (1) : 81-90. doi: 10.3934\/dcds.2011.29.81 [5] Qiaoyi Hu, Zhijun Qiao. Analyticity, Gevrey regularity and unique continuation for an integrable multi-component peakon system with an arbitrary polynomial function. Discrete & Continuous Dynamical Systems - A, 2016, 36 (12) : 6975-7000. doi: 10.3934\/dcds.2016103 [6] Yongsheng Mi, Boling Guo, Chunlai Mu. Well-posedness and blow-up scenario for a new integrable four-component system with peakon solutions. Discrete & Continuous Dynamical Systems - A, 2016, 36 (4) : 2171-2191. doi: 10.3934\/dcds.2016.36.2171 [7] Xiuting Li, Lei Zhang. The Cauchy problem and blow-up phenomena for a new integrable two-component peakon system with cubic nonlinearities. Discrete & Continuous Dynamical Systems - A, 2017, 37 (6) : 3301-3325. doi: 10.3934\/dcds.2017140 [8] Oksana Koltsova, Lev Lerman. Hamiltonian dynamics near nontransverse homoclinic orbit to saddle-focus equilibrium. Discrete & Continuous Dynamical Systems - A, 2009, 25 (3) : 883-913. doi: 10.3934\/dcds.2009.25.883 [9] \u00c1lvaro Pelayo, San V\u0171 Ng\u1ecdc. First steps in symplectic and spectral theory of integrable systems. Discrete & Continuous Dynamical Systems - A, 2012, 32 (10) : 3325-3377. doi: 10.3934\/dcds.2012.32.3325 [10] Miguel-C. Mu\u00f1oz-Lecanda. On some aspects of the geometry of non integrable distributions and applications. Journal of Geometric Mechanics, 2018, 10 (4) : 445-465. doi: 10.3934\/jgm.2018017 [11] Ernest Fontich, Pau Mart\u00edn. Arnold diffusion in perturbations of analytic integrable Hamiltonian systems. Discrete & Continuous Dynamical Systems - A, 2001, 7 (1) : 61-84. doi: 10.3934\/dcds.2001.7.61 [12] Carles Sim\u00f3, Dmitry Treschev. Stability islands in the vicinity of separatrices of near-integrable symplectic maps. Discrete & Continuous Dynamical Systems - B, 2008, 10 (2&3, September) : 681-698. doi: 10.3934\/dcdsb.2008.10.681 [13] V. Barbu. Periodic solutions to unbounded Hamiltonian system. Discrete & Continuous Dynamical Systems - A, 1995, 1 (2) : 277-283. doi: 10.3934\/dcds.1995.1.277 [14] Eric Beno\u00eet. Bifurcation delay - the case of the sequence: Stable focus - unstable focus - unstable node. Discrete & Continuous Dynamical Systems - S, 2009, 2 (4) : 911-929. doi: 10.3934\/dcdss.2009.2.911 [15] Sebasti\u00e1n Ferrer, Francisco Crespo. Parametric quartic Hamiltonian model. A unified treatment of classic integrable systems. Journal of Geometric Mechanics, 2014, 6 (4) : 479-502. doi: 10.3934\/jgm.2014.6.479 [16] Rongmei Cao, Jiangong You. The existence of integrable invariant manifolds of Hamiltonian partial differential equations. Discrete & Continuous Dynamical Systems - A, 2006, 16 (1) : 227-234. doi: 10.3934\/dcds.2006.16.227 [17] Colin Rogers, Tommaso Ruggeri. q-Gaussian integrable Hamiltonian reductions in anisentropic gasdynamics. Discrete & Continuous Dynamical Systems - B, 2014, 19 (7) : 2297-2312. doi: 10.3934\/dcdsb.2014.19.2297 [18] Alicia Cordero, Jos\u00e9 Mart\u00ednez Alfaro, Pura Vindel. Bott integrable Hamiltonian systems on $S^{2}\\times S^{1}$. Discrete & Continuous Dynamical Systems - A, 2008, 22 (3) : 587-604. doi: 10.3934\/dcds.2008.22.587 [19] Fuzhong Cong, Jialin Hong, Hongtian Li. Quasi-effective stability for nearly integrable Hamiltonian systems. Discrete & Continuous Dynamical Systems - B, 2016, 21 (1) : 67-80. doi: 10.3934\/dcdsb.2016.21.67 [20] Daniel Fusca. The Madelung transform as a momentum map. Journal of Geometric Mechanics, 2017, 9 (2) : 157-165. doi: 10.3934\/jgm.2017006\n\n2018\u00a0Impact Factor:\u00a00.525","date":"2019-08-24 10:03:36","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\": 0.7683512568473816, \"perplexity\": 2931.6598571998047}, \"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-2019-35\/segments\/1566027320156.86\/warc\/CC-MAIN-20190824084149-20190824110149-00463.warc.gz\"}"}
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This tutorial will teach you how to make a simple winter scene, including some fir trees and a snowman. It is supposed to show you how to use the most common Blender functionalities (i.e. modelling, adding materials and textures) and shortcuts.
Today we'll be creating a hillside scene in Blender.
Part 1 of this tutorial will demonstrate how to create basic cartoon eyeballs in Blender Creator 2.23 and deform the eyeballs with a lattices.
The goal of this tutorial is to make a Pixar-looking eye.
One of Blender's features is its ability to use animations as a tool for modeling. Strange as this may sound, it is often a very powerful technique. In this tutorial I describe building a spiral stair. Using traditional methods, this would require many steps to model, but once you grasp how a moving object can represent the final result, it becomes a lot easier.
I have always been fond of castles. I simply can't resist the urge to create one in Blender every now and then. This tutorial will introduce you to basic modeling techniques in Blender.
In this tutorial you'll be making your own cartoon style mouse head like you see on the image, using the blender Nurbs feature.
In this one, I'll be explaining the skinning tool of Blender. The idea is simple: using a number of cross-sections, I will create an organic-looking cave.
In this tutorial, I'll explain how you can create a die in Blender using only meshes.
Creating a model of a dolphin by using nurbs and deformations.
I first modeled this Toyota Celica to learn the modeling of one of my favorite machines and to increase my modeling skills. Most will agree that modeling a complete car for the first time is quite complex. What follows is based upon that modeling experience, with the hope that it will benefit others.
The techniques described here are not the only way to create a chain, but it is an effective method. This tutorial requires a working knowledge of blender.
In this tutorial, we'll be playing around with curves and procedural objects a bit. Starting off with only a handfull of shapes, we will make an animation of a ride through an abandoned mine. The technique used will allow you to make changes to the shape of the path easily and not have to worry about recreating stuff like the motion from scratch.
Using mesh editing and subsurfaces to model a racing car.
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Call for Papers - istorians without Borders: Writing Histories of International Organizations | Calling All Papers!
Ever since the paradigm of 'globalization' has found its way into the field of history, ways of writing histories beyond borders have proliferated. Today, historians no longer need to justify enlarging their geographical scope beyond the national, but it can nonetheless be a daunting task to decide on how to do this. While we are going beyond borders, the choice for a translocal, transnational, transregional or global history still reveals our preference for a certain scale. Methodologically, our toolbox now offers us concepts such as comparisons, transfers, connections, entanglements and circulations. As different approaches focus on different concepts, choosing one approach often entails a rejection of other possible approaches. Transnational historians will distance themselves from comparative history; global history, as any global historian will tell you, is not the same as world history. The further we seem to get in advancing the call for breaking with our 'methodological nationalism', the more we seem to split up into different subfields, where fruitful dialogue becomes increasingly difficult. The purpose of this workshop is to open up this dialogue, to see what specific advantages different approaches can offer and how they can be best put to use.
What specific advantages do different approaches bring to the history of international organizations?
Are these approaches mutually exclusive, or do we need to combine different perspectives and concepts?
What are some of the methodological challenges in writing the history of international organizations, in terms of analyzing connections, entanglements, comparisons, etc.?
What are some of the practical challenges in writing the history of international organizations, in terms of mobility, language barriers, cultural sensitivity, etc.?
How can we deal with the fact that levels can be used both as analytical concepts (used by the historian) and as historical concepts (used by the historical actors)?
How can we deal with different uses of terms like international, national, local, e.g. as level, geographical or spatial unit or loyalty of a historical actor?
How can we deal with the (hidden) hierarchy of terms or levels like global, national, etc.?
The workshop will offer a combination of a master class, keynote lectures, and roundtable discussions. It will start on 22 March in the afternoon, with a master class by Davide Rodogno (The Graduate Institute, Geneva), followed by a keynote lecture by Corinne Pernet (University of Geneva). The second day (23 March) will consist of roundtable sessions, where participants present their research and enter into discussion. Senior researchers will chair these sessions and Kiran Patel(Maastricht University) will deliver a closing keynote.
Please send an abstract of max. 500 words and a short CV to the following email address: rethinkingdisability@hum.leidenuniv.nl by 13 November 2017. Questions to the organizers can be sent using the same address. Authors will be notified regarding the acceptance of their contribution by 20 November. Invited participants will be expected to submit a short draft version of a more substantial paper two weeks prior to the event, which will be circulated among all other participants. Participants who are accepted to present their paper are also automatically accepted to participate in the master class. If you are unable or do not wish to attend the master class, kindly indicate this in your application.
The workshop is initiated and hosted by the research team of the ERC project 'Rethinking Disability: the Global Impact of the International Year of Disabled Persons (1981) in Historical Perspective', based in the Institute for History at Leiden University. It is supported by the Huizinga Institute, the national Dutch research network for Cultural History.
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\section{\label{}}
The family of new superconductors $A$Os$_2$O$_6$ ($A$ = K, Rb, and Cs) with the
pyrochlore structure exhibits many exotic properties. For instance, the
superconducting transition temperature $T_c$ increases substantially with
decreasing radius of $A$ ions ($T_c$ = 3.3, 6.3, and 9.6 K for $A$ = Cs, Rb, and K,
respectively \cite{Yonezawa1,Yonezawa2,Bruhwiler1,Yonezawa3}) and shows
non-monotonic pressure dependence \cite{Muramatsu}, in contradiction to what would
be expected from the change of density of states alone. Among this family
KOs$_2$O$_6$ is most anomalous. The electrical resistivity shows strong concave
$T$-dependence down to $T_c$ \cite{Yonezawa1,Hiroi2}, in contrast to the normal
$T^2$-behavior in Rb and Cs compounds \cite{Yonezawa2,Bruhwiler1,Yonezawa3}. The
$T$-linear coefficient of the specific heat $\gamma $ = 70~mJ/K$^2$mol \cite{Hiroi3,Bruhwiler2} is highly
enhanced over the value obtained from band calculations 10-11~mJ/K$^2$mol \cite{Kuns1,Saniz}.
These anomalies imply large density of low energy excitations, which was originally ascribed to magnetic
frustration inherent to the pyrochlore lattice, a network of corner-shared tetrahedra formed by Os atoms.
However, the Pauli-like susceptibility \cite{Hiroi3} excludes local moments on the Os sites. How the frustrated lattice
geometry affects properties of itinerant electrons is still an open issue. On the other hand, there is now
a growing body of experimental evidence for anharmonic motion of isolated alkaline ions in an oversized
cage of Os-O network. The specific heat data show existence of low frequency Einstein modes in addition
to the electronic and the Debye contributions for all members of $A$Os$_2$O$_6$ \cite{Hiroi3,Bruhwiler2,Hiroi1}.
The X-ray results revealed huge atomic displacement parameter for the K sites in KOs$_2$O$_6$ \cite{Yamaura}.
Recently a second phase transition at $T_p$ = 7.5 K was observed for high quality single crystals of
KOs$_2$O$_6$ \cite{Hiroi2}. Since the transition is of first order and persists into the normal state
in high magnetic fields, it is likely a structural transition associated with the K ion dynamics.
Calculation of effective ionic Hamiltonian also indicates instability of K ions in a highly anharmonic
potential \cite{Kuns2}, which might be the origin of various anomalies. Oscillations of isolated ions in a
large space have been discussed also in other materials such as clathrate \cite{Keppens} and
skutterudite \cite{Paschen} compounds and commonly called ``rattling''.
In this Letter, we report results of nuclear magnetic resonance (NMR) experiments on the $^{39,41}$K
nuclei (nuclear spin $I$ = 3/2) and $^{17}$O nuclei ($I$ = 5/2) in KOs$_2$O$_6$.
We found that the nuclear relaxation at the K sites is caused entirely by phonons via the electric
quadrupole interaction. From a phenomenological analysis we propose that rattling phonons undergo
a crossover from underdamped to overdamped behavior with increasing temperature.
Strong electron-phonon coupling is indicated by rapid reduction of the damping below $T_c$.
On the other hand, the relaxation rate at the O sites is sensitive to electronic excitations. Substantial
damping of quasiparticles due to rattling phonons near $T_c$ is indicated by suppression of the
Hebel-Slichter peak at the O sites. A part of the $^{39}$K NMR results has been discussed in terms of spin
fluctuations in a previous publication~\cite{Arai}, which must now be discarded.
Two powder samples of KOs$_2$O$_6$ were used in the present experiment. The sample A was prepared by heating
a mixture of KO$_2$ and OsO$_2$ with Ag$_2$O as an oxidizing agency~\cite{Yonezawa1}.
The sample B was enriched with $^{17}$O by heating KO$_2$ and Os metal in two steps in oxygen atmosphere
containing 45 \% $^{17}$O. Both samples show identical $T_c$ of 9.6 K. Note that the second transition
at 7.5~K has never been observed in powder samples. The nuclear spin-lattice relaxation
rate $1/T_1$ at the $^{39,41}$K sites was measured by the saturation recovery method. Although the K sites
have tetrahedral (T$_d$) symmetry, hence should not suffer quadrupole broadening, the $^{39}$K NMR spectra
consist of a narrow central line (0.3 kHz HWHM at 8.5T) and a slightly broad satellite line (10 kHz HWHM)
probably due to imperfections. The line shape shows no appreciable temperature ($T$) dependence
above $T_{c}$. The entire spectrum was easily saturated by rf comb-pulses,
resulting in single exponential recovery of the spin echo intensity. The $^{17}$O NMR spectra show a quadrupole
broadened powder pattern. To determine $1/T_1$, the recovery of the spin-echo intensity of the central line
as a function of time $t$ after the inversion pulse was fit to the form \cite{Narath}
$M_{eq}-M_{0}\{U\exp(-t/T_1)+V\exp(-6t/T_1)+(1-U-V)\exp(-15t/T_1)\}$ with $U$ and $V$ fixed to the same values
at all temperatures.
The $T$-dependences of $1/(T_1T)$ at the $^{39}$K sites is shown in Fig.~1. The inset of Fig.~1(a)
displays the isotopic ratio $^{39}T_1/^{41}T_1$ for the sample A. Surprisingly, the isotopic ratio coincides with the
squared ratio of the nuclear quadrupole moments $(^{41}Q/^{39}Q)^2 =$ 1.48 rather than the nuclear magnetic
moments $(^{41}\gamma /^{39}\gamma )^2$ = 0.30 in a wide $T$-range 6-100~K. Thus the relaxation is entirely
caused by fluctuations of electric field gradient (EFG) as opposed to magnetic fluctuations. Since any active electronic
states should be dominantly $s$-like at the K sites with negligibly small quadrupole coupling to K nuclei,
it must be phonons that causes the nuclear relaxation.
\begin{figure}[b]
\includegraphics[width=7.5cm]{fig1
\caption{\label{fig:fig1} $T$-dependence of $1/(T_1T)$ at the $^{39}$K sites for the two samples
at different magnetic fields. The data for a wide $T$-range are shown in (a). The solid line shows
the calculated result of Eq. (5) as described in the text. The inset shows the isotopic ratio
of $1/T_1$. The dotted (dashed) line indicates the squared ratio of the nuclear
quadrupole (magnetic) moments. The data in the low $T$ region are expanded in (b). The arrows
show the superconducting $T_c$ at the respective fields.}
\end{figure}
The prominent features of the data in Fig.~1 are: (1) a peak near 13~K (16~K) for the sample B (A),
(2) approximate constant behavior at high temperatures, (3) rapid decrease at low temperatures,
(4) a clear kink at the superconducting $T_c$ and sudden decrease below $T_c$.
The last point indicates strong influence of superconductivity on the phonon dynamics,
a direct evidence for strong electron-phonon coupling. The two samples show slightly
different results near the peak temperature. Although the reason is not understood clearly, this may arise from
different degrees of imperfection or hydration \cite{Hiroi3}.
For nuclei with spin 3/2, the transition probability $W_q$ between two nuclear levels $|I_z = m \rangle$ and
$|I_z = m \pm q \rangle$ (q = 1 or 2) due to quadrupole coupling is given by the correlation function of EFG as
\cite{Abragam}
\begin{eqnarray}
W_q=\frac{1}{12}\Big(\frac{eQ}{\hbar}\Big)^2\int_{-\infty}^{\infty}\langle [V_{+q}(t),V_{-q}(0)] \rangle e^{iq\omega _Lt}dt,
\end{eqnarray}
where $\omega _L$ is the nuclear Larmor frequency and $[A, B]$ stands for
$(AB+BA)/2$. The irreducible components of the
EFG tensor are defined as $V_{\pm 1} = V_{xz} \pm iV_{yz}$, $V_{\pm 2} = (V_{xx} - V_{yy} \pm 2iV_{xy})/2$, where
$V_{xy} = \partial^2 V/\partial x\partial y$ etc. is the second derivative of the electrostatic potential at the nucleus. For simplicity we assume spherically symmetric
EFG fluctuations. Then $W_1 = W_2 \equiv W$ and the relaxation rate is given as $1/T_1 = 2W$.
In diamagnetic insulators phonons are usually the dominant source of relaxation for quadrupolar nuclei.
The ionic motion modulates the EFG,
\begin{eqnarray}
V_{\pm q}=V_{\pm q,0} + \frac{\partial V_{\pm q}}{\partial u}u +
\frac{1}{2}\frac{\partial ^2V_{\pm q}}{\partial u^2}u^2+\cdot \cdot \cdot ,
\end{eqnarray}
where $u$ is the ionic displacement from the equilibrium position expressed as a linear combination of the phonon
creation ($a^\dagger $) and annihilation ($a$) operators. The second term inserted into Eq.~(1) leads to
the \textit{direct process} \cite{Abragam}, which is expressed by the one phonon correlation
function or the imaginary part of the susceptibility,
\begin{eqnarray}
W_q \propto k_BT\frac{\mathrm{Im}\chi (q\omega _L)}{q\hbar \omega _L},
\end{eqnarray}
provided $q \hbar \omega_{L} \ll k_{B}T$. For harmonic phonons with infinite life time, the contribution from the direct process
is negligible since $\omega _L$ ($\sim$ 10 MHz) is many orders of magnitude smaller than the typical phonon frequency and the
phonon density of states at $\omega _L$ is practically zero.
On the other hand, the third term in Eq.~(2) leads to the \textit{two phonon Raman process}, in which a nuclear transition occurs by
absorbing one phonon and emitting another phonon \cite{Abragam}. Usually this is by far the dominant process expressed
as $1/T_1 \propto \int \{ \rho (\omega) \}^{2} n(\omega) (n(\omega) + 1) |A(\omega)|^{2} d\omega$, where $n(\omega)$ is the
Bose factor, $\rho (\omega)$ is the phonon density of states, and $|A(\omega)|$ is the transition matrix element. One can
see $1/T_1 \propto T^2$ at high temperatures where $n(\omega) \sim T/\omega $. At low temperatures, $1/T_1$
decreases monotonically and approaches either to $T^7$-dependence for acoustic phonons or activated behavior for optical
phonons \cite{Abragam}. Our results in Fig.~1 are, however, in strong contradiction with such behavior.
Since dominance of phonons for nuclear relaxation is extremely rare in metals, we suppose that rattling makes a special case
for KOs$_2$O$_6$. The distinct features of rattling are its low energy scale and strong anharmonicity,
which should lead to damping. We expect that damping-induced broadening of the phonon spectrum peaked at a very
low frequency, will bring substantial spectral weight at zero frequency so that the direct process (Eq.~3) becomes important.
The peak in $1/(T_1T)$ is then naturally explained as follows. Let us assume a low frequency
Einstein mode at $\omega _0$. At low temperatures, the spectrum $\mathrm{Im}\chi(\omega )/\omega $ is sharp, therefore, the weight
at $\omega _L \approx 0$ should be small. As the spectral width increases with increasing temperature, the weight at $\omega = 0$
will grow and take a maximum when the width becomes comparable to $\omega _0$. Further broadening, however, will spread
the spectrum over higher frequencies and reduce the weight at $\omega = 0$. (See the inset of Fig.~2.) Thus the peak in $1/(T_1T)$
can be reproduced if the rattling undergoes a crossover from underdamped to overdamped behavior with increasing temperature.
The contribution from the Raman process should be also influenced by damping \cite{Rubinstein}. However, we could not reproduce the peak in $1/(T_1T)$ by the
Raman process without fine tuning the $T$-dependence of the damping. We consider
that the Raman process is unimportant. The details will be discussed elsewhere \cite{Yoshida}.
\begin{figure}[b]
\includegraphics[width=7.5cm]{fig2
\caption{\label{fig:fig2} The normalized damping $ \gamma = \Gamma /\omega _0$ is plotted against $1/T$.
The arrows correspond to $T_c$ at 5~T and 8.5~T. The inset shows the phonon line shape Im$\chi (\omega )/\omega$
calculated from Eq.~(4) for several values of $\gamma$.}
\end{figure}
In order to quantify the above picture, we employ a suitable phenomenological model describing strongly damped oscillators.
A commonly used expression is the damped harmonic oscillator model,
$\mathrm{Im}\chi (\omega )/\omega = \Gamma /\{(\omega _0^2 - \omega ^2)^2 + \Gamma ^2\omega ^2\}$, which leads to
$1/(T_1T)\propto \Gamma /\omega _0^4$ by taking the limit $\omega _L \rightarrow 0$ in Eq. (3). However, this model is unable
to reproduce the observed peak for any monotonic change of the damping $\Gamma $. In the damped harmonic oscillator model,
only the friction or the momentum relaxation is considered as the sources of damping. It was argued, however, that in
the collision dominated regime one must also consider the jump of the spatial coordinate or the amplitude relaxation \cite{Silverman,Takagi}.
One then obtains the van Vleck-Weisskopf formula \cite{VanVleck}, which should be suitable for
strongly damped anharmonic phonons,
\begin{eqnarray}
\frac{\mathrm{Im}\chi (\omega )}{\omega } = \frac{\Gamma /\omega _0 }{(\omega + \omega _0)^2+\Gamma ^2} +
\frac{\Gamma /\omega _0 }{(\omega - \omega _0)^2+\Gamma ^2}.
\end{eqnarray}
From Eqs.~(4) and (3) we obtain in the limit $\omega _L \rightarrow 0$,
\begin{eqnarray}
\frac{1}{T_1T} \propto \frac{1}{\omega _0^2}\frac{\gamma }{1+\gamma ^2},
\end{eqnarray}
where $\gamma = \Gamma /\omega _0$. For a fixed value of $\omega _0$, this formula has a maximum at $\gamma = 1$, which
corresponds to the observed peak. By identifying the $T$-ranges below or above the peak temperature with
the underdamped ($\gamma \leq 1$) or the overdamped ($\gamma \geq 1$) regions and assuming that $\omega _0$ is constant,
we extracted the T-dependence of $\gamma $ from the experimental data of $1/(T_1T)$ as shown in Fig. 2. The result above
$T_c$ is well represented by an activation law $\gamma = \gamma _0$exp$(-E/k_BT)$ with $\gamma _0 = 5$ and $E =$ 20 K (the solid line).
Thus our analysis predicts extremely overdamped behavior at high temperatures. Note that the phonon frequency $\omega _0$ cannot be
determined from our analysis. The solid line in Fig. 1 shows the $T$-dependence of $1/(T_1T)$ calculated from Eq. (5) and
this activation law. A possible interpretation of the activated behavior is that $\Gamma $ represents the hopping frequency
of the K ions among different metastable positions in highly anharmonic potential \cite{Kuns2} and the activation energy
corresponds to the potential barrier between them.
\begin{figure}[b]
\includegraphics[width=7.5cm]{fig3
\caption{\label{fig:fig3} $T$-dependence of $1/(T_1T)$ at the $^{17}$O
sites at 2 T (solid circles) and 8.5 T (open squares). The arrows indicate $T_c$ at the respective fields.
The inset shows the fitting to a power-law $1/T_1\propto T^\alpha $ with $\alpha = 4.7$ (solid line)
and an activation law $1/T_1\propto \exp(\Delta /T)$ with $\Delta = 17$ K
(dashed line) for the data at 2 T below $T_c$.}
\end{figure}
We have also measured the relaxation rate for $^{85}$Rb and $^{87}$Rb
nuclei in RbOs$_2$O$_6$ and separated the phononic and magnetic contributions \cite{Yoshida}. The phonon
contribution to $1/(T_1T)$ in RbOs$_2$O$_6$ does not show a peak, indicating underdamped behavior in the whole $T$-range
below the room temperature. The $T$-dependence of $\gamma$ in RbOs$_2$O$_6$ obtained from similar analysis
is also compatible with an activation law with $E$=12 - 20~K, although the accuracy is limited.
A remarkable feature of the results in Fig. 2 is the sudden reduction of $\gamma $ below the superconducting $T_c$.
This indicates that the damping of rattling phonons near $T_c$ is primarily caused by electron-phonon interaction
and the phonon life time is enhanced by opening of the superconducting gap. Therefore, $\omega_{0}$ should be
smaller than 2$\Delta \sim 50$~K \cite{Hiroi3}, where $\Delta$ is the superconducting gap.
We now discuss the results at the O sites. Figure 3 shows the $T$-dependence of $1/(T_1T)$ at the O sites at the fields
of 2 T and 8.5 T. Unlike the result for the K sites, $1/(T_1T)$ is only weakly $T$-dependent above $T_c$, approximately
consistent with the Korringa relation. Thus the spin dynamics of conduction electrons are probed at the O sites.
This is consistent with the Os-5$d(t_{2g})$ and O-2$p$ hybridized nature of the conduction band \cite{Kuns2,Saniz}.
However, the weak $T$-dependence of $1/(T_1T)$ in the normal state is still anomalous and suggests development of
modest spin correlation.
Below $T_{c}$, it is known that $1/(T_1T)$ shows a peak near $T=0.9T_{c}$ (the Hebel-Slichter peak \cite{Hebel})
in weak coupling superconductors with an isotropic gap. This is not the case for the data in Fig. 3; $1/(T_1T)$ decreases
gradually below $T_{c}$, indicating that the Hebel-Slichter peak is strongly suppressed.
At lower temperatures, $T$-dependence of $1/T_1$ can be fit either by a power law $1/T_1 \propto T^\alpha $
with $\alpha \sim 5$ or by an activation law $1/T_1 \propto \mathrm{exp}(-\Delta /T)$ with $\Delta = 17 K$.
Since the upper critical field $H_{c2}$ is about 30 T at $T = 0$ \cite{Ohmichi,Shibauchi}, pair breaking effects
of magnetic field should not be strong at 2 T \cite{Pennington}.
The behavior at low temperatures rules out highly anisotropic gap structure such as line nodes, which leads to $T^3$ dependence
of $1/T_1$. This is consistent with the field independent thermal conductivity showing an isotropic gap \cite{Kasahara}.
Therefore, suppression of the Hebel-Slichter peak cannot be ascribed to a highly anisotropic gap. It appears more likely
due to inelastic scattering of quasiparticles by phonons, which broadens the superconducting density of states as was
proposed for the case of TlMo$_6$Se$_{7.5}$ \cite{Kitaoka}.
This has been also predicted by numerical calculations based on the Eliashberg theory \cite{Allen,Akis}.
The calculation by Akis {\it et al}. \cite{Akis}, for example, shows that the Hebel-Slichter peak disappears when
$k_BT_c/\hbar \omega _{ln}$ is larger than $0.2 \sim 0.3$, where $\omega _{ln}$ is the Allen-Dynes parameter representing
the typical phonon frequency. This is compatible with our previous conclusion that the frequency of the rattling phonon is
smaller than $2\Delta \sim 5k_BT_c$. Whether the rattling works as a pairing mechanism or acts as a pair breaker is still
an open question.
In conclusion, our analysis of the $1/(T_1T)$ data at the K sites indicates a crossover of rattling from the low $T$ underdamped to
high $T$ overdamped behavior near 13 - 16~K. The rapid decrease of $1/(T_1T)$ at the K sites below $T_c$ indicates
strong electron-phonon coupling and enhancement of the lifetime of rattling phonons. Hence its frequency must be smaller
than $2\Delta $. The results at the O sites rule out strongly anisotropic gap. The absence of the Hebel-Slichter peak at the O
sites is also compatible with strong electron-phonon coupling.
We thank T. Kato, Y. Takada, H. Maebashi K. Ishida, H. Harima, H. Fukuyama, Y. Matsuda, and
S. Tajima for valuable discussions. This work was supported by a Grant-in-Aid for Scientific
Research on Priority Areas
(No. 16076204 ``Invention of Anomalous Quantum Materials'') from the MEXT Japan.
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{
"redpajama_set_name": "RedPajamaArXiv"
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With a complimentary airport shuttle, continental breakfast and wireless internet access, it's no wonder the Holiday Inn Express LaGuardia Airport is one of the area's most popular hotels among our guests. The six-story Holiday Inn Express has 81 rooms, most of them non-smoking. Expect up-to-date amenities like flat-panel TVs with cable channels and free Wi-Fi connections. Well-lighted desks and coffeemakers transform your room into a virtual office. A compact fitness center has cardio machines so you can stick to your workout plan. The hotel's signature cinnamon rolls and other hot options make the free buffet-style breakfast a treat. If you're in a hurry, grab a to-go bag of morning munchies. There is a free shuttle to the airport. Proximity to the airport is why many people choose the Holiday Inn Express, but the nearby Van Wyck, Whitestone and Long Island expressways can get you anywhere in the area. Flushing Meadows, one of the city's most expansive parks, is a stone's throw from the hotel. Arthur Ashe Stadium, the venue for the U.S. Open, and Citi Field, home of the New York Mets, are both about two miles north. LaGuardia Airport is four miles north, JFK is nine miles south and Newark Airport is 22 miles west. "Right on the freeway," say our guests, who praise the "moderate" prices at Holiday Inn Express LaGuardia Airport and appreciate the "quiet" atmosphere and the "comfortable" rooms.
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It's that time of year again. No, not the holidays; end-of-the-year salary surveys, of course. Like clock work, the Information Architecture Institute released their 2007 IAI Salary Survey and it looks like the gender pay gap is narrowing, if ever so slightly. This is the first year the IAI chose to include a question on gender, which they say "provided the most interesting result coming out of our salary survey to date."Of the 575 respondents, not only were there almost as many females as males defining themselves as Information Architects, female IAs on average make slightly more than males. Even when upper and lower salary categories were removed from the calculation, women still held a lead in salary earned; yet not enough to be considered "statistically significant." The IAI giveth and the IAI taketh away. The survey also indicated that in the top salary categories, men outnumber women 2:1. If on a whole, women as a group are making more money as IAs than men are why are there more men in the top salary group? It is suggested that advanced education makes men more profitable as well as having an entrepreneurial spirit. Of respondents whose titles were President or Principal, eight were men and only three were women. Next year, the IAI plans to ask about the academic areas the higher degrees represent, and how many people in these higher positions are running their own businesses. Despite the awkward findings, it's still heartwarming to see women competitive in the field of IA, especially when we've learned that other fields are losing women. So, do your part. Visit our content management industry job board today and climb the career ladder to success! For all the juicy details, you can access the entire full 2007 IAI Salary Survey results and analysis on the IAI website.
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"redpajama_set_name": "RedPajamaC4"
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The Application has failed to start because side-by-side configuration is incorrect when trying to run Microsoft Office.
How to setup and configure Exchange 2013 to accept Email for your specified domain.
How to recover lost iPhone, iPad and any data you have lost from your Apple device even if you have not taken a backup – with Dr Fone.
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"redpajama_set_name": "RedPajamaC4"
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Colonel Thomas Wildman (1787 – 1859) was a British Army officer during the Napoleonic Wars, a draftsman, and landowner.
Life
He was the eldest son of Thomas Wildman of Bacton Hall, Suffolk, by Sarah, daughter of Henry Hardinge, of Durham. A nephew of the political reformer John Horne Tooke and friend of Lord Byron at Harrow, Wildman purchased a cornetcy in the 7th Light Dragoons in 1808 and later the same year he was promoted lieutenant without purchase. At the Battle of Waterloo, he was an extra aide-de-camp to Lord Uxbridge. His letter after the battle described Uxbridge's wounding at the end of the battle (grapeshot to the knee) and the subsequent amputation. Wildman himself was slightly wounded in the battle. In 1816, he purchased a majority in the 2nd West India Regiment, and later transferred to the 9th Light Dragoons. In 1828, he became captain of the Mansfield Troop of the Nottinghamshire Yeomanry and a few months later became major-commandant of the Sherwood Rangers Yeomanry. He was promoted colonel in the Army in 1837. In 1840, he transferred to be lieutenant-colonel of the 5th Dragoon Guards.
He recorded his service during the Peninsular War in a diary, which was subsequently published.
The Wildman family had obtained Quebec Estate, a large sugar plantation in Jamaica, from William Beckford, who was having financial problems. The wealth generated from this plantation provided Wildman with the means to purchase Newstead Abbey in 1818 for £95,000 /> The Abbey was owned by his friend and old schoolmate, Lord Byron who, like Beckford, was having financial difficulties. Byron had been trying to sell the Abbey since 1812. Of the sale, Byron's half-sister Augusta said Wildman had "soul enough to value the dear Abbey..."
Although Wildman's purchase ended almost four centuries of Byron family ownership of the Abbey, he was considered to be the man who saved Byron's home. He spent £100,000 restoring it, hiring the architect John Shaw to make improvements. He also amassed a large collection of Byron memorabilia there. He served as High Sheriff of Nottinghamshire for 1821–22.
The Wildmans entertained many guests who wished to visit the home of Lord Byron, including Franz Liszt and Washington Irving. The Duke of Sussex visited annually for a six-week holiday with his chaplain. After Wildman's death, Louisa sold the Abbey to William Frederick Webb.
Personal life
In 1816, he married a Swiss woman, Louisa Preisig. They had no children.
References
1787 births
1859 deaths
People educated at Harrow School
West Indies merchants
Henry Paget, 1st Marquess of Anglesey
7th Queen's Own Hussars officers
West India Regiment officers
9th Queen's Royal Lancers officers
5th Dragoon Guards officers
British Army personnel of the Napoleonic Wars
Sherwood Rangers Yeomanry officers
High Sheriffs of Nottinghamshire
19th-century British businesspeople
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{
"redpajama_set_name": "RedPajamaWikipedia"
}
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Q: ElementTree does not seem to get some texts/elements in the tree I have a wikipedia dump that I want to parse, and I am having some difficulties/mysterious problems while using Python xml parser, ElementTree.
My recent problem is, ElementTree does not seem to find the texts that are actually there. This is an example data:
<page>
<title>Cengiz Han</title>
<ns>0</ns>
<id>10</id>
<revision>
<id>20337884</id>
<parentid>20218916</parentid>
<timestamp>2019-01-29T14:02:43Z</timestamp>
<contributor>
<username>CommonsDelinker</username>
<id>31545</id>
</contributor>
<comment>China_11b.jpg dosyası Map_of_China_1142.jpg ile değiştirildi</comment>
<model>wikitext</model>
<format>text/x-wiki</format>
<text xml:space="preserve">
...some long Genghis Khan stuff...
</text>
</page>
Now when I parse it with this:
for event, elem in et.iterparse('dataset/wiki_test', events=('start', 'end', 'start-ns', 'end-ns')):
if event == 'start':
if elem.tag == 'page':
if len(list(elem)) == 0:
continue
title = elem.find('title').text
if title == None or 'MediaWiki' in title:
elem.clear()
continue
wiki_id = elem.find('id')
if wiki_id == None:
elem.clear()
continue
wiki_id = wiki_id.text
revision = elem.find('revision')
if revision != None:
print(list(revision))
text = revision.find('text').text
print(text)
if text != None:
count += 1
titles += title + '\n'
page = {'wiki_id': wiki_id, 'title': title, 'text': text.text}
pages += json.dumps(page, ensure_ascii=False) + '\n'
elem.clear()
revision.find('text').text line seems to find no text for some elements, including the one above, and that some makes like one seventh of my data, which is annoying. This was also the case for page->id for some other entries, in which it claimed that element does not exist at all. I solved that problem by ignoring that ones, but I don't really want to do that, also this error does not make sense to me at all.
Here is another page, which works totally fine.
<page>
<title>Mustafa Suphi</title>
<ns>0</ns>
<id>22</id>
<revision>
<id>20077185</id>
<parentid>20017115</parentid>
<timestamp>2018-10-14T08:31:32Z</timestamp>
<contributor>
<username>Vikiçizer</username>
<id>90501</id>
</contributor>
<comment>/* top */düzeltme [[Vikipedi:AWB|AWB]] ile</comment>
<model>wikitext</model>
<format>text/x-wiki</format>
<text xml:space="preserve">
...some Mustafa Suphi stuff...
</text>
<sha1>m5finh6h2kr8h2fbtmsatp5fhz1siwq</sha1>
</revision>
</page>
What am I doing wrong?
A: You have posted two examples: "working" and "not working".
In the "not working" one there is no
</revision>
Are you sure this the XML you have or it is just copy & paste mistake.
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{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 3,898
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Dr. Schneider is a diagnostic cardiologist board certified in cardiology, echocardiography and nuclear cardiology. He graduated from the Tufts University School of Medicine in 1989 and continued his postgraduate education in internal medicine at the Lemuel Shattuck Hospital and Faulkner Hospitals in Boston. After serving as Chief Medical Resident at the Faulkner Hospital, he completed his cardiovascular medicine fellowship at the Lahey Clinic, where he concentrated in noninvasive imaging. During his fellowship, Dr. Schneider participated in research projects looking at the use of echocardiography to evaluate cardiac performance. His research has been presented at a number of national meetings, as well as being published in major cardiology journals.
Since joining Cardiovascular Associates of Rhode Island in 1996, he has continued to pursue his interest in non-invasive testing. Dr. Schneider is a fellow of the American College of Cardiology, American Society of Nuclear Cardiology, and American Society of Echocardiography.
His personal interests include spending time with his family, photography, hiking, and physical fitness.
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"redpajama_set_name": "RedPajamaC4"
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{"url":"https:\/\/tex.stackexchange.com\/questions\/345086\/xelatex-produces-gibberish-with-devanagari-document-in-texshop","text":"# XeLaTeX produces gibberish with Devanagari document in TeXShop\n\nI was able to process devanagari-documents into PDF successfully a couple of years ago on my Mac. I tried processing a simple document with xelatex but it doesn't work. My document is:\n\n\\documentclass{article}\n\\usepackage{fontspec}\n\\begin{document}\n\\setmainfont[Script=Devanagari]{Nakula}\n\u0928\u092e\u0938\u094d\u0924\u0947 \u0930\u0941\u0926\u094d\u0930 \u092e\u0928\u094d\u092f\u0935\u0947\n\\end{document}\n\n\nBelow is an image of all the fonts that are installed on my Mac (running Sierra 10.12.1)\n\nHere's log file displayed in TeXshop's console:\n\nThis is XeTeX, Version 3.14159265-2.6-0.99992 (TeX Live 2015) (preloaded format=xelatex)\nrestricted \\write18 enabled.\nentering extended mode\n(.\/Untitled.tex\nLaTeX2e <2015\/01\/01>\nBabel <3.9l> and hyphenation patterns for 79 languages loaded.\n(\/usr\/local\/texlive\/2015\/texmf-dist\/tex\/latex\/base\/article.cls\nDocument Class: article 2014\/09\/29 v1.4h Standard LaTeX document class\n(\/usr\/local\/texlive\/2015\/texmf-dist\/tex\/latex\/base\/size10.clo))\n(\/usr\/local\/texlive\/2015\/texmf-dist\/tex\/latex\/fontspec\/fontspec.sty\n(\/usr\/local\/texlive\/2015\/texmf-dist\/tex\/latex\/l3kernel\/expl3.sty\n(\/usr\/local\/texlive\/2015\/texmf-dist\/tex\/latex\/l3kernel\/expl3-code.tex)\n(\/usr\/local\/texlive\/2015\/texmf-dist\/tex\/latex\/l3kernel\/l3unicode-data.def)\n(\/usr\/local\/texlive\/2015\/texmf-dist\/tex\/latex\/l3kernel\/l3xdvipdfmx.def))\n(\/usr\/local\/texlive\/2015\/texmf-dist\/tex\/latex\/l3packages\/xparse\/xparse.sty)\n(\/usr\/local\/texlive\/2015\/texmf-dist\/tex\/latex\/fontspec\/fontspec-patches.sty)\n(\/usr\/local\/texlive\/2015\/texmf-dist\/tex\/latex\/fontspec\/fontspec-xetex.sty\n(\/usr\/local\/texlive\/2015\/texmf-dist\/tex\/latex\/base\/fontenc.sty\n(\/usr\/local\/texlive\/2015\/texmf-dist\/tex\/latex\/euenc\/eu1enc.def)\n(\/usr\/local\/texlive\/2015\/texmf-dist\/tex\/latex\/euenc\/eu1lmr.fd))\n(\/usr\/local\/texlive\/2015\/texmf-dist\/tex\/xelatex\/xunicode\/xunicode.sty\n(\/usr\/local\/texlive\/2015\/texmf-dist\/tex\/latex\/tipa\/t3enc.def\n(\/usr\/local\/texlive\/2015\/texmf-dist\/tex\/latex\/euenc\/eu1lmss.fd))\n(\/usr\/local\/texlive\/2015\/texmf-dist\/tex\/latex\/graphics\/graphicx.sty\n(\/usr\/local\/texlive\/2015\/texmf-dist\/tex\/latex\/graphics\/keyval.sty)\n(\/usr\/local\/texlive\/2015\/texmf-dist\/tex\/latex\/graphics\/graphics.sty\n(\/usr\/local\/texlive\/2015\/texmf-dist\/tex\/latex\/graphics\/trig.sty)\n(\/usr\/local\/texlive\/2015\/texmf-dist\/tex\/latex\/latexconfig\/graphics.cfg)\n(\/usr\/local\/texlive\/2015\/texmf-dist\/tex\/xelatex\/xetex-def\/xetex.def))))\n(\/usr\/local\/texlive\/2015\/texmf-dist\/tex\/latex\/fontspec\/fontspec.cfg)))\n(.\/Untitled.aux) (\/usr\/local\/texlive\/2015\/texmf-dist\/tex\/latex\/tipa\/t3cmr.fd)\n[1] (.\/Untitled.aux) )\nOutput written on Untitled.pdf (1 page).\nSyncTeX written on Untitled.synctex.gz.\nTranscript written on Untitled.log.\n\n\nI also tried setting the font type to \"Devanagari MT\" but that didn't work either.\n\nUpdate:\n\nI downloaded and installed texlive-2016 and re-ran my file through xelatex. Now it produces a meaningful error Font 'Nakula' does not contain script 'Devanagari'.\n\nThis is XeTeX, Version 3.14159265-2.6-0.99996 (TeX Live 2016) (preloaded format=xelatex)\nrestricted \\write18 enabled.\nentering extended mode\n(.\/Untitled.tex\nLaTeX2e <2016\/03\/31>\nBabel <3.9r> and hyphenation patterns for 83 language(s) loaded.\n(\/usr\/local\/texlive\/2016\/texmf-dist\/tex\/latex\/base\/article.cls\nDocument Class: article 2014\/09\/29 v1.4h Standard LaTeX document class\n(\/usr\/local\/texlive\/2016\/texmf-dist\/tex\/latex\/base\/size10.clo))\n(\/usr\/local\/texlive\/2016\/texmf-dist\/tex\/latex\/fontspec\/fontspec.sty\n(\/usr\/local\/texlive\/2016\/texmf-dist\/tex\/latex\/l3kernel\/expl3.sty\n(\/usr\/local\/texlive\/2016\/texmf-dist\/tex\/latex\/l3kernel\/expl3-code.tex)\n(\/usr\/local\/texlive\/2016\/texmf-dist\/tex\/latex\/l3kernel\/l3xdvipdfmx.def))\n(\/usr\/local\/texlive\/2016\/texmf-dist\/tex\/latex\/l3packages\/xparse\/xparse.sty)\n(\/usr\/local\/texlive\/2016\/texmf-dist\/tex\/latex\/fontspec\/fontspec-xetex.sty\n(\/usr\/local\/texlive\/2016\/texmf-dist\/tex\/latex\/base\/fontenc.sty\n(\/usr\/local\/texlive\/2016\/texmf-dist\/tex\/latex\/euenc\/eu1enc.def)\n(\/usr\/local\/texlive\/2016\/texmf-dist\/tex\/latex\/euenc\/eu1lmr.fd))\n(\/usr\/local\/texlive\/2016\/texmf-dist\/tex\/xelatex\/xunicode\/xunicode.sty\n(\/usr\/local\/texlive\/2016\/texmf-dist\/tex\/latex\/tipa\/t3enc.def\n(\/usr\/local\/texlive\/2016\/texmf-dist\/tex\/latex\/euenc\/eu1lmss.fd))\n(\/usr\/local\/texlive\/2016\/texmf-dist\/tex\/latex\/graphics\/graphicx.sty\n(\/usr\/local\/texlive\/2016\/texmf-dist\/tex\/latex\/graphics\/keyval.sty)\n(\/usr\/local\/texlive\/2016\/texmf-dist\/tex\/latex\/graphics\/graphics.sty\n(\/usr\/local\/texlive\/2016\/texmf-dist\/tex\/latex\/graphics\/trig.sty)\n(\/usr\/local\/texlive\/2016\/texmf-dist\/tex\/latex\/graphics-cfg\/graphics.cfg)\n(\/usr\/local\/texlive\/2016\/texmf-dist\/tex\/xelatex\/xetex-def\/xetex.def\n(\/usr\/local\/texlive\/2016\/texmf-dist\/tex\/generic\/oberdiek\/infwarerr.sty)\n(\/usr\/local\/texlive\/2016\/texmf-dist\/tex\/generic\/oberdiek\/ltxcmds.sty)))))\n(\/usr\/local\/texlive\/2016\/texmf-dist\/tex\/latex\/fontspec\/fontspec.cfg)))\n(.\/Untitled.aux) (\/usr\/local\/texlive\/2016\/texmf-dist\/tex\/latex\/tipa\/t3cmr.fd)\n*************************************************\n* fontspec warning: \"script-not-exist\"\n*\n* Font 'Nakula' does not contain script 'Devanagari'.\n*************************************************\n[1] (.\/Untitled.aux) )\nOutput written on Untitled.pdf (1 page).\nSyncTeX written on Untitled.synctex.gz.\nTranscript written on Untitled.log.\n\n\nOutput:\n\n\u2022 Somehow it doesn't make sense that the output contains six question marks: the input has 19 Unicode codepoints (two of them spaces) or 8 glyphs (Devanagari aksharas) (plus two spaces). Where does 6 come from: are you sure you're looking at the correct output file? :-) (For example, \"\u0928\u092e\u0938\u094d\u0924\u0947\" has exactly 6 Unicode codepoints\u2026). \u2013\u00a0ShreevatsaR Dec 21 '16 at 19:54\n\u2022 Yes it does. Please see dropbox.com\/s\/d1fmeoitquxnj04\/zoom_0.mp4?dl=0 \u2013\u00a0linuxfan Dec 21 '16 at 20:13\n\u2022 Try clicking on \"Trash Aux Files\" to delete the aux files; also remove the \\expandafter line (and\/or hit Enter at the prompt in TeXShop's console to make the compilation complete). \u2013\u00a0ShreevatsaR Dec 21 '16 at 20:31\n\u2022 Thanks for trying to help :=) dropbox.com\/s\/d1fmeoitquxnj04\/after-trashing-aux.mp4?dl=0 \u2013\u00a0linuxfan Dec 21 '16 at 20:43\n\u2022 Arrgghh. I found the solution. TexShop was converting my file to plain-text. When I cat the file, I saw a bunch of ??? instead of devanagari letters. I added a line at the top of my file (as per tex.stackexchange.com\/questions\/46199\/\u2026) and now I finally see devanagari in my pdf! Thanks @ShreevatsaR and all others for your help \u2013\u00a0linuxfan Dec 21 '16 at 21:05\n\nThe problem is that TeXShop apparently defaults to saving documents as latin1, rather than utf-8 as a modern editor would.\n\nAlthough adding the !TS line in every document is an option, the best thing to do is to alter the default encoding for files in TeXShop's preferences so that utf-8 is used by default. A !TS line can then be used to override this in the unlikely event that it is ever necessary.\n\nThis happened to one of my students a couple of weeks ago and I was astonished that TeXShop does not default to utf-8. But it certainly doesn't - this was a brand new install and there it was: latin1 encoding for all documents as default.\n\n\u2022 I did as you suggested and agree that it is more efficient than applying a workaround to every .tex file. Thanks \u2013\u00a0linuxfan Dec 22 '16 at 2:48\n\nThe real problem was with TexShop. It saved my file as ASCII and not unicode:\n\n$file zz.tex zz.tex: LaTeX 2e document text, ASCII text$ cat zz.tex\n\\documentclass{article}\n\\usepackage{fontspec}\n\\newfontscript{Devanagari}{deva}\n\\setmainfont{Devanagari MT}\n\n\\begin{document}\n\n?????? ????? ??????\n\\end{document}\n\n\nThis gave me a hint that the file-type was incorrect. Following TeXShop doesn't remember file encoding, I added the following as the first line of my .tex file:\n\n% !TEX encoding = UTF-8 Unicode\n\n\nand everything worked just fine.\n\n\\$ file zz.tex\nzz.tex: LaTeX 2e document text, UTF-8 Unicode text\n\n\u2022 This will work, but please see my answer below. This happened to one of my students a couple of weeks ago and I was astonished that TeXShop does not default to this. But it certainly doesn't - this was a brand new install and there it was: latin1 encoding for all documents as default. \u2013\u00a0cfr Dec 22 '16 at 2:29\n\nToo long for a comment.\n\nThe following document\n\n\\documentclass{article}\n\\usepackage{fontspec}\n\\setmainfont[Path=.\/,Extension=.ttf,Script=Devanagari]{nakula}\n\n\\begin{document}\n\n\u0928\u092e\u0938\u094d\u0924\u0947 \u0930\u0941\u0926\u094d\u0930 \u092e\u0928\u094d\u092f\u0935\u0947\n\n\\expandafter\\show\\the\\font\n\n\\end{document}\n\n\nhalts XeLaTeX (as expected) and displays the information\n\n*************************************************\n* fontspec warning: \"script-not-exist\"\n*\n* Font 'nakula' does not contain script 'Devanagari'.\n*************************************************\n(.\/nak.aux) (\/usr\/local\/texlive\/2016\/texmf-dist\/tex\/latex\/tipa\/t3cmr.fd)\n> \\EU1\/nakula(0)\/m\/n\/10=select font \"[.\/nakula.ttf]\/OT:script=deva;language=DFL\nT;mapping=tex-text;\".\n<inserted text> \\EU1\/nakula(0)\/m\/n\/10\n\nl.9 \\expandafter\\show\\the\\font\n\n\nThus you can see that the warning message is spurious, which seems to be a small bug in fontspec. Indeed, if I do\n\n\\documentclass{article}\n\\usepackage{fontspec}\n\\newfontscript{Devanagari}{deva}\n\n\\setmainfont[Path=.\/,Extension=.ttf,Script=Devanagari]{nakula}\n\n\\begin{document}\n\n\u0928\u092e\u0938\u094d\u0924\u0947 \u0930\u0941\u0926\u094d\u0930 \u092e\u0928\u094d\u092f\u0935\u0947\n\n\\expandafter\\show\\the\\font\n\n\\end{document}\n\n\nno warning is issued and the font information is exactly the same. As you can see, the script is correctly loaded.\n\nThe output on my system is\n\nI can't read the Devanagari script, so I can't judge whether it's correct or not. However, other software shows\n\n\u2022 Errors after applying changes that you suggested: kpathsea:make_tex: Invalid fontname [.\/Nakula.ttf]\/OT', contains '[' .\/Untitled.tex:5: fontspec error: \"font-not-found\" ! The font \"Nakula\" cannot be found. l.5 ...\/,Extension=.ttf,Script=Devanagari]{Nakula} kpathsea:make_tex: Invalid fontname [.\/Nakula.ttf]\/OT', contains '[' .\/Untitled.tex:5: Font EU1\/Nakula(0)\/m\/n\/10=[.\/Nakula.ttf]\/OT:mapping=tex-text; at 10.0pt not loadable: Metric (TFM) file or installed font not found. <to be read again> relax l.5 ...\/,Extension=.ttf,Script=Devanagari]{Nakula} \u2013\u00a0linuxfan Dec 21 '16 at 18:40\n\u2022 Your output looks perfect. Is there a way to get the correct names of all fonts installed on my system? \u2013\u00a0linuxfan Dec 21 '16 at 18:44\n\u2022 Try with \\setmainfont{Nakula} -- you probably don't have Nakula in your current folder. \u2013\u00a0Ulrike Fischer Dec 21 '16 at 18:51\n\u2022 No luck even after making the change suggested by Ulrike. All the additional fonts (Chandas, Charis SIL, Nakula etc.) are under \"User\" in my fontbook. Even if that were a reason for this failure, \"Devanagari MT\", installed under \"All Fonts\" doesn't work. \u2013\u00a0linuxfan Dec 21 '16 at 18:55\n\u2022 But it did work in your first example. So why should xelatex no longer find your font? \u2013\u00a0Ulrike Fischer Dec 21 '16 at 19:01\n\n(This is just \"could not reproduce\", but too long for a comment.) I installed Nakula from its website and ran exactly the file in the question\n\n\\documentclass{article}\n\\usepackage{fontspec}\n\\begin{document}\n\\setmainfont[Script=Devanagari]{Nakula}\n\u0928\u092e\u0938\u094d\u0924\u0947 \u0930\u0941\u0926\u094d\u0930 \u092e\u0928\u094d\u092f\u0935\u0947\n\\end{document}\n\n\nthrough xelatex (on Mac OS X Yosemite 10.10.5, texlive-2015) and got the expected output:\n\nSo I imagine it's something odd about your installation of Nakula that makes XeLaTeX not find it properly.\n\n\u2022 I can probably find a mac running a different version of OSX and check if xelatex works in that environment. \u2013\u00a0linuxfan Dec 21 '16 at 19:35\n\u2022 @linuxfan Also I strongly suggest uninstalling Nakula and trying with a fresh install from the website as above. \u2013\u00a0ShreevatsaR Dec 21 '16 at 19:40\n\u2022 I already did a reinstall but it has no impact on the result. \u2013\u00a0linuxfan Dec 21 '16 at 19:48","date":"2020-01-19 18:07:49","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 1, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.6747979521751404, \"perplexity\": 6467.205367290384}, \"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\/1579250594705.17\/warc\/CC-MAIN-20200119180644-20200119204644-00236.warc.gz\"}"}
| null | null |
\section{Introduction}
Modern natural language processing (NLP) models pose a challenge due to the massive size of the training data they require to perform well. For resource-rich languages such as Chinese, English, French, and Spanish, collections of texts from open sources such as Wikipedia \shortcite{wikiorg}, variations of Common Crawl data \shortcite{commoncrawl}, and other open-source corpora such as the BooksCorpus \citep{zhu15} are generally used. When researchers at Google released their Bidirectional Encoder Representations from Transformers (BERT) model, they trained it on a huge corpus of 16GB of uncompressed text (3,300M words) \citep{devlin18}. Later research has shown that the corpus size might have even been too small, and when Facebook released its Robustly Optimized BERT (RoBERTa), it showed a considerable gain in performance by increasing the corpus to 160GB \citep{liu19}.
Norwegian is spoken by just 5 million people worldwide. The reference publication \textit{Ethnologue} lists the 200 most commonly spoken native languages, and it places Norwegian as number 171. The Norwegian language has two different varieties, both equally recognized as written languages: Bokm{\aa}l and Nynorsk. The number of Wikipedia pages written in a certain language is often used to measure its prevalence, and in this regard, Norwegian Bokm{\aa}l ranges as number 23 and Nynorsk as number 55. However, there exist more than 100 times as many English Wikipedia pages as there are Norwegian Wikipedia pages \shortcite{metawiki}. When it comes to building large text corpora, Norwegian is considered a minor language, with scarce textual resources. So far, it has been hard to train well-performing transformer-based models for such languages.
As a governmental entity, the National Library of Norway (NLN) established in 2006 a mass digitization program for its collections. The Language Bank, an organizational unit within the NLN, provides text collections and curated corpora to the scholarly community \cite{sprakbanken}. Due to copyright restrictions, the publicly available Norwegian corpus consists mainly of Wikipedia pages and online newspapers, and it is around 5GB (818M words) in size (see Table \ref{tab1}). However, in this study, by adding multiple sources only accessible from the NLN, we were able to increase that size up to 109GB (18,438M words) of raw, deduplicated text. While such initiatives may produce textual data that can be used for the large-scale pre-training of transformer-based models, relying on text derived from optical character recognition (OCR)--based pipelines introduces new challenges related to the format, scale, and quality of the necessary data. On these grounds, this work describes the effort to build a pre-training corpus and to use it to train a BERT-based language model for Norwegian.
\subsection{Previous Work}
Before the advent of transformer-based models, non-contextual word and document embeddings were the most prominent technology used to approach general NLP tasks. In the Nordic region, the Language Technology Group at the University of Oslo, as part of the joint Nordic Language Processing Laboratory, collected a series of monolingual resources for many languages, with a special emphasis on Norwegian \citep{kutuzov17}. Based on these resources, they trained and released collections of dense vectors using word2vec and fastText (both with continuous skip-gram and continuous bag-of-words architectures) \citealt{mikolov13,bojanowski2017}, and even using an Embeddings from Language Models (ELMo)--based model with contextual capabilities \citep{peters18}. Shortly thereafter, Devlin et al. \shortcite{devlin18} introduced the foundational work on the monolingual English BERT model, which would later be extended to support 104 different languages including Norwegian Bokm{\aa}l and Norwegian Nynorsk, Swedish, and Danish. The main data source used was Wikipedia \shortcite{wikiorg}. In terms of Norwegian, this amounted to around 0.9GB of uncompressed text (140M words) for Bokm{\aa}l and 0.2GB (32M words) for Nynorsk \shortcite{metawiki}. While it is generally agreed that language models acquire better language capabilities by pre-training with multiple languages \cite{pires19,wudr20}, there is a strong indication that this amount of data might have been insufficient for the multilingual BERT (mBERT) model to learn high-quality representations of Norwegian at a level comparable to, for instance, monolingual English models \citep{pires19}.
In the area of monolingual models, the Danish company BotXO trained BERT-based models for a few of the Nordic languages using corpora of various sizes. Their repository \cite{botxo} lists models trained mainly on Common Crawl data for Norwegian (5GB), Danish (9.5GB), and Swedish (24.7GB). Unfortunately, we were unable to make the Norwegian models work, as they seem to be no longer updated. Similarly, the KBLab at the National Library of Sweden trained and released a BERT-based model and an A Lite BERT (ALBERT) model, both trained on approximately 20GB of raw text from a variety of sources such as books, news articles, government publications, Swedish Wikipedia, and internet forums \citep{malmsten20}. They claimed significantly better performance than both the mBERT and the Swedish model by BotXO for the tasks they evaluated.
At the same of the release of our model, the Language Technology Group at the University of Oslo released a monolingual BERT-based model for Norwegian named NorBERT. It was trained on around 5GB of data from Wikipedia and the Norsk aviskorpus \shortcite{nlpl}. We were unable to get sensible results when finetuning version 1.0 of their model. However, they released a second version shortly thereafter (1.1) fixing some errors \cite{norbert}. We have therefore included the evaluation results of this second version of the model in our benchmarking. They have also evaluated their and our model themselves \cite{KutBarVel21} with consistent results.
\section{Building a Colossal Norwegian Corpus}
As the main Norwegian memory institution, the NLN has the obligation to preserve and give access to all published information in Norway. A large amount of the traditional collection is now available in digital format. As part of the current legal deposit, many born-digital documents are also available as digital documents in the collection. The texts in the NLN collection span hundreds of years and exhibit varied uses of texts in society. All kinds of historical written materials can be found in the collections, although we found that the most relevant resources for building an appropriate corpus for NLP were books, magazines, journals, and newspapers (see Table \ref{tab1}). As a consequence, the resulting corpus reflects the variation in the use of the Norwegian written language, both historically and socially.
\begin{figure*}
\centering
\includegraphics[width=0.9\textwidth]{media/fig1}
\caption{The general corpus-building process.}\label{fig1}
\end{figure*}
Texts in the NLN have been subject to a large digitization operation in which digital copies were created for long-term preservation. The NLN employs METS/ALTO\footnote{Metadata Encoding and Transmission Schema and Analyzed Layout and Text Object \cite{mets,alto}} as the preferred format for storing digital copies. As the digitized part of the collection conforms to standard preservation library practices, the format in which the texts are stored is not suitable for direct text processing; thus, they needed to be pre-processed and manipulated for use as plain text. One major challenge was the variation in the OCR quality, which varied both over time and between the types of materials digitized. This limited the number of usable resources and introduced some artifacts that affected the correctness of the textual data.
The basic inclusion criterion for our corpus was that as long as it was possible for a human to infer the meaning from the text, it should be included. However, the amount of text involved in building the model meant that this needed to be determined automatically. The METS/ALTO files contain information from the OCR process regarding the confidence of every word (from 0 for no confidence to 1 for certainty), so we used this assessment to calculate the average confidence for paragraphs and pages. Setting the minimum paragraph confidence to 0.8 and the minimum page confidence to 0.9 allowed us to filter out a significant part of the text with the lowest quality. We also noticed that in the period of digitization from the beginning of 2006 until the end of 2008, the quality of the OCR was low and the estimated confidence values were too optimistic. We ended up excluding all text scanned in this period.
To further filter out erroneous textual information, we calculated the number of words in the documents and averaged the number of words per paragraph. Establishing a threshold of at least 20 words per document and an average of 6 words per paragraph, we could filter out text sources that had little value for training, such as cartoons and picture books. We estimated the language composition using various methods, including metadata tags in the collection and counting the frequency of words of certain types (e.g., personal pronouns). Our estimate is that 83$\%$ of the text is in Norwegian Bokm{\aa}l and 12$\%$ is in Nynorsk. Close to 4$\%$ of the texts are written in English, and the 1$\%$ left is a mixture of Sami, Danish, Swedish, and a few traces from other languages.
The aforementioned process was carefully orchestrated, with data moving from preservation storage, through error correction and quality assessment, and ending up as text in the corpus. As shown in Figure \ref{fig1}, after filtering, OCR-scanned documents were added to the other digital sources. After this step, the data went through the cleaning process, in which we ensured the consistency of the text encoding and special characters used. In the deduplication stage, all duplicated paragraphs in the entire collection were removed. Finally, we drew out two pre-training-sets: one with a sequence length of 128 tokens, and one with a sequence length of 512 tokens.
\begin{table*}[ht]
\begin{center}
\begin{tabular}{llrr}
\hline
\textbf{Sources}&\textbf{Period}&\textbf{Words (Millions)}&\textbf{Text (GB)}\\
\hline
Books (OCR) & 1814--2020 & 11,820 & 69.0\\
Newspaper Scans (OCR) & 2015--2020 &3,350 & 20.0\\
Parliament Documents$^{a}$ (OCR) & 1814--2014 & 809 &5.1\\
Common Crawl OSCAR & 1991--2020 & 799 & 4.9\\
Online Bokm{\aa}l Newspapers & 1998--2019 & 678 &4.0\\
Periodicals (OCR) & 2010--2020&317&1.9\\
Newspaper Microfilms (OCR) & 1961, 1971, 1981, 1998--2007 & 292 & 1.8\\
Bokm{\aa}l Wikipedia & 2001--2019 & 140 &0.9\\
Public Reports$^{b}$ (OCR) & 1814--2020 &91&0.6\\
Legal Collections$^{c}$ & 1814--2004 &63&0.4\\
Online Nynorsk Newspapers & 1998--2019 & 47 & 0.3\\
Nynorsk Wikipedia & 2001--2019 & 32 & 0.2\\
\hline
Total (After Deduplication) && 18,438 & 109.1\\
\hline
\multicolumn{4}{l}{
$^{a}$\footnotesize{Stortingsforhandlingene.}
$^{b}$\footnotesize{Evalueringsrapporter.}
$^{c}$\footnotesize{Lovdata CD/DVD.}
} \\
\end{tabular}
\end{center}
\caption{\label{tab1} The composition of the Colossal Norwegian Corpus.}
\end{table*}
\section{Pre-training a Norwegian BERT model}
In order to build our own pre-trained language model for Norwegian, we decided to use the original BERT architecture pre-trained with a masked-language model (MLM) objective, as published by Devlin et al. \shortcite{devlin18}. We evaluated the effect of changes in hyperparameters in terms of MLM performance and of the fine-tuning of the pre-trained models on various downstream tasks. All pre-training work was run on a v3-8 TPU (128GB) provided by the TPU Research Cloud, while the evaluation was done on in-house machines with a single NVIDIA Quadro RTX6000 (24GB).
Our goal was to build a solid model that would perform well on all types of Norwegian language tasks, ranging from old to modern text, and including texts that might be mixed with foreign languages like English. We therefore chose to initiate the model from the pre-trained mBERT weights \cite{tfhub}. The mBERT model was trained on 104 languages, including both Norwegian varieties (Bokm{\aa}l and Nynorsk). The model uses a 119,547-token vocabulary, and its pre-trained weights might also benefit from cross-lingual transfer. Our assumption is that using the mBERT weights for Norwegian should result in a better-performing model in comparison to starting with random weights. It might also keep some of its multilingual abilities, making it more robust when dealing with new words and texts containing fragments of other languages \citep{wudr20}.
\subsection{Improving the Model Beyond mBERT}
All subsequent training runs followed the findings by You et al. \shortcite{ylrh19}, who showed that the pre-training of a BERT model could be improved by increasing the batch size but that, at the same time, an increase in the learning rate could lead to instability, especially when using the adaptive moment estimation (Adam) optimizer. When training on large batch sizes, You et al. suggested using their layer-wise adaptive moments base (LAMB) optimizer instead. We confirmed these results on our dataset when pre-training for 100,000 steps on a batch size of 2,048 sequences, which is very close to the optimum size for our v3-8 TPU (128GB) setup (see Figure \ref{fig2}).
\begin{figure}
\centering
\includegraphics[scale=.18]{media/fig2}
\caption{Comparison of Adam and LAMB optimizers (learning rate: 4e-4; batch size: 2,048).}\label{fig2}
\end{figure}
The basic pre-training strategy was to use the largest possible batch size on our TPU and to increase the learning rate as long as it showed stability. An evaluation of the learning rate was done for 100,000 steps, but because we used decay, we expected the stability to be maintained even after this point. Devlin et al. \shortcite{devlin18} trained for 128-length sequences for approximately 90$\%$ of the training examples, then trained for 512-length sequences for 10$\%$ . Due to memory limits on our TPUs, we needed to reduce the batch size (by a factor of approximately 7) for the 512 sequences in the pre-training data; we also increased the number of pre-training steps for the long sequences to resemble the same distribution of short and long sequences that were used in training the BERT model. To investigate the effect of this, we experimented with two different setups in our model (version A and version B). Both were initialized from the same mBERT weights and trained identically for the first 1,750,000 steps. In the last steps, version A followed the training schedule used in the BERT model where roughly 10$\%$ of the total training time was used on long sequences (step 3a) and then an additional step (3b) on shorter sequences. Version B reduced the training on short sequences and instead trained almost 30$\%$ of the time on long sequences. The setup was chosen for making the total training time roughly the same for both models (see Table \ref{tab2}).
\begin{table*}[ht]
\begin{center}
\resizebox{\textwidth}{!}{
\begin{tabular}{lrrrrrr}
\hline
\multicolumn{4}{l}{}& \multicolumn{2}{c}{\textbf{Version A}} & {\textbf{Version B}} \\
& {\textbf{Warmup}} & {\textbf{Step 1}} &{\textbf{Step 2 }} &
{\textbf{Step 3a}} &{\textbf{Step 3b}} & {\textbf{Step 3}} \\
\hline
{Steps} & {50k} & {700k} &{1M} &{1.2M} &{1.2M} &{2M} \\
{Batch Size} &{2760} &{2760} &{384} &{384} &{2760} &{384} \\
{Examples} &{138M} &{1,938M} &{384M} &{460M} &{3,312M} &{768M}\\
{Sequence Length} &{128} &{128} &{512} &{512} &{128} &{512} \\
{Learning Rate} &{0 $ \rightarrow $ 4e-4 } &{4e-4} &{4e-4} &{ 4e-4 $ \rightarrow $ 2e-4 } & {2e-4 $ \rightarrow $ 0} & {4e-4 $ \rightarrow $ 0} \\
\hline
\end{tabular}}
\end{center}
\caption{\label{tab2} Training schedule for our models. }
\end{table*}
\section{Evaluation}
While pre-trained language models also can be used for direct MLM-predition and feature extractions, the most common use is to fine-tune it on a specific task. The base procedure for fine-tuning was described by \citet{vaswani17}, and it consists of training for a small number of epochs (typically 4), with a warmup of around 10$\%$ of the training steps; subsequently, a linear decay to zero is used. Devlin et al. \shortcite{devlin18} based their work on the same procedure and selected the best learning rate among 5e-5, 3e-5, and 2e-5, according to the performance of the model on the validation set. The optimal learning rate and number of epochs mainly depend on the size of and variance in the training corpus, but they can also be affected by the properties of the pre-trained model. To get optimal performance out of a pre-trained model, the hyperparameters in the fine-tuning should be adapted. However, in this work, we are not primarily interested in optimization but in a comparison of the performance of our models against the mBERT model.
\subsection{Token Classification}
A common way to evaluate language models is by fine-tuning the models on token classification tasks such as named-entity recognition (NER) and part-of-speech (POS) tagging. For Norwegian, the Norwegian Dependency Treebank \citep[{NDT,}][]{norwegian21} by the Spr{\aa}kbanken at the NLN and the Language Technology Group at the University of Oslo provide text that has been manually annotated with morphological features, syntactic functions, and hierarchical structures. The morphological annotation mainly follows the Oslo-Bergen tagger \citep{johannessen12}, and with a few exceptions, the syntactic analysis follows the Norwegian Reference Grammar \citep{faarlund1997norsk}.\ With the help of Schibsted Media Group, the same group recently published Norwegian Named Entities (NorNE) \cite{JorAasHus20}, an extension of NDT that includes named-entity annotations for more than 300,000 tokens.
Moreover, with the goal of testing being the retaining or vanishing of the multilingual abilities of our model, we also considered NER datasets in both languages included in our corpus and in languages of which there is little to no evidence in our corpus. Specifically, we used CoNLL-2003 for English \citep{tjong-kim-sang-2003-introduction}, Webbnyheter 2012 for Swedish \citep{webbnyheter2012}, DaNE for Danish \citep{hvingelby20}, CoNLL-2002 for Spanish \citep{tjong-kim-sang-2002-introduction}, and FiNER for Finnish \citep{ruokolainen19}. While the number and specificity of the tag sets vary across datasets, rendering the comparison between languages useless, we could still compare the performance of our model against that of English-only and multilingual BERT models. We decided to leave out NER datasets built using automated or semi-automated annotations processes.
\subsection{Sequence Classification}
For sequence classification, we chose another commonly used task: sentiment classification. We used a version of the Norwegian Review Corpus (NoReC) \citep{ovrelid-etal-2020-fine}, a fine-grained sentiment dataset \citep{nosent} for Norwegian created by the Nordic Language Processing Laboratory. Moreover, we defined a second sequence-classification task to capture idiosyncrasies and nuances of the Norwegian language. In this case, we generated a balanced corpus of 6,000 text speeches that had been spoken at the Norwegian Parliament (Storting) between 1998 and 2016 by members of the two major parties, Fremskrittspartiet and Sosialistisk Venstreparti \cite{LapSoyVel18}. The dataset is annotated with the party the speaker was associated with at the time, and the source data was made publicly available by the Norwegian parliament. The classification task is to determine the political affiliation of the transcribed speech segment.
\section{Results}
To evaluate the performance of our model, we searched for the optimal set of fine-tuning hyperparameters for each downstream task by running a small grid search (see Table \ref{tab3}) on the mBERT model. The search space was the same for all tasks and included learning rates ranging from 2e-5 to 5e-5, with the number of training epochs being 3 or 4. We did the same for the warmup ratio and weight decay. The performance was generally best using a warmup ratio of 0.1 and weight decay of 0, so we applied this universally to limit the grid complexity.
For the token classification tasks, we selected the best-performing hyperparameters based on the seqeval \shortcite{seqeval18} F1 micro score on the validation set for Bokm{\aa}l after fine-tuning an mBERT model. For sequence classification, we used the F1 macro score.
\begin{table}[ht]
\begin{center}
\resizebox{\columnwidth}{!}{
\begin{tabular}{lrrrr}
\hline
&{\textbf{NER}} & {\textbf{POS}} & {\textbf{Sentiment}} &{\textbf{Political}} \\
\hline
{Learning Rate} & 2e-5 & 3e-5 & 3e-5 & 2e-5 \\
{Number of Epochs} & 3 & {3} & {3} & {3} \\
\hline
\end{tabular}}
\end{center}
\caption{\label{tab3} Optimal fine-tuning hyperparameters for the mBERT model using the validation datasets.}
\end{table}
We then used the optimal fine-tuning parameters from the mBERT model for the validation dataset on our model and on the NorBERT model. Last, we compared all the models based on their results in relation to the test dataset.
Version B of our model---the version with the extended training-sequence length---performed slightly better on all four tasks than did version A. To simplify the results presented here, we therefore report only the results from version B, which we are naming \textit{NB-BERT}.
\begin{table*}[ht]
\begin{center}
\begin{tabular}{lrrrrrr}
\hline
& \multicolumn{2}{c}{NER} & \multicolumn{2}{c}{POS} & {Sentiment} & {Political}\\
\hline
& {\textbf{Bokm{\aa}l}} & {\textbf{Nynorsk}} & {\textbf{Bokm{\aa}l}} & {\textbf{Nynorsk}} & {\textbf{Bokm{\aa}l $\&$ Nynorsk}} & {\textbf{Bokm{\aa}l}} \\
\hline
{mBERT} & {83.8} & {85.6} & {98.3} & {98.0} & {69.7} & {78.4} \\
{NorBERT} & {89.9} & {86.1} & {98.5} & {98.4} & {81.7} & {78.2} \\
{NB-BERT (ours)} & {\textbf{91.2}} & {\textbf{88.9}} & {\textbf{98.8}} & {\textbf{98.8}} & {\textbf{86.4}} & {\textbf{81.8}} \\
\hline
\end{tabular}
\end{center}
\caption{\label{tab4} Evaluation results from the test dataset (version B of the model; F1 micro in token classifications and F1 macro in sequence classifications; best scores in bold).}
\end{table*}
As can be seen in the Table \ref{tab4}, the NB-BERT model performed significantly better than did the mBERT model for both Bokm{\aa}l and Nynorsk, and on both token and sequence classification. The improvement was the smallest for the POS dataset, with an improvement from 98.3 to 98.8 for Bokm{\aa}l and from 98.0 to 98.8 for Nynorsk. However, POS datasets such as this always contain some ambiguity, and it is hard to tell how much more improvement is possible there. In addition, the NER task improved from 83.8 to 91.2 for Bokm{\aa}l and from 85.6 to 88.9 for Nynorsk. The sequence classification improved from 69.7 to 86.4 in terms of sentiment classification and from 78.4 to 81.8 for political classification. We also tested the release 1.1 of the NorBERT model that is uploaded to Hugging Face \citep{norbert}. The performance of this model lays in between that of NB-BERT and mBERT for Bokm{\aa}l and Nynorsk, but it generally performs worse on all non-Norwegian tasks.\par
As shown in Table \ref{tab5}, our model was able to outperform the English-only and multilingual BERT for both Norwegian Bokm{\aa}l and Nynorsk, as well as for Swedish and Danish, which are languages with a shared tradition with Norwegian. For English, our results are also marginally better than those obtained using the English-only BERT model. For Spanish and Finnish, for which there is no close relationship with Norwegian nor documented occurrences of text in such languages in our corpus, the mBERT model outperformed both the English-only BERT and our model, suggesting that our model is deteriorating for the languages not included in the corpus.\par
\begin{table*}[ht]
\begin{center}
\begin{tabular}{lrrrrrrr}
\hline
&{\textbf{Bokm{\aa}l}} &{\textbf{Nynorsk}} & {\textbf{English}} & {\textbf{Swedish}} & {\textbf{Danish}} & {\textbf{Spanish}} & {\textbf{Finnish}} \\
\hline
{English BERT} & {75.1} & {77.8} & {91.3} & {82.5} & {73.9} & {81.8} & {82.9} \\
{mBERT} & {83.8} & {85.6} & {90.8} & {85.3} & {83.4} & {\textbf{87.6}} & {\textbf{88.7}} \\
{NorBERT} & {89.9} & {86.1} & {87.8} & {83.4} & {80.7} & {79.3} & {81.5} \\
{NB-BERT (ours)} & {\textbf{91.2}} & {\textbf{88.9}} & {\textbf{91.3}} & {\textbf{85.9}} & {\textbf{85.1}} & {85.8} & {85.8} \\
\hline
\end{tabular}
\end{center}
\caption{\label{tab5} Evaluation results (F1 micro) of different monolingual NER datasets using the English-only BERT, mBERT, NorBERT, and our model (best scores in bold).}
\end{table*}
\section{Discussion}
The majority of the training corpora used today for training transformer models are built using mainly open web sources. A major motivation for this project was to investigate whether the digital collections at the NLN could be used to create a suitable corpus to train state-of-the-art transformer language models. The texts available through the library are heterogeneous in nature, including cartoons, novels, news articles, poetry, and government documents published over time and in different contexts. As our results suggest, this seems to be a strength rather than a weakness, in that it enables us to build high-performance transformer models for small languages, such as Norwegian. Consequently, our Norwegian corpus is not only richer in diversity but also significantly larger in size than is any other Norwegian corpus, and it even rivals the size of previous work on a major language such as English. The Norwegian part of the mBERT model consists of around 1GB of text \citep{wudr20}, while the English-only BERT model was trained on 16GB of text \citep{devlin18} mainly based on English Wikipedia and Open Book Corpus. When Facebook developed the first version of its RoBERTa, it added Common Crawl data and Open WebText to the BERT corpus and ended up with 160GB of text \citep{liu19}. Our clean corpus of Norwegian-only text is 109GB in size.
For the target languages Norwegian Bokm{\aa}l and Norwegian Nynorsk, the model performs significantly better than does the mBERT model on both token classifications (POS and NER) as well as on the two sequence classification tasks. In the Bokm{\aa}l NER task, the level of improvement was +7.4 F1 points. Because none of the datasets have been benchmarked against human performance, it is hard to measure how close this is to the theoretical maximum.
The results show that our corpus is a valid training source, and this is by no means surprising. All research points to the possibility of improving transformer models' performance by training them on larger text corpora. However, the novelty of our results lies in that we were able to increase the performance on our domain-specific tasks while maintaining a lot of the multilingual properties of the mBERT model. This was unexpected because English only comprised around 4$\%$ of the training set. Still, we were able to improve the English capabilities of the model up to the level of the monolingual English model. Part of the reason for this might be that we applied some training techniques that were not available when the English-only model was trained and released, most notably the use of larger batch sizes and the LAMB optimizer.
We were also able to significantly improve the scores for Swedish and Danish, though it is hard to pinpoint how much of this was caused by the close linguistic similarities between the languages and how much by the fact that they were represented in the corpus to some degree.
It should not be surprising that the capabilities of the model in relation to languages that were not included in the training corpus (i.e., Spanish and Finnish) did deteriorate. However, the drop in performance was not radical, and the results above indicate that we might have been able to prevent this by adding just a small portion of these languages to the large corpus.
Overall, our results suggest that collections such as the digital collection at the NLN, even if they contain occational OCR-errors, may contribute significantly toward the creation of well-performing language models by providing large training corpora. As discussed earlier, there are OCR errors in the included materials. An exhaustive removal of all OCR artifacts would either have required us to do a major reduction of the size of the corpus, or to invest an unmanageable amount of manual work. We have not seen any indication that the OCR errors negatively impacted the performance. We might speculate that the model has learned to distinguish OCR errors from ordinary text, indicating that quantity is more important than quality when building such corpora. All in all, size matters.
\section{Conclusion and Future Work}
In this work, we have investigated the feasibility of building a large Norwegian-only corpus for the training of well-performing transformer-based language models. We relied on the collections of the NLN, and our model outperformed the existing multilingual alternatives. In the process, while the corpus produced might lack the cleanness of other textual resources, we proved that using somewhat noisy but available sources is an effective way to grow the ecosystem of resources for languages with fewer resources and for which enough open text in a digital format simply does not exist. As part of an effort to democratize the use of technology and digital resources at the NLN, we are releasing our trained BERT-based model \citep{NbAiLab21a} and will be releasing other models based on the same corpus in the future. Moreover, we are also releasing the set of tools and code we used so that others seeking similar results can easily reuse them \citep{NBAiLab21b}.
Although our work may indicate that OCR errors in corpora have little to no impact on the quality of the resulting transformer model, this has not been explicitly proven in the current study. More systematic studies are needed to investigate the real effect of OCR noise and artifacts.
Another important aspect is that, to benefit from the pre-trained mBERT weights, we used a 119,547-token multilingual vocabulary, of which only a small fraction pertained to Norwegian. A natural follow up would be to investigate the performance gains of using only a tailored Norwegian vocabulary.
The decision to use a BERT-based architecture as our target was guided by its simplicity to train and benchmark. However, newer and better-performing models have been released since the original BERT work a few years ago. The current corpus could be used for training such models as well studying the differences between architectural styles and training objectives. While it is already large in size, there is still potential to grow our 109GB corpus to the limits of the extant Norwegian holdings at the NLN, which presents itself as an opportunity to release even larger models.
\section*{Funding}
This research was supported by Cloud TPUs from Google's TPU Research Cloud (TRC).
\section*{Acknowledgment}
We would like to thank KBLab at the National Library of Sweden (Kungliga biblioteket) for its pioneering work on BERT in memory institutions and for the valuable and inspiring discussions. We also appreciate the feedback from and discussions with Andre Kaasen of the Language Bank at the National Library of Norway.
\bibliographystyle{acl_natbib}
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 4,048
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Q: Insert Attachment into Visualforce Email Template this question: says that it is "hard" to put a component inside a visualforce email template because ' needs to be a child of component.'. Yet, is there a way to make this work?
We also need to stick to using the visualforce email template and cannot send the email in a different way. The object on which the email template is running has attachments. How can this be done?
A: I don't think you can do it with pure email template, you'll need some Apex, as suggested in other answers.
<messaging:emailTemplate> complains as soon as you'd try to stick <apex:repeat> in there (for example to create <messaging:attachment> nodes in a loop).
Even if you could - try referring to {!relatedTo.Attachments[0].Body} somewhere (can be in the plaintextBody tag), you'll get an error:
Binary fields cannot be selected in join queries
I think the closest it can get is to create the links to attachments (and if I recall correctly - SF converts attachments > 3 MB to links anyway).
<messaging:emailTemplate subject="hello" recipientType="Contact" relatedToType="Opportunity">
<messaging:htmlEmailBody >
<p>This will list link to attachments... it's not a perfect solution but it's something.</p>
<apex:repeat value="{!relatedTo.Attachments}" var="a">
<p><a href="{!URLFOR($Action.Attachment.Download, a)}">{!a.Name}</a> ({!a.BodyLength} B)</p>
</apex:repeat>
</messaging:htmlEmailBody>
<messaging:plainTextEmailBody >
<apex:repeat value="{!relatedTo.Attachments}" var="a">
{!a.Name} {!a.ContentType} {!a.Body} <!-- this "Body" reference causes the compilation to fail -->
</apex:repeat>
</messaging:plainTextEmailBody>
<!--
<messaging:attachment filename="{!relatedTo.Attachments[0].Name}" renderAs="{!relatedTo.Attachments[0].ContentType}">
{!relatedTo.Attachments[0].Body} :(
</messaging:attachment>
-->
</messaging:emailTemplate>
A: user670186,
Not sure if this is the answer you're looking for but here's my thoughts:
If you have a VF template ready, and if consider using messaging.singleemailmessage methods to send the email, you can
- Use mail.setTemplateId to take the visualforce as template
- Use Messaging.EmailFileAttachment to attach the file in the email.
A: see this http://www.salesforce.com/us/developer/docs/pages/Content/pages_email_templates_with_apex.htm.
Hope it may helpful
|
{
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}
| 624
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Lionel Robert Ashburner (1827 in Tasmania – 26 January 1907 in Marylebone, London) was the Acting governor of Bombay during the British Raj from 13 March 1880 to 28 April 1880.
Biography
Lionel, son of William Page Ashburner and his wife Hester Maria, was a civil servant in India, where he became a director of the Great Indian Peninsular Railway. In the 1871 Birthday Honours, he was appointed a Companion of the Order of the Star of India.
On 14 July 1873 he married Emily Caroline Haggard at Kirkee, Bombay.
References
Governors of Bombay
1827 births
1907 deaths
Companions of the Order of the Star of India
|
{
"redpajama_set_name": "RedPajamaWikipedia"
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| 9,198
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The 2006 Mid-American Conference women's basketball tournament was the post-season basketball tournament for the Mid-American Conference (MAC) 2005–06 college basketball season. The 2006 tournament was held March 4–11, 2006. Regular season champion Bowling Green won their second straight championship over Kent State. Ali Mann of Bowling Green was the MVP.
Format
The top two seeds in each division received byes into the quarterfinals. The first round was played at campus sites. All other rounds were held at Quicken Loans Arena.
Bracket
All-Tournament Team
Tournament MVP – Ali Mann, Bowling Green
References
Mid-American Conference women's basketball tournament
2005–06 Mid-American Conference women's basketball season
MAC women's basketball tournament
MAC women's basketball tournament
Basketball competitions in Cleveland
College basketball tournaments in Ohio
Women's sports in Ohio
2000s in Cleveland
|
{
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| 637
|
{"url":"https:\/\/byjus.com\/parallel-plate-capacitor-formula","text":"## Parallel Plate Capacitor Formula\n\nWhen two parallel plates are connected across a battery, the plates are charged and an electric field is established between them, and this setup is known as the parallel plate capacitor. The direction of the electric field is defined as the direction in which the positive test charge would flow.Capacitance is the limitation of the body to store the electric charge. Every capacitor has its capacitance. The typical parallel-plate capacitor consists of two metallic plates of area A, separated by the distance d.\n\nThe parallel plate capacitor formula is given by,\n\n$c=k\\varepsilon&space;_{0}\\frac{A}{d}$\n\nWhere,\n\n\u03f5o = permittivity of space (8.54 \u00d7 10\u221212 F\/m),\n\nk = relative permittivity of dielectric material,\n\nd = separation between the plates,\n\nA = area of plates.\n\nExample 1\n\nA parallel plate capacitor is kept in the air has an area of 0.50m2 and separated from each other by distance 0.04m. Calculate the parallel plate capacitor.\n\nSolution:\n\nGiven:\n\nArea A = 0.50 m2,\n\nDistance d = 0.04 m,\n\nrelative permittivity k = 1,\n\n\u03f5o = 8.54 \u00d7 10\u221212 F\/m\n\nThe parallel plate capacitor formula is expressed by,\n\nC = oA \/ d\n\n= 8.54\u00d710\u221212 \u00d7 0.50 \/ 0.04\n\n= 4.27 x 10\u221212 \/ 0.04\n\nTherefore, C = 106.75 x 10\u221212 F\n\nExample 2:\n\nDetermine the area of parallel plate capacitor in the air if the capacitance is 25 nF and separation between the plates is 0.04m.\n\nSolution:\n\nGiven:\n\nCapacitance = 25 nF,\n\nDistance d = 0.04 m,\n\nRelative permittivity k = 1,\n\n\u03f5o = 8.54 \u00d7 10\u221212 F\/m\n\nThe parallel plate capacitor formula is expressed by,\n\nC = oA \/ d\n\nA = dc \/ k\u03f5o\n\n= 0.04 \u00d7 25\u00d7109 \/ 1\u00d78.54\u00d71012\n\nA = 1 x109 \/ 8.54 \u00d71012\n\nTherefore, area of parallel plate capacitor is 117.09 m2.","date":"2018-04-21 03:59:23","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 1, \"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\": 0.7649720311164856, \"perplexity\": 1447.9170565155928}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2018-17\/segments\/1524125944982.32\/warc\/CC-MAIN-20180421032230-20180421052230-00441.warc.gz\"}"}
| null | null |
Intertwined: Serena Mitnik-Miller and Stephen Ormandy
Submitted by info@joshualine... on 27 September 2019 - 8:04pm
Exhibition Type:
How many artists:
How many exhibition works:
Thursday, 10 October 2019 to Saturday, 9 November 2019
Thursday, 3 October 2019 -
StephenOrmandy
Serenamitnikmiller
Joshua Liner Gallery is pleased to present Intertwined, a two-person exhibition featuring the works of California based artist, Serena Mitnik-Miller and Sydney, Australia based artist, Stephen Ormandy. Intertwined opens on October 10 and will remain on view through November 9, 2019. Both artists will attend the opening reception.
Intertwined juxtaposes Mitnik-Miller's serene geometric watercolors alongside Ormandy's hard-edge oil on linen canvases and marble sculptures. Using altogether different mediums, both artists explore the effects of layering, motion, and infinity through undulating forms and lines. This fascination with undulation can perhaps be traced to their shared affinity for the ocean. Never far from the water, Mitnik-Miller grew up in Hawaii and currently lives in the Malibu. As the artist explains, "I have always been most inspired by the natural world, the deserts, the mountains, the ocean." Not only does Mitnik-Miller's medium of choice, watercolor, pigment suspended in an aqueous solution, require water, but her curvy compositions echo the rhythm of the waves. Likewise, Ormandy's adoration for the water is clear when he asserts, "My love for the ocean is always with me." The conjoining bands of white marble in Twist and Tower, fold in on themselves as they crest, conjuring the swelling of the waves.
Fashioned by interconnecting patterns of color and concentric shapes, Mitnik-Miller's watercolors continue to "evoke a balance between chaos and control, a tension between the organic nature of the watercolors against the confined geometric walls." Whereas previous works were characterized by subdued transparent hues, the artist's newest watercolors feature a bolder, more saturated color palette. Beginning with a composition that is first laid out in pencil, Mitnik-Miller outlines the overall structure. Without the use of tape or stencils, the artist then manipulates watercolor, a medium known for its unpredictable nature, into impeccable geometric, linear arrangements. The color subtly blends and changes from one inch to the next, as the artist pushes and pulls the pigment through her patterns, giving the work a sense of perpetual motion.
Similarily, Ormandy's hard-edge paintings examine the tension that arises from overlapping curvy, abstracted forms. Born of the subconscious mind and inspired by the rhythm of shape, space, color, and design, Ormandy's hard-edge paintings often resemble abstracted figures and landscapes. His forms recall 21st-century modernist painters and sculptors, such as Hans Arp, Henri Matisse, Henry Moore, and Clement Meadmore, infused with the magnetism of 1960's Pop art. The artist uses contrasting colors and shapes to achieve a push-pull effect, which Ormandy revers to as hug and repel. Ormandy states, "I'm looking for vibration and rhythm, the play of line creating positive and negative space, searching for tonal balance through contrast or harmony, while developing chroma relationships that hug or repel." Pulsating with color, Spinning Top includes bold geometries comprised of puzzle piece-like forms that sinuously connect together, creating a harmonious composition.
Serena Mitnik-Miller lives and works in Malibu, CA. She was born in 1981 in Massachusetts but spent most of her childhood in Hawaii. Mitnik-Miller received her BA in photography and printmaking from the University of California in 2008. The artist has exhibited domestically and internationally, presenting solo exhibitions at the Bolinas Museum, Bolinas, CA; Chandran Gallery, San Francisco, CA; Park Life Gallery, San Francisco, CA; Curators Cube, Tokyo, Japan; and Art Park Gallery, Byron Bay, Australia. In 2014, Mitnik-Miller completed a large-scale mural at the Facebook Headquarters in San Francisco. Mitnik-Miller and husband, Mason St. Peter, own General Store located in San Francisco and Los Angeles. This is the artist's fourth solo exhibition with the gallery.
Born in Melbourne, Stephen Ormandy graduated from The City Art Institute (now known as UNSW Art and Design) in Sydney in 1985. During his time at the college, he met Louise Olsen and Liane Rossler and in their final year of study they established the design company Dinosaur Designs, which has grown to be one of Australia's leading design brands specializing in resin and metallic jewelry and homewares. The artist has exhibited at Melbourne Museum, Melbourne, Australia; MCA Sydney, Sydney, Australia; Galerie Bessieres; Paris, France; PKM Gallery, Seoul, South Korea; and Olsen Gruin, New York, NY. Ormandy's work is held in public collections across Australia, including Artbank, The Sir William Dobell Art Foundation, The Powerhouse Museum, The National Gallery of Victoria, and The Art Gallery of Western Australia.
Artist ( Name ):
Stephen Ormandy
Serena Mitnik-Milller
http://joshualinergallery.com/
540 W 28th St, New York, NY 10001
Other shows from Joshua Liner Gallery
Wayne White: I DON'T KNOW
Wayne White: I DON'T...
Bounty: Kris Kuksi and Mark Wagner
Bounty: Kris Kuksi and Mar...
Arno Beck: Screengrabs
Intertwined: Serena Mitnik...
On-The-Go Sketching & Painting Workshop with Artist Misuk Goltz
On-The-Go Sketching &...
New Dimensions
Realism 2021 | Art Awards
Susan Wahlrab – 'Garden in the Forest'
Susan Wahlrab – 'Garden in...
Ashburnham
Creative Connections Gallery
Keira Kotler: Light of Day
Vera Molnar: Est-ce Une Ligne?
Vera Molnar: Est-ce Une Li...
Vintage Galéria
|
{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
| 5,144
|
using CERental.Core.Enums;
using System.Collections.Generic;
namespace CERental.Core.Domain
{
public class Equipment : Entity
{
public EquipmentType Type { get; set; }
public string Name { get; set; }
/* Navigational properties */
public virtual ICollection<EquipmentInRent> EquipmentInRent { get; set; }
}
}
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 592
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Mira II Ws Light is the women's version of Jaure II Light, with an adapted women's last. This boot is a lightweight version of Lundhags most technical boot.
Mira II Ws Light is the women's version of Jaure II Light, with an adapted women's last. This boot is a lightweight version of Lundhags most technical boot. Durable and light Certech® Exp-technology in the lower part. A Vibram® Curcuma outsole for low weight and rugged traction. The shaft is made of Heinen Terracare nubuck leather.
|
{
"redpajama_set_name": "RedPajamaC4"
}
| 1,310
|
International International Politics News Report
British educators to boycott Israel
British college teachers decided to boycott Israeli academic institutions and pressure the European Union to curtail educational and cultural programs with Israel in a vote last night in the seaside town of Bournemouth, UK during the first conference of the British University and College Union (UCU). The delegates of the union ignored their leader Sally Hunt who urged them to drop their boycott intentions, and passed the resolution with 158 in favor and 99 against.
Union members deplored Israel's 40-year occupation of Palestine and the violation of Palestinian educational rights by curfews, checkpoints, and other restrictions of movement, reports said. Shootings and arrests of Palestinian students and professors have gravely damaged the educational environment in the region, according to delegates.
UCU officials said that the vote yesterday would not enforce an immediate boycott because of union bureaucracy that manages implementation of such resolutions. All union members, totaling some120, 000 British college educators, will be briefed on the issue and will have the opportunity to take a stand, according to UCU rules.
Similar academic boycotts against Israel have been attempted twice before. The Association of University Teachers (AUT) and the National Association of Teachers in Further and Higher Education (NATFHE) passed resolutions in 2005 and in 2006, respectively. Later, the two organizations formed the UCU.
While Jewish organizations in Britain and Israel have ardently protested possible boycott since yesterday, support for Palestinian education came from presidents of four Israeli universities and three Israeli authors who appealed to Israel's defense minister, Amir Peretz, to allow Palestinian students to travel from Gaza to the West Bank for their studies.
Israeli army invades several villages near Jenin and kidnaps five Palestinians
Israeli army closes the local vegetable market in Beta town near Nablus
|
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| 1,562
|
Esajas Kirke også kaldt Esajaskirken er en kirke, der ligger i Malmøgade på Indre by i Københavns Kommune, tæt på Østre Anlæg og Sortedams Sø.
Historie
Det københavnske Kirkefond stillede grunden til rådighed, mens kirken blev opført for penge indsamlet af menigheden. Kirken blev opført i perioden 1903-1912 og tegnet af arkitekten Thorvald Jørgensen. Først blev krypten bygget, som fungerede som kirkerum fra 1903. Resten af kirken stod færdig i 1912, hvorefter gudstjenesten rykkede op til det øvre kirkerum, mens krypten blev brugt af menighedssamfundet.
Fra 1932 til i hvert fald 1990 var kirkens menighed præget af at være delt i to lejre med hver sin præst.
Kirkebygningen og inventar
Kirken deler med blandt andet dens massive facadetårne visse stiltræk med Tveje Merløse Kirke.
Kirken er opført i senromansk stil som en korskirke i røde mursten med granitkvadre i som fundament. Over korsskæringen er et centraltårn med pyramidespir, i østgavlen er en lille apsis, og mod vest to tårne med pyramidespir. Hovedindgangen i vest er ad en granitportal, der flankeres af to ærkeengle udført af Anders Bundgaard. Kirkeskibet er tredelt. Midterskibet og korsarmene har tøndehvælv, korsskæring og sideskibe har krydshvælv.
En glasmosaik fra 1914 af Joakim Skovgaard, med korsfæstelsen som motiv, fungerer som alterudsmykning. Orgelet har 16 stemmer og blev bygget af Frobenius Orgelbyggeri i 1958.
Kirkebygningen
Referencer
Literatur
Storbyens virkeliggjorte længsler ved Anne-Mette Gravgaard. Kirkerne i København og på Frederiksberg 1860-1940. Foreningen til Gamle Bygningers Bevaring 2001.
Eksterne kilder og henvisninger
Esajas Kirke hos nordenskirker.dk
Esajas Kirke hos KortTilKirken.dk
Esajas Kirke i Østervold Sogn
Kirker i Københavns Kommune
Kirker i Københavns Amt
Kirker i Københavns Stift
Østerbro
|
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| 5,379
|
# YOU GOTTA HAVE GIRLFRIENDS
## A Post-Fifty Posse Is Good for Your Health
### SUZANNE BRAUN LEVINE
# CONTENTS
Chapter 1: You Gotta Have Girlfriends
Chapter 2: Those Healthy Hormones. Who Knew?
Chapter 3: Befriending Ourselves
Chapter 4: Unfriending for Better Health
Chapter 5: New Friends at Our Age
Chapter 6: United We Stand
### _Author's note:_
I interviewed and heard from literally hundreds of women (and some men) as I was writing this book. Some were happy to have their full names used, but others were more comfortable with a pseudonym, so I gave them names that I thought reflected their personalities. I also made up names for those online correspondents who posted comments to my blogs under display names such as C mon and noaxe397 because I didn't see any reason to make a distinction between them and other on-line or in-person voices. When I write about my circle of trust I use their first names. I want to thank them all for helping me understand and describe the very special connection among girlfriends.
# CHAPTER 1:
## YOU GOTTA HAVE GIRLFRIENDS
_Time passes._
_Life happens. Distance separates._
_Children grow up._
_Jobs come and go._
_Love waxes and wanes._
_Men don't do what they're supposed to do._
_Hearts break._
_Parents die._
_Colleagues forget favors._
_Careers end._
_But..._
_Sisters are there, no matter how much time and how many miles are between you._
_A girlfriend is never farther away than needing her can reach._
—Internet circular
THE BEST THING A MAN can do for his health is to be married to a woman. One of the best things a woman can do for her health is to nurture her relationships with her girlfriends, especially after the age of fifty. The longer we live, the more important our friends become. We call them our "chosen family" and in times of need they are the most likely to be at the door, on the phone, or in the waiting room. In other words, the post-fifty version of "an apple a day" is "nurture your friendships."
Not only do our girlfriends keep us physically healthy, but their support can also make us well when we are sick. A study of breast cancer patients by researchers at Ohio State University's Comprehensive Cancer Center found that those who were in a supervised support group were 56 percent less likely to die than those who were going it alone. In fact, the stress-reducing influence of their company is thought to contribute to women's longer life expectancy than men. "I had two separate conversations with friends today about how they tend to hold things in, sometimes until they experience physical symptoms," one woman told me. "One has panic attacks, the other had shingles, which she attributes to stress. Both are hesitant to express their emotional pain to their husbands—they just don't seem to understand." But, she adds, "both reiterated how much they need their girlfriends."
Friends are just as vital to our psychological and spiritual well-being. They give us courage, confidence, and understanding; we know that no matter what happens as we age, we are not alone. They give us love, patience, acceptance, and a healthy dose of comic relief for the accumulating absurdities of life. And we empower each other to achieve the well-being of accomplishment beyond the years of past generations. All in all, such cherished friends are a blessing; they make life matter when it counts.
Our mothers' generation wouldn't understand this notion of friends helping friends achieve new levels of accomplishment and vitality beyond the sell-by date of menopause; their friendships remained relatively static as their circles aged together. Our own friendships, on the other hand, shift during different life stages. They may not have been as important to us earlier in our lives as they will become now. Back in our thirties and forties, we were frantically balancing the multiple demands on our days, and friends were just one ingredient in a swirling mix. Many of us neglected some meaningful friendships for those that developed out of common interests (parents whose children were friends with your kids) or shared space (workplace colleagues). In our fifties, we open a new chapter in our lives, and we can't take friends for granted any more. Without a circle of trust, we will have a very hard time turning the page.
We are entering a challenging, promising—and unprecedented—stage of life Gail Sheehy, in her book _Passages_ , has called Second Adulthood. The twenty-plus years after fifty are a statistical gift to our generation, bestowed by the good health, accumulated personal authority, and self-awareness of our first adulthood. An open field of dreams lies between adulthood and old age, where we move from a life in which we have fulfilled many roles—child, student, mother, employee, wife—to one where we are able to write our own scripts. For many of us, it is the most exciting, optimistic, and fulfilling stage of our lives.
I often hear from women who are amazed to find that they are happier now than ever before—that despite sagging boobs and drifting memories, this is the best time of their lives. We feel more confident, more open to new experiences, more appreciative of good times, more thoughtful and self-aware. At the same time, things don't get to us the way they used to; we have literally mellowed with time and simply don't sweat the small stuff. Recent studies by Laura Carstenson at Stanford and others have found that people over age fifty are very happy, happier, according to the researchers, than those respondents in their twenties and thirties. That makes sense when you think about it. We are better able to roll with the punches and many of the conflicting demands on us have receded.
To get there, though, we must pass through a period of upheaval as confusing and extended as adolescence. The two have a lot in common: raging hormones, self-doubt, recklessness, tumultuous relationships, and wrestling with the question of what we will do with the rest of our lives. No matter how independent and resourceful we are, meeting this challenge successfully requires teamwork. And courage. "Many think of courage... as a solitary journey," write Deborah Collins Stephens, Jackie Speier, Michaelene Cristini Risley, and Jan Yanehiro, the friends and co-authors of _This Is Not the Life I Ordered._ "We believe the journey of courage is best walked with women friends who literally and figuratively 'en-courage' us."
The journey takes us through change and doubt, and the destination is not clear. It is easy to get stuck and hard to forge ahead. That is where our encouraging friends come in. As Ellen Goodman and Patricia O'Brien point out in their book, _I Know Just What You Mean_ : "There are times when a friend provides more than the warm soup of empathy. She becomes a catalyst for change. Over a long life, full of disruptions, stops, and start-ups, friends can be the collaborators, the instigators who make change possible. They are often the ones who urge us to take a leap, who jump with us or help us scramble back up the other side." When it gets good, Second Adulthood is about taking leaps, so a loving nudge is essential—especially since the men we know are often heading in the opposite direction, looking to scale back and settle in.
"By the time we reach fifty," Eileen Williams, founder of the website The Feisty Side of Fifty points out, "we know a thing or two about the changes that life can bring. We've lost loved ones to death and gone through the breaking up of relationships. We've endured injured pride, damaged self-esteem, rejections, and crushing disappointments. And we have gone through menopause." Surviving hard times has also taught us a thing or two about what makes the difference between triumph and defeat. I haven't interviewed a single woman about the ups and downs of her recent life who hasn't at some point told me a story that ended with a grateful "I couldn't have gotten through it without my girlfriends."
So, the next time people say (or you say to yourself), "why are you wasting your time having coffee with your girlfriends when you have so many more important things to do?" tell them they couldn't be more wrong! Beyond the fun of it all, and the infusion of energy and optimism, those seemingly meaningless get-togethers are crucial to aging well. Without the kind of human contact and intellectual stimulation we take for granted earlier in our lives, we are likely to literally fade away physically, psychologically, and socially. The failure to maintain quality relationships with others, according to Dr. David Spiegel, a Stanford professor of psychiatry, is as dangerous to our physical health as smoking.
### _Your Circle of Trust: Who Makes the Cut?_
In your gut, you know who your friends are. They are the people you choose over all others to spend your fiftieth birthday with. They root for you and they put up with you. They stand up for you and they stand by you. They patiently teach you how to use your smart phone (and can be trusted not to tell your kids you couldn't figure it out yourself). They listen sympathetically when you need to vent. They know when you are hurt or angry and how to patch things up. And they make you laugh.
I have recently taken stock of the women I call friends and singled out those who I trusted most and who made me feel the best. The list made me smile. These are the women I want on my team as I sort out the rest of my life. What surprised me as I looked over the list was how seamlessly they wove together the chapters of my life so far. It is as if collectively they have been at my side all along, helping me grow up.
Ruthie and I were grade school friends and have now reconnected after a long gap when we probably didn't think of each other at all. We started out in the same place and have ended up with the kind of bond we might have achieved if we had been in constant touch through all that time.
Patricia and I go back almost as far. We went to the same college; she was the one I told about my abortion senior year. The only accommodation to growing up has been that we have gone from calling each other "Patty" and "Susie" to "Patricia" and "Suzanne." While her life has become very different from mine, our understanding of each other is timeless.
I also met Maddy in college; she is my no-nonsense, curmudgeon friend whose reading recommendations are unfailingly rewarding. Because she is "difficult" she gets into tight situations, but I love her for her ability to articulate what emotions and circumstances are at play and her gutsy willingness to face the consequences. She tells it like it is. If I act like Cleopatra, Queen of Denial, Maddy keeps me grounded.
Susie came into my life around the time I got married. In fact, she was at my wedding—enormously pregnant—on the groom's side. Her support and admiration never waver. She is wise in all things family related and has a warm and generous family to prove it. She is incredibly smart and talented, and very funny, too.
My other BFF is actually a posse: five former colleagues who have been having dinner once a month since 1989. We like to try new places, which is a good thing, since I am not sure we would be welcome back to any restaurant after a visit. We generally sit there for three or four hours, order nothing but appetizers, laugh uproariously, and then pay with five credit cards. Collectively, we are more than the sum of our parts. Each one of us brings her own kind of support, encouragement, empathy, and humor to the table. We are in sync and can pretty well predict how any one of us will respond to a situation, even what she will order to eat. (I wonder if our monthly meals have become the postmenopausal equivalent to the synchronized periods that female roommates often have.) As the years go by, we marvel at how many more of us it takes to remember the name of a movie star.
There was one more name on the list, but after due consideration, I crossed her off. I realized that although the deep understanding is there, the commitment is not. We took a course together a few years ago and had lunch after each class. During that time we got down to the real stuff and established a warm and trusting intimacy. As soon as the class was over, so was our regular communion. Now we get together once or twice a year, usually at my instigation. The intimacy clicks in as we catch each other up, but if I don't call her, she never thinks to touch base or give me an interesting piece of information. I am off her radar screen between one encounter and the next. I will call her my out-of-sight-out-of-mind friend but not admit her to my circle of trust.
### _Nurturing Friends over Time_
While the bottom line of friendship is trust, acceptance, and constancy, everyone has her own friendship style, and often a different style with different friends. It is almost as important to be aware of your style as it is to be aware of your true friends. Not only to make sure that you are contributing all the nurturing you can to the relationship, but because as our lives change, so will our needs, and our friendships will have to adapt in order to remain healthy.
Some friends talk or email every day; others get together once or twice a year but stay in touch in other ways. Some do things together—like travel or go to movies or power-walk every morning. Others touch base regularly but not often. I am in that last group, but I never doubt that the tie is there. Every once in a while, I get a call from one dear friend asking if I am "waving or drowning." That line (from a Stevie Smith poem about watching someone drown thinking they were waving, not signaling for help) is our code for "just checking in; I thought of you and wanted to make sure there was nothing I needed to know." It's shorthand that enables us to check in without having to "catch up," a process I must admit I would rather be spared. Headlines about events and honesty about feelings are what I am looking for. My emails are probably the least newsy that anyone I know receives.
Given my telegraphic style, I will have to watch out in the years to come so that I don't miss important concerns and events behind the headlines I ask for. Perhaps I should begin asking more probing questions, even learn to play catch up.
Some friendships form a literal "circle of trust," like my dinner pals. Many reading groups have coalesced into intimate and loyal sororities. Susan created one with some former colleagues when she retired from teaching. They've been together for twelve years, though they rarely get together outside their regular meetings. "We are really devoted to one another in terms of this book group, which we adore and which we protect with our lives," she says. "We are very, very careful about anybody coming into the group that we don't think would fit in."
The time may come when they need to break out of their happy bubble. If, say, one of the women stops showing up, will the rest accept the excuse that she got too busy? And the time may come when a new person would bring a healthy breath of fresh air to the group.
Other women are turned off by the group model. "I have noticed among certain women a tremendous need to bond, which in itself is not a bad thing of course," says Jane, a textile artist. But she is suspicious. "The bad side of bonding is the subtle pressure to conform to the group. I have found that some gossip, or passing on what was said, is really a subtle coercion to conform." Pat Wynn Brown, creator of one-woman shows she calls "hair theater," also finds collective intimacy "uncomfortable. Being together in gangs of girls makes me anxious. I much prefer one-on-one chats or small groups where we can talk." Eileen favors "taking a long walk with one friend and then going to lunch afterwards. There's something about being outdoors in nature, exercising together and then sharing a meal, that elicits feelings of closeness."
Carrie, an executive with a not-for-profit foundation, is a one-to-oner also. But when she needed major surgery, she realized that she would have to mobilize the individual units of her "circle of trust" into a caregiving team. She assigned one friend to take her to the hospital, another to pick her up, a third to organize a meal delivery system. Inevitably they crossed paths and developed an acquaintance with each other. When Carrie was safely recovered, the team did not stay in touch. She didn't encourage it. "I think it is because of the intensity I require from a friendship that I can't bear one relationship being diluted in deference to another," she says. "My friendships are not about doing things together, which is often more fun in a group, but about relating—unique woman to unique woman." Furthermore, she goes on, "if I introduced two friends I would be in high anxiety about whether they liked each other or whether one said something that I knew—but she couldn't possibly—would hurt the other. And what if they didn't like each other? How would I deal with that?" Still, she knows how to adapt her friendshipping style to a crisis.
### _A Sister/Friend_
We have come to call the tight bond among women "sisterhood," but in truth the bond between sisters is both more intense and less easy than that among friends. Still, sisters can serve as witnesses to your life in a way no one else can. It may not be the most accurate account, colored as it is by sibling craziness, but, as Suzanne Gerber puts it on the website Next Avenue, "she remembers every important thing that has ever happened to you."
The part our biological sisters will play in our future well-being is unique, regardless of how the relationships started out. "My sister and I were not close growing up," says Pat. "We literally and physically fought; for years we had to sleep in the same bed. It drove me crazy. She drove me crazy... but now that we are adults, she is my go-to girl. My family has been through much tumult, tension, and sadness, and my sister and I weather it all together, and we share a dark sense of humor. We still have power struggles but we know enough to put the nonessentials aside."
Even if "you can't be open about a lot of stuff," says Alice, one of four sisters, "you're stuck with each other." For better or for worse. "My sister and I are going through a bad phase right now," Suzanne told me. "When that happens we revert to childhood roles. But we know we will get back in sync."
Being connected to a family tree has other benefits for the new stage. "I have no grandchildren," Pat says, "and my sister has three. She lets me share them, and her daughter is especially kind and considerate to me, and I try to be loyal and true and helpful to her. I love having my niece in such a close relationship to me. I love her children as I would love my own grandchildren, and I think that brings some special love to their family."
Most of all, when it comes to coping with failing parents and making peace with them when they are gone, no friend can share that experience as deeply as a sister. "On that heart-wrenching day when you bury a parent," writes Suzanne, "she's the one squeezing your hand, and of the almost 7 billion people you share a planet with, she's the only one thinking the same thing you are: 'Bye-bye Daddy.'"
### _A Husband/Friend_
As I talked to women about their best friends, several insisted on adding the men in their lives to the list. They find the same intimacy, encouragement, and laughter with them that they do with a friend-for-life. Yet most acknowledge a different flavor in the relationship.
"The friendships I developed in college and grad school were mostly dysfunctional and wisely shed," say Lauren as she surveys her past relationships. "Post-college friends have come and gone as we moved, married, had kids, changed jobs. I am in touch with one friend from high school via occasional Facebook exchanges and once-a-year ten-page letters. I wish I had more friends in my current life, but since I don't, I feel all the more grateful for my husband (who is a great husband, but not a BFF in that best-girlfriend sense)."
Pat begs to differ. "My best friend is my husband," she says. "We have been together since I was fourteen and he was fifteen. We will be married forty years this March." She has close women friends too, but "my easiest, loveliest, and most affirming relationship is with my husband. I have the most fun with him and I am most myself with him." Which is not to say, she quickly adds, they are joined at the hip. "He enjoys his nights out and trips with the boys much more even than I enjoy mine with the girls." Pat, on the other hand, enjoys being alone. "Maybe it is the case that the woman I love being with most is me."
"I enjoy my women friends a lot and they mean a great deal to me," she goes on. "But I don't find they understand me more than Steve, or to be more loyal, or to be my top priority. When I would watch _Sex and the City_ I would marvel at the relationship among the four of them—that it was their priority over the men and children in their lives."
One reason Pat's marriage is becoming more intimate with age may be that she and Steve are becoming more compatible. Many heterosexual relationships become increasingly friendly as the pressures of everyday life lift and—interestingly—hormones shift. We all know that women's estrogen goes down in menopause; as a result the testosterone that is always there plays a bigger role and may account, in part, for what anthropologist Margaret Mead has called "postmenopausal zest." At the same time, the levels of testosterone go down for men. This can cause occasional performance problems, but it also seems to make men more emotional and responsive to domesticity and intimacy. That rapprochement could, as Claude Rains famously says in _Casablanca_ , be the beginning of a beautiful friendship.
"Maybe a circle of best girlfriends is in the future for me," Pat wonders. "Who knows? I am open to it. I am open to life and love and anyone who can stand by me in loss. And I welcome all comers when it is time to laugh!"
So what makes a health-protecting friendship-for-life? Novelist Marilyn French answered that question as well as anyone I have encountered, when she wrote about her own circle of trust. "Over the years, we became intimate friends—not in the sense that we spoke every day and knew every detail of the others' lives, but as friends who knew each other's qualities and had a sense of each other's fears and longings, the grooves and velvet folds we were trapped in, our efforts to pull ourselves free; and we were ardent about one another's well-being."
# CHAPTER 2:
## THOSE HEALTHY HORMONES. WHO KNEW?
_What do we live for, if it is not to make life less difficult for each other?_
—George Eliot
WHEN SHE HAD A WORRY or a secret, my grandmother wouldn't dream of discussing it with anyone outside the family. When a family member was in need, home was where they had to take you in. Today, even if we wanted to, we can't always take refuge in family. We are often living—and aging—far from our biological families, and sometimes we feel less connected to them than to our friends. That turns out to be a very healthy turn of events. New clinical research proves that we can literally change one another's chemistry for the better. What scientists have discovered about girlfriends explains some of the mysteries of our bonds.
_Why is it that I can leave for a date with friends, while grumbling that I really don't have the time and I'm not in the mood, yet later when we hug each other goodbye I feel all warm and fuzzy and ready to meet the day?_
Studies show that when we are doing whatever girlfriends do together, our bodies produce oxytocin, also known as the "cuddle hormone" (because it is released in nursing mothers). Unlike husbands or kids, who are often the cause of anxiety, our friends consistently elicit that warm glow, which feels good and soothes anxiety. In fact, while everyone else drives you crazy at one time or another, your friendships may keep you sane. A Swedish study even found that people with broad networks of friends were the lowest risk group for dementia.
Studies of female primates show the same phenomenon: Hanging out with a small but trusted group of other females reduces damaging spikes in stress hormones, reports _New York Times_ science writer Natalie Angier. A circle of trust can, as _she_ puts it, "mop up the cortisol spills that can weaken the immune system," which in turn can support additional years of good health.
_Why is it that when something goes wrong and I start to panic, my friends can talk me through it?_
Again, it is the group hug. Women respond to danger by gathering in a mutually supportive group, while men show a "fight or flight" surge of adrenaline. It used to be thought that all humans exhibited "fight or flight" responses, but recent work (by a team of women scientists at UCLA led by Shelley E. Taylor)) found that women are wired somewhat differently, so that our reaction to a crisis is more likely to be a more diplomatic, "tend and befriend" approach, which again reduces tension.
That conciliatory response may also make women more creative and calm in a crisis, because the "fight or flight" response is produced in the primitive ("reptilian") part of the brain, which shuts down most rational resources in order to concentrate on physical strength and agility.
_When I have a problem to solve, why is it that my husband's immediate reaction—"let's step back and analyze this thing so we can find the right answer"—just makes me more crazy, while huddling with friends who ask questions (basic ones like "what do_ you _want to do?") and laying out various possible approaches (along with the effect of each on the people involved) makes everything seem do-able?_
Swedish researchers set up just such a situation, as recounted by gender scientist Dr. Marianne J. Legato in her book _Eve's Rib_. Women preparing to speak in public (an especially high-stress activity for us) reported that the presence of even a well-meaning male partner made them more anxious, while the presence of a woman friend gave them confidence.
Our friends' ability to "make things better"—or at least _feel_ better—may make a crucial difference in our outlook. When Dr. George E. Vaillant evaluated studies of a group of individuals throughout their lives, he found that "objective good physical health was less important to successful aging than subjective good health. By this I mean," he explained in his book _Aging Well_ , "that it is all right to be ill as long as you do not feel sick."
_How can it be that no matter how bad something is, and no matter how intense the conversation with a girlfriend, we eventually find something to laugh about?_
Laughter may be our most precious gift to each other; it is a powerful elixir (in fact, the act of laughing releases endorphins, those feel-good brain chemicals). It is very rare to spend more than a few minutes with a girlfriend when there isn't a burst of laughter, no matter whatever else is going on. Gestalt therapist Ilana Rubenfeld calls humor "a martial art" because it cuts a frightening situation down to size. In addition, the physical exercise of a hearty laugh, not unlike orgasm, is a good, endorphin-releasing workout. Laughing, Rubenfeld concludes, "improves blood circulation, increases the oxygenation of the blood, enhances digestion, reduces pain... and best of all strengthens the immune system." An Indian doctor, Madan Kataria, has taken this therapy one step further by developing a practice he calls "laughter yoga." Groups of people get together, do some deep breathing, and begin to emit fake laughs. Soon the laughs become real, and after fifteen or twenty minutes, everyone feels great.
Laughter works, no matter how "inappropriate" it may seem. A psychiatrist I talked to is convinced that so-called "black humor" is a uniquely human survival technique. Dr. Andrew Weil puts it nicely in his book _Healthy Aging._ Laughter, he writes, "is a way of seeing the ridiculous side of life, the incongruities and absurdities that can make you laugh even in the midst of misfortune, especially in the midst of misfortune."
_How is it that girlfriends know how to make life a little bit easier for each other? (For example, my dinner group has decided to exchange gifts in January so that coming up with those gifts is not one more item on our stressful pre-holiday to-do lists?) How is it, in other words, that other women just "get it"?_
The phrase "tend and befriend" describes this intuitive understanding nicely. The UCLA scientists who identified it as a distinctly female response first observed the gender-specific behaviors in their own lab. When something went wrong, the men would storm into their offices and slam the doors, while the women would come out of their offices and make coffee. We don't need scientists to tell us that an old-fashioned coffee klatch with the girls is one of the many ways we tend and befriend each other, but it is nice to know that along with our lattes, we are getting a biochemical boost.
We even get a little dessert with our latte. Natalie Angier, writing about how women interact, reports on an experiment in which women were invited to play a game; they had to choose either a cooperative or competitive strategy. Brain imaging showed that the brains lit up most brightly among the women who chose cooperation and increased as cooperation continued. The areas of the brain that lit up were those that, according to Angier, also respond to "chocolate, pretty faces, money, cocaine, and a range of licit and illicit delights."
If all that isn't enough, listen to the prestigious Mayo Clinic; doctors there are so committed to promoting health through friendship that they have developed a list of pointers for building and maintaining a successful relationship. Their tips include:
> **Go easy.** Respect your friends' boundaries.
>
> **Don't compete.**
>
> **Adopt a healthy, realistic self-image.** Vanity and constant self-criticism can be turnoffs to potential friends.
>
> **Listen up.** Ask what's going on in your friends' lives. Avoid talking about your own problems all the time. Try to only give advice when your friends ask for it.
>
> **Don't judge.** Give friends space to change, grow, and make mistakes.
In the "you are not wasting your time" department, I'd add one more practical suggestion: **Make time for your friends**. You may have a built-in reminder—a class you take together or a project you are working on during the weekend—but if not, we are all so busy that it is important to make a time commitment as solid as the emotional commitment. We in my five-woman posse, for example, pull out our calendars at the same time as we pull out our credit cards at the end of each meal, to make sure we schedule the next get-together.
# CHAPTER 3:
## BEFRIENDING OURSELVES
_The only reward of aging is a sense of some honest friendship with yourself, where you get to know yourself—you make peace with the things you are and you aren't._
—Pepper Schwartz, sociologist
AS IMPORTANT AS HAVING FRIENDS is to our well-being, there is a level of understanding and respect that only we can reach for ourselves, and only by becoming as good a friend to our selves as we are to our circle of trust. It may take a little work to get that relationship into shape.
_Authenticity_. When I talk to women about their objectives for their reinvention, that word comes up all the time. Achieving it is, as psychologist James Hillman suggests, a process of liberating _character_ (what you do when you are alone and presumably being true to yourself) from _personality_ (the traits you have developed to navigate society and please other people).
Chipping away at personality in order to reveal character involves unloading some of the baggage from the past, and making peace with the rest. Which "me" did that? Do I want to keep doing it? Why didn't I do that? Do I have the courage to do it now? Which relationships are supportive of the "real me"? What souvenirs from my life are worth keeping and which need to get tossed? How do I put my life in authentic perspective while looking forward—and looking back?
Much of that baggage accumulates during our earliest experiments with relationships. Childhood friends can help sort out our recollections. After all, they know who we were before we became who we are. Indeed, we learned what friendship was about from them. All the drama of those love/hate relationships with classmates was an education in making friends. "No one can teach you what a great friend is, what a fair-weather friend is, what a treacherous and betraying friend is, except to have a great friend, a fair-weather friend, or a treacherous friend," observes psychologist Michael Thompson.
Those first friends are in a unique position. Certainly more objective than family members, they are able to comment on the narrative of our childhood that we have carved in emotional stone. Maybe that humiliating classroom presentation has been forgotten by everyone but you. Maybe everyone thought you were really smart, even though you didn't. Maybe your friend remembers your parents being home more than you do, or remembers your parents being as cold toward you as you hardly dared to think. Maybe she recalls a secret that you have forgotten or never knew that could explain a lot. A need to find witnesses to our lives may explain why we type names from the distant past into a Facebook search and why many women who avoided ten- and twenty-year high school reunions are signing up for them now.
We can also measure how we have grown up in the context of those relationships. Eileen has retained two childhood friends on and off throughout her adult life. They are still teaching her about friendship—and herself. "We reconnected about fifteen years ago and started planning regular get-togethers as a threesome. Back in school, I was much closer to each of them than they were with one another; they both considered me their best friend." But this time around the configuration shifted. "The two of them are actually more like one another than either of them is like me," she realized. "They both tend to be quieter, more internally focused, and share many of the same interests: gardening, cooking, creative pursuits that don't really interest me."
When she found out that the others had arranged activities together that didn't include her, Eileen was struck by a totally new response. "In the old days, I would have been jealous!" But a lot has changed. "Thank goodness that life perspective, menopause, and a needed dose of self-esteem and self-awareness have helped me realize that I don't have to drive myself crazy trying to be everyone's favorite friend." Nowadays, she understands that "relationships change, people evolve, and life can be good just the way it is," without measuring the bestness of friends.
Which is not to say there aren't moments when we revert to the days of cliques and constantly shifting best friends. "I struggle with how to deal with friends who disappoint us when they cancel dates, don't include us in their plans, take too long to return phone calls, forget to send birthday cards, or make friends with other women we introduce them to," Ellyn Cohen, a retired publicist, admits. "What surprises me is that, despite our age, we still react the way we did in junior high school."
It was mutual interests that brought my grade-school friend Ruthie back into my life, and it is our mutual history that has made the relationship uniquely meaningful now. We had twin families, we thought back then: European fathers; upbeat, somewhat flirty mothers (who, unlike other parents, took piano lessons); and pesky little brothers. We even lived in similar houses in the same neighborhood. We spent a lot of time together in one or the other of those houses.
After high school, though, we went our separate ways. She married, had three children, and lived in the Midwest; I went across the country to start my career. Thirty years later, I was beginning to write about women over fifty. A former classmate told me that Ruthie had become a therapist and was leading groups for women called "'Retirement'—or What Next?" that dealt with the same issues of transition, turmoil, and self-doubt that I was writing about. We made a date to visit, and it was as if we had played our favorite card game on the floor of her sun-porch the day before, not decades ago. We talked and talked and seamlessly reentered one another's lives.
Soon after that, both our mothers, who were still living in those houses we grew up in, began to fail. Our sympathy wasn't abstract; we could each picture both mothers in both kitchens where we gathered after school, and could bear witness to the experience of being their daughters. No one else in my life today could go there with me.
### _Shedding Shame_
Deep down in that burdensome baggage, are packed the secrets we never told anyone, because we were mortified by what we did or how we felt. It is time to let them go. With every risk we take, we get stronger and braver, so that what seemed inconceivable earlier is a dare worth accepting now. Even though we know that speaking "the truth will set you free," doing so is one of the hardest challenges we need to face in order to move on. Recently, I saw it happen in a workshop called "Dancing at the Shame Prom." The workshop grew out of a collection of confessional essays by a wide range of women on the subject of shame. After the book was published, the editors, Amy Ferris and Hollye Dexter, heard from women needing to get stories off their chests; as Ferris puts it, they needed to express "feeling not good enough, smart enough, creative enough." They felt they could finally unburden themselves, they wrote, because reading the anthology reassured them they were not, as they had imagined all those years, alone. In response, Ferris and Dexter have developed a workshop, which they described to me, in which women can share their "stories filled with great sorrow and sadness, stories filled with anger and resentment about past mistakes." The experience, says Ferris, enables the teller "to breathe, to exhale, to share, be intimate." And shed tears of relief.
Another kind of tears—tears of laughter—are an antidote to another kind of shame, brought on by the indignities and creeping decrepitude we all have to live with. The trials and tribulations of menopause, and what Eileen calls the "floppin' and droppin'," going on around her provide endless material. Many times I have seen something in the mirror that almost brought me to tears (thinning hair or lips, for example) or been depressed over the newest memory lapse (forgetting whether it is a tablespoonful or a teaspoonful in the time between looking at the recipe and turning toward the mixing bowl) only to find myself overcome with hilarity when I confess it to those who know whereof I speak. As I knew I would.
This is serious business, of course. We have been unrelentingly critical of our bodies, even, as one of my friends puts it, when you had to be on LSD to see the flaws. Even when we should have been proud of what we saw in the mirror. Who hasn't come upon a photo of herself back when she was reluctant to put on a bathing suit and been stunned to find that she looked pretty good? Now that we are becoming increasingly less perfect on the outside, we owe it to each other to appreciate and enhance the strength and wellness within.
### _Visiting the Road Not Taken_
As we look long and hard at whom we have become, the mind wanders toward whom we might have become—and might still aim to be. We can get a glimpse in the lives of friends who took different routes. One major crossroad is parenting or not. Where are those who had children and had to make some hard choices about the rest of their lives? And where are those who chose different commitments and lifestyles?
Rachel, who is a traveling saleswoman, knew she could always count on her friends in a crisis, but she hadn't seen much of them—especially those with families—because she was on the road so much, and they were focusing on home. The stock-taking milestone of her fiftieth birthday made her think about rebuilding her circle of trust. That was not so easy, she found. "Part of it was getting back to good friends and having to deal with some of their annoyance that I was gone." Ultimately, though, she was able to cycle back into those lapsed friendships and found, among other things, that connecting to her friends' children, who were growing up while she was out of the picture, was an unexpected reward, "for five hours, maximum," she adds. As another woman put it, if you are lucky, when you circle back on a friendship, you find that "you haven't dropped a stitch; you just put the knitting down."
Another connection between diverging lives has been membership in an enduring working group. These groups used to be quilting bees; then came consciousness-raising groups; in our time, it has been book groups. Many have lasted for years, and the conversations have created a timeline across the women's lives. Katrina Kenison, author of _The Gift of an Ordinary Day_ , took one path, but has been able to follow the road not taken through the women in her book club. During the decade they have been meeting, "there have been two divorces and both women are dating again," she wrote me. "The conversation around this is always so interesting—those of us in long marriages envy the dates, the spontaneity, the romance, the independence, the second chances. We want to hear all the juicy details. And yet I know my dear friends also envy us in a way, for our old, solid, unmysterious relationships." She goes on, "These two women, ages sixty-one and fifty, both look fantastic, but then I realize they are working really hard to look so good, in a way that us solid-married aren't, because we don't have to. The grass is always greener, I guess. But I can tell you this: the seven married women are definitely living vicariously through our two bold and beautiful friends who are out there risking all in new relationships."
For their part, though, those single women need a little more than green grass. As one woman (not in that group) admitted to me, "one of the difficult things for me is the fact that although I have a wonderful loving caring circle of friends, they are all married. At my age it is difficult to find women who are single and who are on the irreverent edge of life. There is so much to go and do... and laugh about... and have a 'fuck it' attitude one day, yet be able to walk the traditional path the next day."
The paths of some friends have diverged _after_ the family phase. Ann Voorhees Baker, a successful businesswoman, shared the letter she received from her friend Jackie on the eve of a weekend of workshops that Ann had organized for professional women in transition. Jackie traces their friendship back to the "little band of wives and mothers who gathered for countless play dates, dinner parties and holidays during our young mother years." They had so much in common then, but now, when she compares herself to the women attending Ann's workshops, she wonders what she has to offer. Those women "will have had such interesting lives," she wrote, "and mine was/is so not. That's how I've always felt, like who would want to hear my story? (Least of all, my children.) But as I get older, I wonder. We need to tell our story." And she adds, "I still don't know what I want to do when I grow up, my dear."
That's what friends do as we age: Listen to each other's stories and help each other grow up. No matter what choices we make, it's possible for friends to grow separately without growing apart if we have achieved a relationship in which, as British social psychologist Terri Apter puts it, "understanding meant caring about (the other's) experience, rather than sharing it."
### _Maturing Within Our Friendships_
"We have been the peace keepers, the mediators, the selfless workers behind the scenes who have striven to please our parents, our spouses, our friends, our children, our bosses—and for the most part have been quite successful," a woman named Toni points out. But things are changing; she is in shedding mode. "Nowadays I don't have time for all of that. I continually ask myself, what is important for _me_? With the limited amount of time I have left, I want to allow less important things to drop away and focus on the people who matter most."
"I find that my tolerance for 'fluff' or gossip has almost vanished," another woman told me. "I don't care so much what my coworkers make or what vehicles they drive. I've come to think more about the larger picture. And I look for that in my friends, too."
Yet another has discovered that the big picture includes acceptance. "We are so critical and judgmental of ourselves," she explains. "When you are kind to someone else, you learn how to turn it toward yourself. It can be very healing."
Mary, a woman I met not long ago, gave me a list of the qualities in a friend that have become important to her now. They include "the other woman being on a journey and conscious of that" and "common value base (ethics, politics – small p)." And there is one more, a self-protective requirement: "Being psychologically reasonably intact." "I have found it too draining to have to support people if they won't seek professional support when they need it," she explains and then adds, "selfish, perhaps, but honest." Ah, honesty—and that much-maligned expression of self-worth, "selfishness."
Actually, honesty and a healthy respect for our own needs (formerly known as "selfishness") are signs of that authenticity we are after. In the safety of our circle of trust, we can dare to be ourselves without being judged; we can vent unworthy gripes without feeling guilty; we can admit failures and boast about successes (no false modesty here); even accept as much advice as we give. Quite simply, we can say what we really think.
For many women, speaking up and speaking their minds is a totally new experience. One of the hallmarks of this empowering period of self-discovery is the realization that hits each of us at some point: "You know what? I don't care what people think any more!" (I've called that liberating experience "the fuck-you fifties.") Nothing is more exhilarating than hearing yourself utter a resounding "no" to a statement or request you had been acquiescing to all your life. We practice that strong voice on our long-time friends. And we offer that empowered, no-bullshit self to new ones.
### _Making Peace with Weakness, Celebrating Our Strength_
There is no escaping the fact that sooner or later, a circle of trust will be called upon to try to restore a friend's emotional, physical, or financial health, especially since it is statistically likely that most of us will be living alone at some point in the future. The five of us who gather for a life-enhancing dinner every month may well find ourselves called upon to mobilize a different kind of lifesaving team.
Author Marilyn French was a member of just such a group—they called themselves the "coven." When she was stricken with lung cancer, they moved into action; they met with her doctors and evaluated treatments along with her children; they also invoked mystical forces and kept each other informed. When she went into a coma, they organized themselves into a round-the-clock spiritual circle. When doctors believed she was dying, they didn't think so and encouraged their sleeping friend to keep going. When she miraculously emerged from the coma, they got her home and tended to her needs in shifts.
As she got physically stronger, Marilyn discovered a totally new kind of psychological strength has emerged. "When I got home, someone stayed with me every night," she wrote in _A Season in Hell_ , a memoir of those days. "They did this because I asked them to; it was a measure of my desperation that I did so. It was incredibly hard for me to ask anyone to do something for me, but during this period I did." She may not have realized it, but I am sure that her friends were grateful to know what Marilyn needed and wanted, so they could do it, rather than helplessly asking, "is there anything I can do?"
Women of our generation have a very hard time asking for and even accepting help. We pride ourselves on being able to cope with anything, and we are deeply invested in our hard-won independence. By the time we get into our fifties, we have learned a lot about caregiving and rising to the occasion, but know much less about being cared for, what I call "care- _getting_." Dr. Sara S. Auchincloss, a psychiatrist, sees this dynamic all the time. It is, she says, "a challenge for the woman to accept what she needs from someone." We are so used to caregiving, she explains, that we can barely "tolerate it when people give care back to you."
If we are going to address the challenges of aging, some of the trust and honesty that have bound us together will need to shape authentic responses to our circumstances—accepting weakness in ourselves and each other. It is time for a new Golden Rule, Gloria Steinem says: "Do unto yourself as you have been doing unto others." Or to put it another way: "Do unto yourself as you would do unto your friends."
# CHAPTER 4:
## UNFRIENDING FOR BETTER HEALTH
_I've lost friends, some by death... others through sheer inability to cross the street._
—Virginia Woolf
A HEALTHY HUMAN BODY assimilates good nutrition and eliminates toxins at the same time. In the same way we have to shed stale or negative relationships, even as we nurture the high-protein ones. Like much of the "stuff" we discard in this house-cleaning period, many friends have simply been outgrown. But disengaging from friendships that are benign but no longer life-enhancing is a delicate business. Vivian, who moved to the United States from Israel after her divorce, is somber when she describes her decision. "The choices I am making now are much more personal and quiet and involve less people," she says. "And that makes it hard in terms of friendships that have kind of traveled with you. Perhaps the wave that carries them is nostalgia."
Then there are those we called friends who are now and always were unhealthy—belittling our achievements, dismissing our concerns, being disloyal—but somehow became embedded in our lives. They drag us down, hold us back, or just don't understand (or don't care to understand) what is going on. They must go. "I have one friend who I like very much but she always belittles my success," says Jane, the artist. "If I ever told her what she was doing, she would be shocked," she adds, "but I am beginning to think that she is not good for me."
"I recently 'broke up' with one of my best friends," says Candida in a comment on a blog I wrote about unfriending. "It was quite hurtful for me, but I just realized that—in that 'he's not that into you' kind of way—despite how much we care for each other, how close we were, for a long time..." it wasn't working any more. In terms of what Candida needs now, "it is not enough to be called friends. There is something missing, some sort of trust."
"Honesty and trust are important," observes one woman succinctly. "Fool me once, shame on you; fool me twice, shame on me and unfriend you."
### _Unfriending_
The word "unfriending" (some say "defriending") is one of the newest dictionary-worthy terms that, thanks to Facebook, indicates a brutal and final cutoff. In real life, though, you can't just click a friend away. You can have a heart-to-heart; you can have a disagreement and not make up; you can try a white lie such as "I've got to devote my time to my grandchildren/my school work/my job." But any way you look at it, paring down your inner circle can be hurtful, guilt-making, and very hard to initiate.
I've relied on the "drift" technique—fewer calls, slower responses to emails—hoping that distance and silence will dissolve the tie. This tactic has backfired more than once. The worst experience was when I saw an email from the designated friend, opened it, and decided to "keep as new" rather than responding. I let a few days go by and to my surprise got another email from the same person. "I know you are there. You opened my last email. Why didn't you reply?!" Until then I had no idea that it was possible to monitor an email's status. I could have said, "because I would like to downgrade our friendship," but, of course, I didn't—and still haven't.
Whether to "mend it or end it" is a tough call warns psychologist Dr. Irene S. Levine, in _Best Friends Forever._ With a known toxin, it is a good idea to figure out whether you want to have a showdown or just walk away. That will depend on the possible relief such an exit conversation can bring. It may be the case, though, that clearing the air will get the relationship back on track, but make sure that air-clearing doesn't become the ongoing (and only) tie between you.
Clearing the air is easier said than done, especially because frequently the issue involves hurt feelings, not an actual conflict. Moreover, being confrontational is hard for many women—we've been making peace, not war, for so long. There have been many times when I have felt hurt or angry but instead of speaking up, I either harbored my resentment or pulled back, two responses that aren't good for the health of my friendships. I can also vividly remember—will I ever forget?! —a blowup that haunts my dreams to this day. But we owe it to ourselves and to those precious few friends we are nurturing to take each other on when something goes wrong.
It took a man to clarify the friend-mending process for me. He was bemoaning the loss of a lifelong friendship in an essay published in the _New York Times_. "There was no cinematic blowup; it just evaporated. I believe I disappointed or annoyed or let Dan down in some way, and he chose to end the friendship rather than confront me.... I understand. I've done the same thing with other guys," he admits. It's over. He regrets that they can't work things out the way his wife and her friends do. They "hurt each other's feelings. Then they stew and obsess and vent to other friends. Next they engage in a difficult phone call. A few days later they meet and drink wine and work on gently knitting their bond back together. And their friendship not only survives, it is also strengthened."
### _Who's the Boss?_
Perhaps the relationship has soured because the give-and-take has gotten out of whack, or an accepted pattern of one person making all the plans and imposing her opinions is no longer comfortable for the other. Resistance may outrage one, but it will liberate the other. And it may restore equilibrium.
Carole Hyatt, a career coach, is a giver, describing herself as "Lady Bountiful." Her friend Jill is a taker. "Over the last twenty years she had called me at least three times a week to request something," she told me when I interviewed her for _Inventing the Rest of Our Lives_. Once too often, they were sitting in a hot tub and Jill "spent thirty minutes doing this huge 'ask'." Carole finally exploded. "'I have been hearing this conversation for twenty years, and you have never once offered me anything!" Her friend was stunned. "Carole," she said, "you never ask! You look like you have everything. I wouldn't know where to begin giving to you, offering. I'd feel foolish."
That encounter was an epiphany for Carole. "I realized that I never did ask. That I have never 'lowered' myself to ask. That I was Lady Bountiful. And yet mysteriously I felt that she would offer. That everyone would offer. The fact is," she now understands, "no one ever offers—unless you ask!" The revelation lifted Carole's resentment toward Jill, and Jill vowed to become more aware of Carole's needs.
Being "Lady Bountiful" is just what the doctor ordered, writes Sark defiantly. In her popular book _Fabulous Friendship Festival: Loving Wildly, Learning Deeply, Living Fully with Our Friends_ , she relishes the status being a giver. "I actually think I'm more comfortable giving to friends than receiving. Because I feel in control when I'm the one giving. I also get the rewards of being perceived as the 'good' and 'generous' one. It also means that in any score keeping that may occur, I'll be the 'winner.' It feels embarrassing to admit this and also liberating. I know that other people experience this too." Knowing that about herself may enable Sark to continue building relationships that are skewed that way. On the other hand, she may have to make some adjustments as her friends get better at holding their own.
When all is said and done, renewing or ending a friendship is about both sides acknowledging their expectations and weighing them against what's possible. There is another reality to factor in to the equation: human nature. We wish our beloved friends well, but feelings of competitiveness, envy, and pettiness are always lurking. Schadenfreude, that unpleasant better-her-than-me feeling, crops up, right there along with all the compassion and love we feel, at someone's misfortune.
Kate has found that it is almost as hard to know where to stop looking for failings as it is to identify them at the start. She worries that she will set the bar for reconciliation impossibly high. "We all need to learn how to be a good friend, and in my opinion, number one on that list is not to ask too much of our friends, not to pressure them, guilt them, or hint around to them," she says. "We need to behave in such a way that our friends feel supported, and safe in our presence. Otherwise they will begin to distance themselves—and rightly so."
### _Getting Dumped_
Finding yourself on the other side of an unfriending is understandably painful and traumatic. Novelist Jacquelyn Mitchard lost her most intimate friend over some harsh words to the friend's son, which weren't forgiven. "For nearly 18 years we two had been one—what Carson McCullers called 'the we of me,'" she wrote in _More_ magazine. "Our friendship was as powerful as any love affair. So losing her was, I believe, as crushing as a divorce, and harder to face in midlife than a divorce." There are no conventions for consoling a grieving friend, she found. "Your marriage ends; someone dies. It's horrific. It's unbearable. And yet, quickly a circle of compassion surrounds you. People offer condolences, companionship and casseroles. You lose a friend, and unless you tell, no one even knows. If you do tell, no one much cares."
For her, recovery was long and painful, but she learned something from the breakup of this friendship that might make her other relationships healthier. "I wasn't being the friend I wanted to have." Looking back, she wonders, "was I more often the one who made the call or the one who returned it? As we exchanged news, did I listen first, and listen well, no matter how much my own update quivered on the tip of my tongue? Was I the one to postpone the coffee date because my life was too busy? Not always, but too often, the answers were uncomfortable."
Divorce and widowhood unsettle a wide circle of relationships, including long-standing friendships. Although many step forward and become a devoted support group, some friends take sides or pull back. Some couples can't handle socializing with half of a former couple. Others find that their lives are incompatible with that of a newly single friend. I heard from many women who were dumped when their circumstances changed, just when they needed those friends most. When one asked what the problem was, why wasn't she being invited to their homes anymore, she got a straight answer from her brother-in-law: "You aren't an 'and.'" At first she didn't get it, so he explained. "You aren't part of a Dick _and_ Jane or John _and_ Mary."
Imagine being unfriended even before you walk in the door. That's what happened to Ann when, at fifty-two, she remarried and moved into her new husband's community. She was looking forward to making new friends there and was instead dismayed to find herself snubbed by his old friends. "With a few dear-to-me exceptions, the women never invited me to get a drink, meet for coffee, take a walk—and they ignored my similar invitations," she recalls. "I feel terribly alone when they reminisce about their pasts together, often including stories about my husband and his ex-wife." And they show no interest in her life story. After seven years, she finds herself doubly cut off. "They do not include me in their group email lists, so I only hear second-hand about parties, illnesses, etc."
Late writer Carolyn Heilbrun, known for her curmudgeonly manner, suggested that you might want to unfriend _yourself_ briefly in order to protect your friendships from your own excesses. Known for her curmudgeonly manner, crotchety and impatient by nature, she delighted in letting off steam online. "If one sometimes feels compelled, as we all do, to complain about any dimension or all dimensions of one's life but does not do so because all the people one sees are sick of it and will visit even less often if complaints or criticism are forced upon them," she wrote in _The Last Gift of Time: Life Beyond Sixty_ , _"_ well, there are people out there who will be happy to exchange complaints and perhaps even help to talk us out of them, or counter them with other, strange grievances."
Finally, there are sad and frightening circumstances in which emergency treatment is called for in order to protect both you and your friend from herself—even if the friendship does not survive the "intervention." Dr. Levine has a checklist for those circumstances in which a friend should speak out.
* Your friend is self-destructive or being abusive to someone else.
* She is being abused.
* Her health, emotional well-being, or safety is at risk.
* She is breaking the law and putting herself in legal jeopardy.
* She is compromising your integrity or reputation (by asking you to do something dishonest or illegal).
* You just can't take it anymore.
That final item is the saddest realization of all, but the most necessary to act on—for your own survival.
# CHAPTER 5:
## NEW FRIENDS AT OUR AGE
_Each friend represents a world in us, a world possibly not born until they arrive, and it is only by this meeting that a new world is born._
—Anaïs Nin
AS WE REMODEL OUR LIVES, every door we open becomes unhinged. When we get through it, we may find that the floorboards are rickety, too. Each move we make has an impact on the makeup of our circle of trust. We leave jobs where we had friends and take jobs where we have none, or move from neighborhoods where we raised our kids—to be near our children and grandchildren—to ones where we don't know anyone else. Or we take off for parts unknown, leaving friends behind. Each new experience brings new needs and interests in the friend department.
Even if we stay put, the process of reinventing ourselves brings regular reminders that, as I point out in my books, we are not who we were, only older. You may not be the woman who, say, loved to travel with her pals, or sit at home and play Scrabble, or run things, or let other people run things. We are each a work in progress. So it figures that, since we see so much of ourselves reflected in our friendships, it would be invigorating to add to the circle who knew us back when, by making some new friends who know us as we are now. But how do you make friends, anyway? Especially at our age.
There are plenty of women who never really learned how to strike up a friendship. Remember, many of us grew up at the end of an era when women were taught that other women were not to be trusted, because they were rivals for the one accomplishment that mattered: snagging a man. My mother was an extreme case. She had barely any friends throughout her life, partly because she had a suspicious nature, but mostly, I am sure, because she was beautiful and the envy of other women, who had grown up believing that beauty and trust were not compatible.
When I was in high school, there was an unspoken covenant that any appointment with a girlfriend could be broken if you got an offer from a boy. Later, when women broke into male-only professions, others of our generation found themselves cut off from other women simply by being the only female in a given workplace. Again, other women were the competition, in this case for the token slot. At first it may have been fun to be singled out, but it wasn't long before those pioneers realized they were being shut out, too. No matter how well they did their jobs, they began to see that the rules of the game remained mysterious and the granting of power took place in back rooms (or male-only clubs or men's rooms).
Eventually, of course, it became clear that only by working together with other women, regardless of their rank, would we be able to force change. As women found strength in numbers, they forged friendships akin to battlefield comrades; they could be trusted to watch each other's backs. Those professional alliances were the first non-competitive female relationships for many of us.
This was Margaret's experience as well. Growing up, she had never really learned how to form intimate friendships. She thinks that issue goes back to her grade school years, when she was the only Catholic child in a predominantly Jewish student body and was also very tall. When she went out into the world, she entered an all-male corporate environment, where she was also the odd man out. It was only when she and other aggrieved women within the company took on a toxic boss that she found herself part of a tight team, despite the differences in their ages and job levels. Even though she is no longer in that organization, Margaret considers them her core group of friends, and they get together every couple of months.
Over the years, the lives of these girlfriends have changed in ways that are mutually enlightening. While she chose to forego having children, Margaret has gained some understanding of what is involved in parenting from the mothers in the group. She has learned how the workplace has changed from the younger women who are still within the corporate culture she helped change. Through them, she has kept in touch with the kinetic urban life, even after she and her husband moved to a rural community.
Within that community, she is once again feeling her lack of friendship skills. She still thinks of herself as a person who doesn't fit in. Even when she opens up to someone, she is always on the lookout for rejection; recently, she heard about a gathering of women in the community, including some of the women she sees often, and was hurt because she hadn't been invited. As it turned out, hearing about it through word of mouth was all the invitation anyone got. She misread or mistrusted the signals that a more experienced or confident friend might have interpreted differently. She needs more practice, she realizes, and is making a point of getting together one-on-one with the women she has met in the hopes that friendships click in.
Pat, the "hair theater" performer who calls herself "the Countess of Spit Curls," didn't grow up surrounded by girlfriends either. "There were no girls my age in my neighborhood, so I played all of the boy sports and games—and loved doing it. I also learned to fight, play tackle football, and hold my own in a confrontation. And," she adds, "I had no dolls." Growing up a tomboy in midcentury America, as Pat did, and I did too, was to feel you had little in common with girls, especially those who were operating within the traditional pink and dolls-and-jump-rope culture. For us, the time for trust-building came in the 1960s, when we defied those stereotypes alongside other women. But how do we connect with other women now?
### _Breaking the Ice_
Mary had just moved to a suburb and was working part-time from home. Cut off from her old comrades, she set out to make new ones. She signed up for a yoga class, where she saw a woman who looked interesting. "I wanted to get to know her," she says, "but I didn't have a clue how to break the ice. I felt as tongue-tied as if I was asking someone out for a first date."
The analogy is apt. The same advice we heard about "meeting men" applies here too. Do things that interest you: Go places you want to see. If you want to learn, take a course (surveys of women over fifty who have gone back to school report that one of the top reasons is to meet other women). If you want to serve the community, find a volunteer program. If you are at the same job or a new one, start a water-cooler conversation with someone you don't know. If you are setting up a business, seek out others who are new entrepreneurs as well. Or just take a chance.
I made a lively new friend by doing something I had never done before: I followed up on a fan letter from a woman who had interviewed me on the radio. I remembered how smart and energetic and feminist and funny she sounded on air, and so, when I found myself in her home city with a couple of hours free, I thought, _what the hell?_ and invited her for lunch. When I arrived at the restaurant, I recognized her right away. She looked just like she sounded: open, enthusiastic, unabashedly overjoyed to meet me. It was a little awkward at first since she was gushing—a personality trait we now laugh about—and I was uncomfortable with being put on a pedestal. But her enthusiasm for changing her life was so real, and I had so much fun sharing my experiences with her, that our meeting turned out to be a great success. The conversation hasn't stopped since. We email, visit, and do radio interviews regularly; our on-air discussions have evolved from Q-and-A into girlfriend talk. For the more intimate stuff, I call in several minutes before we go on the air, so we can have some private time. We have become friends.
Amy was particularly resourceful. When she got divorced, she stayed in the suburban house she and her ex-husband had lived in. She soon discovered that things had changed socially. "I think the women worried that I was after their husbands," she says. She wasn't after their husbands though; it was her women friends that she missed. In desperation, she decided to attend a singles night—a notoriously unsatisfactory place to meet men, but it suited her purposes. "I met some really great women there who were in the same situation I am," she says. "We go out together all the time now and really enjoy ourselves."
My favorite finding-new-friends story is the saga of "Jewelia." Several years ago _People_ magazine reported on a group of women who had come together to invest in and share a diamond necklace they all coveted. (The jeweler who had observed their separate interest and wanted to make the sale came up with the idea.) The arrangement was that they would transfer the necklace, which they named Jewelia, at monthly meetings. There were only two provisos: If one of the women was going to Paris she would get to take the necklace even if it wasn't her turn, and all of them had to make love wearing it at least once. It didn't take long before the meetings became more precious than the necklace they were set up to pass along. A woman whose sister had died said that the group enabled her "to reconnect with life again." Another, who was going through a divorce, was sure the group's support got her through it. And a self-described "biker chick" learned something about women's friendships. She couldn't believe that thirteen women could "share such a gorgeous piece of jewelry and not fight over it."
Liz, who manages a thrift store, didn't have to look very far for a new friend. She has her thirty-eight-year-old daughter-in-law, Ginny. "I totally relate to her," Liz says with delight. "It's clear to me that she is of a different generation, but that makes it interesting. I see her as a young woman." They talk every day, usually on the way to or from work, about marriage—including both of theirs—and work and life. And they talk a lot about the family they share. "It's like when you sense a particular kind of affinity with a certain woman, you can talk freely to her," Liz adds. "You don't have to think about what you say."
### _I'll Catch You Later—on the Internet_
The Internet, that fertile ground for romance, has produced some solid friendships, too. There are endless websites designed specifically to make those introductions. Meetup.com, for one, connects people with others who live in the same area and who have similar interests. Chris is an enthusiastic consumer of its offerings. Now fifty-nine, she was recently divorced when she joined. She currently belongs to several hiking groups, a movie group, a book club, a museum group, a scrabble group, and a coffee shop group. Some she attends regularly, others occasionally. "It's a safe way to get out and try new activities," she says, "without a big commitment of time and money." But best of all, she has made some very good friends and lots of acquaintances of all ages and lifestyles. Often when she meets someone, she says, "they mention a group they belong to and before I know it I am trying a new activity."
The Internet may not be the ideal format, though, for making the profound and empathetic connections that women require for a real friendship. Certain crucial elements are missing: making eye contact, reading body language, laughing so hard you have to hang on to each other to keep from falling over, picking up vibes. Eye contact is an essential requirement when we share our news over a cup of coffee; the absence of it, in fact, has been an ongoing problem between men and women. When men look away or only want to have a serious conversation when they are driving with eyes straight ahead, women tend to feel shut out. Sharing thoughts online has some of that distance-making quality.
Psychiatrist Michael Civin has studied what he calls "Internet-mediated relationships" and poses some provocative questions. "From the nearly infinite inside to the nearly infinite outside, we can reach out, we can be in touch," he writes in his book _Male Female e-mail._ "But to what are we reaching, and what is doing the reaching? What does it mean to be in touch when neither side can touch?" We know one thing: Words, or even an image, on a screen do not activate the soothing dose of oxytocin that in-person interaction does.
The Internet is wonderful, though, for the more practical elements of friendship: making arrangements, checking in, reporting on experiences and events with several friends at once, staying in touch with far-flung friends in inconvenient time zones, and especially finding long-lost friends. In the past, you really had to want to track someone down—and often failed to do it—by tracing addresses and phone numbers; you couldn't impulsively Google them. Linda Yellen described on Facebook how she "reconnected with members of my Campfire Girls troop (arguably the un-coolest girls on earth); corresponded all fakey-fakey friendly with a guy who once broke my heart." "Your children are lovely," she wrote him, but added, to herself, _they should have been mine_. She also "spent an entire day immersed in a conversation about salmon patties."
Cyberspace is also the place to mount a dress rehearsal of behaviors and personas that you think might be more authentic without committing to them. You can also see how your emerging self plays among strangers. You can ask questions that you may be ashamed to ask even your intimates. (There is no quicker resource for information on health or legal issues, for example.) No matter what you are looking for online, and no matter what you reveal, the response is that most reassuring message: You are not alone.
Getting online is also a great way for making an initial foray into a new flesh-and-blood community. If I ever doubted that the longing to connect was there, but sometimes stymied, I was resoundingly refuted by "Diva Night" in Milford, Pennsylvania. A local woman wondered why the amazing women she knew in the area didn't seem to know each other. So she posted a notice on a community website inviting women to come meet their neighbors over a $20 supper at a local hotel. The group, she announced, would meet once a month. The first meeting attracted a handful of women who had a wonderful time. By the second, the word had spread and fifteen women showed up. The third brought in sixty women. And when I attended the fourth, there were even more. What was amazing was that, for the most part, the women who greeted each other so enthusiastically were not long-lost friends, but newfound friends, women who lived among other interesting women but hadn't known how to meet them. The energy in that room was absolutely exhilarating.
### _Bonding over Misfortune_
Karin had made a few sort-of friends when she moved to a new state several years earlier but missed the "circle of trust" she left behind. Then, in memory of her mother, who died of cancer, she signed up for a sixty-kilometer fund-raising walk. To get in shape, she joined a group of women who train almost every day; all are either cancer survivors or caregivers. Over the past three years, they have walked hundreds of miles and talked for hundreds of hours about the experiences only they can truly understand, sharing the expertise that they have accumulated. Of course, Karin adds, "that's only when we are not talking about flowers or food or our children." After all, they have become friends, and that's what friends talk about.
Sarah found a most unlikely new friend—a woman twenty years older—during a crisis with a grown child. "It all began when my son was having his meltdown," she explains. "I was having lunch with a friend, and when I told her what was happening she said, 'I probably shouldn't be telling you this, but my mother is going through the same thing with my brother.'" Desperate for someone who really understood the helplessness she felt, Sarah asked her friend "if she thought I could write to her mother, she said yes, so I emailed her saying 'I gather we are both having the same heartbreak.'" A channel of profound intimacy was opened, almost without the usual getting-to-know-you phase, between people who rarely see each other or talk on the phone. Now, four years later, they email every day and share so much about their lives beyond their problem children that Sarah has stopped journaling. When she is "talking" to her friend, she is talking to herself. There are no barriers.
For Wendy, too, the connection with a true girlfriend was instigated by a family crisis—her husband's depression. Over decades of married life, she and her husband socialized with other couples, Noah's Ark style, in which everyone in the foursome had to be compatible. The couples got to know what was going on in one another's lives, and they never doubted that they could count on each other. But, she admits, "when couples get together it is on a different level—less intimate." One-on-one with a girlfriend "you might talk about yourself or your partner, but you're certainly not going to talk about your partner when they are there and make believe they are not there."
When Wendy's husband went into an extended depression and stopped talking altogether, she really needed someone to talk to about her partner. She met a woman in her aerobics class who had gone through a depression herself, and was very forthcoming and understanding. They now have lunch once a month and "really pour out our hearts to one another." Unlike Wendy, this woman is not married, and though she has a twenty-year-long relationship, her social life is not limited to couples. "I think that sometimes you can be intimate with a woman when she travels in a somewhat different circle," says Wendy.
This new friend arrived just in time. "I feel like I have been saved these last several months by having lunch with this dear woman. I cannot tell you how helpful and supportive it's been, where I can either vent, or I can get my mind off of stuff, and I don't feel like I'm going to be judged or criticized." She is learning something about woman-to-woman friendship. "Women know how to listen and how to respond, they just do."
### _Disappointments_
Of course, new friends may not be all you expect. I recently got all excited about a feisty and creative woman I met. We had so much in common and seemed to bring the same kind of intensity to the conversation. We had lunch several times and it felt as though we were on the same wavelength. After a while, however, I began to feel our conversations were getting short-circuited. It sounds silly, I know, but I found myself pulling back, because she didn't laugh at my jokes. Truly. If we couldn't laugh together, I was sure the friendship was going nowhere.
Trish had high hopes for a new friendship, too. "A few years ago, I made a concerted effort to become friends with a woman I met through my work. She was going through a divorce, contemplating leaving the company and starting a new career. I became her cheerleader, encouraging her to date, improve her CV, and work on her qualifications for a new job. She finally succeeded in all those things. Then she dumped me. I still don't know why." The former new friend probably didn't want to have any reminders of the bad old days. Not nice.
Friend-making is a matter of trial and error, risk-taking, and building trust. We proceed by degrees. A good example is Jane's account of the close/closer/closest sifting process she went through when she relocated to a new town. Given her work in textiles, she selected this particular place because it was known for its artist community. Soon after she arrived, she was invited to join a group of women artists who met once a week to talk about their work (close). Soon, though, the group evolved into what Jane calls "a therapy group." So she stopped going to the meetings, but stayed in touch with three of the women (closer). Two of them enjoy going places and doing things that neither Jane nor the third one do. Jane and the third woman have become real friends (closest). "I trust her," says Jane. "I can tell her anything."
# CHAPTER 6:
## UNITED WE STAND
_The one thought I wish to express is how little my friend and I could accomplish alone... I never expected to know any joy in this world equal to that of going up and down, getting good editorials written, engaging halls and advertising Mrs. Stanton's speeches. After that is through with, I don't expect any more joy... I never could have done the work I have if I had not had that woman at my right hand._
—Susan B. Anthony, describing fellow suffragist Elizabeth Cady Stanton
WE KNOW THAT HAVING GIRLFRIENDS is good for our overall health and that working together is a catalyst for building strength and reducing stress for every woman in the group. But it gets even better. On a larger scale, our collaborative power can have a healing impact on society. I learned both of those lessons when I worked at _Ms._ magazine—an experience that changed my life while our efforts helped change the world.
Workplace friendships are unlike others in that they exist in a context where we, and others, are able to see ourselves independently from how we are at home. It is also a place where we find routine, rewards, and teamwork—qualities we didn't necessarily find in the rest of our lives, especially during those hectic balancing-act years. Those collaborative rewards are hard to replace.
When I made the change from editing in a bustling magazine office to writing in a solitary cubicle, I found that the most difficult adjustment was the absence of the ongoing-chatter, information-sharing, problem-solving, life-changing (I quit smoking with a group of colleagues), morale-boosting atmosphere of the _Ms._ operation. Not to mention the pride we took in the impact of our work.
Now that we are at the point in our lives when so many women are moving the pieces of their lives around, we have to consider how we are going to enhance, replicate, or replace the collaborative experience—an important source of oxytocin, that uplifting hormone.
There isn't always a choice about working or the work itself. But even if the circumstances stay the same, there will be changes. Many women keep working at the same pace at the same job, but the team may turn over. Others change or cut back on work, which means establishing or revising relationships in a culture where ageism may be a source of new tensions. The same is true of those reentering the job market for the first time in years. Not only does a woman joining the workforce again have to develop new skills, but she has to gird herself for the difficult task of selling herself to a prospective employer and then to new colleagues. Others retire and need to look in new directions for community and collaboration.
Still another group is looking to refocus their skills on the bigger societal picture. "Millions are looking for productive engagement that is not only meaningful but also means something beyond themselves," writes Marc Freedman, founder of Encore.org ("second acts for the greater good"). His organization has generated a movement promoting work options for people over fifty, and celebrating the expertise and commitment to social justice that we have to offer. "People over fifty," Freedman adds, "are our most underused civic resource."
Thinking big is one of the healthiest and most rewarding characteristics that emerge in later life. According to psychiatrist Erik Erikson, who pioneered the study of society and the life cycle, successful aging involves "generativity"—making connections between your life and future generations—because, as he put it, "I am what survives of me." Some find generativity in work, or volunteering, or engagement with grandchildren or other young people. But for women, whatever work we do, we feel most productive when doing it together.
When Ellyn was downsized out of her marketing job, she was at a loss. Then she read a book she found particularly helpful in her situation, and she thought how much she would like to meet the author. She decided to take the plunge and invite the author to speak at her local community center. The evening was a great success and became the first in an ongoing series. "I stopped waiting for someone to walk into my kitchen to offer me great projects. Instead I created one," she says proudly. That impulsive venture also revealed a change in her attitude toward work: It "made a light bulb go off in my head—that I could work in projects without making a commitment for the rest of my life." But, she adds, she couldn't have done it alone. The joy for her was the "terrific group of bright, creative women" she has assembled. "We are so happy to have found each other. The project provided us with new friends and a new focus."
A retired English teacher who volunteers once a week at a medical library has replaced the water-cooler bond she had with her former colleagues with a strong connection to her fellow volunteers. "Every Wednesday we work together, and we have become amazingly attached to one another. We share books and we've had a couple of outings, and we keep saying to each other, 'Isn't it amazing how we all just found each other?' There are volunteers there every day, but the day we are there we just clicked as a group."
### _Transitioning from "What I Do" to "Who I Am"_
It isn't only co-workers that we miss when we leave a familiar workplace. Any woman who has had professional success becomes invested in that identity. When she pulls back, the challenge is to replace "what I do" with "who I am." The Transition Network (TTN), a ten-year-old organization whose motto is "Embracing Change After 50," brings women together who are making their way from their previous work lives on to "what's next?"
Robyn Yale, a social worker in San Francisco, joined the Bay Area chapter of TTN several years ago as she prepared to retire. "I didn't want to wake up one day with no activities," she recalls, and she figured that the TTN members were looking for the same thing, "new activities, new ways of doing things, new attitudes." She explored her emerging interests and needs in the discussion groups, get-togethers, outings, volunteer projects, and lectures that TTN offers. She found a circle of compatible and supportive "colleagues" and ultimately a new friend. She went to a networking meeting determined to take a risk. "I decided I would be the initiator. I picked this one person who looked interesting and approached her," she says. The two talked for a while and decided to have lunch to learn more about each other. That was four years ago. Since then they have become fixtures in one another's lives—and in the group effort that brought them together (which recently involved, for Robyn, participating in a panel discussion of women's friendships).
### _Mentoring in Our Own Way_
Mentoring is one of the most effective strategies we have developed to navigate a world that isn't always welcoming. It is also a satisfying, collaborative undertaking within a group as small as two. But from the start, the version created by women has been different from the established man-to-man formula that initiates new generations in the mysteries of climbing up the hierarchy. Women's objective is not so much to transfer power to a protégée as it is to create a power center within the system that would pave the way for future female employees, either by sharing information and political know-how with up-and-coming young women or by violating "class" barriers to mobilize a force for change. Interestingly, these diverging approaches reflect a wider pattern. Pollster Ethel Klein found that when asked why they go into politics, men say it is to gain power and influence, while women say they do so to make change. As Margaret discovered, the best way to confront the powerful hierarchy in her accounting firm was to reach beyond her solitary rank to all the women in the organization. Needless to say, those alliances have opened up avenues of shared intimacy among the courageous pioneers.
Nowadays, the mentoring connection is not limited to work or the workplace. I have met many women who are engaged in long-running relationships they found through volunteering at social service agencies, or simply pursuing compatibility with a young relative or daughter of a friend. For those mentors, many of whom do not have children of their own, the relationship is a unique collaboration.
Betsy made a major career shift—from the corporate world to the not-for-profit world—but has not shifted her commitment to mentoring young immigrant women. It's been twenty years, and some of her first mentees are still in her life. As she sees it, the set-up is different from other friendships right from the start; both women make a conscious commitment to build something between them. Over time, Betsy finds that she has entered the younger woman's life—consulting on family and romantic problems—and, to her amazement, the younger woman plays a supportive role in hers. "I can vent to her more freely than with anyone else," she says of one longtime friend. Getting to know her mentees' families, as she does, offers a personal pleasure as well; she has, she says, "always been curious about people and places" she doesn't know. As Betsy sees it, the real bond has nothing to do with age. It is the shared understanding that "I'm going to be me, and you are going to be you."
Earlier in my life, I lived in a generational no-woman's-land for a time, where a kind of mutual mentoring was lifesaving. When my daughter was born, I was forty-four, old enough to be her grandmother. When she went to school, I was old enough to be her teachers' mother. My contemporaries had long since forgotten about coping with babies and young children—they were on to the joys of grandchildren. My most meaningful cohort, outside of work, was other women with children my children's age. I needed the expertise and empathy—the mentoring—only they could offer. And I think they benefited from my more laid-back approach to organizing our children's lives. (I had lived long enough to have experienced John Lennon's words of wisdom: "Life is what happens when you are busy making other plans.")
As I move ahead, I look for that kind of common ground with women in their forties who are involved with the political community I found so exhilarating at _Ms._ and are approaching the stage of life I am chronicling. My friend Courtney and I can't be pals; I will never be as hip as she is, and she isn't going to be as world-weary as I am. But we can compare our lives. Our conversational pattern is mostly inquisitive. I want to know how activism works in a technological age. She wants to know what it was like back when you had to spend hours on the phone mobilizing a write-to-your-congressman action. I want her to know she can trust me. She wants me to know she is vulnerable. She even uses the term "mentoring" to describe the wisdom she is looking for. "There is the sense out there," she explains, "that we're so sophisticated, that we know it all; we might pretend we know it all, but in fact we are very hungry for mentoring, someone to tell me 'OK, here's the big picture, hang in there.'" I leave those conversations delighted to affirm that we can make each other laugh over the vagaries of family life and feminist politics in our different decades.
There is one difference, though: My generation is confronting a new form of exclusion. Historically, older women were involuntarily retired from the action at menopause and expected to compliantly erase themselves from the picture. We can't even imagine doing that. We are most definitely in the picture, even holding the paintbrush; we are as engaged in the workplace, the community, and the world as younger women. But the pressure to get lost is still there. As Alice Barden put it when describing the genesis of _Alice and Elizabeth's One Woman Show_ , a play she wrote about her "thrilling and rocky" lifelong friendship, "by the time we were circling our 40s we were getting along famously, when suddenly we weren't happy about the way society was treating us." They were, she went on, "two single career women who had never compromised our dreams in work or relationships; we were suddenly hearing a chorus of 'game over' from every area of our lives." They had no intention of retreating.
And neither do the rest of us. Our well-being is at stake. The forces of "game over" are not good for our health. It would be literally sickening to suppress all of those ambitions we nurture in each other, all of those risks we are prepared to take, all we are poised to contribute to society. Now that we are at the height of our powers—no more bullshit; no more acting a part; no more sweating the small stuff when there are so many more important things to get on with—we are ready to take on this challenge to our self-respect. And we'll do it the way we do best: together.
When the chips are down, we are formidable. In collective outrage, we become Thelma and Louise defiantly meeting their fate; or the First Wives Club, singing "You Don't Own Me" as they gleefully stick it to their feckless husbands. Or we become the women of _9 to 5_ , in which—contrary to stereotype—the Sexy One, the Smart One, and the Loyal Secretary join forces, bring their overlords to justice, and transform the workplace. Anthropologists call this kind of strength "mastery," a gut feeling that comes with age, a sense that you know what you are doing and you can do it. When we mobilize our numbers, we are fierce and effective.
An anonymous and hilarious post surfaced on the Internet soon after the United States went into Afghanistan in 2001, arguing that older women should be enlisted to fight terrorism because, among other reasons:
> We've spent years tracking down husbands or lovers in bars, hardware stores, or sporting events... finding bin Laden in some cave will be no problem.
>
> Between us, we've divorced enough husbands to know every trick there is for how they hide, launder, or cover up bank accounts and money sources. We know how to find that money and we know how to seize it... with or without government help.
"Let us go and fight," it concluded. "The Taliban hates women. Imagine their terror as we crawl like ants with hot-flashes over their godforsaken terrain."
The Taliban is not the only culture that "hates women." As our recent presidential election campaign has revealed, there are forces in our country crusading to roll back the achievements and the hard-won freedoms we have won over the past forty years. To their dismay, as the gender gap in that election proved, women are continuing to change the course of history.
But legislation isn't the only front on which we are being besieged. The more we flex our muscles, the more threatened cultural throwbacks feel; our accruing power is a challenge to the world as they know it. Their panic drives a creeping backlash effort to diminish our lives, to push us back "where we belong." For example, we are seeing the reemergence of a defense mechanism we used to call "the D and C," divide and conquer. This tactic emerged when, for example, groups that opposed both women's rights and civil rights, and feared the combined power of the two social justice movements, tried to turn them against each other. By promoting the notion that only one could prevail, the anti forces hoped the groups would fight among themselves and neither would succeed.
In its current form, the D and C effort tries to undermine our collective power by persuading us to mistrust each other, the way we did back in the Good Old Days. It is no accident that as counterforces make themselves heard, we see the emergence of cultural phenomena like the _Housewives_ series, the so-called reality shows that reinforce their subversive narrative: Watch out for your so-called friends, girls. They can't be trusted farther than you can throw them. The housewives turn on each other like scorpions in a bottle out of fear—fear of aging, fear of losing their social success, and fear of the deviousness in each of them. They don't build each other up; they tear their "friends" down. Those relationships don't look very real to any woman who knows how a circle of trust works. _Sex and the City_ , despite its Manolo Blahnik obsession, was a more recognizable reflection of the loyalty, laughter, and concern for each other's well-being that we know we can count on from our girlfriends. To the extent that it is humanly possible, we know that they will keep us healthy and sane.
Every friend is different. Every friendship is different and evolves over time. Especially as the effects of time accumulate and our physical and emotional needs intensify. At the same time, as our physical limitations increase, we emerge from a lifetime of traditional and psychological limitations into an exciting new world of possibilities and exhilarating freedom. For better and for worse, the friends we commit to now are signing up for the long haul. In sickness and in health, we will help each other become stronger, more vital, braver than we could ever be alone. We will defend our friendships fiercely against scourges large and small, natural and cultural, because our own lives depend on it. Individually and collectively, we are defining and promoting a totally new stage of life and wellness by living it to the fullest.
# About the Author
Suzanne Braun Levine's recent book, _How We Love Now: Women Talk About Intimacy After Fifty_ , continues her exploration of a new stage of life for women that began with _Inventing the Rest of Our Lives_ and _Fifty Is the New Fifty_. Levine was the first editor of _Ms._ magazine and editor of the _Columbia Journalism Review_. She produced the Peabody Award–winning documentary _She's Nobody's Baby: American Women in the 20 th Century._ She blogs for Huff/Post50, AARP, Next Avenue, and Vibrant Nation, and has appeared on _The Oprah Show_ , _Today_ , and _Charlie Rose_. She has also presented at the TEDxWomen event.
All rights reserved under International and Pan-American Copyright Conventions. By payment of the required fees, you have been granted the non-exclusive, non-transferable right to access and read the text of this ebook onscreen. No part of this text may be reproduced, transmitted, downloaded, decompiled, reverse engineered, or stored in or introduced into any information storage and retrieval system, in any form or by any means, whether electronic or mechanical, now known or hereinafter invented, without the express written permission of the publisher.
Copyright © 2004 by Suzanne Braun Levine
Cover design by Andrea C. Uva
978-1-4804-1485-3
Published in 2013 by Open Road Integrated Media
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Of 1,000 healthcare workers in the United Kingdom polled in the survey, 43 per cent were also so concerned about patient care that they thought their organisation could be at the centre of new scandal such as the one that hit the Mid Staffordshire NHS Foundation Trust.
Just one in four respondents said they have confidence in their senior managers, the research, carried out by the Chartered Institute of Personnel and Development (CIPD), found.
Article The Independent: Read more…………………….
The purpose of this paper is to raise further concerns about medical practitioners visited by medical misfortune.
The paper describes how social medicine focuses on the social determinants of health but it rarely focuses on the social determinants of the health of medical practitioners who may be visited by ill health or medical misfortune leading to disciplinary processes and ill health.
There is little written formally about the epidemiology of medical misfortune but we know that it may occur unexpectedly challenging the practitioner and demonstrating that in these situations the practitioner may well be on his or her own, frequently deserted by colleagues.
In 2004, in the United Kingdom, sanctions were made against 315 doctors; 82 erasures, 116 suspensions (64 on health grounds) and 117 conditions on registration (64 on health grounds). In 2003, of 214 doctors subject to GMC supervision, nine died, giving a case fatality rate of 4.2%, twice the death rate from coronary artery by-pass surgery. After the deaths of four soldiers at Deepcut Barracks in Surrey, a mortality of 0.03 per cent (throughput of 12,000 soldiers), Amnesty International called for a Public Inquiry. The trend has been downward since the National Audit Office documented the £40m spent on suspensions (the most frequent challenge) to the Public Accounts Committee..
If the morbidity and mortality amongst such medical practitioners is not addressed, more health service resources will be wasted, morale amongst medical practitioners will continue to fall and recruitment to the medical profession may adversely be affected.
1. General Medical Council. Annual Report 2004. General Medical Council. London, 2004.
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\section{Introduction}
\subsection{Log-correlated Gibbs measures}
\label{SUBSEC:1.1}
In this paper, we study
the Gibbs measure
$\rho$
on
the $d$-dimensional torus on $\mathbb{T}^d = (\mathbb{R}/2\pi\mathbb{Z})^d$, formally written as\footnote{In this introduction, we keep our discussion at a formal level and do not worry about renormalizations.
While we keep the following discussion only to the real-valued setting,
our results also hold in the complex-valued setting,
where $k \ge 4$ is an even integer and
$u^k$ in \eqref{Gibbs1}
is replaced by
$|u|^k$.
See Footnote \ref{FT1}.
Hereafter,
we use $Z$, $Z_{N}$, etc. to denote various normalization constants.}
\begin{align}
d\rho(u) = Z^{-1} \exp \bigg(\frac{\sigma}{k} \int_{\mathbb{T}^d} u^k dx\bigg) d\mu(u),
\label{Gibbs1}
\end{align}
\noindent
where $k \geq 3$ is an integer and $\sigma \in \mathbb{R}\setminus \{0\}$.
Here,
$\mu$ is the log-correlated Gaussian free field on~$\mathbb{T}^d$,
formally given by
\begin{align}
d \mu
= Z^{-1} e^{-\frac 12 \| u\|_{H^{d/ 2} }^2 } du
& = Z^{-1} \prod_{n \in \mathbb{Z}^d}
e^{-\frac 12 \jb{n}^{d} |\widehat u(n)|^2}
d\widehat u(n) ,
\label{gauss0}
\end{align}
\noindent
where
$\jb{\,\cdot\,} = (1+|\cdot|^2)^\frac{1}{2}$
and $\widehat u(n)$ denotes the Fourier coefficient of $u$.
When $d = 2$,
$\mu$ corresponds to
the massive Gaussian free field on $\mathbb{T}^2$.
Recall that this Gaussian measure $\mu$ is
nothing but the induced probability measure under the map:\footnote{By convention,
we endow $\mathbb{T}^d$ with the normalized Lebesgue measure $dx_{\mathbb{T}^d}= (2\pi)^{-d} dx$
so that we do not need to worry about the factor $2\pi$ in various places.}
\begin{equation}
\o\in \O \longmapsto u(\o) = \sum_{n \in \mathbb{Z}^d } \frac{ g_n(\o)}{\jb{n}^\frac d2} e_n,
\label{IV2}
\end{equation}
\noindent
where $e_n=e^{i n\cdot x}$
and
$\{ g_n \}_{n \in \mathbb{Z}^d}$
is a sequence of mutually independent standard complex-valued
Gaussian random variables on
a probability space $(\O,\mathcal{F},\mathbb{P})$
conditioned that
$g_{-n} = \overline{g_n}$.
See Remark \ref{REM:corr}.
It is well known that a typical function $u$
in the support of $\mu$ is merely a distribution
and thus a renormalization
on the potential energy $\frac \sigma k \int_{\mathbb{T}^d} u^k dx$
is required
for the construction of the Gibbs measure $\rho$.
Our main goal in this paper is to study
the Gibbs measure $\rho$ in \eqref{Gibbs1}
in the focusing case.\footnote{In this paper, by ``focusing'', we mean ``non-defocusing''.
Namely,
$\sigma > 0$ or $k$ is odd in \eqref{Gibbs1}.}
In particular,
we prove
the non-normalizability of the focusing Gibbs measure $\rho$
with the quartic interaction ($\sigma > 0$ and $k = 4$). See Theorem \ref{THM:1}.
We also present a brief discussion on the construction of the Gibbs measure
with the cubic interaction.
See Theorem \ref{THM:2}.
Before proceeding further, let us first go over the defocusing case:
$\sigma < 0$ and $k \geq 4$ is an even integer.
When $d = 2$,
the defocusing Gibbs measure $\rho$ in \eqref{Gibbs1}
corresponds
to the well-studied $\Phi^k_2$-measure
whose construction
follows
from
the hypercontractivity of the Ornstein-Uhlenbeck semigroup
(see Lemma \ref{LEM:hyp}) and Nelson's estimate~\cite{Nelson}.
See \cite{Simon, GlimmJ87, DT, OTh}.
For a general dimension $d \geq 1$,
the same argument allows us to construct
the defocusing Gibbs measure~$\rho$ in~\eqref{Gibbs1}
for any $\sigma < 0$ and any even integer $k \ge 4$.
Let us briefly go over the procedure.
Given $N \in \mathbb{N}$, we define the frequency projector\footnote{We may also proceed with
regularization via mollification.} $\pi_N$ by
\begin{align*}
\pi_N f =
\sum_{ |n| \leq N} \widehat f (n) e_n.
\end{align*}
\noindent
For $u$ as in \eqref{IV2},
set $u_N = \pi_N u$.
Then, for each fixed $x \in \mathbb{T}^d$,
$u_N(x)$ is
a mean-zero real-valued Gaussian random variable with variance
\begin{align}
\sigma_N = \mathbb{E}\big[u_N^2(x)\big] = \sum _{|n| \le N} \frac1{\jb{n}^d}
\sim \log N \longrightarrow \infty,
\label{sigma1}
\end{align}
\noindent
as $N \to \infty$. Note that $\sigma_N$ is independent of $x \in \mathbb{T}^d$.
We then define the renormalized power (= Wick power)
$:\! u_N^k\!:$
by setting
\begin{align*}
:\! u_N^k (x) \!: \, \stackrel{\text{def}}{=} H_k(u_N(x); \sigma_N),
\end{align*}
\noindent
where
$H_k(x;\sigma)$ is the Hermite polynomial of degree $k$
with a variance parameter $\sigma$.\footnote{\label{FT1}In the complex-valued setting
(with even $k$), we use the Laguerre polynomial
$c_k L_{\frac k2} (|u_N|^2; \sigma_N)$ to define the Wick renormalization.
See \cite{OTh}.}
We then define the following renormalized truncated potential energy:
\begin{align}
\begin{split}
R_N(u)=\frac \sigma k\int_{\mathbb{T}^d} :\! u_N^k \!: dx.
\end{split}
\label{Wick0}
\end{align}
\noindent
A standard computation allows
us to show that
$\{R_N \}_{N \in \mathbb{N}}$
forms a Cauchy sequence in $L^p(\mu)$
for any finite $p \geq 1$,
thus converging to some
random variable $R(u)$:
\begin{equation}
\lim_{N\rightarrow\infty} R_N(u)=R(u)
\qquad \text{in } L^p(\mu).
\label{conv1}
\end{equation}
\noindent
See, for example, Proposition 1.1 in \cite{OTh}.
Define the renormalized truncated Gibbs measure $\rho_{N}$ by
\begin{align*}
d \rho_{N} (u) = Z_{N}^{-1} e^{R_N(u)} d \mu(u).
\end{align*}
\noindent
Then,
a standard application of Nelson's estimate\footnote{One may also prove the uniform
exponential integrability bound \eqref{exp1}
via the variational approach as in \cite{BG},
using the Bou\'e-Dupuis variational formula (Lemma \ref{LEM:var3}).
When $k$ is large, however, the combinatorial complexity
for the variational approach
may be cumbersome,
while there is no such combinatorial issue in
the approach of~\cite{DT, OTh}.
} yields
the following
uniform exponential integrability of the density;
given any finite $ p \ge 1$,
there exists $C_{p, d} > 0$ such that
\begin{equation}
\sup_{N\in \mathbb{N}} \Big\| e^{R_N(u)}\Big\|_{L^p( \mu)}
\leq C_{p,d} < \infty.
\label{exp1}
\end{equation}
\noindent
See, for example, Proposition 1.2 in \cite{OTh}.
Then, the uniform bound \eqref{exp1}
together with
softer convergence in measure (as a consequence of \eqref{conv1})
implies the following $L^p$-convergence of the density:
\begin{equation*}
\lim_{N\rightarrow\infty}e^{ R_N(u)}=e^{R(u)}
\qquad \text{in } L^p( \mu).
\end{equation*}
\noindent
See, for example, Remark 3.8 in \cite{Tz08}.
This allows us to construct the defocusing Gibbs measure:
\begin{align*}
d\rho(u)= Z^{-1} e^{R(u)}d\mu(u)
\end{align*}
\noindent
as a limit of the truncated defocusing Gibbs measure $\rho_N$.
As mentioned above, our main goal is to study the Gibbs measure
$\rho$ with the log-correlated Gaussian field $\mu$
in the focusing case.
Before doing so, we present a brief discussion
on dynamical problems
associated with these Gibbs measures
in Subsection \ref{SUBSEC:PDE}.
We then present the non-normalizability
of the focusing log-correlated Gibbs measure
with the quartic interaction (Theorem \ref{THM:1})
and the construction of the focusing log-correlated Gibbs measure
with the cubic interaction (Theorem \ref{THM:2}).
\begin{remark}
\label{REM:corr}
\rm
Recall from
\cite[(4,2)]{AS} that
the Green's function $G_{\mathbb{R}^d}$ for $(1 - \Delta)^\frac{d}{2}$
on $\mathbb{R}^d$
satisfies
\begin{align}
G_{\mathbb{R}^d}(x) = -c_d \log|x| + o(1)
\label{G1}
\end{align}
\noindent
as $x \to 0$ for some $c_d > 0$.
It is a smooth function on $\mathbb{R}^d \setminus \{0\}$
and decays exponentially as $|x| \to \infty$;
see \cite[Proposition 1.2.5]{Gra2}.
Now,
let $G$ be the Green's function
for $(1-\Delta)^\frac{d}{2}$ on $\mathbb{T}^d$.
Then,
we have
\begin{align}
G \stackrel{\textup{def}}{=} (1-\Delta)^{-\frac{d}{2}} \delta_0 = \sum_{n\in\mathbb{Z}^d}\frac 1{\jb{n}^{d}}e_n
= \lim_{N \to \infty} \sum_{|n|\leq N}\frac 1{\jb{n}^{d}}e_n.
\label{G1a}
\end{align}
\noindent
Then, by applying the Poisson summation formula
(\cite[Theorem 3.2.8]{Gra1} with a frequency truncation $\pi_N$ and taking $N \to \infty$)
together with the asymptotics \eqref{G1},
we conclude that there exists
a smooth function $ R$ such that
\begin{align}
G(x) = - c_d \log|x| + R(x)
\label{G2}
\end{align}
\noindent
for any $x \in \mathbb{T}^d \setminus \{0\}$.
See \cite[Section 2]{ORSW} for a related discussion.
Finally, from \eqref{IV2}, \eqref{G1a}, and~\eqref{G2},
we obtain
\begin{align*}
\mathbb{E}_\mu \big[u(x) u(y)\big] = G(x-y) =
- c_d \log|x-y| + R(x-y)
\end{align*}
\noindent
for any $x, y \in \mathbb{T}^d$ with $x\ne y$.
\end{remark}
\subsection{Dynamical problems associated with
the log-correlated Gibbs measures}
\label{SUBSEC:PDE}
From the viewpoint
of mathematical physics such as
Euclidean quantum field theory,
the construction of the Gibbs measures $\rho$
in \eqref{Gibbs1} is of interest in its own right.
In this subsection, we briefly discuss
some examples of dynamical problems
associated with these log-correlated Gibbs measures.
These examples show the importance of studying the log-correlated
Gibbs measure $\rho$ in \eqref{Gibbs1}
from the (stochastic) PDE point of view.
The associated energy functional\footnote{Once again, we do not worry about renormalizations
in this formal discussion.}
for the Gibbs measure $\rho$ in \eqref{Gibbs1}
is given by
\begin{align}
E(u)
= \frac 12 \int_{\mathbb{T}^d} |(1-\Delta)^\frac{d}{4} u|^2 dx
- \frac \sigma k \int_{\mathbb{T}^d} u^{k} dx.
\label{Gibbs2}
\end{align}
\noindent
The study of the Gibbs measures for Hamiltonian PDEs, initiated by \cite{Fried, LRS, BO94, McKean, BO96},
has been an active field of research over the last decade.
We first list examples of
the Hamiltonian PDEs
generated by this energy functional $E(u)$ in \eqref{Gibbs2}
along with the references.
\begin{itemize}
\item[(i)]
fractional nonlinear Schr\"odinger equation
(for complex-valued $u$):
\begin{align}
i\partial_t u + (1-\Delta)^{\frac d2}u -\sigma | u |^{k-2}u=0.
\label{H1}
\end{align}
\noindent
The equation \eqref{H1} corresponds to
the nonlinear half-wave equation
(also known as
the semi-relativistic nonlinear Schr\"odinger equation)
when $ d= 1$,
to
the well-studied cubic nonlinear Schr\"odinger equation (NLS)
when $d = 2$ (\cite{BO96, OTh, DNY}),
and to
the biharmonic nonlinear Schr\"odinger equation
when $d = 4$.
In Appendix \ref{SEC:A}, we also provide a brief discussion
on the Gibbs measure
for the Zakharov system when $d = 2$.
\smallskip
\item[(ii)] fractional nonlinear wave equation:\footnote{For \eqref{H2},
we need to add $\frac 12 \int_{\mathbb{T}^d} (\partial_t u)^2 dx$ to the energy functional $E(u)$ in \eqref{Gibbs2}.}
\begin{align}
\partial_t^2 u + (1- \Delta)^\frac{d}{2} u - \sigma u^{k-1} = 0.
\label{H2}
\end{align}
\noindent
The equation \eqref{H2} corresponds to
the nonlinear wave equation (NLW)
(or the nonlinear Klein-Gordon equation)
when $d = 2$ (\cite{OTh2}),
and to the nonlinear beam equation
when $d = 4$.
\smallskip
\item[(iii)] generalized Benjamin-Ono equation (with $d = 1$):\footnote{For \eqref{H3},
the coefficient of the potential energy in \eqref{Gibbs2} is slightly different.
Thanks to the conservation of the spatial mean $\int_\mathbb{T} u dx$
under the generalized Benjamin-Ono equation~\eqref{H3}, we can work on the mean-zero functions.
In this case, we consider the Gibbs measure associated
with the massless log-correlated Gaussian field
by replacing $(1-\partial_x^2)^\frac{1}{4}$ in \eqref{Gibbs2} with
$(-\partial_x^2)^\frac{1}{4}$.}
\begin{align}
\partial_t u+\mathcal{H}\partial_x^2u -\sigma \partial_x (u^{k-1})=0.
\label{H3}
\end{align}
\noindent
The equation \eqref{H3} is known
as the Benjamin-Ono equation when $k = 3$ (\cite{TzBO, Deng})
and the modified Benjamin-Ono equation when $k = 4$.
\end{itemize}
\smallskip
\noindent
Next, we list stochastic PDEs associated with
the Gibbs measure $\rho$ in \eqref{Gibbs1}.
\begin{itemize}
\item[(iv)] parabolic stochastic quantization equation
\cite{PW}:
\begin{align}
\partial_t u + (1 - \Delta)^\frac{d}{2} u -\sigma u^{k-1} = \sqrt 2\xi.
\label{H4}
\end{align}
\noindent
Here, $\xi$ denotes the space-time white noise on $\mathbb{T}^d \times \mathbb{R}_+$.
When $d = 2$ and $\sigma < 0$, \eqref{H4}
corresponds to the standard parabolic $\Phi^k_2$-model
(\cite{DPD, RZZ, TsW}).
\smallskip
\item[(v)] canonical stochastic quantization equation
\cite{RSS}:
\begin{align}
\partial_t^2 u + \partial_t u + (1 - \Delta)^\frac{d}{2} u -\sigma u^{k-1} = \sqrt{2} \xi.
\label{H5}
\end{align}
\noindent
The equation \eqref{H5} corresponds to
the stochastic damped nonlinear wave equation
when $d = 2$ (\cite{GKO, GKOT, Tolomeo3}),
and to the stochastic damped nonlinear beam equation
when $d = 4$.
\end{itemize}
\smallskip
\noindent
When $d = 2$,
the conservative stochastic Cahn-Hilliard equation is known to
(formally) preserve the Gibbs measure $\rho$ in \eqref{Gibbs1} (\cite{RYZ}).
For the equations listed above,
once
we establish local well-posedness almost surely
with respect to the Gibbs measure initial data,
Bourgain's invariant measure argument \cite{BO94, BO96}
allows us to construct almost sure global dynamics
and to prove invariance of the Gibbs measure.
Due to the low regularity\footnote{For example, the $L^p$-based Sobolev spaces $W^{s, p}(\mathbb{T}^d)$ only for $ s < 0$ with any $1 \le p \le \infty$.}
of the support of the
log-correlated Gibbs measure~$\rho$ in \eqref{Gibbs1},
however,
there are only a handful of the well-posedness results
\cite{BO96, Deng, OTh2, GKO, DNY}
for the Hamiltonian PDEs mentioned above (including \eqref{H5}).
\begin{remark}\rm
We point out that as long as we can construct the Gibbs measure,
a compactness argument
with invariance of the truncated Gibbs measures
and Skorokhod's theorem allows
us to construct (non-unique) global-in-time dynamics
along with invariance of the Gibbs measure in some mild sense.
See \cite{AC, DPD2, BTTz, OTh, ORTh}.
In our current setting, this almost sure global existence
result holds
for (i) the defocusing case ($\sigma < 0$ and even $k \geq 4$; see the discussion in Subsection~\ref{SUBSEC:1.1})
and (ii) the quadratic nonlinearity (i.e.~$k = 3$).
See Theorem \ref{THM:2}
for the latter case.
\end{remark}
\subsection{Non-normalizability of the focusing Gibbs measure}
\label{SUBSEC:1.3}
We now turn our attention to the focusing case.
In this subsection, we study the Gibbs measure $\rho$ in
\eqref{Gibbs1}
with the focusing quartic interaction
($\sigma > 0$ and $k = 4$).
In this case, we prove the following
non-normalizability
of the (renormalized) focusing Gibbs measure~$\rho$.
\begin{theorem}\label{THM:1}
Let $\sigma > 0$ and $k = 4$.
Then, given any $K > 0$, we have
\begin{align}
\begin{split}
\sup_{N \in \mathbb{N}} \mathbb{E}_{\mu} \Big[ e^{ R_N (u)} \Big]
& \ge
\sup_{N \in \mathbb{N}} \mathbb{E}_{\mu} \Big[ \mathbf 1_{\{\int_{\mathbb{T}^d} \, :u_N^2: \, dx \, \leq K\}} e^{ R_N (u)} \Big]\\
& \ge
\sup_{N \in \mathbb{N}} \mathbb{E}_{\mu} \Big[ \mathbf 1_{\{|\int_{\mathbb{T}^d} \, :u_N^2: \, dx| \, \leq K\}} e^{ R_N (u)} \Big] = \infty,
\end{split}
\label{Gibbs4}
\end{align}
\noindent
where $R_N$ is the renormalized potential energy defined in \eqref{Wick0}
with $k = 4$.
In particular, the focusing Gibbs measure \textup{(}even with a Wick-ordered $L^2$-cutoff\textup{)}
can not be defined as a probability measure.
\end{theorem}
When $d = 2$,
Theorem \ref{THM:1} provides an alternative proof of the non-normalizability
result
for
of the focusing $\Phi^4_2$-measure
due to Brydges and Slade \cite{BS}
whose proof
is based on analysis of a model closely related to
the Berlin-Kac spherical model.
Our proof of Theorem \ref{THM:1} is based on the variational approach
due to Barashkov and Gubinelli~\cite{BG}.
More precisely, we will rely on
the Bou\'e-Dupuis variational formula \cite{BD, Ust}; see Lemma~\ref{LEM:var3}.
Our main task is to
construct a drift term which achieves the desired divergence~\eqref{Gibbs4}.
Our argument is based on recent works
by the third author with Weber~\cite{TW}
and by the first and third authors with Okamoto
\cite{OOT, OOTolo}.
In particular, our presentation closely follows that in \cite{OOT},
where an analogous non-normalizability is shown
for Gibbs measures with quartic interaction of Hartree-type.
We point out that the argument in \cite{OOT}
shows non-normalizability only for large $K \gg 1$
and thus we need to refine the argument to prove the divergence~\eqref{Gibbs4}
for {\it any} $K > 0$.
We also mention related works
\cite{LRS, BS, Rider, BB14, OST}
on the non-normalizability (and other issues)
for focusing Gibbs measures.
\begin{remark}\rm
(i) An analogous non-normalizability result
holds for a focusing Gibbs measure with
the quartic interaction
even if we endow it with taming by the Wick-ordered $L^2$-norm.
See Remark \ref{REM:NLW}.
\smallskip
\noindent
(ii)
By controlling combinatorial complexity,
we can extend the non-normalizability result in Theorem \ref{THM:1}
to the higher order interactions $k \geq 5$ in the focusing case
(i.e.~either $k$ is odd or $\sigma > 0$ when $k$ is even).
\smallskip
\noindent
(iii)
In terms of dynamical problems,
Theorem \ref{THM:1} states
that Gibbs measures
associated with
the equations
listed in Subsection \ref{SUBSEC:PDE}
do not exist for (i) $\sigma> 0$ and $k \ge 4$
or (ii) odd $k \ge 5$.
This list in particular includes
\smallskip
\begin{itemize}
\item
the focusing $L^2$-(super)critical fractional NLS \eqref{H1}
(including the focusing (super)cubic NLS on $\mathbb{T}^2$),
\smallskip
\item
the focusing $L^2$-(super)critical fractional NLW \eqref{H2}
(including the focusing (super)cubic NLW on $\mathbb{T}^2$
and the focusing (super)cubic nonlinear beam equation on $\mathbb{T}^4$),
\smallskip
\item
the focusing modified Benjamin-Ono equation \eqref{H3}
(and
the focusing generalized Benjamin-Ono equation with $k \ge 5$).
\smallskip
\end{itemize}
See also Appendix \ref{SEC:A} for a brief discussion
on the two-dimensional Zakharov system.
\end{remark}
\subsection{Gibbs measure with the cubic interaction}
\label{SUBSEC:1.4}
Let us first go over the focusing Gibbs measure construction in the two-dimensional setting.
In \cite{BO95},
Bourgain reported Jaffe's construction of a $\Phi^3_2$-measure endowed with a Wick-ordered
$ L^2$-cutoff:
\begin{align}
d\rho(u) = Z^{-1}
\mathbf 1_{\{\int_{\mathbb{T}^2} :\,u^2: \, dx\, \leq K\}}
e^{ \int_{\mathbb{T}^2} :u^3: \, dx }d \mu(u) .
\label{Gibbs5}
\end{align}
\noindent
Note that
the measure in \eqref{Gibbs5}
is not suitable to generate any NLS\,/\,NLW\,/\,heat dynamics since
(i)~the renormalized cubic power $:\!u^3 \!:$ makes sense only
in the real-valued setting and hence is not suitable for the Schr\"odinger equation
and (ii) NLW and the heat equation do not preserve the $L^2$-norm of a solution
and thus are incompatible with the Wick-ordered $L^2$-cutoff.
In \cite{BO95},
Bourgain instead proposed to consider the Gibbs measure
of the form:
\begin{align}
d\rho(u ) = Z^{-1}
e^{ \int_{\mathbb{T}^2} :u^3: \, dx - A
\big(\int_{\mathbb{T}^2} :\,u^2: \, dx\big)^2} d \mu( u )
\label{Gibbs6}
\end{align}
\noindent
(for sufficiently large $A>0$)
in studying NLW dynamics on $\mathbb{T}^2$.\footnote{For the NLW dynamics,
we need to couple $\rho$ on the $u$-component with the white noise measure on the $\partial_t u$-component.}
We now extend the construction of the Gibbs measures in \eqref{Gibbs5}
and \eqref{Gibbs6} to a general dimension $d \geq 1$.
Given $N \in \mathbb{N}$, let
\begin{align}
\begin{split}
\mathcal{R}_N (u)
&= \frac \sigma 3\int_{\mathbb{T}^d} :\! u_N^3 \!: dx
- A \, \bigg( \int_{\mathbb{T}^d} :\! u_N^2 \!: dx\bigg)^2
\end{split}
\label{K2}
\end{align}
\noindent
and
define the truncated renormalized Gibbs measure $\rho_{N}$ by
\begin{align}
d \rho_{N} (u) = Z_{N}^{-1} e^{\mathcal{R}_N(u)} d \mu(u).
\label{GibbsN1}
\end{align}
\noindent
Then, we have the following result
for the focusing Gibbs measure with a cubic interaction.
\begin{theorem} \label{THM:2}
Let $\sigma > 0$.
Given any finite $ p \ge 1$,
there exists sufficiently large $A = A(\sigma, p) > 0$
such that
$\mathcal{R}_N $ in \eqref{K2}
converges to some limit $\mathcal{R}$
in $L^p(\mu)$.
Moreover,
there exists $C_{p, d, A} > 0$ such that
\begin{equation}
\sup_{N\in \mathbb{N}} \Big\| e^{\mathcal{R}_N(u)}\Big\|_{L^p(\mu)}
\leq C_{p,d, A} < \infty.
\label{exp3}
\end{equation}
\noindent
In particular, we have
\begin{equation}
\lim_{N\rightarrow\infty}e^{ \mathcal{R}_N(u)}=e^{\mathcal{R}(u)}
\qquad \text{in } L^p(\mu).
\label{exp4}
\end{equation}
\noindent
As a consequence,
the truncated renormalized Gibbs measure $\rho_{N}$ in \eqref{GibbsN1} converges, in the sense of \eqref{exp4},
to the focusing Gibbs measure $\rho$ given by
\begin{align*}
d\rho(u)= Z^{-1} e^{\mathcal{R}(u)}d\mu(u).
\end{align*}
\noindent
Furthermore,
the resulting Gibbs measure $\rho$ is equivalent
to the log-correlated Gaussian field~$\mu$.
\end{theorem}
As for the convergence of $\mathcal{R}_N$, we omit details since
the argument is standard.
See, for example,
\cite[Proposition 1.1]{OTh},
\cite[Proposition 3.1]{OTz},
\cite[Lemma 4.1]{GOTW},
and \cite[Lemma 5.1]{OOT} for related details.
As mentioned in Subsection \ref{SUBSEC:1.1},
the main task is to prove
the uniform integrability bound \eqref{exp3}.
Once this is done, the rest follows from a standard argument.
In Section \ref{SEC:4},
we establish the bound \eqref{exp3}
by using the variational formulation.
\begin{remark} \label{REM:cub}\rm
Note that
\begin{align}
\mathbf 1_{\{|\,\cdot \,| \le K\}}(x) \le \exp\big( - A |x|^\gamma\big) \exp(A K^\gamma)
\label{H6}
\end{align}
\noindent
for any $K, A , \gamma > 0$.
Then,
the following
uniform bound
for the focusing cubic interaction:
\begin{equation*}
\sup_{N\in \mathbb{N}} \Big\|
\mathbf 1_{\{|\int_{\mathbb{T}^d} :\,u^2: \, dx| \leq K\}}
e^{R_N(u)}\Big\|_{L^p( \mu)}
\leq C_{p,d, K} < \infty
\end{equation*}
\noindent
for any $K > 0$
follows as a direct consequence
of the uniform bound \eqref{exp3} and \eqref{H6},
where
$R_N$ is as in \eqref{Wick0} with $\sigma > 0$ and $k = 3$.
This allows us to construct the log-correlated Gibbs measure
with the cubic interaction (with a Wick-ordered
$ L^2$-cutoff):
\begin{align*}
d\rho(u) = Z^{-1}
\mathbf 1_{\{|\int_{\mathbb{T}^d} :\,u^2: \, dx| \leq K\}}
e^{ \frac \sigma 3\int_{\mathbb{T}^d} :u^3: \, dx }d \mu(u)
\end{align*}
\noindent
as a limit of its truncated version
(for any $\sigma , K > 0$).
\end{remark}
\begin{remark} \rm
In \cite{TzBO}, Tzvetkov constructed the Gibbs
measure (with a Wick-ordered
$ L^2$-cutoff) for the Benjamin-Ono equation \eqref{H3} with $k = 3$.
Theorem \ref{THM:2} and Remark \ref{REM:cub} provide
an alternative proof of the construction of the Gibbs
measure
for the Benjamin-Ono equation.
\end{remark}
\begin{remark}\label{REM:NLW} \rm
(i) It follows from Theorem \ref{THM:1} and \eqref{H6} that
an analogue of Theorem \ref{THM:2} fails
for the quartic interaction ($k = 4$).
More precisely, we have
\begin{equation*}
\sup_{N\in \mathbb{N}} \bigg\|
\exp \bigg(
\frac \sigma 4 \int_{\mathbb{T}^d} :\! u_N^4 \!: dx
- A \, \Big| \int_{\mathbb{T}^d} :\! u_N^2 \!: dx\Big|^\gamma
\bigg)\bigg\|_{L^p(\mu)}
= \infty
\end{equation*}
\noindent
for any $\sigma, A, \gamma > 0$.
\smallskip
\noindent
(ii)
If we consider
a smoother base Gaussian
measure $\mu_\alpha$, then
we can prove the following uniform exponential integrability bound;
given any $\sigma > 0$, $\alpha > \frac d2$,
and finite $p \geq 1$,
there exists
sufficiently large $A = A(\sigma, \alpha, p) > 0$
and $\gamma = \gamma(\alpha) > 0$
such that
\begin{equation}
\sup_{N\in \mathbb{N}} \bigg\|
\exp \bigg(
\frac \sigma 4 \int_{\mathbb{T}^d} u_N^4 dx
- A \, \Big( \int_{\mathbb{T}^d} u_N^2 dx\Big)^\gamma
\bigg)\bigg\|_{L^p(\mu)}
\leq C_{p, d, A} < \infty.
\label{Q1}
\end{equation}
\noindent
Here, $\mu_\alpha$ denotes the Gaussian measure
with a formal density
\begin{align}
d \mu_\alpha
= Z^{-1} e^{-\frac 12 \| u\|_{H^{\alpha} }^2 } du.
\label{gauss1}
\end{align}
\noindent
See Appendix \ref{SEC:B}
for the proof of \eqref{Q1}.
The bound \eqref{Q1} allows us to construct
the focusing Gibbs measure
with a focusing quartic interaction
of the form:
\begin{align}
d\rho_\alpha = Z^{-1}
e^{ \frac \sigma 4 \int_{\mathbb{T}^d} u^4 dx - A
\big(\int_{\mathbb{T}^d} u^2 dx\big)^\gamma} d \mu_\alpha .
\label{Q2}
\end{align}
\noindent
Moreover, in view of \eqref{H6},
we can also construct the following
the focusing Gibbs measure with an $L^2$-cutoff:
\begin{align}
d\rho_\alpha = Z^{-1}
\mathbf 1_{\{\int_{\mathbb{T}^d} |u|^2 dx \leq K\}}
e^{ \frac \sigma 4 \int_{\mathbb{T}^d} |u|^4 dx
} d \mu_\alpha
\label{Q3}
\end{align}
\noindent
for any $K > 0$.
In \cite{STz1, STz2},
Sun and Tzvetkov recently studied the following
fractional NLS on $\mathbb{T}$:
\begin{align}
i\partial_t u + (1-\partial_x^2)^{\alpha}u -\sigma | u |^{2}u=0
\label{Q4}
\end{align}
\noindent
in the defocusing case ($\sigma < 0$).
They
proved almost sure local well-posedness
of \eqref{Q4} with respect to the Gaussian measure $\mu_\alpha$
in \eqref{gauss1}
for $\alpha > \frac {31- \sqrt{233}}{28} \approx 0.562 \ ( \, > \frac 12)$,
which in turn yielded almost sure global well-posedness
with respect to the defocusing Gibbs measure
(namely, $\rho_\alpha$ in~\eqref{Q3} without an $L^2$-cutoff)
and invariance of the defocusing Gibbs measure.
Since their local result also holds
in the focusing case ($\sigma>0$),
our construction of the focusing Gibbs measure
$\rho_\alpha$ in \eqref{Q3}
implies
almost sure global well-posedness of \eqref{Q4}
with respect to the focusing Gibbs measure $\rho_\alpha$ in~\eqref{Q3}
and its invariance under the dynamics of \eqref{Q4}
for the same range of $\alpha$.
\smallskip
\noindent
(iii)
Theorem \ref{THM:1} and Part (ii) of this remark show that
in the case of the focusing cubic nonlinearity,
there is no bifurcation, depending on the value of $\sigma > 0$.
Compare this with the situation in \cite{OOT, OOTolo},
where such a bifurcation\footnote{Namely, normalizability for small $\sigma > 0$
and non-normalizability for $\sigma \gg 1$.} was established in the critical case.
It may be of interest to pursue
the issue of a possible bifurcation
for a higher order nonlinearity.
\end{remark}
\section{Preliminary lemmas}
\label{SEC:2}
In this section, we recall basic lemmas to be used in this paper.
\subsection{Deterministic estimates}
We first recall the following interpolation
and fractional Leibniz rule.
\begin{lemma}\label{LEM:prod}
The following estimates hold.
\noindent
\textup{(i) (interpolation)}
For $0 < s_1 < s_2$, we have\footnote{We use the convention that the symbol $\lesssim$ indicates that inessential constants are suppressed in the inequality.}
\begin{equation*}
\| u \|_{H^{s_1}} \lesssim \| u \|_{H^{s_2}}^{\frac{s_1}{s_2}} \| u \|_{L^2}^{\frac{s_2-s_1}{s_2}}.
\end{equation*}
\smallskip
\noindent
\textup{(ii) (fractional Leibniz rule)} Let $0\le s \le 1$. Suppose that
$1<p_j,q_j,r < \infty$, $\frac1{p_j} + \frac1{q_j}= \frac1r$, $j = 1, 2$.
Then, we have
\begin{equation}
\| \jb{\nabla}^s (fg) \|_{L^r(\mathbb{T}^d)}
\lesssim \Big( \| f \|_{L^{p_1}(\mathbb{T}^d)}
\| \jb{\nabla}^s g \|_{L^{q_1}(\mathbb{T}^d)} + \| \jb{\nabla}^s f \|_{L^{p_2}(\mathbb{T}^d)}
\| g \|_{L^{q_2}(\mathbb{T}^d)}\Big),
\label{bilinear+}
\end{equation}
\noindent
where $\jb{\nabla} = \sqrt{1 - \Delta}$.
\end{lemma}
As for the second estimate \eqref{bilinear+},
see \cite[Lemma 3.4]{GKO}.
\subsection{Tools from stochastic analysis}
Next, we recall the Wiener chaos estimate (Lemma~\ref{LEM:hyp}).
For this purpose, we first recall
basic definitions
from stochastic analysis;
see \cite{Bog, Shige}.
Let $(H, B, \mu)$ be an abstract Wiener space.
Namely, $\mu$ is a Gaussian measure on a separable Banach space $B$
with $H \subset B$ as its Cameron-Martin space.
Given a complete orthonormal system $\{e_j \}_{ j \in \mathbb{N}} \subset B^*$ of $H^* = H$,
we define a polynomial chaos of order
$k$ to be an element of the form $\prod_{j = 1}^\infty H_{k_j}(\jb{x, e_j})$,
where $x \in B$, $k_j \ne 0$ for only finitely many $j$'s, $k= \sum_{j = 1}^\infty k_j$,
$H_{k_j}$ is the Hermite polynomial of degree $k_j$,
and $\jb{\cdot, \cdot} = \vphantom{|}_B \jb{\cdot, \cdot}_{B^*}$ denotes the $B$--$B^*$ duality pairing.
We then
denote the closure of
polynomial chaoses of order $k$
under $L^2(B, \mu)$ by $\mathcal{H}_k$.
The elements in $\mathcal{H}_k$
are called homogeneous Wiener chaoses of order $k$.
We also set
\begin{align}
\mathcal{H}_{\leq k} = \bigoplus_{j = 0}^k \mathcal{H}_j
\notag
\end{align}
\noindent
for $k \in \mathbb{N}$.
Let $L = \Delta -x \cdot \nabla$ be
the Ornstein-Uhlenbeck operator.\footnote{For simplicity,
we write the definition of the Ornstein-Uhlenbeck operator $L$
when $B = \mathbb{R}^d$.}
Then,
it is known that
any element in $\mathcal H_k$
is an eigenfunction of $L$ with eigenvalue $-k$.
Then, as a consequence
of the hypercontractivity of the Ornstein-Uhlenbeck
semigroup $U(t) = e^{tL}$ due to Nelson \cite{Nelson},
we have the following Wiener chaos estimate
\cite[Theorem~I.22]{Simon}.
\begin{lemma}\label{LEM:hyp}
Let $k \in \mathbb{N}$.
Then, we have
\begin{equation*}
\|X \|_{L^p(\O)} \leq (p-1)^\frac{k}{2} \|X\|_{L^2(\O)}
\end{equation*}
\noindent
for any $p \geq 2$
and any $X \in \mathcal{H}_{\leq k}$.
\end{lemma}
\section{Non-normalizability of
the focusing Gibbs measure with the quartic interaction}
\label{SEC:foc2}
In this section, we present the proof of
the non-normalizability of the log-correlated Gibbs
measure
with the focusing quartic interaction
(Theorem \ref{THM:1}).
\subsection{Variational formulation}
\label{SUBSEC:var}
In order to prove \eqref{Gibbs4},
we use a variational formula for the partition function
as in \cite{TW, OOT}.
Let us first introduce some notations.
Let $W(t)$ be a cylindrical Brownian motion in $L^2(\mathbb{T}^d)$.
Namely, we have
\begin{align}
W(t) = \sum_{n \in \mathbb{Z}^d} B_n(t) e_n,
\label{P1}
\end{align}
\noindent
where
$\{B_n\}_{n \in \mathbb{Z}^d}$ is a sequence of mutually independent complex-valued\footnote{By convention, we normalize $B_n$ such that $\text{Var}(B_n(t)) = t$. In particular, $B_0$ is a standard real-valued Brownian motion.} Brownian motions such that
$\overline{B_n}= B_{-n}$, $n \in \mathbb{Z}^d$.
Then, define a centered Gaussian process $Y(t)$
by
\begin{align}
Y(t)
= \jb{\nabla}^{-\frac d2}W(t).
\label{P2}
\end{align}
\noindent
Note that
we have $\Law(Y(1)) = \mu$,
where $\mu$ is the log-correlated Gaussian measure in \eqref{gauss0}.
By setting $Y_N = \pi_NY $,
we have $\Law(Y_N(1)) = (\pi_N)_*\mu$,
i.e.~the pushforward of $\mu$ under $\pi_N$.
In particular,
we have $\mathbb{E} [Y_N^2(1)] = \sigma_N$,
where $\sigma_N$ is as in~\eqref{sigma1}.
Next, let $\mathbb{H}_a$ denote the space of drifts,
which are progressively measurable processes
belonging to
$L^2([0,1]; L^2(\mathbb{T}^d))$, $\mathbb{P}$-almost surely.
We now state the Bou\'e-Dupuis variational formula \cite{BD, Ust};
in particular, see Theorem 7 in \cite{Ust}.
\begin{lemma}\label{LEM:var3}
Let $Y$ be as in \eqref{P2}.
Fix $N \in \mathbb{N}$.
Suppose that $F:C^\infty(\mathbb{T}^d) \to \mathbb{R}$
is measurable such that $\mathbb{E}\big[|F(\pi_NY(1))|^p\big] < \infty$
and $\mathbb{E}\big[|e^{-F(\pi_NY(1))}|^q \big] < \infty$ for some $1 < p, q < \infty$ with $\frac 1p + \frac 1q = 1$.
Then, we have
\begin{align}
- \log \mathbb{E}\Big[e^{-F(\pi_N Y(1))}\Big]
= \inf_{\theta \in \mathbb H_a}
\mathbb{E}\bigg[ F(\pi_N Y(1) + \pi_N I(\theta)(1)) + \frac{1}{2} \int_0^1 \| \theta(t) \|_{L^2_x}^2 dt \bigg],
\label{P3}
\end{align}
\noindent
where $I(\theta)$ is defined by
\begin{align*}
I(\theta)(t) = \int_0^t \jb{\nabla}^{-\frac d2} \theta(t') dt'
\end{align*}
\noindent
and the expectation $\mathbb{E} = \mathbb{E}_\mathbb{P}$
is an
expectation with respect to the underlying probability measure~$\mathbb{P}$.
\end{lemma}
In the following,
we construct a drift $\theta$ depending on $Y$
and
the Bou\'e-Dupuis variational formula
(Lemma \ref{LEM:var3}) is suitable for this purpose
since
an expectation in \eqref{P3} is taken
with respect to the underlying probability $\mathbb{P}$.
Compare this with the variational formula
in \cite{GOTW},
where an expectation is taken
with respect to a shifted measure.
Before proceeding to the proof of Theorem \ref{THM:1},
we state a lemma on the pathwise regularity bounds of
$Y(1)$ and $I(\theta)(1)$.
\begin{lemma} \label{LEM:Dr}
\textup{(i)}
Let $\varepsilon > 0$. Then, given any finite $p \ge 1$,
we have
\begin{align*}
\begin{split}
\mathbb{E}
\Big[ & \|Y_N(1)\|_{W^{-\varepsilon,\infty}}^p
+ \|:\!Y_N(1)^2\!:\|_{W^{-\varepsilon,\infty}}^p
+
\big\| :\! Y_N(1)^3 \!: \big\|_{W^{-\varepsilon,\infty}}^p
\Big]
\leq C_{\varepsilon, p} <\infty,
\end{split}
\end{align*}
\noindent
uniformly in $N \in \mathbb{N}$.
\smallskip
\noindent
\textup{(ii)} For any $\theta \in \mathbb{H}_a$, we have
\begin{align}
\| \hspace{0.5mm}\text{I}\hspace{0.5mm}(\theta)(1) \|_{H^{\frac d2}}^2 \leq \int_0^1 \| \theta(t) \|_{L^2}^2dt.
\label{CM}
\end{align}
\end{lemma}
Part (i) of Lemma \ref{LEM:Dr} follows
from a standard computation and thus we omit details.
See, for example,
\cite[Proposition 2.3]{OTh2}
and
\cite[Proposition 2.1]{GKO}
for related results when $d = 2$.
As for Part (ii),
the estimate \eqref{CM}
follows from Minkowski's and Cauchy-Schwarz' inequalities.
See the proof of Lemma 4.7 in \cite{GOTW}.
\subsection{Proof of Theorem \ref{THM:1}}
\label{SUBSEC:3.2}
In this subsection, we present the proof of Theorem \ref{THM:1}.
Namely, we prove
the divergence
\eqref{Gibbs4}
for any $K>0$.
Note that
it suffices to prove
the following divergence:
\begin{align}
\lim_{L \to \infty} \liminf_{N \to \infty}
\mathbb{E}_\mu\Big[\exp\big(\min{( R_N(u),L)} \big)
\cdot \mathbf 1_{\{ |\int_{\mathbb{T}^d} \, : \, u_N^2 :\, dx | \le K\}} \Big] = \infty,
\label{pax}
\end{align}
\noindent
where $R_N(u)$ is as in \eqref{Wick0} with $\sigma > 0$ and $k = 4$.
Noting that
\begin{align}
\begin{split}
\mathbb{E}_\mu\Big[ & \exp\big(\min{( R_N(u),L)} \big)
\cdot\mathbf 1_{\{ |\int_{\mathbb{T}^d} \, : u_N^2 :\, dx | \le K\}} \Big]\\
&\ge \mathbb{E}_\mu\Big[\exp\Big(\min{( R_N(u),L)}
\cdot \mathbf 1_{\{ |\int_{\mathbb{T}^d} \, : u_N^2 :\, dx | \le K\}}\Big) \Big]
- 1,
\end{split}
\label{pax1}
\end{align}
\noindent
the divergence \eqref{pax} (and thus \eqref{Gibbs4}) follows once we prove
\begin{align}
\lim_{L \to \infty} \liminf_{N \to \infty} \mathbb{E}_\mu\Big[\exp\Big(\min{( R_N(u),L)}
\cdot \mathbf 1_{\{ |\int_{\mathbb{T}^d} \, : u_N^2 :\, dx | \le K\}}\Big) \Big] = \infty.
\label{pa0}
\end{align}
By the Bou\'e-Dupuis variational formula
(Lemma \ref{LEM:var3}), we have
\begin{align}
\begin{split}
- \log & {\mathbb{E}_\mu \Big[\exp\Big(\min{( R_N(u),L)} \cdot \mathbf 1_{\{ |\int_{\mathbb{T}^d} \, : u_N^2 :\, dx | \le K\}}\Big) \Big]} \\
&= \inf_{\theta \in \mathbb H_a} \mathbb{E}\bigg[ -\min\big( R_N(Y(1) + I(\theta)(1)),L\big)\\
&\hphantom{XXXXX} \times \mathbf 1_{\{ |\int_{\mathbb{T}^d} : (\pi_N Y(1))^2:
+ 2 (\pi _N Y(1)) (\pi_N I(\theta)(1)) + (\pi_N I(\theta)(1))^2 dx | \le K\}} \\
&\hphantom{XXXXX}
+ \frac 12 \int_0^1 \| \theta(t)\|_{L^2_x} ^2 dt \bigg],
\label{DPf}
\end{split}
\end{align}
\noindent
where $Y(1)$ is as in \eqref{P2}.
Here, $\mathbb{E}_\mu$ and $\mathbb{E}$ denote expectations
with respect to the Gaussian field~$\mu$ in \eqref{gauss0}
and the underlying probability measure $\mathbb{P}$, respectively.
In the following, we show that the right-hand side
of \eqref{DPf} tends to $-\infty$ as $N, L \to \infty$.
The main idea is to construct a drift $\theta$
such that
$I(\theta)$ looks like ``$- Y(1) + $
a perturbation'', where the perturbation term is bounded in $L^2(\mathbb{T}^d)$
but has a large $L^4$-norm.\footnote{While we do not make use of solitons in this paper,
one should think of this perturbation as something like a soliton.}
We first construct a perturbation term in the next lemma.
Fix a large parameter $M \gg 1$.
Let $f: \mathbb{R}^d \to \mathbb{R}$ be a real-valued Schwartz function
with $\|f\|_{L^2} = 1$
such that
the Fourier transform $\widehat f$ is a radial smooth function
supported on $\big\{\frac 12 < |\xi| \le 1 \big\}$.
Define a function $f_M$ on $\mathbb{T}^d$ by
\begin{align}
f_M(x) = M^{-\frac d2} \sum_{|n| > M/2 } \widehat f\Big( \frac nM \Big) e_n, \label{fMdef}
\end{align}
\noindent
where $\widehat f$ denotes the Fourier transform on $\mathbb{R}^d$ defined by
\[ \widehat f(\xi) = \frac{1}{(2\pi)^\frac{d}{2}} \int_{\mathbb{R}^d} f(x) e^{-i\xi\cdot x} dx.\]
\noindent
Then, a direct computation yields the following lemma.
See Lemma 5.12 in \cite{OOT}.
\begin{lemma}\label{LEM:leo1}
Let $\alpha > 0$. Then, we have
\begin{align}
\int_{\mathbb{T}^d} f_M^2 dx &= 1 + O( M^{-\alpha}), \label{fM0} \\
\int_{\mathbb{T}^d} (\jb{\nabla}^{-\alpha} f_M)^2 dx &\lesssim M^{-2\alpha}, \label{fm2} \\
\int_{\mathbb{T}^d} f_M^4 dx &\sim M^d \label{fM1}
\end{align}
\noindent
for any $M \gg 1$.
\end{lemma}
Lemma \ref{LEM:leo1} follows
from a systematic modification of the proof of Lemma 5.12 in \cite{OOT}
and thus we omit details.
See the proof of Lemma 5.12 in \cite{OOT}
for the details.
In the next lemma, we construct
an approximation $Z_M$ to $Y$ in \eqref{P2}
by solving stochastic differential equations.
Note that, in \cite{OOT}, such an approximation
of $Y(1)$
was constructed essentially by (a suitable frequency truncation of)
$Y(\frac 12)$,
which was sufficient to prove a divergence analogous
to \eqref{pa0} for large $K \gg1 $.
In order to prove the divergence \eqref{pa0} for any $K > 0$,
we need to establish a more refined approximation argument.
For simplicity, we denote $Y(1)$ and $\pi_N Y(1)$ by $Y$ and $Y_N$,
respectively, in the following.
\begin{lemma} \label{LEM:leo2}
Given $ M\gg 1$, define $Z_M$ by its Fourier coefficients
as follows.
For $|n| \leq M$, $\widehat Z_M(n, t)$ is a solution of the following differential equation\textup{:}
\begin{align}
\begin{cases}
d \widehat Z_{M}(n, t) = \jb{n}^{-\frac d2} M^\frac d2 (\widehat Y(n, t)- \widehat Z_{M}(n, t)) dt \\
\widehat Z_{M}|_{t = 0} =0,
\end{cases}
\label{ZZZ}
\end{align}
\noindent
and we set $\widehat Z_{M}(n, t) \equiv 0$ for $|n| > M$.
Then, $Z_M(t)$ is a centered Gaussian process in $L^2(\mathbb{T}^d)$, which is frequency localized on $\{|n| \le M \}$,
satisfying
\begin{align}
&\mathbb{E} \big[ |Z_M(x)|^2 \big] \sim \log M,\label{NRZ0}\\
&\mathbb{E}\bigg[ 2 \int_{\mathbb{T}^d} Y_N Z_M dx - \int_{\mathbb{T}^d} Z_M^2 dx \bigg] \sim \log M, \label{NRZ1}\\
&\mathbb{E} \bigg[ \Big| \int_{\mathbb{T}^d} :\! ( Y_N-Z_M)^2 \!: dx \Big|^2 \bigg] \lesssim M^{-d}\log M, \label{NRZ3}\\
&\mathbb{E}\bigg[\Big( \int_{\mathbb{T}^d} Y_N f_M dx \Big)^2\bigg]
+ \mathbb{E}\bigg[\Big( \int_{\mathbb{T}^d} Z_M f_M dx \Big)^2\bigg] \lesssim M^{-d}, \label{NRZ5}\\
&\mathbb{E}\bigg[\int_0^1 \Big\| \frac d {ds} Z_M(s) \Big\|^2_{H^\frac d 2}ds\bigg] \lesssim M^d \label{NRZ6}
\end{align}
\noindent
for any $N \ge M \gg 1$,
where $Z_M =Z_M|_{t = 1}$
and
\begin{align}
:\! ( Y_N-Z_M)^2 \!: \, =
( Y_N-Z_M)^2 - \mathbb{E}\big[ ( Y_N-Z_M)^2 \big].
\label{ZZZ2}
\end{align}
\noindent
Here, \eqref{NRZ0} is independent of $x \in \mathbb{T}^d$.
\end{lemma}
We present the proof of Lemma \ref{LEM:leo2} in
Subsection \ref{SUBSEC:LEM}.
We now define $ \alpha_{M, N}$ by
\begin{align}
\alpha_{M, N}= \frac {\mathbb{E} \bigg[ 2 \int_{\mathbb{T}^d}Y_NZ_M dx-\int_{\mathbb{T}^d}Z_M^2 dx \bigg]}{\int_{\mathbb{T}^d} f_M^2 dx}.
\label{fmb1}
\end{align}
\noindent
for $N\ge M \gg 1$.
Then, from \eqref{fM0} and \eqref{NRZ1}, we have
\begin{align}
\alpha_{M, N} \sim \log M
\label{logM}
\end{align}
\noindent
for any $N \ge M\gg 1$.
We are now ready to prove the divergence \eqref{pa0}.
For $M \gg 1$,
we set $f_M$, $Z_M$, and $ \alpha_{M, N}$ as in \eqref{fMdef}, Lemma \ref{LEM:leo2}, and \eqref{fmb1}.
For the minimization problem \eqref{DPf},
we set a drift $\theta= \theta^0$ by
\begin{align*}
\begin{split}
\theta^0 (t)
& = \jb{\nabla}^{\frac d2} \bigg( -\frac{d}{dt} Z_M(t) + \sqrt{ \alpha_{M, N}} f_M \bigg)
\end{split}
\end{align*}
\noindent
such that
\begin{align}
\Theta^0 = I(\theta^0)(1)
= \int_0^1 \jb{\nabla}^{-\frac d2} \theta^0(t) \, dt = - Z_M + \sqrt{ \alpha_{M, N}} f_M.
\label{paa0}
\end{align}
\noindent
We also define $Q(u$) by
\begin{align}
Q(u) = \frac 14 \int_{\mathbb{T}^d} u^4 dx.
\label{paa1}
\end{align}
Let us first make some preliminary computations.
By Cauchy's inequality, we have
\begin{align}
\begin{split}
|Z_M(\sqrt{ \alpha_{M, N}} f_M)^3| &\le \frac \delta 4 \alpha_{M, N}^2 f_M^4 +\frac { 1}{\delta} \alpha_{M, N} Z_M^2 f_M^2, \\
|Z_M^3 \sqrt{ \alpha_{M, N}} f_M | &\le \frac \dl4 Z_M^4+\frac{ 1}{\delta} \alpha_{M, N}Z_M^2 f_M^2
\end{split}
\label{YO}
\end{align}
\noindent
for any $0<\delta<1$.
Then, from
\eqref{paa0}, \eqref{paa1}, and \eqref{YO}, we have
\begin{align}
\begin{split}
Q(\Theta^0) & - \alpha_{M, N}^2 Q(f_M) \\
&= - \int_{\mathbb{T}^d} Z_M(\sqrt{ \alpha_{M, N}} f_M)^3 dx
+ \frac32 \int_{\mathbb{T}^d} Z_M^2 (\sqrt{ \alpha_{M, N}} f_M)^2 dx \\
&\quad
- \int_{\mathbb{T}^d} Z_M^3 \sqrt{ \alpha_{M, N}} f_M dx + Q(Z_N) \\
&\ge -\delta \alpha_{M, N}^2 Q(f_M)
- C_\delta \alpha_{M, N}\int_{\mathbb{T}^d} Z_M^2 f_M^2 dx +(1-\delta) Q(Z_N) \\
&\ge -\delta \alpha_{M, N}^2 Q(f_M) - C_\delta \alpha_{M, N} \int_{\mathbb{T}^d}
Z_M^2f_M^2 dx
\end{split}
\label{pa1}
\end{align}
\noindent
for any $0<\delta<1$.
From Lemmas \ref{LEM:leo1}, \ref{LEM:leo2}, and \eqref{logM},
we have
\begin{align}
\begin{split}
\mathbb{E} \bigg[ \alpha_{M, N} \int_{\mathbb{T}^d} Z_M^2 f_M^2 dx \bigg]
&\sim (\log M)^2 \| f_M \|_{L^2}^2
\lesssim (\log M )^2
\end{split}
\label{pa2}
\end{align}
\noindent
for any $N \ge M \gg 1$.
Therefore,
it follows from
\eqref{pa1}, \eqref{pa2}, \eqref{logM}, and \eqref{fM1}
that
for any measurable set $E$ with $\mathbb{P}(E) \ge \frac 12 $
and any $L \gg \sigma \cdot \alpha_{M, N}^2 Q(f_M)$, we have
\begin{align}
\begin{split}
\mathbb{E} \Big[\min\big(\tfrac \sigma 2 Q(\Theta^0), L\big) \cdot \mathbf 1_E\Big]
&\ge \frac \sigma 2 (1-\delta) \alpha_{M, N}^2 Q(f_M) \mathbb{P}(E) - \frac \s8 C'_\delta (\log M )^2 \\
&\gtrsim \sigma M^d(\log M )^2
\end{split}
\label{paa2}
\end{align}
\noindent
for any $N \ge M \gg 1$.
%
Recall that $\widehat f_M$ is supported on $\{|n|\leq M\}$.
Then,
by Lemma \ref{LEM:Dr}, \eqref{paa0},
and Lemmas \ref{LEM:leo1} and \ref{LEM:leo2}
with \eqref{logM}, we have
\begin{align}
\begin{split}
\mathbb{E} \big[ \| \Theta^0\|_{H^\frac d2}^2 \big]
&\le \mathbb{E} \bigg[ \int_0^1 \|\theta^0(t)\|_{L^2}^2 dt \bigg] \\
&\lesssim \mathbb{E}\bigg[\int_0^1 \Big\| \frac d {ds} Z_M(s) \Big\|^2_{H^\frac d 2}ds\bigg]
+ M^d \alpha_{M,N} \|f_M \|_{L^2}^2 \\
&\lesssim M^d \log M.
\end{split}
\label{pa4}
\end{align}
Lastly, from \eqref{Wick0} with $k = 4$
and the identity (see \cite{GKO}):
\begin{align}
H_k(x+y; \sigma )
& =
\sum_{\l = 0}^k
\begin{pmatrix}
k \\ \l
\end{pmatrix}
x^{k - \l} H_\l(y; \sigma),
\label{Herm}
\end{align}
\noindent
we have
\begin{align}
\begin{split}
R_N (Y + \Theta^0) & =
\frac \s4\int_{\mathbb{T}^d} :\! Y_N^4 \!: dx
+ \sigma \int_{\mathbb{T}^d} :\! Y_N^3 \!: \Theta^0 dx+\frac {3\sigma}2\int_{\mathbb{T}^d} :\! Y_N^2 \!: (\Theta^0)^2 dx
\\
&\hphantom{X}
+ \sigma\int_{\mathbb{T}^d} Y_N (\Theta^0)^3 dx
+ \frac \s4\int_{\mathbb{T}^d} (\Theta^0)^4 dx,
\end{split}
\label{Y00}
\end{align}
\noindent
where we used
\begin{align}
\pi_N \Theta^0 = \Theta^0
\label{YY0a}
\end{align}
for $N \ge M \ge 1$.
We now state a lemma, controlling the second, third, and fourth
terms on the right-hand side of~\eqref{Y00}.
We present the proof of this lemma in Subsection \ref{SUBSEC:LEM}.
\begin{lemma} \label{LEM:Dr1}
There exist small $\varepsilon>0$ and a constant $c_0=c_0(\varepsilon) >0$
such that for any $\delta>0$, we have
\begin{align}
\begin{split}
\bigg| \int_{\mathbb{T}^d} :\! Y_N^3 \!: \Theta^0dx \bigg|
&\le c(\delta) \| :\! Y_N^3 \!: \|_{W^{-\varepsilon,\infty}}^2
+ \delta
\| \Theta^0\|_{ H^{\frac d2}}^2,
\end{split}
\label{Y1}\\
\bigg| \int_{\mathbb{T}^d} :\! Y_N^2 \!: (\Theta^0)^2 dx \bigg|
&\le c(\delta) \| :\! Y_N^2 \!: \|_{W^{-\varepsilon,\infty}}^4
+ \delta \Big(
\| \Theta^0\|_{ H^{\frac d2}}^2 +
\| \Theta^0 \|_{L^4}^4 \Big),
\label{Y2} \\
\begin{split}
\bigg| \int_{\mathbb{T}^d} Y_N (\Theta^0)^3 dx\bigg|
&\le c(\delta) \| Y_N \|_{W^{-\varepsilon,\infty}}^{c_0}+ \delta \Big(
\| \Theta^0\|_{ H^{\frac d2}}^2 +
\| \Theta^0 \|_{L^4}^4 \Big),
\end{split}
\label{Y3}
\end{align}
\noindent
uniformly in $N \in \mathbb{N}$.
\end{lemma}
Then, from \eqref{Y00} and Lemmas \ref{LEM:Dr1},
we have
\begin{align}
\begin{split}
R_N(Y + \Theta^0)
&\ge \frac \sigma 2 Q(\Theta^0) \\
& \quad - c\sigma \Big(
\| :\! Y_N^3 \!: \|_{W^{-\varepsilon,\infty}}^2
+ \|:\! Y_N^2 \!:\|_{W^{-\varepsilon,\infty}}^4
+ \|Y_N\|_{W^{-\varepsilon,\infty}}^{c_0}
\Big) \\
&\quad
-c \sigma \|\Theta^0\|_{H^\frac d2}^2- |R_N(Y)|.
\end{split}
\label{paa}
\end{align}
We are now ready to put everything together.
With \eqref{YY0a} in mind,
suppose that for any $K>0$, there exists $M_0=M_0(K) \geq 1$ such that
\begin{align}
\mathbb{P}\bigg( \Big|\int_{\mathbb{T}^d} (:{Y_N^2}: + 2 Y_N \Theta^0 + (\Theta^0)^2) dx \Big| \le K \bigg)
\ge \frac 12,
\label{pa5}
\end{align}
\noindent
uniformly in $N \ge M \ge M_0$.
Then,
it follows from
\eqref{DPf}, \eqref{paa}, \eqref{paa2},
Lemma \ref{LEM:Dr}, \eqref{pa4}, and~\eqref{conv1} (controlling $|R_N(Y)|$, uniformly in $N \in \mathbb{N}$)
with \eqref{YY0a}
that
there exist constants $C_1, C_2, C_3 > 0 $ such that
\begin{align*}
-\log & \, \mathbb{E}_\mu\Big[\exp\Big(\min{( R_N(u),L)} \cdot \mathbf 1_{\{ |\int_{\mathbb{T}^d} \, : u_N^2 :\, dx | \le K\}}\Big) \Big]\notag \\
&\le \mathbb{E}\bigg[ -\min\big( R_N(Y + \Theta^0),L\big)\cdot
\mathbf 1_{\{ |\int_{\mathbb{T}^d} ( :{Y_N^2}: + 2 Y_N \Theta^0 + (\Theta^0)^2) dx | \le K\}} + \frac 12 \int_0^1 \| \theta^0(t)\|_{L^2_x} ^2 dt \bigg] \notag \\
&\le \mathbb{E}\bigg[ -\min\big(\tfrac \sigma 2 Q(\Theta^0),L\big)\cdot \mathbf 1_{\{ |\int_{\mathbb{T}^d} ( :{Y_N^2}: + 2 Y_N \Theta^0 + (\Theta^0)^2) dx | \le K\}} \notag \\
&\hphantom{XXX}
+ c(\sigma) \Big(
\| :\! Y_N^3 \!: \|_{W^{-\varepsilon,\infty}}^2
+ \|:\! Y_N^2 \!:\|_{W^{-\varepsilon,\infty}}^4
+ \|Y_N\|_{W^{-\varepsilon,\infty}}^{c_0}
\Big) \notag \\
&\hphantom{XXX}
+ c\sigma\|\Theta^0\|_{H^\frac d2}^2
+ |R_N(Y)|
+ \frac 12 \int_0^1 \| \theta^0(t)\|_{L^2_x} ^2 dt \bigg] \notag \\
&\le - C_1 \sigma M^d (\log M)^2+ C_2 \sigma M^d \log M+ C_3
\end{align*}
\noindent
for any $N \ge M \ge M_0(K)$
and $L \gg
\sigma \cdot \alpha_{M, N}^2 Q(f_M)\sim \sigma M^d (\log M)^2$.
Therefore, we obtain
\begin{align*}
\begin{split}
\lim_{L \to \infty} & \liminf_{N \to \infty} \mathbb{E}_\mu\Big[\exp\Big(\min{(\sigma R_N(u),L)} \Big)
\mathbf 1_{\{ \left|\int_{\mathbb{T}^d} \, : u_N^2 :\, dx \right| \le K\}} \Big]\\
\ge&~ \exp\Big( C_1 \sigma M^d (\log M)^2 - C_2 \sigma M^d (\log M ) -C_3(\sigma)\Big) \longrightarrow \infty,
\end{split}
\end{align*}
\noindent
as $M \to \infty$.
This proves \eqref{pa0} by assuming \eqref{pa5}.
We conclude this subsection by proving
\eqref{pa5} for any $K>0$.
From \eqref{paa0}, we have
\begin{align}
\begin{split}
&\mathbb{E} \bigg[ \Big|\int_{\mathbb{T}^d} \Big( :{Y_N^2}: + 2 Y_N \Theta^0 + (\Theta^0)^2 \Big) dx \Big|^2 \bigg]\\
&= \mathbb{E} \bigg[ \Big|\int_{\mathbb{T}^d} :{Y_N^2}: dx - 2 \int_{\mathbb{T}^d} Y_N Z_M dx+\int_{\mathbb{T}^d} Z_M^2 dx+ \alpha_{M, N} \int_{\mathbb{T}^d} f_M^2 dx \\
&\hphantom{X}
+ 2 \sqrt{ \alpha_{M, N}} \int_{\mathbb{T}^d} (Y_N-Z_M)f_M dx \Big|^2 \bigg].
\end{split}
\label{Pr1}
\end{align}
\noindent
From \eqref{logM} and \eqref{NRZ5} in Lemma \ref{LEM:leo2}, we have
\begin{align}
\mathbb{E} \bigg[ \Big| \sqrt{ \alpha_{M, N}} \int_{\mathbb{T}^d} (Y_N-Z_M)f_M dx \Big|^2 \bigg] \lesssim M^{-d} \log M.
\label{Pr4}
\end{align}
\noindent
On other hand,
from \eqref{fmb1} and \eqref{ZZZ2}, we have
\begin{align}
\begin{split}
\int_{\mathbb{T}^d} & :\!Y_N^2\!: dx - 2 \int_{\mathbb{T}^d} Y_N Z_M dx+\int_{\mathbb{T}^d} Z_M^2 dx+ \alpha_{M, N} \int_{\mathbb{T}^d} f_M^2 dx \\
& =\int_{\mathbb{T}^d} (Y_N-Z_M)^2 -\mathbb{E} \big[(Y_N-Z_M )^2 \big] dx \\
& =\int_{\mathbb{T}^d} :\! ( Y_N-Z_M)^2 \!: dx.
\end{split}
\label{Pr2}
\end{align}
\noindent
Hence, from \eqref{Pr1}, \eqref{Pr4}, and \eqref{Pr2} with \eqref{NRZ3}
in Lemma \ref{LEM:leo2}, we obtain
\begin{align*}
\mathbb{E} \bigg[ \Big|\int_{\mathbb{T}^d} \Big( :{Y_N^2}: + 2 Y_N \Theta^0 + (\Theta^0)^2 \Big) dx \Big|^2 \bigg]\lesssim M^{-d }\log M.
\end{align*}
\noindent
Therefore, by Chebyshev's inequality,
given any $K > 0$, there exists $M_0 = M_0(K) \geq 1$ such that
\begin{align*}
\mathbb{P}\bigg( \Big|\int_{\mathbb{T}^d} (:{Y_N^2}: + 2 Y_N \Theta^0 + (\Theta^0)^2) dx \Big| > K \bigg)
&\le C\frac{M^{-d} \log M}{K^2}
< \frac 12
\end{align*}
\noindent
for any $M \ge M_0 (K)$.
This proves \eqref{pa5}.
\subsection{Proof of the auxiliary lemmas}
\label{SUBSEC:LEM}
In this subsection, we present the proofs of
Lemmas~\ref{LEM:leo2} and~\ref{LEM:Dr1}.
We first present the proof of the approximation lemma (Lemma~\ref{LEM:leo2}).
\begin{proof}[Proof of Lemma \ref{LEM:leo2}]
Let
\begin{align}
X_n(t)=\widehat Y_N(n, t)- \widehat Z_{M}(n, t),
\quad |n|\le M.
\label{ZZ1}
\end{align}
Then, from \eqref{P2} and \eqref{ZZZ},
we see that $X_n(t)$ satisfies
the following stochastic differential equation:
\begin{align*}
\begin{cases}
dX_n(t)=-\jb{n}^{-\frac d 2}M^\frac d2 X_n(t) dt +\frac{1}{\jb{n}^\frac d2}dB_n(t)\\
X_n(0)=0
\end{cases}
\end{align*}
\noindent
for $|n|\le M$.
By solving this stochastic differential equation, we have
\begin{align}
X_n(t)=\frac{1}{\jb{n}^\frac d2}\int_0^t e^{-\jb{n}^{-\frac d 2}M^\frac d2(t-s)}dB_n(s).
\label{ZZ2}
\end{align}
\noindent
Then, from \eqref{ZZ1} and \eqref{ZZ2}, we have
\begin{align}
\widehat Z_{M}(n, t)= \widehat Y_N(n, t)-\frac{1}{\jb{n}^\frac d2 }\int_0^t e^{-\jb{n}^{-\frac d 2}M^\frac d2 (t-s)}dB_n(s)
\label{SDE1}
\end{align}
\noindent
for $|n|\le M$. Hence, from \eqref{SDE1}, the independence of $\{B_n \}_{n \in \mathbb{Z}^d}$,\footnote{Here, we are referring to the independence modulo the condition $\overline{B_n} = B_{-n}$, $n \in \mathbb{Z}^d$.
Similar comments apply in the following.}
Ito's isometry, and \eqref{P2}, we have
\begin{align}
\begin{split}
\mathbb{E} \big[ |Z_M(x)|^2 \big]&=\sum_{|n| \le M} \bigg( \mathbb{E} \big[ | \widehat Y_N(n) |^2 \big]
-\frac 2{\jb{n}^d}\int_0^1 e^{-\jb{n}^{-\frac d 2}M^\frac d2 (1-s)}ds \\
& \hphantom{XXXXX}
+\frac{1}{\jb{n}^d}\int_0^1 e^{-2\jb{n}^{-\frac d 2}M^\frac d2 (1-s)}ds \bigg)\\
& \sim \log M + \sum_{|n|\le M }\frac 1{\jb{n}^\frac d2 }\cdot \frac{1}{M^\frac d2}\\
& \sim \log M
\end{split}
\label{ZZ3}
\end{align}
\noindent
for any $M\gg 1$.
This proves \eqref{NRZ0}.
By Parseval's theorem, \eqref{SDE1}, \eqref{NRZ0}, and
proceeding as in \eqref{ZZ3}, we have
\begin{align*}
\mathbb{E}\bigg[ 2 \int_{\mathbb{T}^d} Y_N & Z_M dx - \int_{\mathbb{T}^d} Z_M^2 dx \bigg]
=\mathbb{E} \bigg[ 2 \sum_{|n| \le M }\widehat Y_N(n) \overline{ \widehat Z_M(n)} -\sum_{|n |\le M }| \widehat Z_M(n) |^2 \bigg]\\
&=\mathbb{E} \bigg[ \sum_{|n | \le M } | \widehat Z_M(n) |^2+\sum_{|n|\leq M } \bigg( \frac 2{\jb{n}^\frac d2 } \int_0^1 e^{-\jb{n}^{-\frac d 2}M^\frac d2 (1-s) }dB_n(s) \bigg) \overline{\widehat Z_M(n)} \bigg]\\
&\sim \log M+\sum_{|n| \le M } \frac 1{\jb{n}^\frac d2}\cdot \frac 1{M^\frac d2}\\
&\sim \log M
\end{align*}
\noindent
for any $N\ge M\gg 1$.
This proves \eqref{NRZ1}.
Note that $\widehat Y(n)-\widehat Z_M(n)$ is a mean-zero Gaussian random variable.
Then, from \eqref{SDE1}
and Ito's isometry,
we have
\begin{align}
\begin{split}
\mathbb{E} \bigg[ \Big( |\widehat Y_N(n)- & \widehat Z_M(n)|^2-\mathbb{E}\big[ |\widehat Y(n)-\widehat Z_M(n)|^2 \big] \Big)^2 \bigg]
\lesssim
\Big(\mathbb{E}\big[ |\widehat Y_N(n)-\widehat Z_M(n)|^2 \big] \Big)^2 \\
& = \frac 1 {\jb{n}^{2d} } \bigg(\int_0^1 e^{-2\jb{n}^{-\frac d 2}M^{\frac d2}(1-s) } ds \bigg)^2
\sim \frac 1 {\jb{n}^{d} } \cdot \frac 1 {M^d}.
\end{split}
\label{ZZ4}
\end{align}
\noindent
Hence,
from Plancherel's theorem, \eqref{ZZZ2},
the independence of $\{B_n \}_{n \in \mathbb{Z}^d}$,
the independence
of
$\big\{ |\widehat Y_N(n) |^2- \mathbb{E} \big[ | \widehat Y_N(n)|^2 \big]\big\}_{M < |n|\le N}$
and
\[\big\{ |\widehat Y_N(n)-\widehat Z_M(n)|^2-\mathbb{E}\big[ |\widehat Y_N(n)-\widehat Z_M(n)|^2\big]\big\}_{|n|\le M},\]
\noindent
\eqref{P2},
and \eqref{ZZ4}, we have
\begin{align*}
\mathbb{E} \bigg[ & \Big| \int_{\mathbb{T}^d} :\! ( Y_N-Z_M)^2 \!: dx \Big|^2 \bigg] \notag \\
&= \sum_{M<|n|\le N } \mathbb{E} \bigg[ \Big( |\widehat Y_N(n) |^2- \mathbb{E} \big[ | \widehat Y_N(n)|^2 \big] \Big)^2 \bigg]\notag \\
&\hphantom{XX}+ \sum_{|n|\le M} \mathbb{E} \bigg[ \Big( |\widehat Y_N(n)-\widehat Z_M(n)|^2-\mathbb{E}\big[ |\widehat Y_N(n)-\widehat Z_M(n)|^2 \big] \Big)^2 \bigg]\notag \\
&\lesssim \sum_{M<|n|\le N}\frac 1{\jb{n}^{2d}}
+\sum_{|n|\le M }
\frac 1{\jb{n}^{d}}
\frac1 {M^d}
\lesssim M^{-d}\log M.
\end{align*}
\noindent
This proves \eqref{NRZ3}.
From \eqref{fm2} and \eqref{P2}, we have
\begin{align}
\begin{split}
\mathbb{E}\bigg[ \Big( \int_{\mathbb{T}^d} Y_N f_M dx \Big)^2\bigg]
&= \mathbb{E} \bigg[ \Big( \sum_{|n| \le M} \widehat Y_N(n) \overline{ \widehat f_M(n)} \Big)^2 \bigg]
= \sum_{|n| \le M} \frac 1{\jb{n}^d} |\widehat f_M(n)|^2 \\
&\le \int_{\mathbb{T}^d} \big(\jb{\nabla}^{-\frac d2} f_M (x)\big)^2 dx
\lesssim M^{-d}.
\end{split}
\label{app4}
\end{align}
\noindent
From \eqref{ZZ2}, Ito's isometry, and \eqref{fm2}, we have
\begin{align}
\begin{split}
\mathbb{E} \bigg[ \Big( \sum_{|n| \le M} X_n(1) \overline{ \widehat f_M(n)} \Big)^2 \bigg]
&=\mathbb{E} \Bigg[ \bigg|\sum_{|n|\leq M} \bigg( \frac 1{\jb{n}^\frac d2} \int_0^1 e^{-\jb{n}^{-\frac d 2} M^{\frac d2}(1-s) } dB_n(s) \bigg) \widehat f_M(n) \bigg|^2 \Bigg]\\
&\lesssim M^{-\frac d2}\sum_{|n| \leq M} \frac 1{\jb{n}^\frac d2 }| \widehat f_M(n)|^2\\
& \lesssim M^{-d}.
\end{split}
\label{app5}
\end{align}
\noindent
Hence, \eqref{NRZ5} follows
from \eqref{app4} and \eqref{app5}
with \eqref{SDE1}.
Lastly, from \eqref{ZZZ}, \eqref{ZZ1}, \eqref{ZZ2}, and Ito's isometry, we have
\begin{align*}
\mathbb{E}\bigg[\int_0^1 \Big\| \frac d {ds} Z_M(s) \Big\|^2_{H^\frac d 2}ds\bigg] &= M^d \mathbb{E}\bigg[\int_0^1 \Big\| \pi_M(Y_N(s)) - Z_M(s) \Big\|^2_{L^2}ds\bigg] \\
&= M^d \mathbb{E}\bigg[ \int_0^1 \Big(\sum_{|n| \le M} |X_n(s)|^2\Big) ds\bigg]\\
&= M^d \sum_{|n| \le M} \int_0^1\mathbb{E}\big[|X_n(s)|^2\big] ds \\
&=M^d \sum_{|n| \le M} \frac 1 {\jb{n}^{d}} \int_0^1 \int_0^s e^{-2\jb{n}^{-\frac d 2}M^\frac d2(s-s')} d s' ds \\
& \lesssim M^d \sum_{|n| \le M } \frac 1{\jb{n}^\frac d2}\cdot \frac 1{M^\frac d2} \\
&\lesssim M^d,
\end{align*}
\noindent
yielding \eqref{NRZ6}.
This completes the proof of Lemma~\ref{LEM:leo2}.
\end{proof}
Next, we
present the proof of Lemma \ref{LEM:Dr1}.
\begin{proof}[Proof of Lemma \ref{LEM:Dr1}]
From Cauchy's inequality,
we have
\begin{align}
\begin{split}
\bigg| \int_{\mathbb{T}^d} :\! Y_N^3 \!: \Theta^0 dx \bigg|
&\le \| :\! Y_N^3 \!:\|_{W^{-\varepsilon,\infty}} \| \Theta^0\|_{W^{\varepsilon,1} }\le \| :\! Y_N^3 \!:\|_{W^{-\varepsilon,\infty}}
\| \Theta^0\|_{H^\frac d2}\\
&\le c(\delta)\| :\! Y_N^3 \!:\|_{W^{-\varepsilon,\infty}}^2 + \delta \| \Theta^0\|_{H^\frac d2}^2.
\end{split}
\label{ZZZ3}
\end{align}
\noindent
This yields \eqref{Y1}.
From the fractional Leibniz rule (Lemma \ref{LEM:prod}\,(ii)), we have
\begin{align}
\begin{split}
\bigg| \int_{\mathbb{T}^d} :\! Y_N^2 \!: (\Theta^0)^2 dx \bigg|
& \le \| :\! Y_N^2 \!:\|_{W^{-\varepsilon,\infty}}
\|(\Theta^0)^2\|_{W^{\varepsilon, 1}}\\
& \leq
\| :\! Y_N^2 \!:\|_{W^{-\varepsilon,\infty}}
\|(\Theta^0)^2\|_{W^{\varepsilon, \frac{4}{3}}}\\
& \lesssim \| :\! Y_N^2 \!:\|_{W^{-\varepsilon,\infty}}\|\Theta^0\|_{H^\frac d2}\|\Theta^0\|_{L^4}.
\end{split}
\label{ZZZ4}
\end{align}
\noindent
Then, the second estimate \eqref{Y2}
follows from Young's inequality.
Lastly, we consider \eqref{Y3}.
From the fractional Leibniz rule
(Lemma \ref{LEM:prod}\,(ii))
(with $\frac{1}{1+\delta} = \frac 1{2+\delta_0} + \frac 14 + \frac 14$
for small $\delta, \delta_0 > 0$),
Sobolev's inequality,
and the interpolation (Lemma \ref{LEM:prod}\,(i)),
we have
\begin{align}
\begin{split}
\bigg| \int_{\mathbb{T}^d} Y_N (\Theta_N^0)^3 dx \bigg|
&\le \| Y_N \|_{W^{-\varepsilon,\infty}} \| \jb{\nabla}^{\varepsilon} (\Theta_N^0)^3\|_{L^{1+\delta}}\\
&\lesssim\| Y_N \|_{W^{-\varepsilon,\infty}}
\| \Theta_N^0 \|_{W^{\varepsilon, 2+\delta_0}} \| \Theta_N^0 \|_{L^4}^2\\
&\lesssim\| Y_N \|_{W^{-\varepsilon,\infty}}
\| \Theta_N^0 \|_{H^\frac{d}{2}}^\beta \| \Theta_N^0 \|_{L^4}^{3-\beta}
\end{split}
\label{ZZZ2a}
\end{align}
\noindent
for some small $\beta > 0$.
Then, the third estimate \eqref{Y3}
follows from Young's inequality
since $\frac{\beta}2 + \frac {3-\beta}4 < 1$ for small $\beta > 0$.
This completes the proof of Lemma \ref{LEM:Dr1}.
\end{proof}
\section{Construction of the Gibbs measure with the cubic interaction}
\label{SEC:4}
In this section, we present the proof of Theorem \ref{THM:2}.
We prove the uniform exponential integrability
\eqref{exp3}
via the variational formulation.
Since the argument is identical for any finite $p \geq 1$, we only present details for the case $p =1$.
Moreover, the precise value of $\sigma \in \mathbb{R}\setminus \{0\}$ does not play any role
and thus we set $\sigma = 3$ in the following.
In view of the Bou\'e-Dupuis formula (Lemma \ref{LEM:var3}),
it suffices to establish a lower bound on
\begin{equation}
\mathcal{W}_N(\theta) = \mathbb{E}
\bigg[-\mathcal{R}_N(Y(1) + I(\theta)(1)) + \frac{1}{2} \int_0^1 \| \theta(t) \|_{L^2_x}^2 dt \bigg],
\label{v_N0}
\end{equation}
\noindent
uniformly in $N \in \mathbb{N}$ and $\theta \in \mathbb{H}_a$.
We set $Y_N = \pi_N Y = \pi_N Y(1)$ and $\Theta_N = \pi_N \Theta = \pi_N I(\theta)(1)$.
From \eqref{K2} and \eqref{Herm}, we have
\begin{align}
\begin{split}
\mathcal{R}_N (Y + \Theta) & =
\int_{\mathbb{T}^d} :\! Y_N^3 \!: dx
+ 3\int_{\mathbb{T}^d} :\! Y_N^2 \!: \Theta_N dx+3 \int_{\mathbb{T}^d} Y_N \Theta_N^2 dx
\\
&\hphantom{X}
+ \int_{\mathbb{T}^d} \Theta_N^3 dx
- A \bigg\{ \int_{\mathbb{T}^d} \Big( :\! Y_N^2 \!: + 2 Y_N \Theta_N + \Theta_N^2 \Big) dx \bigg\}^2.
\end{split}
\label{Y0}
\end{align}
\noindent
Hence, from \eqref{v_N0} and \eqref{Y0}, we have
\begin{align}
\begin{split}
\mathcal{W}_N(\theta)
&=\mathbb{E}
\bigg[
-\int_{\mathbb{T}^d} :\! Y_N^3 \!: dx
-3\int_{\mathbb{T}^d} :\! Y_N^2 \!: \Theta_N dx
-3\int_{\mathbb{T}^d} Y_N \Theta_N^2 dx
\\
&\hphantom{XXX}
-\int_{\mathbb{T}^d} \Theta_N^3 dx
+ A \bigg\{ \int_{\mathbb{T}^d} \Big( :\! Y_N^2 \!: + 2 Y_N \Theta_N + \Theta_N^2 \Big) dx \bigg\}^2\\
&\hphantom{XXX}
+ \frac{1}{2} \int_0^1 \| \theta(t) \|_{L^2_x}^2 dt
\bigg].
\end{split}
\label{v_N0a}
\end{align}
In the following, we first state
a lemma, controlling the terms appearing in \eqref{v_N0a}.
We present the proof of this lemma at the end of this section.
\begin{lemma} \label{LEM:Dr2}
\textup{(i)}
There exist small $\varepsilon>0$ and a constant $c >0$ such that
\begin{align}
\bigg| \int_{\mathbb{T}^d} :\! Y_N^2 \!: \Theta_N dx \bigg|
&\le c \| :\! Y_N^2 \!: \|_{W^{-\varepsilon,\infty}}^2
+ \frac 1{100}
\| \Theta_N \|_{H^\frac d2}^2,
\label{YY2} \\
\bigg| \int_{\mathbb{T}^d} Y_N \Theta_N^2 dx \bigg|
&\le c
\| Y_N \|_{W^{-\varepsilon,\infty}}^{6} + \frac 1{100} \Big(
\| \Theta_N \|_{H^\frac d2}^2 + \| \Theta_N \|_{L^2}^4 \Big),
\label{YY3}\\
\bigg| \int_{\mathbb{T}^d} \Theta_N^3 dx \bigg|
&\le \frac 1{100} \| \Theta_N \|_{H^\frac d2}^2
+ \frac{A}{100} \| \Theta_N \|_{L^2}^{4 }
\label{YY14}
\end{align}
\noindent
for any sufficiently large $A>0$,
uniformly in $N \in \mathbb{N}$.
\smallskip
\noindent
\textup{(ii)}
Let $A> 0$. Given any small $\varepsilon > 0$,
there exists $c = c(\varepsilon, A)>0$ such that
\begin{align}
\begin{split}
A\bigg\{ \int_{\mathbb{T}^d}& \Big( :\! Y_N^2 \!: + 2 Y_N \Theta_N + \Theta_N^2 \Big) dx \bigg\}^2 \\
&\ge \frac A4 \| \Theta_N \|_{L^2}^4 - \frac 1{100} \| \Theta_N \|_{H^\frac d2}^2
- c\bigg\{ \| Y_N \|_{W^{-\varepsilon,\infty}}^c
+ \bigg( \int_{\mathbb{T}^d} :\! Y_N^2 \!: dx \bigg)^2 \bigg\},
\end{split}
\label{YY5}
\end{align}
\noindent
uniformly in $N \in \mathbb{N}$.
\end{lemma}
As in \cite{BG, GOTW, ORSW2, OOT},
the main strategy is
to establish a pathwise lower bound on $\mathcal{W}_N(\theta)$ in~\eqref{v_N0a},
uniformly in $N \in \mathbb{N}$ and $\theta \in \mathbb{H}_a$,
by making use of the
positive terms:
\begin{equation}
\mathcal{U}_N(\theta) =
\mathbb{E} \bigg[\frac A 4\| \Theta_N\|_{L^2}^4 + \frac{1}{2} \int_0^1 \| \theta(t) \|_{L^2_x}^2 dt\bigg].
\label{v_N1}
\end{equation}
\noindent
coming from \eqref{v_N0a} and \eqref{YY5}.
From \eqref{v_N0a} and \eqref{v_N1} together with Lemmas \ref{LEM:Dr2} and \ref{LEM:Dr}, we obtain
\begin{align}
\inf_{N \in \mathbb{N}} \inf_{\theta \in \mathbb{H}_a} \mathcal{W}_N(\theta)
\geq
\inf_{N \in \mathbb{N}} \inf_{\theta \in \mathbb{H}_a}
\Big\{ -C_0 + \frac{1}{10}\mathcal{U}_N(\theta)\Big\}
\geq - C_0 >-\infty.
\label{YYY5}
\end{align}
\noindent
Then,
the uniform exponential integrability \eqref{exp3}
follows from
\eqref{YYY5} and Lemma \ref{LEM:var3}.
This proves
Theorem \ref{THM:2}.
\medskip
We conclude this section by
presenting the proof of Lemma \ref{LEM:Dr2}.
\begin{proof}[Proof of Lemma \ref{LEM:Dr2}]
(i)
The estimate \eqref{YY2} follows
from replacing $:\! Y_N^3 \!:$ in \eqref{ZZZ3}
by $:\! Y_N^2 \!:$.
With small $\delta > 0$,
it follows from
the fractional Leibniz rule
(Lemma \ref{LEM:prod}\,(ii))
and Sobolev's inequality
as in \eqref{ZZZ2a} that
\begin{align*}
\bigg| \int_{\mathbb{T}^d} Y_N \Theta_N^2 dx \bigg|
& \le \| Y_N\|_{W^{-\varepsilon,\infty}}
\|\Theta_N^2\|_{W^{\varepsilon, 1+\delta}}\\
& \leq
\| Y_N\|_{W^{-\varepsilon,\infty}}
\|\Theta_N\|^2_{H^\varepsilon}\\
& \lesssim \| Y_N\|_{W^{-\varepsilon,\infty}}
\|\Theta_N\|_{H^\frac d2}^{\beta} \|\Theta_N\|_{L^2}^{2-\beta}
\end{align*}
\noindent
for some small $\beta> 0$.
Then,
the second estimate \eqref{YY3} follows
from Young's inequality since
$\frac{\beta} 2 + \frac{2-\beta}4 < 1$.
As for the third estimate
\eqref{YY14},
it follows from
Sobolev's inequality, Lemma \ref{LEM:prod}\,(i), and Cauchy's inequality that
\begin{align*}
\begin{split}
\bigg| \int_{\mathbb{T}^d} \Theta_N^3 dx \bigg|
&\le C \| \Theta_N \|_{H^\frac d6}^3
\le C \| \Theta_N \|_{H^\frac d2}\| \Theta_N \|_{L^2}^2 \\
&\le \frac 1{100}\| \Theta_N \|_{H^\frac d2}^2 + \frac A{100} \| \Theta_N \|_{L^2}^4,
\end{split}
\end{align*}
\noindent
where $A>0$ is sufficiently large.
\medskip
\noindent
(ii) The bound \eqref{YY5} follows from a slight modification
of Lemma 5.8 in \cite{OOT}.
Noting that
\begin{align*}
(a+b+c)^2
\ge \frac 12 c^2 - 2 (a^2+b^2)
\end{align*}
\noindent
for any $a,b,c \in \mathbb{R}$,
we have
\begin{align}
\begin{split}
A\bigg\{ & \int_{\mathbb{T}^d} \Big( :\! Y_N^2 \!: + 2 Y_N \Theta_N + \Theta_N^2 \Big) dx \bigg\}^2 \\
&\ge \frac A2 \bigg( \int_{\mathbb{T}^d} \Theta_N^2dx \bigg)^2
- 2A \bigg\{ \bigg( \int_{\mathbb{T}^d} :\! Y_N^2 \!: dx \bigg)^2
+ \bigg( \int_{\mathbb{T}^d} Y_N \Theta_N dx \bigg)^2 \bigg\}.
\end{split}
\label{YZ3}
\end{align}
\noindent
From Lemma \ref{LEM:prod}\,(i)
and Young's inequality, we have
\begin{align}
\begin{split}
\bigg| \int_{\mathbb{T}^d} Y_N \Theta_N dx \bigg|^2
&\le \| Y_N \|_{W^{-\varepsilon,\infty}}^2 \| \Theta_N \|_{W^{\varepsilon,1}}^2 \le \| Y_N \|_{W^{-\varepsilon,\infty}}^2 \| \Theta_N \|_{H^\varepsilon}^2\\
&\lesssim \| Y_N \|_{W^{-\varepsilon,\infty}}^2 \| \Theta_N \|_{L^2}^{2-\frac{4\varepsilon}{d} } \| \Theta_N \|_{H^\frac d2}^{\frac {4\varepsilon}d} \\
&\le c \| Y_N \|_{W^{-\varepsilon,\infty}}^{\frac{2d}{d-2\varepsilon}} + \frac 1{8} \| \Theta_N \|_{L^2}^4 + \frac1{200 A} \| \Theta_N \|_{H^\frac d2}^2.
\end{split}
\label{YZ4}
\end{align}
\noindent
Hence, \eqref{YY5} follows from \eqref{YZ3} and \eqref{YZ4}.
\end{proof}
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Helping companies to effectively manage integrity risks around human capital is one of four areas of focus for The Red Flag Group. It is sometimes said that a company is only as good as its employees, and even that it is only as ethical as its senior management. When an employee engages in conduct that is so antithetical to the company, swift and decisive action is needed. I thought of all of the above when reading the sad story of the now former manager of the English national football team Sam Allardyce, who recently lost what he termed his "dream job".
As reported by Rory Smith in The New York Times, in a piece entitled 'England manager undone by a film', Allardyce was filmed in an undercover sting operation instituted by United Kingdom newspaper The Telegraph. According to The New York Times, he provided an 'explanation of how to circumvent rules on third party ownership, a model in which companies or individuals own some or all of the economic rights of players, which was banned by the Premier League in 2008 and by FIFA in 2014. In the video, Allardyce explained that he knew of a group that had been "doing it for years" before outlining how potential investors would seek to make a return'.
The organisation that heads English football, the Football Association (FA), acted swiftly in sacking Allardyce, even though he had only had the position since June and indeed was undefeated (1-0) in international play. Unfortunately, as the article noted, 'the FA will have to face an uncomfortable truth. Three of the last seven managers the FA appointed left the job for nonsporting reasons: Glenn Hoddle, who made deeply unpleasant comments about the disabled; Fabio Capello, who left amid a controversy over John Terry; and now Allardyce. Another, Sven-Goran Eriksson, had a reign mired in scandal'.
The article then dryly noted, 'There is a pattern here'. Admittedly, managing the English national team will always draw intense scrutiny. However, it is more than simply the bright lights that caused Allardyce to be sacked. Such a high profile position requires an organisation to look closely into not only the background of a candidate but also whether that candidate's values are aligned with those of the organisation. The FA prided itself on standing above the world's soccer scandals. As Smith noted in his piece, 'For the FA, claiming to be the world's policeman while turning a blind eye to apparent offences at home would have been the rankest hypocrisy'.
Investing in human capital requires a full range of commitments, including the commitment to find and recruit employees who model the values of the organisation.
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\section{Introduction}
The core of neutron stars is generally believed to be composed of microscopically uniform nuclear (or quark) matter (e.g., \citealt*{hpy07}). However, it is not the case for the crust, where nuclear matter is clumped into clusters (nuclei, ions), located on the background of almost uniform degenerate electron gas and, except for the outer crust, additional background of unbound neutrons (e.g., \citealt{ch08}). For typical temperatures of neutron stars ($T<10^9$~K) the Coulomb interaction between these nuclei is much larger than the thermal energy and nuclei become ordered into a lattice and crust solidifies.
As for terrestrial conditions, solidification allows for elastisity: solid crust supports shear stresses, and,
according to the state-of-art models, this effect indeed affects observations. Crustal elasticity is supposed to be responsible for quasi-periodic oscillations after giant flares of magnetars (e.g., \citealt{hc80_torsOsc,st83_tors_osc,McDermott_ea88,Strohmayer_etal91,Gabler_ea11,Gabler_ea12,Gabler_ea13,Gabler_ea18,Sotani_ea18,KY20}),
it can support static asymmetry in the mass distribution in the crust (mountains), which can lead to emission of gravitational waves (see e.g., \citealt{Ushomirsky_ea00,hja06_mountains,Horowitz10_lowmass,McDaniel_Owen13,gaj20,KM22_Mountain,mh22}
for the models and \citealt{LIGO_VIRGO20_MSPelipt,LIGO21_Pulsars,LIGO22_GW_isotated} for recent observational constraints).
Assuming that the matter in the neutron star crust is macroscopically isotropic (e.g., it is polycrystalline), the coupling of the shear stress with deformation can be described by the shear modulus,
which should be calculated theoretically (the matter of neutron star crust cannot be obtained and directly studied in the laboratory).
This problem were analyzed in a long list of papers (e.g., \citealt{Strohmayer_etal91,hh08,hk09,Baiko11,Baiko12,kp15,Baiko15,Kozhberov19_elast,C21_elastCoins, Kozhberov22,Chugunov22_elast_screen}).
Typically, these works replace nuclei by point-like massive particles with Coulomb or screened Coulomb (Yukawa) interaction, arrange these particles into lattice (or polycrystalline system)
and consider energy change (or stresses) which arises in this system as a result of deformation.
The results for static one-component Coulomb crystals with body- and face-centred lattices were obtained almost a century ago by \cite{Fuchs36} and more recent works improve them, analyzing also multicomponent lattices and screening (e.g.\ \citealt{Kozhberov19_elast}, see also \citealt{C21_elastCoins,Chugunov22_elast_screen} for recent approximate, but universal formulae), as well as quantum and thermal motion of the particles (\citealt{Strohmayer_etal91,hh08,Baiko12}).
To replace nuclei with point-like massive particles seems very natural approximation, at least if nuclei sizes are much smaller than internuclear distances.
It is indeed true for the major part of the crust, except for the innermost crustal layers, where nuclei (proton) radius $R$
can exceed half of the nuclei cell radius $a=\left(4\pi n_\mathrm N/3\right)^{-1/3}$ (note, the distance to the nearest neighbour for body-centered cubic lattice is $(n_\mathrm N/2)^{-1/3}\approx 2 a$). Here $n_\mathrm N$ is number density of nuclei.
However, if one one neglects electron screening and assumes that nuclei charge (proton) density is spherically symmetric, the electrostatic theory suggests that Coulomb interaction energy of two nuclei should be the same as for the point-like charges (up to effects associated with overlap of proton density profiles).
Thus, the shear modulus, determined by energy change associated with deformation, should be unaffected by finite nuclei size, being the same as for Coulomb crystal, if nuclei remain spherically symmetric during deformation.
Here we demonstrate that the latter assumption generally is not true, if one considers deformation of neutron star crust.
Namely, even if nuclei are spherically symmetric for undeformed crust, deformation would induce the quadrupole asymmetry. Taken into account, this effect reduces the energy in deformed state (in the opposite case, it is not energetically favourable and would not occur) and, thus, the shear modulus.
In this paper we limit ourselves to consideration of the toy problem, which is based on compressible liquid drop model (CLDM) with approximation of primitive lattice cell in undeformed state by a sphere (e.g.\ supplementary material of \cite{GC20_DiffEq} for description of the applied CLDM).
Namely, we model infinitesimal deformation of the crust by respective deformation of the spherical cell, which becomes a spheroid (ion spheroid model).
We apply CLDM to calculate energy of this spheroidal cell, allowing for deformation of nucleus in the center of the cell.
The deformation parameter of the cell $\varepsilon$ serves as a driving parameter for nuclei deformation $\varepsilon_\mathrm{p}$.
In this way we demonstrate that:
(1) If nucleus remains spherical ($\varepsilon_\mathrm{p}=0$), our model reproduces shear modulus, as obtained by \cite{C21_elastCoins} using ion sphere approximation. It serves as justification of our model;
(2) Minimization of the cell energy leads to $\varepsilon_\mathrm{p}\propto \varepsilon$ and reduces shear modulus by a factor $1-u^{5/3}/(2+3\,u-4\,u^{1/3})$ in the case of the ground-state crust.
The maximal reduction by $\sim 25\%$ takes place at the inner layers of the neutron star crust, where the filling factor is maximal $u\sim 0.2$. As shown by \cite{KY20}, these layers are the most important for torsional modes oscillation spectrum, and therefore it is crucial to know the shear modulus accurately to interpret observed quasi-periodic oscillation of magnetars correctly.
\section{Crust elasticity within ion spheroid model}
Generally, elastic properties can be calculated by considering the energy change associated with deformation of solid. If one neglects nuclei motion and consider static lattice, it is enough to calculate the energy change of primitive cell, imposing appropriate boundary conditions (for Coulomb crustal is it periodic boundary conditions for electrostatic potential).
Here, as discussed in the introduction, we apply a toy model. Namely, we follow a well-known spherical Wigner-Seitz cell approximation and replace the primitive cell in undeformed state (truncated octahedron in the case of body-centered cubic crystal) by a sphere.
After that, we apply deformation for this spherical cell and calculate energy change associated with deformation.
To justify this approach, we demonstrate that it reproduces the results of accurate calculations of effective shear modulus for Coulomb crystal well enough.
We also discuss the approaches, which can be applied to make accurate calculations in the last section (Summary and discussion).
To calculate the energy of deformed cell we generalize CLDM by \cite{GC20_DiffEq}.
The first step was done in \cite{ZC22_stability}, where we consider deformed (spheroidal) nucleus in spherical cell and conclude that spherical nuclei should be stable with respect to infinitesimal deformation, if their number density $n_\mathrm N$ is not too small for a given baryon number density (in particular, spherical nuclei in ground state and accreted crust was shown to be stable with respect to considered deformation at arbitrary filling factor).
Here we make the next step and assume infinitesimal deformation of the initially spherical cell.
As long as our model problem is spherically symmetric in undeformed state, its elastic properties can be described with just two constants: bulk modulus $K$ and shear modulus $\mu$. Bulk modulus $K$ is determined by the equation of state, and therefore is generally well-known. Here we consider only $\mu$ and determine it, applying volume-conserving deformation,%
\footnote{Such method helps to avoid complications of finite pressure elasticity theory, see \cite{Chugunov22_elast_screen} for brief discussion and references for details.}
which deforms cell to the spheroid with axis $a(1+\varepsilon)$ and $a/\sqrt{1+\varepsilon}$.
According to definition of $\mu$, this deformation should change the cell energy by the amount
\begin{equation}
\delta E=\frac{3}{2} \mu \varepsilon^2 V_\mathrm c,
\label{dE_mu}
\end{equation}
where cell volume $V_\mathrm c =1/n_\mathrm N=4\pi a^3/3$.
Below we calculate the energy change $\delta E$ and apply (\ref{dE_mu}) to quantify $\mu$.
Within our CLDM the nucleus is assumed to be a spheroid with axes $R(1+\varepsilon_\mathrm{p})$ and $R/\sqrt{1+\varepsilon_\mathrm{p}}$, which is coaxial with the cell (note, parameter $\varepsilon_\mathrm{p}$ does not affect the nucleus volume).
As a result, the parameter space of CLDM by \cite{GC20_DiffEq} [neutron and proton number density inside nucleus ($n_\mathrm{ni}$ and $n_\mathrm{pi}$ respectively), neutron number density outside nucleus ($n_\mathrm{no}$; we assume that protons are absent outside nucleus), surface density of adsorbed neutrons $\nu_\mathrm{s}$, nucleus (proton) radius $R$, and cell volume $V_\mathrm{c}=4\pi a^3/3\equiv n_\mathrm{N}^{-1}$] is supplemented with two additional parameters: $\varepsilon$ and $\varepsilon_\mathrm{p}$.
Within this paper these two are treated as infinitesimal.
To make the equations compact, it is useful to use total baryon number density $n_\mathrm{b}$ instead of $\nu_\mathrm{s}$, filling factor $u=R^3/a^3$ instead of $R$, and $n_\mathrm{N}$ instead of $V_\mathrm{c}$.
Denoting resulting parameter set of spherically symmetric CLDM as $X=\{n_\mathrm{b}, n_\mathrm{N}, n_\mathrm{ni}, n_\mathrm{pi},n_\mathrm{no},u\}$, we write down the cell energy (see Appendix for derivation details):
\begin{equation}
E=E^0(X)+f_1(X)\varepsilon^2+f_2(X)\varepsilon_\mathrm{p}^2+f_3(X)\varepsilon\varepsilon_\mathrm{p},
\label{E}
\end{equation}
where $E^0(X)$ is the energy of spherical cell with spherical nucleus (see \citealt{GC20_DiffEq}),
\begin{eqnarray}
f_1&=&\frac{9}{50}\frac{Z^2 e^2}{a},\label{f1}\\
f_2&=&\frac{8}{5}\pi R^2\sigma+\frac{3}{50}\frac{Z^2 e^2}{R}(5u-2),
\label{f2}\\
f_3&=&-\frac{9}{25}\frac{Z^2 e^2}{R}u.\label{f3}
\end{eqnarray}
The first term in the right-hand side of (\ref{f2}) corresponds to the change of the surface energy ($\sigma$ is surface tension), associated with nucleus deformation, while the second is related to the change of the Couloumb energy. Note, that the latter has a correction with respect to a well-known result for isolated nucleus (e.g., \citealt{BW39_fission})
associated with the presence of the neutralizing electron background in the cell (see \citealt{ZC22_stability} for more detailed discussion).
For completeness we do not assume ground state crust composition and allow for non-equlibrium number density of nuclei $n_\mathrm N$.
As discussed by \cite{GC20_DiffEq}, it can be the case, e.g., for accreted crust.
First, let us consider a simplified problem and assume that the nucleus is spherical, $\varepsilon_\mathrm{p}=0$.
The parameter set $X$ should be determined by minimization of energy at a given baryon number density $n_\mathrm{b}$, nuclei number density $n_\mathrm{N}$, and deformation $\varepsilon$.
As long as $\varepsilon$ is infinitesimal we can present $X$ as $X=X^0+\delta X$, where $X^0$ corresponds to undeformed crust: they are energetically optimal parameters set for the same $n_\mathrm{b}$ and $n_\mathrm N$, but $\varepsilon=0$. It is easy to show that $\delta X\propto \varepsilon^2$.
Since $X^0$ corresponds to a minimum of $E^0$, up to the second order in $\varepsilon$ the energy change associated with deformation is $\delta^\mathrm{sp} E=E(X,\varepsilon_\mathrm{p}=0,\varepsilon)-E(X^0,\varepsilon_\mathrm{p}=0,\varepsilon=0)=f_1(X^0)\varepsilon^2$ (here, the upper index `sp' indicates that it is obtained assuming nuclei to have spherical shape).
Using Eq.\ (\ref{dE_mu}), we obtain
\begin{equation}
\mu^\mathrm{sp}=\frac{2}{3V_c} f_1=\frac{3}{25}\frac{Z^2 e^2}{a}n_\mathrm N.
\label{mu_sp}
\end{equation}
This result coincides with the estimate obtained by \cite{C21_elastCoins} using ion sphere model, and, as pointed in that work, it differs only by $\sim 0.5\%$ from the accurate calculations of Voigt averaged shear modulus.
It is also worth noting, that, following the discussion in the introduction, this result does not depend on the filling factor $u$ and neither on nuclei size.
Let us now get rid of $\varepsilon_\mathrm{p}=0$ assumption and treat it as a free parameter.
In this case, not only parameters $X$, but also $\varepsilon_\mathrm{p}$ should be determined by minimization of energy at a given baryon number density $n_\mathrm{b}$, nuclei number density $n_\mathrm N$, and deformation $\varepsilon$. For $\varepsilon_\mathrm{p}$ it leads
\begin{equation}
\varepsilon_\mathrm{p}=-\frac{f_3(X)}{2f_2(X)}\varepsilon
=\frac{u}{2-4u^{1/3}+3u+\tilde \eta_\mathrm N}\varepsilon,
\label{epsilon_p}
\end{equation}
where $\tilde \eta_\mathrm N =20\eta_\mathrm N R/(3 Z^2 e^2)$ and $\eta_\mathrm N$ is energy, required to create an additional cluster from existing baryons ($\eta_\mathrm N=0$ for ground state matter, i.e.\ equilibrium value of $n_\mathrm N$).
The term $\tilde \eta_\mathrm N$ arises in the denominator of (\ref{epsilon_p}) as a result of exclusion of surface tension using relation for undeformed crust
\begin{equation}
3\eta_\mathrm N=4\pi \sigma R^2-\frac{6}{5}\frac{Z^2 e^2}{R}\left(1-\frac{3}{2}u^{1/3}+\frac{1}{2}u\right).
\label{vir}
\end{equation}
It is generalization of the virial theorem and allows to calculate $\eta_\mathrm N$ for crust with non-equilibrium $n_\mathrm N$ (all other parameters of CLDM are adjusted to minimize energy, which leads to set of equations that can be interpreted as beta-equilibrium, chemical and mechanical equilibrium conditions for the cell, see supplementary material in \citealt{GC20_DiffEq} for details).%
\footnote{\cite{GC20_DiffEq} denote $\eta_\mathrm N$ as $\mu_\mathrm N$, but we do not follow this notation to avoid confusion with shear modulus, also denoted as $\mu$.}
As for the $\varepsilon_\mathrm{p}=0$ case discussed above, $X$ can be presented as $X^0+\delta X$, where $\delta X\propto \varepsilon^2$. Substituting (\ref{epsilon_p}) into (\ref{E}) and keeping only the second-order terms in $\varepsilon$, it is straightforward to write down an explicit expression for $\mu$ using (\ref{dE_mu}):
\begin{equation}
\mu=\frac{3}{25}\frac{Z^2 e^2}{a}n_\mathrm N\left(1-\frac{u^{5/3}}{2-4u^{1/3}+3u+\tilde \eta_\mathrm N}\right).
\label{mu}
\end{equation}
This equation differs from the result for the point-like nuclei (Eq.\ \ref{mu_sp}) only by the factor in the brackets, which is referred below as `correction factor' and denoted $C\equiv 1-u^{5/3}/(2-4u^{1/3}+3u+\tilde \eta_\mathrm N) $.
\begin{figure}
\begin{center}
\leavevmode
\includegraphics[width=\columnwidth]
{Eps_p2.pdf}
\end{center}
\caption{$\varepsilon_\mathrm{p}/\varepsilon$ ratio as function of filling factor $u$ for ground-state crust.
At the main plot $u$ is limited to physically motivated region $u\le0.2$. The inset represents the same function for $u\le 1$.}
\label{Fig_eps_ratio}
\end{figure}
\begin{figure}
\begin{center}
\leavevmode
\includegraphics[width=\columnwidth]
{mu_cor2.pdf}
\end{center}
\caption{Correction factor for the shear modulus as function of filling factor $u$ for ground state crust for physically motivated region $u<0.2$.
The inset represents correction factor for $u\le 1$
}
\label{Fig_mu_cor}
\end{figure}
The ratio $\varepsilon_\mathrm{p}/\varepsilon$ is shown in Fig.\ \ref{Fig_eps_ratio} for ground-state crust ($\eta_N=0$) and physically motivated region $u\le0.2$.
It is worth to stress that
this figure does not depend on the microphysical model, applied to quantify parameters of CLDM (bulk energy density and surface tension).
One can see that the $\varepsilon_\mathrm{p}/\varepsilon$ ratio increases monotonically from zero at $u=0$.
It confirms that for small filling factors nucleus deformation is negligible. For $u=0.2$ the ratio $\varepsilon_\mathrm{p}/\varepsilon$ reaches $\approx 0.75$.
For the sake of completeness, the inset demonstrates $\varepsilon_\mathrm{p}/\varepsilon$ ratio for $u\le 1$.
In this region the ratio becomes non-monotonic and has a maximum at filling factor $u\sim 0.4$, where deformation of nucleus $\varepsilon_\mathrm{p}$ is larger than the crust deformation $\varepsilon$. However, it is unlikely that there are any nuclei with so large filling factors in realistic models of the crust.
The dependence of the shear modulus correction factor $C$
on $u$ is shown in Fig.\ \ref{Fig_mu_cor}, taken ground-state crust as an example ($\tilde \eta_\mathrm N=0$).
As for Fig.\ \ref{Fig_eps_ratio}, this plot does not depend on the numerical parameters of CLDM.
As expected, $C\approx 1$ (no correction), if nuclei size is negligible ($u\ll1$), but for $u\sim0.1$ correction factor for nuclei deformation becomes noticeable ($C\approx 0.95$) and for $u=0.2$, which is typical for the innermost crystal layers, where spherical nuclei are present (e.g., \citealt{ZC22_stability} and references therein), shear modulus is reduced by $25\%$ ($C\approx 0.75$).
The inset demonstrates the behaviour of the correction factor in the whole mathematically possible region, up to $u=1$. One can see that the correction factor becomes negative for $u\gtrsim 0.6$, however filling factors that large are not expected for the crust and our model is possibly oversimplified to analyze this region.
\begin{figure}
\begin{center}
\leavevmode
\includegraphics[width=\columnwidth]
{mu2.pdf}
\end{center}
\caption{Shear modulus of the densest region of inner crust as function of pressure $P$.
Dotted line and short dashes represent shear modulus neglecting correction for nuclei deformation ($\mu^\mathrm{sp}$ as given by Eq.\ \ref{mu_sp}) for ground state (GS) crust and fully accreted crust (FAC), respectively.
Solid and long dashes represent shear modulus, corrected for nuclei deformation according to Eq.\ (\ref{mu}) for GS and FAC, respectively.
Numerical results corresponds to SLy4 energy-density functional, shell effects were neglected.
The lines end at the crust-core interface
}
\label{Fig_mu}
\end{figure}
To illustrate the effect of non-equilibrium nuclei number density $n_\mathrm{N}$ we take fully accreted crust model by \cite{GC20_DiffEq} as an example.
This model takes into account neutron hydrostatic/diffusion (nHD) equlibrium condition and
based on SLy4 nucleon-nucleon potential (\citealt{Chabanat_ea98_SLY4,Chabanat_ea98_SLY4_p2_nuclei}).
The model neglects shell effects, being thus uniquely specified for chousen set of CLDM microphysical parameters (\citealt{GC20_DiffEq,GC21_HeatReleaze}).
Resulting dependencies of the shear modulus on pressure for ground state and fully accreted crust models are shown in Fig.\ \ref{Fig_mu}.
High-pressure end of lines corresponds to the crust-core interface (see \citealt{GC20_DiffEq} for details).
If nuclei deformation is neglected, shear modulus $\mu=\mu^\mathrm{sp}$ is predicted to be almost linear function of pressure for ground state (GS) crust composition (dotted line in Fig.\ \ref{Fig_mu}).
If nuclei deformation is taken into account, the shear modulus decreases substantially (solid line at Fig.\ \ref{Fig_mu}).
For fully accreted crust
model by \cite{GC20_DiffEq}, the nuclei charge $Z$ is just a bit lower than for ground-state crust up to the densest regions of inner crust, where the difference starts to increase (see Fig.\ 2 there). As a result, shear modulus of accreted crust is almost the same as for the ground-state crust up to $P\approx 10^{32}$~erg\,cm$^{-3}$, but becomes noticeably lower for larger pressure, even if nuclei deformation is neglected (short dashed line at Fig.\ \ref{Fig_mu}). Account for nuclei deformation reduces the shear modulus, but this effect is weaker than for ground state crust. It happens because for fully accreted crust $\tilde \eta_\mathrm N>0$ and reduces a correction factor according to Eq.\ (\ref{mu}).
It should be noted, that shell effects, which are not now taken into account here, can modify composition of fully accreted crust substantially, leading to even lower nuclei charges (see \citealt{GC21_HeatReleaze}).
As a result, the shear modulus of accreted crust can be substantially reduced.
\section{Summary and discussion}
We analyze nuclei finite size effects for shear modulus of neutron star crust and conclude that they become non-negligible for the innermost layers, where the filling factor $u\sim 0.1$.
Our results are based on CLDM and spheroidal primitive cell approximation for the deformed crustal lattice. This model allows one to analyze the problem analytically and obtain a general expression for the correction factor for effective shear modulus $C$ (Eq.\ \ref{mu_sp}). This correction reduces the shear modulus ($C<1$).
In the case of ground-state composition of the crust, which is generally believed to be a reasonable approximation for isolated neutron stars (e.g., \citealt{hpy07}, see, however, \citealt{cfg20,pc21}), it has especially simple form $C=1-u^{5/3}/(2+3\,u-4\,u^{1/3})$.
The reduction of the shear modulus has simple physical nature: deformed lattice induces asymmetric quadrupolar electrostatic field in a vicinity of lattice sites, where nuclei are located. Similarly to the tidal field in the case of neutron star mergers or Earth's tides, the nuclei deformation becomes energetically favourable in the presence of this field. Accounting for this effect
decreases the energy in the deformed state and thus estimate for shear modulus is reduced.
Clearly, this work presents a simplified model. An accurate calculations should include precise description of the induced quadrupole electric field, which generally depends on the lattice type as well as on the type of deformation. Detailed (quantum) description of the nuclear response for this field is also required.
However, accurate treatment for infinitesimal deformations should lead to energy dependence in form of Eq.\ (\ref{E}). We believe that our model gives a reasonable (at least by order of magnitude) estimate for parameters of this equation: $f_1$, $f_2$, and $f_3$, and thus for the final result.
Namely, for $f_1$, which is determined by deformation of lattice with spherical (or point-like) nuclei we have a good approximation of accurately calculated Voigt averaged effective shear modulus.
The parameter $f_2$, which describes energy change associated with nuclei deformation, is estimated within CLDM model, which should be reasonable for nuclei with large number of nucleons.
Finally, for parameter $f_3$, which is responsible for coupling of lattice and nucleus deformation, we obtain an estimate with natural dependence on the parameters (filling factor, nuclei charge, and radius), which should give a correct order of magnitude value.
We plan to get rid of a spheroidal cell approximation and analyze nuclei finite size effects for realistic lattice types in subsequent publications.
Please, be aware that our calculations neglect electron screening and nuclei motion.
The latter effects were analyzed in literature only in approximation of point-like particles (e.g., \citealt{Baiko12}).
However, for practical applications one needs an approach, which combines all necessary effects.
For this aim we would like to suggest to multiply the results of \cite{Baiko12} (or, e.g., \citealt{oi90} supplemented with the electron screening correction from \citealt{Chugunov22_elast_screen}) by the correction factor (\ref{mu_sp}).
It would not affect shear modulus in the outer crustal region, where point-like approximation is applicable, while in the innermost crustal regions the shear modulus will be corrected for nuclei finite size effects.
Note, that the melting temperature for the innermost layers of neutron star crust is expected to be much larger than the typical neutron star temperatures and nuclei motion effects can be neglected.
Finally, the suggested approach would protect against artificial instabilities ($\mu<0$), which potentially can arise at extreme parameters, if electron screening and nuclei motion corrections are applied to reduce shear modulus, given by Eq.\ (\ref{mu}).
\section*{Acknowledgements}
The work of AIC was partially supported
by The Ministry of Science and Higher
Education of the Russian Federation (Agreement with Joint Institute for High Temperatures RAS No 075-15-2020-785 dated September 23, 2020).
\section*{DATA AVAILABILITY}
The data underlying this paper are available in the paper.
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An obscure database could be instrumental in helping to identify some owners of the roughly 1,400 works seized last month from the Munich apartment of Cornelius Gurlitt. Mary Lane has details on Lunch Break. Photo: AP.
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See e.g., In re Man Mach. Interface Techs. LLC, 822 F.3d 1282, 1286 (Fed. Cir. 2016) ("We have noted that the phrase 'adapted to' generally means 'made to,' 'designed to,' or 'configured to,' though it can also be used more broadly to mean 'capable of' or 'suitable for.'") (citations omitted).
|
{
"redpajama_set_name": "RedPajamaC4"
}
| 2,698
|
Measurements for the bronze Laver, (the wash basin) is not given. The bronze Laver was made of brass mirrors from the serving woman, the base of the basin is made of bronze. It was used to wash the hands and feet of the priests before they entered into the Tabernacle.
|
{
"redpajama_set_name": "RedPajamaC4"
}
| 4,271
|
Mrs.Foss has not voted in the 2017 Cape Cod A-List.
I never got a chance to visit the salon in person. The bride ordered our shoes and dresses from the shop, so I only spoke with them on the phone. We had an issue with a pair of shoes, they were way to big and had a hole in the seam. We shipped the shoed back on our dime, and they wanted us to pay the shipping back. That to me is not good customer service! Living 4 hours away and speing $65 on apair of shoes and then spending another 34 for shipping costs! Not cool!!
Other then that any time I called the Salon the women were very kind and helpful in any way they could! But the shoe thing kinda put a sour taste in my mouth!
Mrs.Foss has not cast any votes in the 2017 Cape Cod A-List.
|
{
"redpajama_set_name": "RedPajamaC4"
}
| 5,873
|
Q: How do I reject a snapshot during recovery? I'm currently attempting to implement snapshotting to an event sourced application using Akka Persistence. The strategy that I'm attempting to take here is:
*
*Save a snapshot every X events
*Maintain a schemaVersion property on my state object. The purpose of the schemaVersion is to allow me to evolve/migrate my state if I change the way I'm replying my events.
*During recovery, if I receive a SnapshotOffer whose state.schemaVersion is less than my current schema version, discard that snapshot and replay events from the beginning
*Delete all snapshots earlier than my latest valid snapshot after recovery complete
However, I'm finding that I'm not able to discard snapshots during recovery. If there is a snapshot in the snapshot store, earlier events will not be offered to me.
I'm having a hard time reading up on how this should be handled. What's the proper approach here?
A: There is one thing I don't get, what is the state object? And from where it's getting the schemaVersion? Isn't the state exactly the state of your actor that is reconstructed from the snapshot / events?
In any case, you can't delete and/or skip a snapshot once it's offered. Instead, you can do the following:
*
*add a config drop-snapshot (defaults to false)
*on preStart
override def preStart(): Unit = {
super.preStart()
if (dropSnapshot) deleteSnapshots(SnapshotSelectionCriteria.Latest)
}
*override recovery method
override def recovery = {
if (dropSnapshot) {
Recovery(fromSnapshot = SnapshotSelectionCriteria.None)
} else {
Recovery(fromSnapshot = SnapshotSelectionCriteria.Latest)
}
}
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 5,379
|
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